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useful for all sorts of math problems. estigaaaaationtiontiontiontion — Transforming Functions estigestig estig pter 8 Invvvvvestig pter 8 In ChaChaChaChaChapter 8 In pter 8 In pter 8 In 435435435435435 436436436436436436436436436436 Appendixes Glossary................................................... 438 Formula Sheet................................................... 440 Index................................................... 442 437437437437437 Glossary Symbols • infinity < > ∴ therefore ⊥ is less than is greater than is perpendicular to is parallel to is less than or equal to is greater than or equal to is not equal to the irrational numbers ΩΩΩΩΩΩΩΩΩΩ £ ≥ π I N the natural numbers Q the rational numbers Δ the empty set « the intersection of sets » the union of sets Ã is a subset of À is not a subset of Œ is an element of œ is not an element of ﬁ implies R the real numbers W the whole numbers Z the integers A aaaaabsolute v alue alue bsolute v bsolute v alue the distance between zero and a number on the bsolute value alue bsolute v number line (the absolute value of a is written \|a\|) ession aic exprxprxprxprxpression ession aic e algalgalgalgalgeeeeebrbrbrbrbraic e aic e ession a mathematical expression containing at ession aic e least one variable tion) tion) ultiplica ultiplica dition and m ties (of ad ad ad ad addition and m dition and m ties (of ties (of oper oper associatititititivvvvve pre pre pre pre proper associa associa tion) ultiplication) dition and multiplica operties (of tion) ultiplica dition and m ties (of oper associa associa for any a, b, c: a + (b + c) = (a + b) + c a(bc) = (ab)c B base base base in the expression bx, the base is b base base binomial binomial binomial a polynomial with two terms binomial binomial C losed intervvvvval al al al al an interval that includes its end
points ccccclosed inter losed inter losed inter losed inter losure pre pre pre pre proper dition and dition and umber ad umber ad eal-n eal-n ties (of r r r r real-n ties (of ties (of oper oper losur losur ccccclosur operties (of dition and umber addition and eal-number ad dition and umber ad eal-n ties (of oper losur mmmmmultiplica tion) tion) ultiplica ultiplica tion) when two real numbers are added or ultiplication) tion) ultiplica multiplied, the result is also a real number actor actor common f common f actor a number or expression that is a factor of two common factor actor common f common f or more other numbers or expressions commcommcommcommcommutautautautautatititititivvvvve pre pre pre pre proper tion) tion) ultiplica ultiplica dition and m ties (of ad ad ad ad addition and m dition and m ties (of ties (of oper oper tion) for ultiplication) dition and multiplica operties (of tion) ultiplica dition and m ties (of oper any a, b: a + b = b + a and ab = ba completing the squareeeee the process of changing a quadratic completing the squar completing the squar completing the squar completing the squar expression into a perfect square trinomial compound inequality compound inequality compound inequality two inequalities combined using either compound inequality compound inequality “and” (a conjunction) or “or” (a disjunction) D a monomial) a monomial) ee (of ee (of dededededegggggrrrrree (of a monomial) the sum of the powers of the variables ee (of a monomial) a monomial) ee (of ynomial ynomial a pol a pol dededededegggggrrrrree of ee of ee of ynomial the largest degree of a polynomial’s a polynomial ee of a pol ynomial a pol ee of terms tor tor denomina denomina tor the bottom expression of a fraction denominator tor denomina denomina tions tions equa equa pendent system of dededededependent system of pendent system
of tions a system of equations with equations pendent system of equa tions pendent system of equa infinitely many possible solutions discriminant discriminant discriminant for a quadratic equation ax2 + bx + c = 0, the discriminant discriminant discriminant is b2 – 4ac distrib distrib distributiutiutiutiutivvvvve pr e pre proper e pre pr oper oper ty (of ty (of ty (of m m m m multiplica ultiplica ultiplica tion o tion o tion ovvvvver ad er ad er ad dition) dition) operty (of ultiplication o er addition) dition) distrib distrib oper ty (of ultiplica tion o er ad dition) for any a, b, c: a(b + c) = ab + ac 438438438438438 Glossar Glossaryyyyy Glossar Glossar Glossar tion or function) a relaelaelaelaelation or function) tion or function) a r a r domain (of domain (of tion or function) the set of all possible domain (of a r tion or function) a r domain (of domain (of “inputs” of a relation or function E tions tions alent equa equivvvvvalent equa alent equa equi equi tions equations that have the same solution alent equations tions alent equa equi equi set equivvvvvalent fr actions actions alent fr alent fr equi equi actions fractions are equivalent if they have the alent fractions actions alent fr equi equi same value xponent eeeeexponent xponent xponent in the expression bx, the exponent is x xponent F fffffactoring actoring actoring actoring writing a polynomial as a product of two or more actoring factors fffffactor actor actor actor a number or expression that can be multiplied to get actor another number or expression — for example, 2 is a factor of 6, because 2 × 3 = 6 function function function a rule for transforming an “input” into a unique “output” function function G actor (GCF) actor (GCF) test common f gggggrrrrreaeaeaeaeatest common f test common f actor (GCF) largest expression that is a test common factor (GCF) actor (GCF) test common
f common factor of two or more other expressions; all other common factors will also be factors of the GCF ouping symbols gggggrrrrrouping symbols ouping symbols ouping symbols symbols that show the order in which ouping symbols mathematical operations should be carried out — such as parentheses and brackets I tion) tion) ultiplica ultiplica dition and m identities (of ad ad ad ad addition and m dition and m identities (of identities (of tion) ultiplication) dition and multiplica tion) ultiplica dition and m identities (of identities (of the additive identity is 0 (zero) — 0 can be added to any other number without changing it; the multiplicative identity is 1 — any number can be multiplied by 1 without changing tions tions equa equa inconsistent system of inconsistent system of tions a system of equations with equations inconsistent system of equa tions equa inconsistent system of inconsistent system of no solutions inteinteinteinteintegggggererererersssss the numbers 0, ±1, ±2, ±3,...; the set of all integers is denoted Z sets) sets) section (of section (of inter inter sets) the intersection of two or more sets is section (of sets) intersection (of sets) section (of inter inter the set of elements that are in all of them; intersection is denoted by « inininininvvvvvererererersessessessesses a number’s additive inverse is the number that can be added to it to give 0 (the additive identity); a number’s multiplicative inverse is the number that it can be multiplied by to give 1 (the multiplicative identity) umbersssss the set of all numbers that cannot be umber umber tional n iririririrrrrrraaaaational n tional n tional number umber tional n written as a fraction p q, where pŒZ and qŒN; the set of all irrational numbers is denoted I L least common m least common m ultiple (L ultiple (L ultiple (LCM)CM)CM)CM)CM) the smallest expression that least common multiple (L least common m least common m ultiple (L has two or more other expressions as factors likliklikliklike ter e ter e ter e termsmsmsmsms two or more terms that contain the
same variables, e ter and where each variable is raised to the same power in every term and and tion in tion in linear equa linear equa and y an equation that can be written in the tion in x and linear equation in and tion in linear equa linear equa form Ax + By = C (where A and B are not both zero) M monomial monomial monomial an expression with a single term monomial monomial N turtur umbersssss the set of numbers 1, 2, 3,...; umber turtural nal nal nal nal number umber nananananatur umber the set of all natural numbers is denoted N nnnnnumer tor umeraaaaator tor umer umer tor the top expression of a fraction umer tor ession umeric exprxprxprxprxpression ession umeric e umeric e nnnnnumeric e ession a number or an expression containing only ession umeric e numbers (and therefore no variables) O open intervvvvvalalalalal an interval that does not contain its endpoints open inter open inter open inter open inter ed pair orororororderderderderdered pair ed pair ed pair a pair of numbers or expressions written in the ed pair form (x, y); they can be used to identify a point in the coordinate plane P allel lines parparparparparallel lines allel lines allel lines lines with equal slopes; lines in the same plane allel lines that never meet e trinomial e trinomial ect squar perfperfperfperfperfect squar ect squar e trinomial a trinomial that can be written as the ect square trinomial e trinomial ect squar square of a binomial pendicular lines pendicular lines perperperperperpendicular lines pendicular lines lines whose slopes multiply together to give pendicular lines the product –1; lines that intersect at 90° point-slope fororororormmmmmulaulaulaulaula an equation of a line of the form point-slope f point-slope f point-slope f point-slope f y – y1 = m(x – x1), where m is the slope and (x1, y1) are the coordinates of a particular point lying on that line polynomial polynomial polynomial a monomial or sum of monomials polynomial
polynomial actorizationtiontiontiontion a factorization of a number where each actoriza actoriza prime f prime f prime factoriza actoriza prime f prime f factor is a prime number umber umber prime n prime n umber a number that can only be divided by itself and 1 prime number prime n umber prime n oduct oduct prprprprproduct oduct the result of multiplying numbers or expressions oduct together equality equality ties of ties of oper prprprprproper oper equality ties of equality operties of oper equality ties of addition property of equality: if a = b, then a + c = b + c multiplication property of equality: if a = b, then ac = bc subtraction property of equality: if a = b, then a – c = b – c division property of equality: if a = b, then a ÷ c = b ÷ c Q tic equationtiontiontiontion a polynomial equation of degree 2 tic equa tic equa quadraaaaatic equa quadr quadr tic equa quadr quadr R rrrrradical adical adical adical an expression written with a radical symbol adical (for example, 2 ) adicand adicand rrrrradicand adicand the number inside a radical symbol adicand tion or function) tion or function) a relaelaelaelaelation or function) a r a r e (of e (of rrrrrangangangangange (of tion or function) the set of all possible e (of a r tion or function) a r e (of “outputs” of a relation or function rrrrraaaaational e tional e tional e tional exprxprxprxprxpression ession ession ession an expression written as a fraction tional e ession umbersssss the set of all numbers that can be written as umber umber tional n rrrrraaaaational n tional n tional number umber tional n p q numbers is denoted Q, where p∈Z and q∈N; the set of all rational a fraction umbersssss denoted R, all numbers of the number line rrrrreal n umber umber eal n eal n eal number umber eal n rrrrrecipr ocal ocal ecipr ecipr ocal the
multiplicative inverse of an expression eciprocal ocal ecipr rrrrrelaelaelaelaelationtiontiontiontion any set of ordered pairs (the first number or expression in each pair can be thought of as the relation’s “input,” the second as the relation’s “output”) an equationtiontiontiontion an equation’s roots are its solutions an equa an equa rrrrroots of oots of oots of oots of an equa an equa oots of S umber sign of a n a n a n a n a number umber sign of sign of umber whether a number is positive or negative sign of umber sign of slope slope slope the steepness of a line; the ratio of the vertical “rise” to the slope slope horizontal “run” between any two points on a line slope-intercececececept fpt fpt fpt fpt fororororormmmmmulaulaulaulaula an equation of a line of the form slope-inter slope-inter slope-inter slope-inter y = mx + b, where m is the slope and b is the y-intercept square re re re re rootootootootoot if p2 = q, then p is a square root of q; if p is positive squar squar squar squar it is the principal square root of q, but if p is negative it is the minor square root of q subset subset subset a subset of a set is a set whose elements are all subset subset contained in the set sumsumsumsumsum the result of adding numbers or expressions together tions tions equa equa system of system of tions two or more equations equations system of equa tions equa system of system of T tertermsmsmsmsms the parts that are added to form an expression terter ter trinomial trinomial trinomial a polynomial with three terms trinomial trinomial U sets) sets) union (of union (of sets) the union of two or more sets is the set of union (of sets) sets) union (of union (of elements that are in at least one of them; union is denoted by » V vvvvvariaariaariaariaariabbbbblelelelele a letter that is used in place of a number W umbersssss the set of
numbers 0, 1, 2, 3,...; umber umber hole n hole n wwwwwhole n hole number umber hole n the set of all whole numbers is denoted W X -intercececececeptptptptpt the x-coordinate of a point where a graph meets -inter -inter x-inter -inter the x-axis Y -intercececececeptptptptpt the y-coordinate of a point where a graph meets -inter -inter y-inter -inter the y-axis Z zzzzzererererero pro pro pro pro product pr oduct pr oduct pr oper oper opertytytytyty if the product of two factors is zero, oduct proper oduct pr oper then at least one of the factors must itself be zero Glossaryyyyy Glossar Glossar Glossar Glossar 439439439439439 Formula Sheet Axioms of the Real Number System For any real numbers a, b, and c, the following properties hold: PrPrPrPrProper ty Name ty Name oper oper ty Name operty Name ty Name oper Closure Property: Identity Property: Inverse Property: Commutative Property: Associative Property: Distributive Property of Multiplication over Addition: AdAdAdAdAddition dition dition dition dition a + b is a real number + (–a) = 0 = –a + b) + c = a + (b + c) a(b + c) = ab + ac and (b + c)a = ba + ca Multiplicationtiontiontiontion Multiplica Multiplica Multiplica Multiplica a × b is a real number –1 = 1 = a–ab)c = a(bc) PrPrPrPrProper Equality Equality ties of ties of oper oper Equality ties of Equality operties of Equality ties of oper If a = b, then a + c = b + c. If a = b, then ac = bc. If a = b, then a – c = b – c. If a =, then b = a c b c. Absolute Value Order of Operations = x ⎧ x if ⎪⎪⎪⎪ ⎪⎪⎪⎪ x ⎨ 0 if − ⎩ x if x x > = < 0 0 0 Perform operations in the following order: 1. Anything in g
ggggrrrrrouping symbols ouping symbols ouping symbols ouping symbols — working from the innermost grouping ouping symbols symbols to the outermost. Exponents Exponents Exponents. 2. Exponents Exponents visions tions and dididididivisions visions tions tions Multiplica Multiplica 3. Multiplica visions, working from left to right. Multiplications tions visions Multiplica actions actions subtr subtr ditions ditions 4. AdAdAdAdAdditions actions, again from left to right. subtractions ditions and subtr actions subtr ditions Using Roots ab = ⋅ a b = a b a b Rules of Exponents x m n mn x 1 = x x x0 1 Fractions Adding and subtracting fractions with the same denominator:: Adding and subtracting fractions wi tth different denominators ad + bc bd a b c − = d ad − bc bd Mult iiplying fractions: a b c ⋅ = d ac bd Dividing fractions: a b 440440440440440 FFFFFororororormmmmmulasulasulasulasulas ÷ = ⋅⋅ =d a b c d c ad bc Applications Formulas InInInInInvvvvvestments estments estments estments estments The return (I) earned in one year when p is invested at an interest rate of r (expressed as a fraction): I = pr The total return (I) earned in one year when p1 is invested at an interest rate of r1 and p2 is invested at an interest rate of r2: I = p1r1 + p2r2 Mixtureseseseses Mixtur Mixtur Mixtur Mixtur concentration = amount of substance total volume percent of an ingredient = amount of ingredient total amount ×1100 The concentration (c) and total volume (v) of a mixture are given by cv = c1v1 + c2v2 where c1 and v1 describe the first ingredient and c2 and v2 the second. Speed and WWWWWororororork Rk Rk Rk Rk Raaaaatetetetete Speed and Speed and Speed and Speed and speed = distance time work rate = work completed time taken Graphs Slope of a line Inequalities \|x\| < m means –m < x < m \|x –
c\| < m means c – m < x < c + m Special Products of Binomials Point-slope form: y – y1 = m(x – x1) Slope-intercept form: y = mx + b Standard form of a linear equation: Ax + By = C Two lines with slopes m1 and m2 are: parallel if m1 = m2 perpendicular if m1 × m2 = –1 Quadratics Basic form of a quadratic equation: ax2 + bx + c = 0 The quadratic formula: − ± b = x 4 ac −2 b 2 a Completing the square: 2 x + + bx ⎛ ⎜⎜⎜ ⎝ b 2 2 ax + + bx ⎛ ⎜⎜⎜ ⎝ 1 a b 2 ⎞ 2 ⎟⎟⎟ ⎠ ⎛ ⎞ 2 ⎟⎟⎟ = + ⎜⎜⎜ x ⎝ ⎠ b 2 ⎛ ⎞ 2 ⎟⎟⎟ = aa x ⎜⎜⎜ ⎝ ⎠ + ⎞ 2 ⎟⎟⎟ ⎠ b 2 a (a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2 (a + b)(a – b) = a2 – b2 Inequality Inequality ties of ties of oper PrPrPrPrProper oper Inequality ties of Inequality operties of Inequality ties of oper For any real numbers a, b, and c: If a < b, then a + c < b + c. If a < b, then a – c < b – c. For any real numbers a, b, and c: If a < b and c > 0, then ac < bc. If a < b and c > 0, then a c < b c. The discriminant: 2 2 22 b b b − − − 4 ac 4 ac 4 ac > ⇒ 0 = ⇒ 0 < ⇒ 0 2 distinct real roots 1 real double root 0 real roots ⎛ ⎜⎜⎜⎜ The vertex of y = ax2 + b
x + c If (x – a)(x – b) = 0, then x = a or x = b. ⎞ ⎟⎟⎟⎟ ⎠ FFFFFororororormmmmmulasulasulasulasulas 441441441441441 Index A abscissa 164 absolute values 19, 21, 128, equations 128-132 inequalities 158-160 addends 14, 21 addition 14, 15, 21, 27, 28 of fractions 52, 53 of polynomials 266, 267, 272 property of equality 76, 77 property of inequalities 147, 148 of rational expressions 403-409 additive identity 16 inverse 16, 266 age-related tasks 99, 251 algebraic expressions 9, 10 annual interest 108, 109 applications 89, 90 of fractional equations 417, 418 of inequalities 150-152 of polynomial division 294, 295 of quadratic equations 359, 360, 379-385 of systems of equations 249-251, 253, 255, 256, 258, 259 associative properties of addition and multiplication 30, 34, 35 axioms 34, 35 B bases 37, 38 binary operations 14, 15, 30 binomials 263 factors 310, 311 multiplication of 276, 281 products of 297-300 squaring of 342 braces 2 as grouping symbols 27 brackets as grouping symbols 27 C canceling common factors 46, 391, 392 fractions 284, 288, 289 closure of sets under addition 14, 15, 21, 22, 34 under multiplication 14, 22, 34 coefficients 12 fractional coefficients 81, 82, 84, 85 coin tasks 92, 93 442442442442442 Indexxxxx Inde Inde Inde Inde combining like terms 67, 68 common denominator 403, 404 common difference 95, 96 common factors 46, 47, 50, 51, 288, 289, 302, 303, 305-308, 313, 314, 322, 323, 329, 330, 391, 392 commutative properties of addition and multiplication 24, 25, 30, 34, 35 completing the square 342-345, 347-353, 368-370 to derive the quadratic formula 355, 356 compound inequalities 155, 156, 158 conjunctions 155, 158 consecutive integer tasks 95, 96, 97, 150, 151 coordinate plane 164, 165 quadrants of 167-169 coordinates 164, 165, 179-181 coordinate of a point on a number line 18 D decimal coefficients 86, 87 degree of a polynomial 264, 278 denominator 46
, 288, 388 common denominator 403, 406-409 dependent systems of linear equations 239, 240 difference 24 difference of two squares 298, 325, 326 discriminant 372-378 disjunctions 156, 159, 160 distance 103-106 distributive laws 31 distributive property 34, 35, 67, 68, 70, 71, 276, 281 use in combining like terms 67 use in getting rid of grouping symbols 70, 84 dividend 291 division 24-26 of fractions 50, 51 of polynomials by monomials 283, 284 of polynomials by polynomials 285-292, 294, 295 property of equality 76-79 property of inequalities 139, 140, 142-145 property of square roots 43, 44 by rational expressions 397-401 by zero 17, 26, 33, 388 divisor 291, 303 domain of a relation 420-434 double root 375 E economic tasks 383-385 elements of sets 2 elimination method for solving systems of equations 244-248, 249-251, 255, 256, 258, 259 empty set 3 endpoints of intervals 137, 138, 155, 156, 158-160 equality in equations 74 of functions 431-433 properties of reflexive 14 symmetric 14 transitive 14 of sets 3 equations 74-76, 171, 194 absolute value equations 128-133 equivalent equations 76-79, 86 fractional equations 412-415, 417, 418 linear equations 74-78, 81, 82, 84-87, 89, 90, 175, 176, 229, 230, 249-250 systems of 229, 230 proof of an equation 74 quadratic equations 333-336, 338-341 setting up equations 95-97, 99-101, 103-106 solving equations 76, 77 of straight lines 170, 171, 173-175, 194, 195 205-207 equivalent equations 76-79, 86, 87 fractions / rational expressions 46-48, 390-392, 406-410 inequalities 139, 140, 142-145 evaluating algebraic expressions 13 exponents 28, 37, 38, 73, 274, 275 fractional 38, 40 negative 286, 287 rules of 37, 38, 274, 275, 283-287 expressions algebraic 13 numeric 9 quadratic 310, 311, 313-317, 318-320, 325-328 F factoring 288, 289 291, 292, 294, 295, 318-320, 322, 323, 333-336 monomials 302-303 polynomials 305-308, 310, 311,
313-317, 318-320, 322, 323, 325-330 polynomials using long division 291, 292 prime factorization 53, 54, 81 quadratic expressions and equations 294, 295, 310-320, 325-328, 333-336, 338-341 third-degree polynomials 322, 323 factors 14, 22, 302, 303 common factors 288, 289, 302, 303, 305-308, 313, 314, 322, 323, 329, 330, 390-392 fractional coefficients 81, 82, 84, 85 fractional equations 412-415, 417, 418 fractional powers 38, 40 fractions (see also rational expressions) 46, 47, 49, 50-54, 81, 82, 84, 85 equivalent 46, 47 functions 424-427, 429-433 equality of 431, 432 function notation 429, 430 G gradient (slope) 189-192, 194, 195, H half-open intervals 138 horizontal lines 173, 174 I identity 16 of addition 16 of multiplication 17 properties 30, 31 inconsistent systems of linear equations 237, 238 inequalities 137, 138 absolute value 158, 159, 160 applications of 139, 140, 142, 143 compound inequalities 155, 156 equivalent inequalities 150-153 graphing on a coordinate plane 212-215, 217-220, 222-225 graphing on a number line 137-140, 155, 156, 158-160 multistep inequalities 147-148 properties of addition 139, 140, 147, 148 division 143, 145, 147, 148 multiplication 142, 144 subtraction 140 regions defined by 212-215, 217-220 input-output tables 422 inputs of binary operations 15 integer problems 153 197-199, 201-203, 205-207, 209-210 consecutive integer tasks 95-97, 150, graph of a number 18 graphing method for solving systems of equations 229, 230, 234 graphing inequalities on a coordinate plane 212-215, 217-220 inequalities on a number line 137-140, 155, 156, 170, 171, 173, 174 quadratic functions 362, 363, 365, 366, 368, 370 relations 420 straight lines 158-160, 175, 176, 186, 187 in the form Ax + By = C 177, 178, 186, 187 systems of equations 229, 230, 237-240 systems of linear inequalities 222, 223 gravity 379-381 greatest common factor (GCF) 46, 47, 305-308, 391 grouping like terms 67, 68
omial 266 order of operations 27-29, 399, 400 ordered pairs 164, 175, 179, 420, 421, 424 ordinate 164 origin 164 P parabolas 362, 363 parallel lines 197-199, 209 parentheses 148 as grouping symbols 27 percent mixture problems 108-111, 113, 114, 116-119, 255, 256 perfect square trinomials 327, 328, 342-345, 251-253, 353 perpendicular lines 201-203, 210 physical problems 379-381, 383-385 plane 164, 165, 167-169 quadrants of the coordinate plane 167 plotting points 170, 177, 186 point-slope form of equation of a line 189-192, 194, 195, 198, 203 points of intersection 229 points on a line 170, 175, 176, 179, 180 points on a number line 18 polynomials 263 adding 266-268 degree of 264, 278 dividing polynomials by monomials 283, 284 prime factors 81, 302 principal square root 40-42 products 22 of polynomials 276-279 of powers 38 special products of binomials 297-300 proof of an equation 55-57, 74, 75 properties of equality 14, 55-57, 76-79 properties of inequalities 139, 140, 142-145 Q quadrants of the coordinate plane 167 quadratic equations 333 graphing 362, 363, 365, 366, 368-370, 381 quadratic expressions 310, 311 factoring 310, 311, 313-316, 325-330 in two variables 318-320 solving by completing the square 338-341 factoring 335, 336 taking square roots 351-353 using the quadratic formula 355-357, 359, 360, 368-370, 372-378 quotients 25, 26 of powers 38 R radicals 40 radicands 40 range of a relation 420-422, 424-427, 429-433 rates 121, 122, 124-126 regions defined by inequalities 212-214, 217-220 relations 420-422, 424, 425 remainders after division 43, 288, 289, 291, 292 removing fractions from equations 81, 82, 84, 85 removing decimals from equations 86, 87 return on investment 108 “rise over run” formula 189-192, 197, 198 roots 372-378 of quadratic equations 76, 362, 363, 365, 366 square roots 40-44, 338-341 rules of exponents 37, 38, 73, 274, 275, 283
, 284, 286 rules of the number system 14-22, 24-35 run 189 S second-degree polynomials 318-320 sequences of integers 95-97 sets 2-6, 420-422, 424-426, 430-433 sign of a number 19 signs of coordinates in different quadrants 167 simplifying algebraic expressions 13, 67, 68, 70, 71, 73-75, 147, 148, 150-153, 263, 264, 266-272, 274, 275, 281, 283-287 inequalities 148 numeric expressions 13 rational expressions 390-392 slope 189-192, 194, 197-199, 201-203, 205-207, 209, 210 slope-intercept form of equation of a line 205-207, 209, 210 dividing polynomials by polynomials rate problems 103-106, 258, 259 solution intervals 137-140, 158-160 288, 289, 291, 292 factoring 305-308, 310, 311, 313-320, 322, 323, 325-330, 333-336 factoring using long division 291, 292 multiplying 276-281 opposite of 266 roots of 372-378 simplifying 264 subtracting 269-271 positive numbers 18 as real numbers 5 postulates 34 powers (exponents) 28, 37, 38, 73, 274, 275, 283, 284 fractional 38, 40 negative 286, 287 rules of 37, 38, 274, 275, 283-287 prime factorization 53, 81 444444444444444 Indexxxxx Inde Inde Inde Inde rational expressions solution sets 155, 160, 174, 212-215 adding and subtracting 403-410 dividing by 397-401 equations with 412-415, 417, 418 equivalent 390-392, 406-408 multiplying by 394-396, 397-401 undefined 388, 389 rational numbers 6 real-life tasks using inequalities 150-153 using quadratic equations 379-381, 383-385 using systems of equations 388-392 real numbers 5, 14-22, 24-35, 433 reciprocals 17, 25, 26, 32, 33, 201, 202, 210, 397-401 of polynomials 285-287 reflexive property of equality 14 solving absolute value inequalities 158-160 compound inequalities 155, 156, 158 conjunctions 155, 156 equations 76 fractional equations 412-415 inequalities 139, 140, 142-145, 147, 148, 150-153 quadratic equations 333-336, 338-
341, 351-353, 355-357 speed 103-106 square roots 40-42 properties of 43, 44 square root method for solving quadratics 338-341 stacking method for multiplying polynomials 278, 279 W whole numbers 5 word problems 89, 90, 92, 93, 95-97, 99-101, 103-106, 108-111, 113, 114, 116-119, 121, 122, 124-126, 150-153, 249-251, 253, 255, 256, 258, 259 work-related tasks 121, 122, 124-126 work rates 122, 124-126 X x-axis 164-169 x-intercepts 182, 183, 362, 363, 365, 366, 372-378 Y y-axis 164, 165 y-intercepts 183, 184, 205-207, 362, 363, 365, 366, 370 Z zero 16, 32, 33 as the additive identity 16 division by 17, 26, 32, 33, 388 as a real number 5 zero product 33, 334-336 straight lines 170, 171, 173-184, 186, 187, 189-192, 194, 195, 197-199, 201-203, 205-207 subsets 3 subsets of the real numbers 5, 6 substitution method for solving systems of equations 232, 233, 235, 238, 240, 242, 249, 250, 253 substitution principle (or property) 15 subtraction 24, 25 of fractions 52, of polynomials 269-272 property of equality 76-78 property of inequalities 140, 147-148 of rational expressions 403-409 sum (of two numbers) 21 symmetric property of equality 14 systems of equations 229, 230, 244-248, 249-251, 253, 255, 256, 258, 259 applications of 249-251, 253, 255, 256, 258, 259 dependent systems 239, 240 inconsistent systems 237, 238 solving by elimination 244-248 solving by graphing 229, 230, 237, 239, 240 solving by substitution 232, 233, 238, 240 T terms in algebraic expressions 67, 68 third-degree polynomials 322, 323 time-related problems 103-106 transitive property of equality 14 trial-and-error method for factoring quadratics 314-317 trinomials 263, 327, 328, 342-344 U undefined rational expressions 388, 389 union of sets 7, 8 universal set 2 unknown quantities 9, 76 V values of coefficients 12, 13
andtwofromgroupB)andwereexcludedfromthestatisticalanalysissincetheyrequestedtheremovaloftheneedles.OnepatientfromgroupAdidnotgiveherconsenttotheimplantofthesemi-permanentneedles.IngroupA,themeannumberofFig.1Theappropriatearea(M)versustheinappropriatearea(S)usedinthetreatmentofmigraineattacksS174NeurolSci(2011)32(Suppl1):S173–S175123 16 CHAPTER 1. DATA COLLECTION 1.2 Data basics You collect data on dozens of questions from all of the students at your school. How would you organize all of this data? Eﬀective presentation and description of data is a ﬁrst step in most analyses. This section introduces one structure for organizing data as well as some terminology that will be used throughout this book. We use loan data from Lending Club and county data from the US Census Bureau to motivate and illustrate this section’s learning objectives. Learning objectives 1. Identify the individuals and the variables of a study. 2. Identify variables as categorical or numerical. Identify numerical variables as discrete or con- tinuous. 3. Understand what it means for two variables to be associated. 1.2.1 Observations, variables, and data matrices Figure 1.3 displays rows 1, 2, 3, and 50 of a data set for 50 randomly sampled loans oﬀered through Lending Club, which is a peer-to-peer lending company. These observations will be referred to as the loan50 data set. Each row in the table represents a single loan. The formal name for a row is a case or observational unit. The columns represent characteristics, called variables, for each of the loans. For example, the ﬁrst row represents a loan of $7,500 with an interest rate of 7.34%, where the borrower is based in Maryland (MD) and has an income of $70,000. GUIDED PRACTICE 1.2 What is the grade of the ﬁrst loan in Figure 1.3? And what is the home ownership status of the borrower for that ﬁrst loan? For these Guided Practice questions, you can check your answer in the footnote.5 In practice, it is especially important to ask clarifying questions to ensure important aspects
of the data are understood. For instance, it is always important to be sure we know what each variable means and the units of measurement. Descriptions of the loan50 variables are given in Figure 1.4. loan amount 7500 25000 14500... 3000 interest rate 7.34 9.43 6.08... 7.96 term 36 60 36... 36 grade A B A... A state MD OH MO... CA total income 70000 254000 80000... 34000 homeownership rent mortgage mortgage... rent 1 2 3... 50 Figure 1.3: Four rows from the loan50 data matrix. 5The loan’s grade is A, and the borrower rents their residence. 1.2. DATA BASICS 17 variable loan amount interest rate term grade state total income homeownership description Amount of the loan received, in US dollars. Interest rate on the loan, in an annual percentage. The length of the loan, which is always set as a whole number of months. Loan grade, which takes values A through G and represents the quality of the loan and its likelihood of being repaid. US state where the borrower resides. Borrower’s total income, including any second income, in US dollars. Indicates whether the person owns, owns but has a mortgage, or rents. Figure 1.4: Variables and their descriptions for the loan50 data set. The data in Figure 1.3 represent a data matrix, which is a convenient and common way to organize data, especially if collecting data in a spreadsheet. Each row of a data matrix corresponds to a unique case (observational unit), and each column corresponds to a variable. When recording data, use a data matrix unless you have a very good reason to use a diﬀerent structure. This structure allows new cases to be added as rows or new variables as new columns. GUIDED PRACTICE 1.3 The grades for assignments, quizzes, and exams in a course are often recorded in a gradebook that takes the form of a data matrix. How might you organize grade data using a data matrix?6 GUIDED PRACTICE 1.4 We consider data for 3,142 counties in the United States, which includes each county’s name, the state in which it is located, its population in 2017, how its population changed from 2010 to 2017, poverty rate, and six additional characteristics. How might these data be organized in a data matrix?7 The data described in Guided Practice 1.4
represents the county data set, which is shown as a data matrix in Figure 1.5. These data come from the US Census, with much of the data coming from the US Census Bureau’s American Community Survey (ACS). Unlike the Decennial Census, which takes place every 10 years and attempts to collect basic demographic data from every resident of the US, the ACS is an ongoing survey that is sent to approximately 3.5 million households per year. As stated by the ACS website, these data help communities “plan for hospitals and schools, support school lunch programs, improve emergency services, build bridges, and inform businesses looking to add jobs and expand to new markets, and more.”8 A small subset of the variables from the ACS are summarized in Figure 1.6. 6There are multiple strategies that can be followed. One common strategy is to have each student represented by a row, and then add a column for each assignment, quiz, or exam. Under this setup, it is easy to review a single line to understand a student’s grade history. There should also be columns to include student information, such as one column to list student names. 7Each county may be viewed as a case, and there are eleven pieces of information recorded for each case. A table with 3,142 rows and 11 columns could hold these data, where each row represents a county and each column represents a particular piece of information. 8https://www.census.gov/programs-surveys/acs/about.html 18 CHAPTER 1. DATA COLLECTION..................... -................2. DATA BASICS 19 1.2.2 Types of variables Examine the unemp rate, pop, state, and median edu variables in the county data set. Each of these variables is inherently diﬀerent from the other three, yet some share certain characteristics. First consider unemp rate, which is said to be a numerical variable since it can take a wide range of numerical values, and it is sensible to add, subtract, or take averages with those values. On the other hand, we would not classify a variable reporting telephone area codes as numerical since the average, sum, and diﬀerence of area codes doesn’t have any clear meaning. The pop variable is also numerical, although it seems to be a little diﬀerent than unemp rate. This variable of the population count can only take whole non
-negative numbers (0, 1, 2,...). For this reason, the population variable is said to be discrete since it can only take numerical values with jumps. On the other hand, the unemployment rate variable is said to be continuous. The variable state can take up to 51 values after accounting for Washington, DC: AL, AK,..., and WY. Because the responses themselves are categories, state is called a categorical variable, and the possible values are called the variable’s levels. Finally, consider the median edu variable, which describes the median education level of county residents and takes values below hs, hs diploma, some college, or bachelors in each county. This variable seems to be a hybrid: it is a categorical variable but the levels have a natural ordering. A variable with these properties is called an ordinal variable, while a regular categorical variable without this type of special ordering is called a nominal variable. To simplify analyses, any ordinal variable in this book will be treated as a nominal (unordered) categorical variable. Figure 1.7: Breakdown of variables into their respective types. EXAMPLE 1.5 Data were collected about students in a statistics course. Three variables were recorded for each student: number of siblings, student height, and whether the student had previously taken a statistics course. Classify each of the variables as continuous numerical, discrete numerical, or categorical. The number of siblings and student height represent numerical variables. Because the number of siblings is a count, it is discrete. Height varies continuously, so it is a continuous numerical variable. The last variable classiﬁes students into two categories – those who have and those who have not taken a statistics course – which makes this variable categorical. GUIDED PRACTICE 1.6 An experiment is evaluating the eﬀectiveness of a new drug in treating migraines. A group variable is used to indicate the experiment group for each patient: treatment or control. The num migraines variable represents the number of migraines the patient experienced during a 3-month period. Classify each variable as either numerical or categorical.9 9The group variable can take just one of two group names, making it categorical. The num migraines variable describes a count of the number of migraines, which is an outcome where basic arithmetic is sensible, which means this is a numerical outcome; more speciﬁcally, since
it represents a count, num migraines is a discrete numerical variable. all variablesnumericalcategoricalcontinuousdiscretenominal(unordered categorical)ordinal(ordered categorical) 20 CHAPTER 1. DATA COLLECTION 1.2.3 Relationships between variables Many analyses are motivated by a researcher looking for a relationship between two or more variables. A social scientist may like to answer some of the following questions: (1) If homeownership is lower than the national average in one county, will the percent of multi-unit structures in that county tend to be above or below the national average? (2) Does a higher than average increase in county population tend to correspond to counties with higher or lower median household incomes? (3) How useful a predictor is median education level for the median household income for US counties? To answer these questions, data must be collected, such as the county data set shown in Figure 1.5. Examining summary statistics could provide insights for each of the three questions about counties. Additionally, graphs can be used to visually explore the data. Scatterplots are one type of graph used to study the relationship between two numerical variables. Figure 1.8 compares the variables homeownership and multi unit, which is the percent of units in multi-unit structures (e.g. apartments, condos). Each point on the plot represents a single county. For instance, the highlighted dot corresponds to County 413 in the county data set: Chattahoochee County, Georgia, which has 39.4% of units in multi-unit structures and a homeownership rate of 31.3%. The scatterplot suggests a relationship between the two variables: counties with a higher rate of multi-units tend to have lower homeownership rates. We might brainstorm as to why this relationship exists and investigate the ideas to determine which are the most reasonable explanations. Figure 1.8: A scatterplot of homeownership versus the percent of units that are in multi-unit structures for US counties. The highlighted dot represents Chattahoochee County, Georgia, which has a multi-unit rate of 39.4% and a homeownership rate of 31.3%. Explore this scatterplot and dozens of other scatterplots using American Community Survey data on Tableau Public. The multi-unit and homeownership rates are said to be associated because the plot shows a discernible pattern. When two variables show some connection with one another, they are called associated variables. Associated variables can also be called dependent variables
and vice-versa. Homeownership Rate020406080100020406080100lPercent of Units in Multi−Unit Structures 1.2. DATA BASICS 21 Figure 1.9: A scatterplot showing pop change against median hh income. Owsley County of Kentucky, is highlighted, which lost 3.63% of its population from 2010 to 2017 and had median household income of $22,736. Explore this scatterplot and dozens of other scatterplots using American Community Survey data on Tableau Public. GUIDED PRACTICE 1.7 Examine the variables in the loan50 data set, which are described in Figure 1.4 on page 17. Create two questions about possible relationships between variables in loan50 that are of interest to you.10 EXAMPLE 1.8 This example examines the relationship between a county’s population change from 2010 to 2017 and median household income, which is visualized as a scatterplot in Figure 1.9. Are these variables associated? The larger the median household income for a county, the higher the population growth observed for the county. While this trend isn’t true for every county, the trend in the plot is evident. Since there is some relationship between the variables, they are associated. Because there is a downward trend in Figure 1.8 – counties with more units in multi-unit structures are associated with lower homeownership – these variables are said to be negatively associated. A positive association is shown in the relationship between the median hh income and pop change in Figure 1.9, where counties with higher median household income tend to have higher rates of population growth. If two variables are not associated, then they are said to be independent. That is, two variables are independent if there is no evident relationship between the two. ASSOCIATED OR INDEPENDENT, NOT BOTH A pair of variables is either related in some way (associated) or not (independent). No pair of variables is both associated and independent. 10Two example questions: (1) What is the relationship between loan amount and total income? (2) If someone’s income is above the average, will their interest rate tend to be above or below the average? $0$20k$40k$60k$80k$100k$120k−10%0%10%20%30%Median Household IncomePopulation Changeover 7 Yearsl 22 CHAPTER 1. DATA COLLECTION Section summary • Researchers often summarize
data in a table, where the rows correspond to individuals or cases and the columns correspond to the variables, the values of which are recorded for each individual. • Variables can be numerical (measured on a numerical scale) or categorical (taking on levels, such as low/medium/high). Numerical variables can be continuous, where all values within a range are possible, or discrete, where only speciﬁc values, usually integer values, are possible. • When there exists a relationship between two variables, the variables are said to be associated or dependent. If the variables are not associated, they are said to be independent. 1.2. DATA BASICS Exercises 23 1.3 Air pollution and birth outcomes, study components. Researchers collected data to examine the relationship between air pollutants and preterm births in Southern California. During the study air pollution levels were measured by air quality monitoring stations. Speciﬁcally, levels of carbon monoxide were recorded in parts per million, nitrogen dioxide and ozone in parts per hundred million, and coarse particulate matter (PM10) in µg/m3. Length of gestation data were collected on 143,196 births between the years 1989 and 1993, and air pollution exposure during gestation was calculated for each birth. The analysis suggested that increased ambient PM10 and, to a lesser degree, CO concentrations may be associated with the occurrence of preterm births.11 (a) Identify the main research question of the study. (b) Who are the subjects in this study, and how many are included? (c) What are the variables in the study? Identify each variable as numerical or categorical. If numerical, state whether the variable is discrete or continuous. If categorical, state whether the variable is ordinal. 1.4 Buteyko method, study components. The Buteyko method is a shallow breathing technique developed by Konstantin Buteyko, a Russian doctor, in 1952. Anecdotal evidence suggests that the Buteyko method can reduce asthma symptoms and improve quality of life. In a scientiﬁc study to determine the eﬀectiveness of this method, researchers recruited 600 asthma patients aged 18-69 who relied on medication for asthma treatment. These patients were randomnly split into two research groups: one practiced the Buteyko method and the other did not. Patients were scored on quality of life, activity, asthma symptoms, and medication reduction on a scale from 0
to 10. On average, the participants in the Buteyko group experienced a signiﬁcant reduction in asthma symptoms and an improvement in quality of life.12 (a) Identify the main research question of the study. (b) Who are the subjects in this study, and how many are included? (c) What are the variables in the study? Identify each variable as numerical or categorical. If numerical, state whether the variable is discrete or continuous. If categorical, state whether the variable is ordinal. 1.5 Cheaters, study components. Researchers studying the relationship between honesty, age and selfcontrol conducted an experiment on 160 children between the ages of 5 and 15. Participants reported their age, sex, and whether they were an only child or not. The researchers asked each child to toss a fair coin in private and to record the outcome (white or black) on a paper sheet, and said they would only reward children who report white.13 (a) Identify the main research question of the study. (b) Who are the subjects in this study, and how many are included? (c) The study’s ﬁndings can be summarized as follows: ”Half the students were explicitly told not to cheat and the others were not given any explicit instructions. In the no instruction group probability of cheating was found to be uniform across groups based on child’s characteristics. In the group that was explicitly told to not cheat, girls were less likely to cheat, and while rate of cheating didn’t vary by age for boys, it decreased with age for girls.” How many variables were recorded for each subject in the study in order to conclude these ﬁndings? State the variables and their types. 11B. Ritz et al. “Eﬀect of air pollution on preterm birth among children born in Southern California between 1989 and 1993”. In: Epidemiology 11.5 (2000), pp. 502–511. 12J. McGowan. “Health Education: Does the Buteyko Institute Method make a diﬀerence?” In: Thorax 58 (2003). 13Alessandro Bucciol and Marco Piovesan. “Luck or cheating? A ﬁeld experiment on honesty with children”. In: Journal of Economic Psychology 32.1 (2011), pp. 73–78. 24 CHAPTER 1. DATA COL
LECTION 1.6 Stealers, study components. In a study of the relationship between socio-economic class and unethical behavior, 129 University of California undergraduates at Berkeley were asked to identify themselves as having low or high social-class by comparing themselves to others with the most (least) money, most (least) education, and most (least) respected jobs. They were also presented with a jar of individually wrapped candies and informed that the candies were for children in a nearby laboratory, but that they could take some if they wanted. After completing some unrelated tasks, participants reported the number of candies they had taken.14 (a) Identify the main research question of the study. (b) Who are the subjects in this study, and how many are included? (c) The study found that students who were identiﬁed as upper-class took more candy than others. How many variables were recorded for each subject in the study in order to conclude these ﬁndings? State the variables and their types. 1.7 Migraine and acupuncture, Part II. Exercise 1.1 introduced a study exploring whether acupuncture had any eﬀect on migraines. Researchers conducted a randomized controlled study where patients were randomly assigned to one of two groups: treatment or control. The patients in the treatment group received acupuncture that was speciﬁcally designed to treat migraines. The patients in the control group received placebo acupuncture (needle insertion at non-acupoint locations). 24 hours after patients received acupuncture, they were asked if they were pain free. What are the explanatory and response variables in this study? 1.8 Sinusitis and antibiotics, Part II. Exercise 1.2 introduced a study exploring the eﬀect of antibiotic treatment for acute sinusitis. Study participants either received either a 10-day course of an antibiotic (treatment) or a placebo similar in appearance and taste (control). At the end of the 10-day period, patients were asked if they experienced improvement in symptoms. What are the explanatory and response variables in this study? 1.9 Fisher’s irises. Sir Ronald Aylmer Fisher was an English statistician, evolutionary biologist, and geneticist who worked on a data set that contained sepal length and width, and petal length and width from three species of iris ﬂowers (setosa, versicolor and virginica). There were 50 ﬂowers
from each species in the data set.15 (a) How many cases were included in the data? (b) How many numerical variables are included in the data? Indicate what they are, and if they are continuous or discrete. (c) How many categorical variables are included in the data, and what are they? List the corresponding levels (categories). Photo by Ryan Claussen (http://ﬂic.kr/p/6QTcuX) CC BY-SA 2.0 license 1.10 Smoking habits of UK residents. A survey was conducted to study the smoking habits of UK residents. Below is a data matrix displaying a portion of the data collected in this survey. Note that “£” stands for British Pounds Sterling, “cig” stands for cigarettes, and “N/A” refers to a missing component of the data.16 1 2 3... 1691 sex Female Male Male... Male marital age Single 42 44 Single 53 Married... 40... Single grossIncome Under £2,600 £10,400 to £15,600 Above £36,400... £2,600 to £5,200 smoke Yes No Yes... Yes amtWeekends 12 cig/day N/A 6 cig/day... 8 cig/day amtWeekdays 12 cig/day N/A 6 cig/day... 8 cig/day (a) What does each row of the data matrix represent? (b) How many participants were included in the survey? (c) Indicate whether each variable in the study is numerical or categorical. If numerical, identify as contin- uous or discrete. If categorical, indicate if the variable is ordinal. 14P.K. Piﬀ et al. “Higher social class predicts increased unethical behavior”. In: Proceedings of the National Academy of Sciences (2012). 15R.A Fisher. “The Use of Multiple Measurements in Taxonomic Problems”. In: Annals of Eugenics 7 (1936), pp. 179–188. 16National STEM Centre, Large Datasets from stats4schools. 1.2. DATA BASICS 25 1.11 US Airports. The visualization below shows the geographical distribution of airports in the contiguous United States and Washington, DC. This visualization was constructed based on a dataset where each observation is an airport.17 (
a) List the variables used in creating this visualization. (b) Indicate whether each variable in the study is numerical or categorical. If numerical, identify as contin- uous or discrete. If categorical, indicate if the variable is ordinal. 1.12 UN Votes. The visualization below shows voting patterns the United States, Canada, and Mexico in the United Nations General Assembly on a variety of issues. Speciﬁcally, for a given year between 1946 and 2015, it displays the percentage of roll calls in which the country voted yes for each issue. This visualization was constructed based on a dataset where each observation is a country/year pair.18 (a) List the variables used in creating this visualization. (b) Indicate whether each variable in the study is numerical or categorical. If numerical, identify as contin- uous or discrete. If categorical, indicate if the variable is ordinal. 17Federal Aviation Administration, www.faa.gov/airports/airport safety/airportdata 5010. 18David Robinson. unvotes: United Nations General Assembly Voting Data. R package version 0.2.0. 2017. url: https://CRAN.R-project.org/package=unvotes. 26 CHAPTER 1. DATA COLLECTION 1.3 Overview of data collection principles How do researchers collect data? Why are the results of some studies more reliable than others? The way a researcher collects data depends upon the research goals. In this section, we look at diﬀerent methods of collecting data and consider the types of conclusions that can be drawn from those methods. Learning objectives 1. Distinguish between the population and a sample and between the parameter and a statistic. 2. Know when to summarize a data set using a mean versus a proportion. 3. Understand why anecdotal evidence is unreliable. 4. Identify the four main types of data collection: census, sample survey, experiment, and obser- vation study. 5. Classify a study as observational or experimental, and determine when a study’s results can be generalized to the population and when a causal relationship can be drawn. 1.3.1 Populations and samples Consider the following three research questions: 1. What is the average mercury content in swordﬁsh in the Atlantic Ocean? 2. Over the last 5 years, what is the average time to complete a degree for Duke undergrads? 3. Does a new drug reduce the number of deaths
in patients with severe heart disease? Each research question refers to a target population. In the ﬁrst question, the target population is all swordﬁsh in the Atlantic ocean, and each ﬁsh represents a case. Often times, it is too expensive to collect data for every case in a population. Instead, a sample is taken. A sample represents a subset of the cases and is often a small fraction of the population. For instance, 60 swordﬁsh (or some other number) in the population might be selected, and this sample data may be used to provide an estimate of the population average and answer the research question. GUIDED PRACTICE 1.9 For the second and third questions above, identify the target population and what represents an individual case.19 19(2) Notice that this question is only relevant to students who complete their degree; the average cannot be computed using a student who never ﬁnished her degree. Thus, only Duke undergrads who have graduated in the last ﬁve years are part of the population of interest. Each such student would represent an individual case. (3) A person with severe heart disease represents a case. The population includes all people with severe heart disease. 1.3. OVERVIEW OF DATA COLLECTION PRINCIPLES 27 We collect a sample of data to better understand the characteristics of a population. A variable is a characteristic we measure for each individual or case. The overall quantity of interest may be the mean, median, proportion, or some other summary of a population. These population values are called parameters. We estimate the value of a parameter by taking a sample and computing a numerical summary called a statistic based on that sample. Note that the two p’s (population, parameter) go together and the two s’s (sample, statistic) go together. EXAMPLE 1.10 Earlier we asked the question: what is the average mercury content in swordﬁsh in the Atlantic Ocean? Identify the variable to be measured and the parameter and statistic of interest. The variable is the level of mercury content in swordﬁsh in the Atlantic Ocean. It will be measured for each individual swordﬁsh. The parameter of interest is the average mercury content in all swordﬁsh in the Atlantic Ocean. If we take a sample of 50 swordﬁsh from the Atlantic Ocean, the average mercury content among just those 50 sword�
�sh will be the statistic. Two statistics we will study are the mean (also called the average) and proportion. When we are discussing a population, we label the mean as µ (the Greek letter, mu), while we label the sample mean as ¯x (read as x-bar ). When we are discussing a proportion in the context of a population, we use the label p, while the sample proportion has a label of ˆp (read as p-hat). Generally, we use ¯x to estimate the population mean, µ. Likewise, we use the sample proportion ˆp to estimate the population proportion, p. EXAMPLE 1.11 Is µ a parameter or statistic? What about ˆp? µ is a parameter because it refers to the average of the entire population. ˆp is a statistic because it is calculated from a sample. EXAMPLE 1.12 For the second question regarding time to complete a degree for a Duke undergraduate, is the variable numerical or categorical? What is the parameter of interest? The characteristic that we record on each individual is the number of years until graduation, which is a numerical variable. The parameter of interest is the average time to degree for all Duke undergraduates, and we use µ to describe this quantity. GUIDED PRACTICE 1.13 The third question asked whether a new drug reduces deaths in patients with severe heart disease. Is the variable numerical or categorical? Describe the statistic that should be calculated in this study.20 If these topics are still a bit unclear, don’t worry. We’ll cover them in greater detail in the next chapter. 20The variable is whether or not a patient with severe heart disease dies within the time frame of the study. This is categorical because it will be a yes or a no. The statistic that should be recorded is the proportion of patients that die within the time frame of the study, and we would use ˆp to denote this quantity. 28 CHAPTER 1. DATA COLLECTION Figure 1.10: In February 2010, some media pundits cited one large snow storm as valid evidence against global warming. As comedian Jon Stewart pointed out, “It’s one storm, in one region, of one country.” —————————– February 10th, 2010. 1.3.2 Anecdotal evidence Consider the following possible responses to the three research questions: 1. A man on the news got mercury poisoning from eating
swordﬁsh, so the average mercury concentration in swordﬁsh must be dangerously high. 2. I met two students who took more than 7 years to graduate from Duke, so it must take longer to graduate at Duke than at many other colleges. 3. My friend’s dad had a heart attack and died after they gave him a new heart disease drug, so the drug must not work. Each conclusion is based on data. However, there are two problems. First, the data only represent one or two cases. Second, and more importantly, it is unclear whether these cases are actually representative of the population. Data collected in this haphazard fashion are called anecdotal evidence. ANECDOTAL EVIDENCE Be careful of making inferences based on anecdotal evidence. Such evidence may be true and veriﬁable, but it may only represent extraordinary cases. The majority of cases and the average case may in fact be very diﬀerent. Anecdotal evidence typically is composed of unusual cases that we recall based on their striking characteristics. For instance, we may vividly remember the time when our friend bought a lottery ticket and won $250 but forget most the times she bought one and lost. Instead of focusing on the most unusual cases, we should examine a representative sample of many cases. 1.3. OVERVIEW OF DATA COLLECTION PRINCIPLES 29 1.3.3 Explanatory and response variables When we ask questions about the relationship between two variables, we sometimes also want to determine if the change in one variable causes a change in the other. Consider the following rephrasing of an earlier question about the county data set: If there is an increase in the median household income in a county, does this drive an increase in its population? In this question, we are asking whether one variable aﬀects another. If this is our underlying belief, then median household income is the explanatory variable and the population change is the response variable in the hypothesized relationship.21 EXPLANATORY AND RESPONSE VARIABLES When we suspect one variable might causally aﬀect another, we label the ﬁrst variable the explanatory variable and the second the response variable. For many pairs of variables, there is no hypothesized relationship, and these labels would not be applied to either variable in such cases. ASSOCIATION DOES NOT IMPLY CAUSATION Labeling variables as explanatory and response does not guarantee the relationship between
the two is actually causal, even if there is an association identiﬁed between the two variables. We use these labels only to keep track of which variable we suspect aﬀects the other. In many cases, the relationship is complex or unknown. It may be unclear whether variable A explains variable B or whether variable B explains variable A. For example, it is now known that a particular protein called REST is much depleted in people suﬀering from Alzheimer’s disease. While this raises hopes of a possible approach for treating Alzheimer’s, it is still unknown whether the lack of the protein causes brain deterioration, whether brain deterioration causes depletion in the REST protein, or whether some third variable causes both brain deterioration and REST depletion. That is, we do not know if the lack of the protein is an explanatory variable or a response variable. Perhaps it is both.22 21Sometimes the explanatory variable is called the independent variable and the response variable is called the dependent variable. However, this becomes confusing since a pair of variables might be independent or dependent, so we avoid this language. 22nytimes.com/2014/03/20/health/fetal-gene-may-protect-brain-from-alzheimers-study-ﬁnds.html might affectexplanatoryvariableresponsevariable 30 CHAPTER 1. DATA COLLECTION 1.3.4 Observational studies versus experiments There are two primary types of data collection: observational studies and experiments. Researchers perform an observational study when they collect data without interfering with how the data arise. For instance, researchers may collect information via surveys, review medical or company records, or follow a cohort of many similar individuals to study why certain diseases might develop. In each of these situations, researchers merely observe or take measurements of things that arise naturally. When researchers want to investigate the possibility of a causal connection, they conduct an experiment. For all experiments, the researchers must impose a treatment. For most studies there will be both an explanatory and a response variable. For instance, we may suspect administering a drug will reduce mortality in heart attack patients over the following year. To check if there really is a causal connection between the explanatory variable and the response, researchers will collect a sample of individuals and split them into groups. The individuals in each group are assigned a treatment. When individuals are randomly assigned to a group, the experiment is called a randomized experiment. For example, each heart attack patient in the drug trial could be randomly assigned
into one of two groups: the ﬁrst group receives a placebo (fake treatment) and the second group receives the drug. See the case study in Section 1.1 for another example of an experiment, though that study did not employ a placebo. EXAMPLE 1.14 Suppose that a researcher is interested in the average tip customers at a particular restaurant give. Should she carry out an observational study or an experiment? In addressing this question, we ask, “Will the researcher be imposing any treatment?” Because there is no treatment or interference that would be applicable here, it will be an observational study. Additionally, one consideration the researcher should be aware of is that, if customers know their tips are being recorded, it could change their behavior, making the results of the study inaccurate. ASSOCIATION = CAUSATION In general, association does not imply causation, and causation can only be inferred from a randomized experiment. Section summary • The population is the entire group that the researchers are interested in. Because it is usually too costly to gather the data for the entire population, researchers will collect data from a sample, representing a subset of the population. • A parameter is a true quantity for the entire population, while a statistic is what is calculated from the sample. A parameter is about a population and a statistic is about a sample. Remember: p goes with p and s goes with s. • Two common summary quantities are mean (for numerical variables) and proportion (for categorical variables). • Finding a good estimate for a population parameter requires a random sample; do not gener- alize from anecdotal evidence. • There are two primary types of data collection: observational studies and experiments. In an experiment, researchers impose a treatment to look for a causal relationship between the treatment and the response. In an observational study, researchers simply collect data without imposing any treatment. • Remember: Correlation is not causation! In other words, an association between two variables does not imply that one causes the other. Proving a causal relationship requires a well-designed experiment. 1.3. OVERVIEW OF DATA COLLECTION PRINCIPLES 31 Exercises 1.13 Air pollution and birth outcomes, scope of inference. Exercise 1.3 introduces a study where researchers collected data to examine the relationship between air pollutants and preterm births in Southern California. During the study air pollution levels were measured by air quality monitoring stations. Length of gestation data were collected on 143,196 births between the years 1989 and 1993, and air
pollution exposure during gestation was calculated for each birth. (a) Identify the population of interest and the sample in this study. (b) Comment on whether or not the results of the study can be generalized to the population, and if the ﬁndings of the study can be used to establish causal relationships. 1.14 Cheaters, scope of inference. Exercise 1.5 introduces a study where researchers studying the relationship between honesty, age, and self-control conducted an experiment on 160 children between the ages of 5 and 15. The researchers asked each child to toss a fair coin in private and to record the outcome (white or black) on a paper sheet, and said they would only reward children who report white. Half the students were explicitly told not to cheat and the others were not given any explicit instructions. Diﬀerences were observed in the cheating rates in the instruction and no instruction groups, as well as some diﬀerences across children’s characteristics within each group. (a) Identify the population of interest and the sample in this study. (b) Comment on whether or not the results of the study can be generalized to the population, and if the ﬁndings of the study can be used to establish causal relationships. 1.15 Buteyko method, scope of inference. Exercise 1.4 introduces a study on using the Buteyko shallow breathing technique to reduce asthma symptoms and improve quality of life. As part of this study 600 asthma patients aged 18-69 who relied on medication for asthma treatment were recruited and randomly assigned to two groups: one practiced the Buteyko method and the other did not. Those in the Buteyko group experienced, on average, a signiﬁcant reduction in asthma symptoms and an improvement in quality of life. (a) Identify the population of interest and the sample in this study. (b) Comment on whether or not the results of the study can be generalized to the population, and if the ﬁndings of the study can be used to establish causal relationships. 1.16 Stealers, scope of inference. Exercise 1.6 introduces a study on the relationship between socioeconomic class and unethical behavior. As part of this study 129 University of California Berkeley undergraduates were asked to identify themselves as having low or high social-class by comparing themselves to others with the most (least) money, most (least) education, and most (least) respected jobs. They were also
presented with a jar of individually wrapped candies and informed that the candies were for children in a nearby laboratory, but that they could take some if they wanted. After completing some unrelated tasks, participants reported the number of candies they had taken. It was found that those who were identiﬁed as upper-class took more candy than others. (a) Identify the population of interest and the sample in this study. (b) Comment on whether or not the results of the study can be generalized to the population, and if the ﬁndings of the study can be used to establish causal relationships. 32 CHAPTER 1. DATA COLLECTION 1.17 Relaxing after work. The General Social Survey asked the question, “After an average work day, about how many hours do you have to relax or pursue activities that you enjoy?” to a random sample of 1,155 Americans. The average relaxing time was found to be 1.65 hours. Determine which of the following is an observation, a variable, a sample statistic (value calculated based on the observed sample), or a population parameter. (a) An American in the sample. (b) Number of hours spent relaxing after an average work day. (c) 1.65. (d) Average number of hours all Americans spend relaxing after an average work day. 1.18 Cats on YouTube. Suppose you want to estimate the percentage of videos on YouTube that are cat videos. It is impossible for you to watch all videos on YouTube so you use a random video picker to select 1000 videos for you. You ﬁnd that 2% of these videos are cat videos. Determine which of the following is an observation, a variable, a sample statistic (value calculated based on the observed sample), or a population parameter. (a) Percentage of all videos on YouTube that are cat videos. (b) 2%. (c) A video in your sample. (d) Whether or not a video is a cat video. 1.4. OBSERVATIONAL STUDIES AND SAMPLING STRATEGIES 33 1.4 Observational studies and sampling strategies You have probably read or heard claims from many studies and polls. A background in statistical reasoning will help you assess the validity of such claims. Some of the big questions we address in this section include: • If a study ﬁnds a relationship between two variables, such as eating chocolate and positive health outcomes, is it reasonable to
conclude eating chocolate improves health outcomes? • How do opinion polls work? How do research organizations collect the data, and what types of bias should we look out for? Learning objectives 1. Identify possible confounding factors in a study and explain, in context, how they could con- found. 2. Distinguish among and describe a convenience sample, a volunteer sample, and a random sample. 3. Identify and describe the eﬀects of diﬀerent types of bias in sample surveys, including under- coverage, non-response, and response bias. 4. Identify and describe how to implement diﬀerent random sampling methods, including simple, systematic, stratiﬁed, and cluster. 5. Recognize the beneﬁts and drawbacks of choosing one sampling method over another. 6. Understand when it is valid to generalize and to what population that generalization can be made. 1.4.1 Observational studies Generally, data in observational studies are collected only by monitoring what occurs, while experiments require the primary explanatory variable in a study be assigned for each subject by the researchers. Making causal conclusions based on experiments is often reasonable. However, making the same causal conclusions based on observational data is treacherous and is not recommended. Observational studies are generally only suﬃcient to show associations. GUIDED PRACTICE 1.15 Suppose an observational study tracked sunscreen use and skin cancer, and it was found people who use sunscreen are more likely to get skin cancer than people who do not use sunscreen. Does this mean sunscreen causes skin cancer?23 Some previous research tells us that using sunscreen actually reduces skin cancer risk, so maybe there is another variable that can explain this hypothetical association between sunscreen usage and skin cancer. One important piece of information that is absent is sun exposure. Sun exposure is what is called a confounding variable (also called a lurking variable, confounding factor, or a confounder). 23No. See the paragraph following the exercise for an explanation. 34 CHAPTER 1. DATA COLLECTION CONFOUNDING VARIABLE A confounding variable is a variable that is associated with both the explanatory and response variables. Because of the confounding variable’s association with both variables, we do not know if the response is due to the explanatory variable or due to the confounding variable. Sun exposure is a confounding factor because it is associated with both the use of sunscreen and the development of skin cancer. People who are out in the sun all day are
more likely to use sunscreen, and people who are out in the sun all day are more likely to get skin cancer. Research shows us the development of skin cancer is due to the sun exposure. The variables of sunscreen usage and sun exposure are confounded, and without this research, we would have no way of knowing which one was the true cause of skin cancer. EXAMPLE 1.16 In a study that followed 1,169 non-diabetic adults who had been hospitalized for a ﬁrst heart attack, the people that reported eating chocolate had increased survival rate over the next 8 years than those that reported not eating chocolate. Also, those who ate more chocolate tended to live longer on average. The researchers controlled for several confounding factors, such as age, physical activity, smoking, and many other factors. Can we conclude that the consumption of chocolate caused the people to live longer? This is an observational study, not a controlled randomized experiment. Even though the researchers controlled for many possible variables, there may still be other confounding factors. (Can you think of any that weren’t mentioned?) While it is possible that the chocolate had an eﬀect, this study cannot prove that chocolate increased the survival rate of patients. EXAMPLE 1.17 The authors who conducted the study did warn in the article that additional studies would be necessary to determine whether the correlation between chocolate consumption and survival translates to any causal relationship. That is, they acknowledged that there may be confounding factors. One possible confounding factor not considered was mental health. In context, explain what it would mean for mental health to be a confounding factor in this study. Mental health would be a confounding factor if, for example, people with better mental health tended to eat more chocolate, and those with better mental health also were less likely to die within the 8 year study period. Notice that if better mental health were not associated with eating more chocolate, it would not be considered a confounding factor since it wouldn’t explain the observed associated between eating chocolate and having a better survival rate. If better mental health were associated only with eating chocolate and not with a better survival rate, then it would also not be confounding for the same reason. Only if a variable that is associated with both the explanatory variable of interest (chocolate) and the outcome variable in the study (survival during the 8 year study period) can it be considered a confounding factor. While one method to justify making causal conclusions from observational studies is to exhaust the search for confounding variables
, there is no guarantee that all confounding variables can be examined or measured. In the same way, the county data set is an observational study with confounding variables, and its data cannot be used to make causal conclusions. sun exposureuse sunscreenskin cancer? 1.4. OBSERVATIONAL STUDIES AND SAMPLING STRATEGIES 35 GUIDED PRACTICE 1.18 Figure 1.8 shows a negative association between the homeownership rate and the percentage of multi-unit structures in a county. However, it is unreasonable to conclude that there is a causal relationship between the two variables. Suggest one or more other variables that might explain the relationship visible in Figure 1.8.24 Observational studies come in two forms: prospective and retrospective studies. A prospective study identiﬁes individuals and collects information as events unfold. For instance, medical researchers may identify and follow a group of similar individuals over many years to assess the possible inﬂuences of behavior on cancer risk. One example of such a study is The Nurses’ Health Study, started in 1976 and expanded in 1989. This prospective study recruits registered nurses and then collects data from them using questionnaires. Retrospective studies collect data after events have taken place, e.g. researchers may review past events in medical records. Some data sets, such as county, may contain both prospectively- and retrospectively-collected variables. Local governments prospectively collect some variables as events unfolded (e.g. retails sales) while the federal government retrospectively collected others during the 2010 census (e.g. county population counts). 1.4.2 Sampling from a population We might try to estimate the time to graduation for Duke undergraduates in the last 5 years by collecting a sample of students. All graduates in the last 5 years represent the population, and graduates who are selected for review are collectively called the sample. The goal is to use information from the sample to generalize or make an inference to the population. In order to be able to generalize, we must randomly select a sample from the population of interest. The most basic type of random selection is equivalent to how raﬄes are conducted. For example, in selecting graduates, we could write each graduate’s name on a raﬄe ticket and draw 100 tickets. The selected names would represent a random sample of 100 graduates. Figure 1.11: In this graphic, ﬁve graduates are randomly selected from the population
to be included in the sample. Why pick a sample randomly? Why not just pick a sample by hand? Consider the following scenario. 24Answers will vary. Population density may be important. If a county is very dense, then this may require a larger fraction of residents to live in multi-unit structures. Additionally, the high density may contribute to increases in property value, making homeownership infeasible for many residents. all graduatessample 36 CHAPTER 1. DATA COLLECTION Figure 1.12: Instead of sampling from all graduates equally, a nutrition major might inadvertently pick graduates with health-related majors disproportionately often. EXAMPLE 1.19 Suppose we ask a student who happens to be majoring in nutrition to select several graduates for the study. What kind of students do you think she might select? Do you think her sample would be representative of all graduates? Perhaps she would pick a disproportionate number of graduates from health-related ﬁelds. Or perhaps her selection would be well-representative of the population. When selecting samples by hand, we run the risk of picking a biased sample, even if that bias is unintentional or diﬃcult to discern. If the student majoring in nutrition picked a disproportionate number of graduates from healthrelated ﬁelds, this would introduce undercoverage bias into the sample. Undercoverage bias occurs when some individuals of the population are inherently less likely to be included in the sample than others, making the sample not representative of the population. In the example, this bias creates a problem because a degree in health-related ﬁelds might take more or less time to complete than a degree in other ﬁelds. Suppose that it takes longer. Since graduates from other ﬁelds would be less likely to be in the sample, the undercoverage bias would cause her to overestimate the parameter. Sampling randomly resolves the problem of undercoverage bias, if the sample is randomly selected from the entire population of interest. If the sample is randomly selected from only a subset of the population, say, only graduates from health-related ﬁelds, then the sample will not be representative of the population of interest. Generalizations can only be made to the population from which the sample is randomly selected. The most basic random sample is called a simple random sample, which is equivalent to using a raﬄe to select cases. This means that each case in the
population has an equal chance of being included and there is no implied connection between the cases in the sample. A common downfall is a convenience sample, where individuals who are easily accessible are more likely to be included in the sample. For instance, if a political survey is done by stopping people walking in the Bronx, this will not represent all of New York City. It is often diﬃcult to discern what sub-population a convenience sample represents. Figure 1.13: Due to the possibility of non-response, surveys studies may only reach a certain group within the population. It is diﬃcult, and often times impossible, to completely ﬁx this problem. all graduatessamplegraduates fromhealth−related fieldspopulation of interestsamplepopulation actuallysampled 1.4. OBSERVATIONAL STUDIES AND SAMPLING STRATEGIES 37 Similarly, a volunteer sample is one in which people’s responses are solicited and those who choose to participate, respond. This is a problem because those who choose to participate may tend to have diﬀerent opinions than the rest of the population, resulting in a biased sample. GUIDED PRACTICE 1.20 We can easily access ratings for products, sellers, and companies through websites. These ratings are based only on those people who go out of their way to provide a rating. If 50% of online reviews for a product are negative, do you think this means that 50% of buyers are dissatisﬁed with the product?25 The act of taking a random sample helps minimize bias; however, bias can crop up in other ways. Even when people are picked at random, e.g. for surveys, caution must be exercised if the non-response is high. For instance, if only 30% of the people randomly sampled for a survey actually respond, then it is unclear whether the results are representative of the entire population. This non-response bias can skew results. Even if a sample has no undercoverage bias and no non-response bias, there is an additional type of bias that often crops up and undermines the validity of results, known as response bias. Response bias refers to a broad range of factors that inﬂuence how a person responds, such as question wording, question order, and inﬂuence of the interviewer. This type of bias can be present even when we collect data from an entire population in what is called a census. Because
response bias is often subtle, one must pay careful attention to how questions were asked when attempting to draw conclusions from the data. EXAMPLE 1.21 Suppose a high school student wants to investigate the student body’s opinions on the food in the cafeteria. Let’s assume that she manages to survey every student in the school. How might response bias arise in this context? There are many possible correct answers to this question. For example, students might respond diﬀerently depending upon who asks the question, such as a school friend or someone who works in the cafeteria. The wording of the question could introduce response bias. Students would likely respond diﬀerently if asked “Do you like the food in the cafeteria?” versus “The food in the cafeteria is pretty bad, don’t you think?” WATCH OUT FOR BIAS Undercoverage bias, non-response bias, and response bias can still exist within a random sample. Always determine how a sample was chosen, ask what proportion of people failed to respond, and critically examine the wording of the questions. When there is no bias in a sample, increasing the sample size tends to increase the precision and reliability of the estimate. When a sample is biased, it may be impossible to decipher helpful information from the data, even if the sample is very large. 25Answers will vary. From our own anecdotal experiences, we believe people tend to rant more about products that fell below expectations than rave about those that perform as expected. For this reason, we suspect there is a negative bias in product ratings on sites like Amazon. However, since our experiences may not be representative, we also keep an open mind. 38 CHAPTER 1. DATA COLLECTION GUIDED PRACTICE 1.22 A researcher sends out questionnaires to 50 randomly selected households in a particular town asking whether or not they support the addition of a traﬃc light in their neighborhood. Because only 20% of the questionnaires are returned, she decides to mail questionnaires to 50 more randomly selected households in the same neighborhood. Comment on the usefulness of this approach.26 1.4.3 Simple, systematic, stratiﬁed, cluster, and multistage sampling Almost all statistical methods for observational data rely on a sample being random and unbiased. When a sample is collected in a biased way, these statistical methods will not generally produce reliable information about the population. The idea of a simple random sample was introduced in the
last section. Here we provide a more technical treatment of this method and introduce four new random sampling methods: systematic, stratiﬁed, cluster, and multistage.27 Figure 1.14 provides a graphical representation of simple versus systematic sampling while Figure 1.15 provides a graphical representation of stratiﬁed, cluster, and multistage sampling. Simple random sampling is probably the most intuitive form of random sampling. Consider the salaries of Major League Baseball (MLB) players, where each player is a member of one of the league’s 30 teams. For the 2019 season, N, the population size or total number of players, is 750. To take a simple random sample of n = 120 of these baseball players and their salaries, we could number each player from 1 to 750. Then we could randomly select 120 numbers between 1 and 750 (without replacement) using a random number generator or random digit table. The players with the selected numbers would comprise our sample. Two properties are always true in a simple random sample: 1. Each case in the population has an equal chance of being included in the sample. 2. Each group of n cases has an equal chance of making up the sample. The statistical methods in this book focus on data collected using simple random sampling. Note that Property 2 – that each group of n cases has an equal chance making up the sample – is not true for the remaining four sampling techniques. As you read each one, consider why. Though less common than simple random sampling, systematic sampling is sometimes used when there exists a convenient list of all of the individuals of the population. Suppose we have a roster with the names of all the MLB players from the 2019 season. To take a systematic random sample, number them from 1 to 750. Select one random number between 1 and 750 and let that player be the ﬁrst individual in the sample. Then, depending on the desired sample size, select every 10th number or 20th number, for example, to arrive at the sample.28 If there are no patterns in the salaries based on the numbering then this could be a reasonable method. EXAMPLE 1.23 A systematic sample is not the same as a simple random sample. Provide an example of a sample that can come from a simple random sample but not from a systematic random sample. Answers can vary. If we take a sample of size 3, then it is possible that we could sample players numbered 1, 2, and 3 in a simple random
sample. Such a sample would be impossible from a systematic sample. Property 2 of simple random samples does not hold for other types of random samples. 26The researcher should be concerned about non-response bias, and sampling more people will not eliminate this issue. The same type of people that did not respond to the ﬁrst survey are likely not going to respond to the second survey. Instead, she should make an eﬀort to reach out to the households from the original sample that did not respond and solicit their feedback, possibly by going door-to-door. 27Multistage sampling is not part of the AP syllabus. 28If we want a sample of size n = 150, it would make sense to select every 5th player since 750/150 = 5. Suppose we randomly select the number 741. Then player 741, 746, 1, 6, 11, · · ·, 731, and 736 would make up the sample. 1.4. OBSERVATIONAL STUDIES AND SAMPLING STRATEGIES 39 Figure 1.14: Examples of simple random sampling and systematic sampling. In the top panel, simple random sampling was used to randomly select 18 cases. In the lower panel, systematic random sampling was used to select every 7th individual. Index●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 40 CHAPTER 1. DATA COLLECTION Sometimes there is a variable that is known to be associated with the quantity we want to estimate. In this case, a stratiﬁed random sample might be selected. Stratiﬁed sampling is a divide-and-conquer sampling strategy. The population is divided into groups called strata. The strata are chosen so that similar cases are grouped together and a sampling method, usually simple random sampling, is employed to select a certain number or a certain proportion of the whole within each stratum. In the baseball salary example, the 30 teams could represent the strata; some teams have a lot more money (we’re looking at you, Yankees). EXAMPLE 1.24 For this baseball example, brieﬂy explain how to select a stratiﬁed random sample of size n = 120. Each team can serve as a stratum, and we could take a simple random sample of 4 players from each of the 30 teams, yielding a sample of 120 players. Stratiﬁed sampling is inherently diﬀerent than simple random sampling. For example, the stratiﬁed sampling approach described would make it impossible for the entire Yankees team to be included in the sample. EXAMPLE 1.25 Stratiﬁed sampling is especially useful when the cases in each stratum are very similar with respect to the outcome of interest. Why is it good for cases within each stratum to be very similar? We should get a more stable estimate for the subpopulation in a stratum if the cases are very similar. These improved estimates for each subpopulation will help us build a reliable estimate for the full population. For example, in a simple random sample, it is possible that just by random chance we could end up with proportionally too many Yankees players in our sample, thus overestimating the true average salary of all MLB players. A stratiﬁed random sample can assure proportional representation from
each team. Next, let’s consider a sampling technique that randomly selects groups of people. Cluster sampling is much like simple random sampling, but instead of randomly selecting individuals, we randomly select groups or clusters. Unlike stratiﬁed sampling, cluster sampling is most helpful when there is a lot of case-to-case variability within a cluster but the clusters themselves don’t look very diﬀerent from one another. That is, we expect individual strata to be homogeneous (self-similar), while we expect individual clusters to be heterogeneous (diverse) with respect to the variable of interest. Sometimes cluster sampling can be a more economical random sampling technique than the alternatives. For example, if neighborhoods represented clusters, this sampling method works best when each neighborhood is very diverse. Because each neighborhood itself encompasses diversity, a cluster sample can reduce the time and cost associated with data collection, because the interviewer would need only go to some of the neighborhoods rather than to all parts of a city, in order to collect a useful sample. Multistage sampling, also called multistage cluster sampling, is a two (or more) step strategy. The ﬁrst step is to take a cluster sample, as described above. Then, instead of including all of the individuals in these clusters in our sample, a second sampling method, usually simple random sampling, is employed within each of the selected clusters. In the neighborhood example, we could ﬁrst randomly select some number of neighborhoods and then take a simple random sample from just those selected neighborhoods. As seen in Figure 1.15, stratiﬁed sampling requires observations to be sampled from every stratum. Multistage sampling selects observations only from those clusters that were randomly selected in the ﬁrst step. It is also possible to have more than two steps in multistage sampling. Each cluster may be naturally divided into subclusters. For example, each neighborhood could be divided into streets. To take a three-stage sample, we could ﬁrst select some number of clusters (neighborhoods), and then, within the selected clusters, select some number of subclusters (streets). Finally, we could select some number of individuals from each of the selected streets. 1.4. OBSERVATIONAL STUDIES AND SAMPLING STRATEGIES 41 Figure 1.15: Examples of stratiﬁed, cluster, and multistage sampling.
In the top panel, stratiﬁed sampling was used: cases were grouped into strata, and then simple random sampling was employed within each stratum. In the middle panel, cluster sampling was used, where data were binned into nine cluster and three clusters were randomly selected. In the bottom panel, multistage sampling was used. Data were binned into the nine clusters, three of the cluster were randomly selected, and then six cases were randomly sampled in each of the three selected clusters. IndexlllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllStratum 1Stratum 2Stratum 3Stratum 4Stratum 5Stratum 6IndexlllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllCluster 1Cluster 2Cluster 3Cluster 4Cluster 5Cluster 6Cluster 7Cluster 8Cluster 9lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllCluster 1Cluster 2Cluster 3Cluster 4Cluster 5Cluster 6Cluster 7Cluster 8Cluster 9 42 CHAPTER 1. DATA COLLECTION EXAMPLE 1.26 Suppose we are interested in estimating the proportion of students at a certain school that have part-time jobs. It is believed that older students are more likely to work than younger students. What sampling method should be
employed? Describe how to collect such a sample to get a sample size of 60. Because grade level aﬀects the likelihood of having a part-time job, we should take a stratiﬁed random sample. To do this, we can take a simple random sample of 15 students from each grade. This will give us equal representation from each grade. Note: in a simple random sample, just by random chance we might get too many students who are older or younger, which could make the estimate too high or too low. Also, there are no well-deﬁned clusters in this example. We wouldn’t want to use the grades as clusters and sample everyone from a couple of the grades. This would create too large a sample and would not give us the nice representation from each grade aﬀorded by the stratiﬁed random sample. EXAMPLE 1.27 Suppose we are interested in estimating the malaria rate in a densely tropical portion of rural Indonesia. We learn that there are 30 villages in that part of the Indonesian jungle, each more or less similar to the next. Our goal is to test 150 individuals for malaria. What sampling method should be employed? A simple random sample would likely draw individuals from all 30 villages, which could make data collection extremely expensive. Stratiﬁed sampling would be a challenge since it is unclear how we would build strata of similar individuals. However, multistage cluster sampling seems like a very good idea. First, we might randomly select half the villages, then randomly select 10 people from each. This would probably reduce our data collection costs substantially in comparison to a simple random sample and would still give us reliable information. ADVANCED SAMPLING TECHNIQUES REQUIRE ADVANCED METHODS The methods of inference covered in this book generally only apply to simple random samples. More advanced analysis techniques are required for systematic, stratiﬁed, cluster, and multistage random sampling. 1.4. OBSERVATIONAL STUDIES AND SAMPLING STRATEGIES 43 Section summary • In an observational study, one must always consider the existence of confounding factors. A confounding factor is a “spoiler variable” that could explain an observed relationship between the explanatory variable and the response. Remember: For a variable to be confounding it must be associated with both the explanatory variable and the response variable. • When taking a sample from a population, avoid convenience samples
and volunteer sam- ples, which likely introduce bias. Instead, use a random sampling method. • Generalizations from a sample can be made to a population only if the sample is random. Furthermore, the generalization can be made only to the population from which the sample was randomly selected, not to a larger or diﬀerent population. • Random sampling from the entire population of interest avoids the problem of undercoverage bias. However, response bias and non-response bias can be present in any type of sample, random or not. • In a simple random sample, every individual as well as every group of individuals has the same probability of being in the sample. A common way to select a simple random sample is to number each individual of the population from 1 to N. Using a random digit table or a random number generator, numbers are randomly selected without replacement and the corresponding individuals become part of the sample. • A systematic random sample involves choosing from of a population using a random starting point, and then selecting members according to a ﬁxed, periodic interval (such as every 10th member). • A stratiﬁed random sample involves randomly sampling from every strata, where the strata should correspond to a variable thought to be associated with the variable of interest. This ensures that the sample will have appropriate representation from each of the diﬀerent strata and reduces variability in the sample estimates. • A cluster random sample involves randomly selecting a set of clusters, or groups, and then collecting data on all individuals in the selected clusters. This can be useful when sampling clusters is more convenient and less expensive than sampling individuals, and it is an eﬀective strategy when each cluster is approximately representative of the population. • Remember: Individual strata should be homogeneous (self-similar), while individual clusters should be heterogeneous (diverse). For example, if smoking is correlated with what is being estimated, let one stratum be all smokers and the other be all non-smokers, then randomly select an appropriate number of individuals from each strata. Alternately, if age is correlated with the variable being estimated, one could randomly select a subset of clusters, where each cluster has mixed age groups. 44 CHAPTER 1. DATA COLLECTION Exercises 1.19 Course satisfaction across sections. A large college class has 160 students. All 160 students attend the lectures together, but the students are divided into 4 groups, each of 40 students, for lab sections administered by di
ﬀerent teaching assistants. The professor wants to conduct a survey about how satisﬁed the students are with the course, and he believes that the lab section a student is in might aﬀect the student’s overall satisfaction with the course. (a) What type of study is this? (b) Suggest a sampling strategy for carrying out this study. 1.20 Housing proposal across dorms. On a large college campus ﬁrst-year students and sophomores live in dorms located on the eastern part of the campus and juniors and seniors live in dorms located on the western part of the campus. Suppose you want to collect student opinions on a new housing structure the college administration is proposing and you want to make sure your survey equally represents opinions from students from all years. (a) What type of study is this? (b) Suggest a sampling strategy for carrying out this study. 1.21 Internet use and life expectancy. The following scatterplot was created as part of a study evaluating the relationship between estimated life expectancy at birth (as of 2014) and percentage of internet users (as of 2009) in 208 countries for which such data were available.29 (a) Describe the relationship between life expectancy and percentage of internet users. (b) What type of study is this? (c) State a possible confounding variable that might explain this relationship and describe its potential eﬀect. 1.22 Stressed out, Part I. A study that surveyed a random sample of otherwise healthy high school students found that they are more likely to get muscle cramps when they are stressed. The study also noted that students drink more coﬀee and sleep less when they are stressed. (a) What type of study is this? (b) Can this study be used to conclude a causal relationship between increased stress and muscle cramps? (c) State possible confounding variables that might explain the observed relationship between increased stress and muscle cramps. 1.23 Evaluate sampling methods. A university wants to determine what fraction of its undergraduate student body support a new $25 annual fee to improve the student union. For each proposed method below, indicate whether the method is reasonable or not. (a) Survey a simple random sample of 500 students. (b) Stratify students by their ﬁeld of study, then sample 10% of students from each stratum. (c) Cluster students by their ages (e.g. 18 years
old in one cluster, 19 years old in one cluster, etc.), then randomly sample three clusters and survey all students in those clusters. 1.24 Random digit dialing. The Gallup Poll uses a procedure called random digit dialing, which creates phone numbers based on a list of all area codes in America in conjunction with the associated number of residential households in each area code. Give a possible reason the Gallup Poll chooses to use random digit dialing instead of picking phone numbers from the phone book. 29CIA Factbook, Country Comparisons, 2014. Percent Internet UsersLife Expectancy at Birth0%20%40%60%80%100%5060708090 1.4. OBSERVATIONAL STUDIES AND SAMPLING STRATEGIES 45 1.25 Haters are gonna hate, study conﬁrms. A study published in the Journal of Personality and Social Psychology asked a group of 200 randomly sampled men and women to evaluate how they felt about various subjects, such as camping, health care, architecture, taxidermy, crossword puzzles, and Japan in order to measure their attitude towards mostly independent stimuli. Then, they presented the participants with information about a new product: a microwave oven. This microwave oven does not exist, but the participants didn’t know this, and were given three positive and three negative fake reviews. People who reacted positively to the subjects on the dispositional attitude measurement also tended to react positively to the microwave oven, and those who reacted negatively tended to react negatively to it. Researchers concluded that “some people tend to like things, whereas others tend to dislike things, and a more thorough understanding of this tendency will lead to a more thorough understanding of the psychology of attitudes.”30 (a) What are the cases? (b) What is (are) the response variable(s) in this study? (c) What is (are) the explanatory variable(s) in this study? (d) Does the study employ random sampling? (e) Is this an observational study or an experiment? Explain your reasoning. (f) Can we establish a causal link between the explanatory and response variables? (g) Can the results of the study be generalized to the population at large? 1.26 Family size. Suppose we want to estimate household size, where a “household” is deﬁned as people living together in the same dwelling, and sharing living accommodations. If we select students at random at an elementary school and
ask them what their family size is, will this be a good measure of household size? Or will our average be biased? If so, will it overestimate or underestimate the true value? 1.27 Sampling strategies. A statistics student who is curious about the relationship between the amount of time students spend on social networking sites and their performance at school decides to conduct a survey. Various research strategies for collecting data are described below. In each, name the sampling method proposed and any bias you might expect. (a) He randomly samples 40 students from the study’s population, gives them the survey, asks them to ﬁll it out and bring it back the next day. (b) He gives out the survey only to his friends, making sure each one of them ﬁlls out the survey. (c) He posts a link to an online survey on Facebook and asks his friends to ﬁll out the survey. (d) He randomly samples 5 classes and asks a random sample of students from those classes to ﬁll out the survey. 1.28 Reading the paper. Below are excerpts from two articles published in the NY Times: (a) An article titled Risks: Smokers Found More Prone to Dementia states the following:31 “Researchers analyzed data from 23,123 health plan members who participated in a voluntary exam and health behavior survey from 1978 to 1985, when they were 50-60 years old. 23 years later, about 25% of the group had dementia, including 1,136 with Alzheimer’s disease and 416 with vascular dementia. After adjusting for other factors, the researchers concluded that pack-a-day smokers were 37% more likely than nonsmokers to develop dementia, and the risks went up with increased smoking; 44% for one to two packs a day; and twice the risk for more than two packs.” Based on this study, can we conclude that smoking causes dementia later in life? Explain your reasoning. (b) Another article titled The School Bully Is Sleepy states the following:32 “The University of Michigan study, collected survey data from parents on each child’s sleep habits and asked both parents and teachers to assess behavioral concerns. About a third of the students studied were identiﬁed by parents or teachers as having problems with disruptive behavior or bullying. The researchers found that children who had behavioral issues and those who were identiﬁed as bullies were twice as likely to
have shown symptoms of sleep disorders.” A friend of yours who read the article says, “The study shows that sleep disorders lead to bullying in school children.” Is this statement justiﬁed? If not, how best can you describe the conclusion that can be drawn from this study? 30Justin Hepler and Dolores Albarrac´ın. “Attitudes without objects - Evidence for a dispositional attitude, its measurement, and its consequences”. In: Journal of personality and social psychology 104.6 (2013), p. 1060. 31R.C. Rabin. “Risks: Smokers Found More Prone to Dementia”. In: New York Times (2010). 32T. Parker-Pope. “The School Bully Is Sleepy”. In: New York Times (2011). 46 CHAPTER 1. DATA COLLECTION 1.5 Experiments You would like to determine if drinking a cup of tea each morning will cause students to perform better on tests. What are diﬀerent ways you could design an experiment to answer this question? What are possible sources of bias, and how would you try to minimize them? The goal of an experiment is to be able to draw a causal conclusion about the eﬀect of a treatment – in this case, drinking tea. If the design is poor, a causal conclusion cannot be drawn, even if you observe an association between drinking tea and performing better on tests. This is why it is crucial to start with a well-designed experiment. Learning objectives 1. Identify the subjects/experimental units, treatments, and response variable in an experiment. 2. Identify the three main principles of experiment design and explain their purpose: direct control, randomization, and replication. 3. Explain placebo eﬀect and describe when and how to implement a single-blind and a double- blind experiment. 4. Identify and describe how to implement the following three experimental designs: completely randomized design, blocked design, and matched pairs design. 5. Explain the purpose of random assignment or randomization in each of the three experimental designs. 6. Explain how to randomize treatments in a completely randomized design using technology or a table of random digits (make sure this is explained). 7. Explain when it is reasonable to draw a causal conclusion about the eﬀect of a treatment. 8. Identify the number of factors in experiment, the number of
levels for each factor and the total number of treatments. 1.5.1 Reducing bias in human experiments In the last section we investigated observational studies and sampling strategies. While these are eﬀective tools for answering certain research questions, often times researchers want to measure the eﬀect of a treatment. In this case, they must carry out an experiment. Just as randomization is essential in sampling in order to avoid selection bias, randomization is essential in the context of experiments to determine which subjects will receive which treatments. If the researcher chooses which patients are in the treatment and control groups, she may unintentionally place sicker patients in the treatment group, biasing the experiment against the treatment. Randomized experiments are essential for investigating cause and eﬀect relationships, but they do not ensure an unbiased perspective in all cases. Human studies are perfect examples where bias can unintentionally arise. Here we reconsider a study where a new drug was used to treat heart attack patients. In particular, researchers wanted to know if the drug reduced deaths in patients. These researchers designed a randomized experiment because they wanted to draw causal conclusions about the drug’s eﬀect. Study volunteers33 were randomly placed into two study groups. One group, the treatment group, received the drug. The other group, called the control group, In an experiment, the explanatory variable is also called a did not receive any drug treatment. It has two levels: yes and no, thus it factor. Here the factor is receiving the drug treatment. 33Human subjects are often called patients, volunteers, or study participants. 1.5. EXPERIMENTS 47 is categorical. The response variable is whether or not patients died within the time frame of the study. It is also categorical. Put yourself in the place of a person in the study. If you are in the treatment group, you are given a fancy new drug that you anticipate will help you. On the other hand, a person in the other group doesn’t receive the drug and sits idly, hoping her participation doesn’t increase her risk of death. These perspectives suggest there are actually two eﬀects: the one of interest is the eﬀectiveness of the drug, and the second is an emotional eﬀect that is diﬃcult to quantify. Researchers aren’t usually interested in the emotional eﬀect, which might bias the study. To circumvent this problem, researchers do not
want patients to know which group they are in. When researchers keep the patients uninformed about their treatment, the study is said to be blind or single-blind. But there is one problem: if a patient doesn’t receive a treatment, she will know she is in the control group. The solution to this problem is to give fake treatments to patients in the control group. A fake treatment is called a placebo, and an eﬀective placebo is the key to making a study truly blind. A classic example of a placebo is a sugar pill that is made to look like the actual treatment pill. Often times, a placebo results in a slight but real improvement in patients. This eﬀect has been dubbed the placebo eﬀect. The patients are not the only ones who should be blinded: doctors and researchers can accidentally bias a study. When a doctor knows a patient has been given the real treatment, she might inadvertently give that patient more attention or care than a patient that she knows is on the placebo. To guard against this bias, which again has been found to have a measurable eﬀect in some instances, most modern studies employ a double-blind setup where researchers who interact with subjects and are responsible for measuring the response variable are, just like the subjects, unaware of who is or is not receiving the treatment.34 GUIDED PRACTICE 1.28 Look back to the study in Section 1.1 where researchers were testing whether stents were eﬀective at reducing strokes in at-risk patients. Is this an experiment? Was the study blinded? Was it double-blinded?35 1.5.2 Principles of experimental design Well-conducted experiments are built on three main principles. Direct Control. Researchers assign treatments to cases, and they do their best to control any other diﬀerences in the groups. They want the groups to be as identical as possible except for the treatment, so that at the end of the experiment any diﬀerence in response between the groups can be attributed to the treatment and not to some other confounding or lurking variable. For example, when patients take a drug in pill form, some patients take the pill with only a sip of water while others may have it with an entire glass of water. To control for the eﬀect of water consumption, a doctor may ask all patients to drink a 12 ounce glass of water with the pill. Direct control refers to variables that the researcher can control,
or make the same. A researcher can directly control the appearance of the treatment, the time of day it is taken, etc. She cannot directly control variables such as gender or age. To control for these other types of variables, she might consider blocking, which is described in Section 1.5.3. Randomization. Researchers randomize patients into treatment groups to account for variables that cannot be controlled. For example, some patients may be more susceptible to a disease than others due to their dietary habits. Randomizing patients into the treatment or control group helps even out the eﬀects of such diﬀerences, and it also prevents accidental bias from entering the study. 34There are always some researchers involved in the study who do know which patients are receiving which treatment. However, they do not interact with the study’s patients and do not tell the blinded health care professionals who is receiving which treatment. 35The researchers assigned the patients into their treatment groups, so this study was an experiment. However, the patients could distinguish what treatment they received, so this study was not blind. The study could not be double-blind since it was not blind. 48 CHAPTER 1. DATA COLLECTION Replication. The more cases researchers observe, the more accurately they can estimate the eﬀect of the explanatory variable on the response. In an experiment with six subjects, even if there is randomization, it is quite possible for the three healthiest people to be in the same treatment group. In a randomized experiment with 100 people, it is virtually impossible for the healthiest 50 people to end up in the same treatment group. In a single study, we replicate by imposing the treatment on a suﬃciently large number of subjects or experimental units. A group of scientists may also replicate an entire study to verify an earlier ﬁnding. However, each study should ensure a suﬃciently large number of subjects because, in many cases, there is no opportunity or funding to carry out the entire experiment again. It is important to incorporate these design principles into any experiment. If they are lacking, the inference methods presented in the following chapters will not be applicable and their results may not be trustworthy. In the next section we will consider three types of experimental design. 1.5.3 Completely randomized, blocked, and matched pairs design A completely randomized experiment is one in which the subjects or experimental units are randomly assigned to each group in the experiment. Suppose we have three
treatments, one of which may be a placebo, and 300 subjects. To carry out a completely randomized design, we could randomly assign each subject a unique number from 1 to 300, then subjects with numbers 1-100 would get treatment 1, subjects 101-200 would get treatment 2, and subjects 201- 300 would get treatment 3. Note that this method of randomly allocating subjects to treatments in not equivalent to taking a simple random sample. Here we are not sampling a subset of a population; we are randomly splitting subjects into groups. While it might be ideal for the subjects to be a random sample of the population of interest, that is rarely the case. Subjects must volunteer to be part of an experiment. However, because randomization is incorporated in the splitting of the groups, we can still use statistical techniques to check for a causal connection, though the precise population for which the conclusion applies may be unclear. For example, if an experiment to determine the most eﬀective means to encourage individuals to vote is carried out only on college students, we may not be able to generalize the conclusions of the experiment to all adults in the population. Researchers sometimes know or suspect that another variable, other than the treatment, inﬂuences the response. Under these circumstances, they may carry out a blocked experiment. In this design, they ﬁrst group individuals into blocks based on the identiﬁed variable and then randomize subjects within each block to the treatment groups. This strategy is referred to as blocking. For instance, if we are looking at the eﬀect of a drug on heart attacks, we might ﬁrst split patients in the study into low-risk and high-risk blocks. Then we can randomly assign half the patients from each block to the control group and the other half to the treatment group, as shown in Figure 1.16. At the end of the experiment, we would incorporate this blocking into the analysis. By blocking by risk of patient, we control for this possible confounding factor. Additionally, by randomizing subjects to treatments within each block, we attempt to even out the eﬀect of variables that we cannot block or directly control. EXAMPLE 1.29 An experiment will be conducted to compare the eﬀectiveness of two methods for quitting smoking. Identify a variable that the researcher might wish to use for blocking and describe how she would carry out a blocked experiment. The researcher should choose the variable that is most likely
to inﬂuence the response variable whether or not a smoker will quit. A reasonable variable, therefore, would be the number of years that the smoker has been smoking. The subjects could be separated into three blocks based on number of years of smoking and each block randomly divided into the two treatment groups. 1.5. EXPERIMENTS 49 Figure 1.16: Blocking using a variable depicting patient risk. Patients are ﬁrst divided into low-risk and high-risk blocks, then each block is evenly separated into the treatment groups using randomization. This strategy ensures an equal representation of patients in each treatment group from both the low-risk and high-risk categories. Numbered patientscreateblocksLow−risk patientsHigh−risk patientsrandomlysplit in halfrandomlysplit in halfControlTreatmentl1l2l3l4l5l6l7l8l9l10l11l12l13l14l15l16l17l18l19l20l21l22l23l24l25l26l27l28l29l30l31l32l33l34l35l36l37l38l39l40l41l42l43l44l45l46l47l48l49l50l51l52l53l54l2l5l6l8l13l16l17l21l23l29l33l34l36l37l39l41l45l46l47l50l53l54l1l3l4l7l9l10l11l12l14l15l18l19l20l22l24l25l26l27l28l30l31l32l35l38l40l42l43l44l48l49l51l52l1l2l3l4l5l6l7l8l9l10l11l12l13l14l15l16l17l18l19l20l21l22l23l24l25l26l27l28l29l30l31l32l33l34l35l36l37l38l39l40l41l42l43l44l45l46l47l48l49l50l51l52l53l54 50 CHAPTER 1. DATA COLLECTION Even in a blocked experiment with randomization, other
variables that aﬀect the response can be distributed unevenly among the treatment groups, thus biasing the experiment in one direction. In a matched pairs A third type of design, known as matched pairs addresses this problem. experiment, pairs of people are matched on as many variables as possible, so that the comparison happens between very similar cases. This is actually a special type of blocked experiment, where the blocks are of size two. An alternate form of matched pairs involves each subject receiving both treatments. Randomization can be incorporated by randomly selecting half the subjects to receive treatment 1 ﬁrst, followed by treatment 2, while the other half receives treatment 2 ﬁrst, followed by treatment. GUIDED PRACTICE 1.30 How and why should randomization be incorporated into a matched pairs design?36 GUIDED PRACTICE 1.31 Matched pairs sometimes involves each subject receiving both treatments at the same time. For example, if a hand lotion was being tested, half of the subjects could be randomly assigned to put Lotion A on the left hand and Lotion B on the right hand, while the other half of the subjects would put Lotion B on the left hand and Lotion A on the right hand. Why would this be a better design than a completely randomized experiment in which half of the subjects put Lotion A on both hands and the other half put Lotion B on both hands?37 Because it is essential to identify the type of data collection method used when choosing an appropriate inference procedure, we will revisit sampling techniques and experiment design in the subsequent chapters on inference. 1.5.4 Testing more than one variable at a time Some experiments study more than one factor (explanatory variable) at a time, and each of these factors may have two or more levels (possible values). For example, suppose a researcher plans to investigate how the type and volume of music aﬀect a person’s performance on a particular video game. Because these two factors, type and volume, could interact in interesting ways, we do not want to test one factor at a time. Instead, we want to do an experiment in which we test all combinations of the factors. Let’s say that volume has two levels (soft and loud) and that type has three levels (dance, classical, and punk). Then, we would want to have experiment groups for each of the six (2 x 3 = 6) combinations: soft dance, soft classical,
soft punk, loud dance, loud classical, loud punk. Each combination is a treatment. Therefore, this experiment will have 2 factors and 6 treatments. To replicate each treatment 10 times, one would need to play the game 60 times. GUIDED PRACTICE 1.32 A researcher wants to compare the eﬀectiveness of four diﬀerent drugs. She also wants to test each of the drugs at two doses: low and high. Describe the factors, levels, and treatments of this experiment.38 As the number of factors and levels increases, the number of treatments become large and the analysis of the resulting data becomes more complex, requiring the use of advanced statistical methods. We will investigate only one factor at a time in this book. 36Assume that all subjects received treatment 1 ﬁrst, followed by treatment 2. If the variable being measured happens to increase naturally over the course of time, it would appear as though treatment 2 had a greater eﬀect than it really did. 37The dryness of people’s skins varies from person to person, but probably less so from one person’s right hand to left hand. With the matched pairs design, we are able control for this variability by comparing each person’s right hand to her left hand, rather than comparing some people’s hands to other people’s hands (as you would in a completely randomized experiment). 38There are two factors: type of drug, which has four levels, and dose, which has 2 levels. There will be 4 x 2 = 8 treatments: drug 1 at low dose, drug 1 at high dose, drug 2 at low dose, and so on. 1.5. EXPERIMENTS Section summary 51 • In an experiment, researchers impose a treatment to test its eﬀects. In order for observed diﬀerences in the response to be attributed to the treatment and not to some other factor, it is important to make the treatment groups and the conditions for the treatment groups as similar as possible. • Researchers use direct control, ensuring that variables that are within their power to modify (such as drug dosage or testing conditions) are made the same for each treatment group. • Researchers randomly assign subjects to the treatment groups so that the eﬀects of uncontrolled and potentially confounding variables are evened out among the treatment groups. • Replication, or imposing the treatments on many subjects, gives more data and decreases the likelihood
that the treatment groups diﬀer on some characteristic due to chance alone (i.e. in spite of the randomization). • An ideal experiment is randomized, controlled, and double-blind. • A completely randomized experiment involves randomly assigning the subjects to the diﬀerent treatment groups. To do this, ﬁrst number the subjects from 1 to N. Then, randomly choose some of those numbers and assign the corresponding subjects to a treatment group. Do this in such a way that the treatment group sizes are balanced, unless there exists a good reason to make one treatment group larger than another. • In a blocked experiment, subjects are ﬁrst separated by a variable thought to aﬀect the response variable. Then, within each block, subjects are randomly assigned to the treatment groups as described above, allowing the researcher to compare like to like within each block. • When feasible, a matched-pairs experiment is ideal, because it allows for the best comparison of like to like. A matched-pairs experiment can be carried out on pairs of subjects that are meaningfully paired, such as twins, or it can involve all subjects receiving both treatments, allowing subjects to be compared to themselves. • A treatment is also called a factor or explanatory variable. Each treatment/factor can have multiple levels, such as yes/no or low/medium/high. When an experiment includes many factors, multiplying the number of levels of the factors together gives the total number of treatment groups. • In an experiment, blocking, randomization, and direct control are used to control for confound- ing factors. 52 CHAPTER 1. DATA COLLECTION Exercises 1.29 Light and exam performance. A study is designed to test the eﬀect of light level on exam performance of students. The researcher believes that light levels might have diﬀerent eﬀects on males and females, so wants to make sure both are equally represented in each treatment. The treatments are ﬂuorescent overhead lighting, yellow overhead lighting, no overhead lighting (only desk lamps). (a) What is the response variable? (b) What is the explanatory variable? What are its levels? (c) What is the blocking variable? What are its levels? 1.30 Vitamin supplements. To assess the eﬀectiveness of taking large doses of vitamin C in reducing the duration of the common cold, researchers recruited 400 healthy volunteers from st
aﬀ and students at a university. A quarter of the patients were assigned a placebo, and the rest were evenly divided between 1g Vitamin C, 3g Vitamin C, or 3g Vitamin C plus additives to be taken at onset of a cold for the following two days. All tablets had identical appearance and packaging. The nurses who handed the prescribed pills to the patients knew which patient received which treatment, but the researchers assessing the patients when they were sick did not. No signiﬁcant diﬀerences were observed in any measure of cold duration or severity between the four groups, and the placebo group had the shortest duration of symptoms.39 (a) Was this an experiment or an observational study? Why? (b) What are the explanatory and response variables in this study? (c) Were the patients blinded to their treatment? (d) Was this study double-blind? (e) Participants are ultimately able to choose whether or not to use the pills prescribed to them. We might expect that not all of them will adhere and take their pills. Does this introduce a confounding variable to the study? Explain your reasoning. 1.31 Light, noise, and exam performance. A study is designed to test the eﬀect of light level and noise level on exam performance of students. The researcher believes that light and noise levels might have diﬀerent eﬀects on graduate and undergraduate students, so wants to make sure both are equally represented in each treatment. The light treatments considered are ﬂuorescent overhead lighting, yellow overhead lighting, no overhead lighting (only desk lamps). The noise treatments considered are no noise, construction noise, and human chatter noise. (a) What is the response variable? (b) How many factors are considered in this study? Identify them, and describe their levels. (c) What is the role of the program type (graduate versus undergraduate) variable in this study? 1.32 Music and learning. You would like to conduct an experiment in class to see if students learn better if they study without any music, with music that has no lyrics (instrumental), or with music that has lyrics. Brieﬂy outline a design for this study. 1.33 Soda preference. You would like to conduct an experiment in class to see if your classmates prefer the taste of regular Coke or Diet Coke. Brieﬂy outline a design for this study. 1.34 Exercise and mental health
. A researcher is interested in the eﬀects of exercise on mental health and he proposes the following study: Use stratiﬁed random sampling to ensure representative proportions of 18-30, 31-40 and 41- 55 year olds from the population. Next, randomly assign half the subjects from each age group to exercise twice a week, and instruct the rest not to exercise. Conduct a mental health exam at the beginning and at the end of the study, and compare the results. (a) What type of study is this? (b) What are the treatment and control groups in this study? (c) Does this study make use of blocking? If so, what is the blocking variable? (d) Does this study make use of blinding? (e) Comment on whether or not the results of the study can be used to establish a causal relationship between exercise and mental health, and indicate whether or not the conclusions can be generalized to the population at large. (f) Suppose you are given the task of determining if this proposed study should get funding. Would you have any reservations about the study proposal? 39C. Audera et al. “Mega-dose vitamin C in treatment of the common cold: a randomised controlled trial”. In: Medical Journal of Australia 175.7 (2001), pp. 359–362. 1.5. EXPERIMENTS 53 Chapter highlights Chapter 1 focused on various ways that researchers collect data. The key concepts are the diﬀerence between a sample and an experiment and the role that randomization plays in each. • Researchers take a random sample in order to draw an inference to the larger population from which they sampled. When examining observational data, even if the individuals were randomly sampled, a correlation does not imply a causal link. • In an experiment, researchers impose a treatment and use random assignment in order to draw causal conclusions about the eﬀects of the treatment. While often implied, inferences to a larger population may not be valid if the subjects were not also randomly sampled from that population. Related to this are some important distinctions regarding terminology. The terms stratifying and blocking cannot be used interchangeably. Likewise, taking a simple random sample is diﬀerent than randomly assigning individuals to treatment groups. • Stratifying vs Blocking. Stratifying is used when sampling, where the purpose is to sample a subgroup from each stratum in order to arrive at a better estimate for the parameter of interest. Blocking
is used in an experiment to separate subjects into blocks and then compare responses within those blocks. All subjects in a block are used in the experiment, not just a sample of them. • Random sampling vs Random assignment. Random sampling refers to sampling a subset of a population for the purpose of inference to that population. Random assignment is used in an experiment to separate subjects into groups for the purpose of comparison between those groups. When randomization is not employed, as in an observational study, neither inferences nor causal conclusions can be drawn. Always be mindful of possible confounding factors when interpreting the results of observation studies. 54 CHAPTER 1. DATA COLLECTION Chapter exercises 1.35 Pet names. The city of Seattle, WA has an open data portal that includes pets registered in the city. For each registered pet, we have information on the pet’s name and species. The following visualization plots the proportion of dogs with a given name versus the proportion of cats with the same name. The 20 most common cat and dog names are displayed. The diagonal line on the plot is the x = y line; if a name appeared on this line, the name’s popularity would be exactly the same for dogs and cats. (a) Are these data collected as part of an experiment or an observational study? (b) What is the most common dog name? What is the most common cat name? (c) What names are more common for cats than dogs? (d) Is the relationship between the two variables positive or negative? What does this mean in context of the data? 1.36 Stressed out, Part II. In a study evaluating the relationship between stress and muscle cramps, half the subjects are randomly assigned to be exposed to increased stress by being placed into an elevator that falls rapidly and stops abruptly and the other half are left at no or baseline stress. (a) What type of study is this? (b) Can this study be used to conclude a causal relationship between increased stress and muscle cramps? 1.37 Chia seeds and weight loss. Chia Pets – those terra-cotta ﬁgurines that sprout fuzzy green hair – made the chia plant a household name. But chia has gained an entirely new reputation as a diet supplement. In one 2009 study, a team of researchers recruited 38 men and divided them randomly into two groups: treatment or control. They also recruited 38 women, and they randomly placed half of these participants into the treatment group and the other half into the
control group. One group was given 25 grams of chia seeds twice a day, and the other was given a placebo. The subjects volunteered to be a part of the study. After 12 weeks, the scientists found no signiﬁcant diﬀerence between the groups in appetite or weight loss.40 (a) What type of study is this? (b) What are the experimental and control treatments in this study? (c) Has blocking been used in this study? If so, what is the blocking variable? (d) Has blinding been used in this study? (e) Comment on whether or not we can make a causal statement, and indicate whether or not we can generalize the conclusion to the population at large. 1.38 City council survey. A city council has requested a household survey be conducted in a suburban area of their city. The area is broken into many distinct and unique neighborhoods, some including large homes, some with only apartments, and others a diverse mixture of housing structures. For each part below, identify the sampling methods described, and describe the statistical pros and cons of the method in the city’s context. (a) Randomly sample 200 households from the city. (b) Divide the city into 20 neighborhoods, and sample 10 households from each neighborhood. (c) Divide the city into 20 neighborhoods, randomly sample 3 neighborhoods, and then sample all households from those 3 neighborhoods. (d) Divide the city into 20 neighborhoods, randomly sample 8 neighborhoods, and then randomly sample 50 households from those neighborhoods. (e) Sample the 200 households closest to the city council oﬃces. 40D.C. Nieman et al. “Chia seed does not promote weight loss or alter disease risk factors in overweight adults”. In: Nutrition Research 29.6 (2009), pp. 414–418. 1.5. EXPERIMENTS 55 1.39 Flawed reasoning. Identify the ﬂaw(s) in reasoning in the following scenarios. Explain what the individuals in the study should have done diﬀerently if they wanted to make such strong conclusions. (a) Students at an elementary school are given a questionnaire that they are asked to return after their parents have completed it. One of the questions asked is, “Do you ﬁnd that your work schedule makes it diﬃcult for you to spend time with your kids after school?” Of the parents who replied,
85% said “no”. Based on these results, the school oﬃcials conclude that a great majority of the parents have no diﬃculty spending time with their kids after school. (b) A survey is conducted on a simple random sample of 1,000 women who recently gave birth, asking them about whether or not they smoked during pregnancy. A follow-up survey asking if the children have respiratory problems is conducted 3 years later. However, only 567 of these women are reached at the same address. The researcher reports that these 567 women are representative of all mothers. (c) An orthopedist administers a questionnaire to 30 of his patients who do not have any joint problems and ﬁnds that 20 of them regularly go running. He concludes that running decreases the risk of joint problems. 1.40 Income and education in US counties. The scatterplot below shows the relationship between per capita income (in thousands of dollars) and percent of population with a bachelor’s degree in 3,143 counties in the US in 2010. (a) What are the explanatory and response variables? (b) Describe the relationship between the two variables. Make sure to discuss unusual observations, if any. (c) Can we conclude that having a bachelor’s degree increases one’s income? 1.41 Eat better, feel better? In a public health study on the eﬀects of consumption of fruits and vegetables on psychological well-being in young adults, participants were randomly assigned to three groups: (1) dietas-usual, (2) an ecological momentary intervention involving text message reminders to increase their fruits and vegetable consumption plus a voucher to purchase them, or (3) a fruit and vegetable intervention in which participants were given two additional daily servings of fresh fruits and vegetables to consume on top of their normal diet. Participants were asked to take a nightly survey on their smartphones. Participants were student volunteers at the University of Otago, New Zealand. At the end of the 14-day study, only participants in the third group showed improvements to their psychological well-being across the 14-days relative to the other groups.41 (a) What type of study is this? (b) Identify the explanatory and response variables. (c) Comment on whether the results of the study can be generalized to the population. (d) Comment on whether the results of the study can be used to establish causal relationships. (e)
A newspaper article reporting on the study states, “The results of this study provide proof that giving young adults fresh fruits and vegetables to eat can have psychological beneﬁts, even over a brief period of time.” How would you suggest revising this statement so that it can be supported by the study? 41Tamlin S Conner et al. “Let them eat fruit! The eﬀect of fruit and vegetable consumption on psychological well-being in young adults: A randomized controlled trial”. In: PloS one 12.2 (2017), e0171206. Percent with Bachelor's Degree$0$20k$40k$60k0%20%40%60%80%Per Capita Income 56 CHAPTER 1. DATA COLLECTION 1.42 Screens, teens, and psychological well-being. In a study of three nationally representative largescale data sets from Ireland, the United States, and the United Kingdom (n = 17,247), teenagers between the ages of 12 to 15 were asked to keep a diary of their screen time and answer questions about how they felt or acted. The answers to these questions were then used to compute a psychological well-being score. Additional data were collected and included in the analysis, such as each child’s sex and age, and on the mother’s education, ethnicity, psychological distress, and employment. The study concluded that there is little clear-cut evidence that screen time decreases adolescent well-being.42 (a) What type of study is this? (b) Identify the explanatory variables. (c) Identify the response variable. (d) Comment on whether the results of the study can be generalized to the population, and why. (e) Comment on whether the results of the study can be used to establish causal relationships. 1.43 Stanford Open Policing. The Stanford Open Policing project gathers, analyzes, and releases records from traﬃc stops by law enforcement agencies across the United States. Their goal is to help researchers, journalists, and policymakers investigate and improve interactions between police and the public.43 The following is an excerpt from a summary table created based oﬀ of the data collected as part of this project. State County Arizona Apaice County Arizona Apaice County Apaice County Arizona Cochise County Arizona Cochise County Arizona Cochise County Arizona · · · Wood County Wood County Wood County · · · Wisconsin Black Wisconsin Hispanic Wisconsin White
Driver’s race Black Hispanic White Black Hispanic White · · · No. of stops per year 266 1008 6322 1169 9453 10826 · · · 16 27 1157 % of stopped cars searched 0.08 0.05 0.02 0.05 0.04 0.02 · · · 0.24 0.04 0.03 drivers arrested 0.02 0.02 0.01 0.01 0.01 0.01 · · · 0.10 0.03 0.03 (a) What variables were collected on each individual traﬃc stop in order to create to the summary table above? (b) State whether each variable is numerical or categorical. If numerical, state whether it is continuous or discrete. If categorical, state whether it is ordinal or not. (c) Suppose we wanted to evaluate whether vehicle search rates are diﬀerent for drivers of diﬀerent races. In this analysis, which variable would be the response variable and which variable would be the explanatory variable? 1.44 Space launches. The following summary table shows the number of space launches in the US by the type of launching agency and the outcome of the launch (success or failure).44 1957 - 1999 2000 - 2018 Failure 13 281 - Success Failure 10 33 5 295 3751 - Success 562 711 65 Private State Startup (a) What variables were collected on each launch in order to create to the summary table above? (b) State whether each variable is numerical or categorical. If numerical, state whether it is continuous or discrete. If categorical, state whether it is ordinal or not. (c) Suppose we wanted to study how the success rate of launches vary between launching agencies and over time. In this analysis, which variable would be the response variable and which variable would be the explanatory variable? 42Amy Orben and AK Baukney-Przybylski. “Screens, Teens and Psychological Well-Being: Evidence from three time-use diary studies”. In: Psychological Science (2018). 43Emma Pierson et al. “A large-scale analysis of racial disparities in police stops across the United States”. In: arXiv preprint arXiv:1706.05678 (2017). 44JSR Launch Vehicle Database, A comprehensive list of suborbital space launches, 2019 Feb 10 Edition. 57 Chapter 2 Summarizing data 2.1 Examining numerical data
2.2 Numerical summaries and box plots 2.3 Normal distribution 2.4 Considering categorical data 2.5 Case study: malaria vaccine (special topic) 58 After collecting data, the next stage in the investigative process is to describe and summarize the data. In this chapter, we will look at ways to summarize numerical and categorical data graphically, numerically, and verbally. While in practice, numerical and graphical summaries are done using computer software, it is helpful to understand how these summaries are created and it is especially important to understand how to interpret and communicate these ﬁndings. For videos, slides, and other resources, please visit www.openintro.org/ahss 2.1. EXAMINING NUMERICAL DATA 59 2.1 Examining numerical data How do we visualize and describe the distribution of household income for counties within the United States? What shape would the distribution have? What other features might be important to notice? In this section, we will explore techniques for summarizing numerical variables. We will apply these techniques using county-level data from the US Census Bureau, which was introduced in Section 1.2, and a new data set email50, that comprises information on a random sample of 50 emails. Learning objectives 1. Use scatterplots to represent bivariate data and to see the relationship between two numerical variables. Describe the direction, form, and strength of the relationship, as well as any unusual observations. 2. Understand what the term distribution means and how to summarize it in a table or a graph. 3. Create univariate displays, including stem-and-leaf plots, dot plots, and histograms, to visualize the distribution of a numerical variable. Be able to read oﬀ speciﬁc information and summary information from these graphs. 4. Identify the shape of a distribution as approximately symmetric, right skewed, or left skewed. Also, identify whether a distribution is unimodal, bimodal, multimodal, or uniform. 5. Read and interpret a cumulative frequency or cumulative relative frequency histogram. 2.1.1 Scatterplots for paired data Sometimes researchers wish to see the relationship between two variables. When we talk of a relationship or an association between variables, we are interested in how one variable behaves as the other variable increases or decreases. A scatterplot provides a case-by-case view of data that illustrates the relationship between two numerical variables. A scatterplot is
shown in Figure 2.1, illustrating the relationship between the number of line breaks (line breaks) and number of characters (num char) in emails for the email50 data set. In any scatterplot, each point represents a single case. Since there are 50 cases in email50, there are 50 points in Figure 2.1. EXAMPLE 2.1 A scatterplot requires bivariate, or paired data. What does paired data mean? We say observations are paired when the two observations correspond to the same case or individual. In unpaired data, there is no such correspondence. In our example the two observations correspond to a particular email. The variable that is suspected to be the response variable is plotted on the vertical (y) axis and the variable that is suspected to be the explanatory variable is plotted on the horizontal (x) axis. In this example, the variables could be switched since either variable could reasonably serve as the explanatory variable or the response variable. 60 CHAPTER 2. SUMMARIZING DATA Figure 2.1: A scatterplot of line breaks versus num char for the email50 data. DRAWING SCATTERPLOTS (1) Decide which variable should go on each axis, and draw and label the two axes. (2) Note the range of each variable, and add tick marks and scales to each axis. (3) Plot the dots as you would on an (x, y) coordinate plane. The association between two variables can be positive or negative, or there can be no association. Positive association means that larger values of the ﬁrst variable are associated with larger values of the second variable. Additionally, the association can follow a linear trend or a curved (nonlinear) trend. EXAMPLE 2.2 What would it mean for two variables to have a negative association? What about no association? Negative association implies that larger values of the ﬁrst variable are associated with smaller values of the second variable. No association implies that the values of the second variable tend to be independent of changes in the ﬁrst variable. EXAMPLE 2.3 Figure 2.2 shows a plot of median household income against the poverty rate for 3,142 counties. What can be said about the relationship between these variables? The relationship is evidently nonlinear, as highlighted by the dashed line. This is diﬀerent from previous scatterplots we’ve seen, which show relationships that do not show much, if any, curvature in
the trend. There is also a negative association, as higher rates of poverty tend to be associated with lower median household income. GUIDED PRACTICE 2.4 What do scatterplots reveal about the data, and how are they useful?1 1Answers may vary. Scatterplots are helpful in quickly spotting associations relating variables, whether those associations come in the form of simple trends or whether those relationships are more complex. Number of Lines010203040506070020040060080010001200llllllllllllllllllllllllllllllllllllllllllllllllllNumber of Characters (in thousands) 2.1. EXAMINING NUMERICAL DATA 61 Figure 2.2: A scatterplot of the median household income against the poverty rate for the county data set. A statistical model has also been ﬁt to the data and is shown as a dashed line. Explore dozens of scatterplots using American Community Survey data on Tableau Public. GUIDED PRACTICE 2.5 Describe two variables that would have a horseshoe-shaped association in a scatterplot (∩ or ∪).2 2.1.2 Stem-and-leaf plots and dot plots Sometimes two variables is one too many: only one variable may be of interest. In these cases we want to focus not on the association between two variables, but on the distribution of a single, or univariate, variable. The term distribution refers to the values that a variable takes and the frequency of these values. Here we introduce a new data set, the email50 data set. This data set contains the number of characters in 50 emails. To simplify the data, we will round the numbers and record the values in thousands. Thus, 22105 is recorded as 22. 22 7 1 2 42 0 1 5 9 17 64 10 43 0 29 10 2 0 5 12 6 7 0 3 27 26 5 3 6 10 25 7 25 26 0 11 4 1 11 0 4 14 9 25 1 14 3 1 9 16 Figure 2.3: The number of characters, in thousands, for the data set of 50 emails. 2Consider the case where your vertical axis represents something “good” and your horizontal axis represents something that is only good in moderation. Health and water consumption ﬁt this description: we require some water to survive, but consume too much and it becomes toxic and can kill a person. If health was represented on
the vertical axis and water consumption on the horizontal axis, then we would create a ∩ shape. 0%10%20%30%40%50%$0$20k$40k$60k$80k$100k$120kPoverty Rate (Percent)Median Household Income 62 CHAPTER 2. SUMMARIZING DATA Rather than look at the data as a list of numbers, which makes the distribution diﬃcult to discern, we will organize it into a table called a stem-and-leaf plot shown in Figure 2.4. In a stemand-leaf plot, each number is broken into two parts. The ﬁrst part is called the stem and consists of the beginning digit(s). The second part is called the leaf and consists of the ﬁnal digit(s). The stems are written in a column in ascending order, and the leaves that match up with those stems are written on the corresponding row. Figure 2.4 shows a stem-and-leaf plot of the number of characters in 50 emails. The stem represents the ten thousands place and the leaf represents the thousands place. For example, 1 \| 2 corresponds to 12 thousand. When making a stem-and-leaf plot, remember to include a legend that describes what the stem and what the leaf represent. Without this, there is no way of knowing if 1 \| 2 represents 1.2, 12, 120, 1200, etc. 0 \| 00000011111223334455566777999 1 \| 0001124467 2 \| 25556679 3 \| 4 \| 23 5 \| 6 \| 4 Legend: 1 \| 2 = 12,000 Figure 2.4: A stem-and-leaf plot of the number of characters in 50 emails. GUIDED PRACTICE 2.6 There are a lot of numbers on the ﬁrst row of the stem-and-leaf plot. Why is this the case?3 When there are too many numbers on one row or there are only a few stems, we split each row into two halves, with the leaves from 0-4 on the ﬁrst half and the leaves from 5-9 on the second half. The resulting graph is called a split stem-and-leaf plot. Figure 2.5 shows the previous stem-and-leaf redone as a split stem-and-leaf. 0 \| 000000111112233344 0 \| 55566777999 1 \| 0001
1244 1 \| 67 2 \| 2 2 \| 5556679 3 \| 3 \| 4 \| 23 4 \| 5 \| 5 \| 6 \| 4 Legend: 1 \| 2 = 12,000 Figure 2.5: A split stem-and-leaf. 3There are a lot of numbers on the ﬁrst row because there are a lot of values in the data set less than 10 thousand. 2.1. EXAMINING NUMERICAL DATA 63 GUIDED PRACTICE 2.7 What is the smallest number in the email50 data set? What is the largest?4 Another simple graph for univariate numerical data is a dot plot. A dot plot uses dots to show the frequency, or number of occurrences, of the values in a data set. The higher the stack of dots, the greater the number occurrences there are of the corresponding value. An example using the same data set, number of characters from 50 emails, is shown in Figure 2.6. Figure 2.6: A dot plot of num char for the email50 data set. GUIDED PRACTICE 2.8 Imagine rotating the dot plot 90 degrees clockwise. What do you notice?5 These graphs make it easy to observe important features of the data, such as the location of clusters and presence of gaps. EXAMPLE 2.9 Based on both the stem-and-leaf and dot plot, where are the values clustered and where are the gaps for the email50 data set? There is a large cluster in the 0 to less than 20 thousand range, with a peak around 1 thousand. There are gaps between 30 and 40 thousand and between the two values in the 40 thousands and the largest value of approximately 64 thousand. Additionally, we can easily identify any observations that appear to be unusually distant from the rest of the data. Unusually distant observations are called outliers. Later in this chapter we will provide numerical rules of thumb for identifying outliers. For now, it is suﬃcient to identify them by observing gaps in the graph. In this case, it would be reasonable to classify the emails with character counts of 42 thousand, 43 thousand, and 64 thousand as outliers since they are numerically distant from most of the data. OUTLIERS ARE EXTREME An outlier is an observation that appears extreme relative to the rest of the data. WHY IT IS IMPORTANT TO LOOK FOR OUTLIERS Examination of data for possible outliers serves many useful purposes, including 1. Identifying asymm
etry in the distribution. 2. Identifying data collection or entry errors. For instance, we re-examined the email pur- ported to have 64 thousand characters to ensure this value was accurate. 3. Providing insight into interesting properties of the data. 4The smallest number is less than 1 thousand, and the largest is 64 thousand. That is a big range! 5It has a similar shape as the stem-and-leaf plot! The values on the horizontal axis correspond to the stems and the number of dots in each interval correspond the number of leaves needed for each stem. llllllllllllllllllllllllllllllllllllllllllllllllllNumber of Characters (in thousands)010203040506070 64 CHAPTER 2. SUMMARIZING DATA GUIDED PRACTICE 2.10 The observation 64 thousand, a suspected outlier, was found to be an accurate observation. What would such an observation suggest about the nature of character counts in emails?6 GUIDED PRACTICE 2.11 Consider a data set that consists of the following numbers: 12, 12, 12, 12, 12, 13, 13, 14, 14, 15, 19. Which graph would better illustrate the data: a stem-and-leaf plot or a dot plot? Explain.7 2.1.3 Histograms Stem-and-leaf plots and dot plots are ideal for displaying data from small samples because they show the exact values of the observations and how frequently they occur. However, they are impractical for larger samples. For larger samples, rather than showing the frequency of every value, we prefer to think of the value as belonging to a bin. For example, in the email50 data set, we create a table of counts for the number of cases with character counts between 0 and 5,000, then the number of cases between 5,000 and 10,000, and so on. Such a table, shown in Figure 2.7, is called a frequency table. Bins usually include the observations that fall on their left (lower) boundary and exclude observations that fall on their right (upper) boundary. This is called left inclusive. For example, 5 (i.e. 5000) would be counted in the 5-10 bin, not in the 0-5 bin. These binned counts are plotted as bars in Figure 2.8 into what is called a histogram or frequency histogram, which resembles the stacked dot plot
shown in Figure 2.6. Characters (in thousands) Count 0-5 5-10 10-15 15-20 20-25 25-30 19 12 6 2 3 5 · · · · · · 55-60 60-65 0 1 Figure 2.7: The counts for the binned num char data. Figure 2.8: A histogram of num char. This histogram uses bins or class intervals of width 5. Explore this histogram and dozens of histograms using American Community Survey data on Tableau Public. 6That occasionally there may be very long emails. 7Because all the values begin with 1, there would be only one stem (or two in a split stem-and-leaf). This would not provide a good sense of the distribution. For example, the gap between 15 and 19 would not be visually apparent. A dot plot would be better here. Number of Characters (in thousands)Frequency01020304050600510152005101520253035404550556065 2.1. EXAMINING NUMERICAL DATA 65 GUIDED PRACTICE 2.12 What can you see in the dot plot and stem-and-leaf plot that you cannot see in the frequency histogram?8 DRAWING HISTOGRAMS 1. The variable is always placed on the horizontal axis. Before drawing the histogram, label both axes and draw a scale for each. 2. Draw bars such that the height of the bar is the frequency of that bin and the width of the bar corresponds to the bin width. Histograms provide a view of the data density. Higher bars represent where the data are relatively more common. For instance, there are many more emails between 0 and 10,000 characters than emails between 10,000 and 20,000 in the data set. The bars make it easy to see how the density of the data changes relative to the number of characters. EXAMPLE 2.13 How many emails had fewer than 10 thousand characters? The height of the bars corresponds to frequency. There were 19 cases from 0 to less than 5 thousand and 12 cases from 5 thousand to less than 10 thousand, so there were 19 + 12 = 31 emails with fewer than 10 thousand characters. EXAMPLE 2.14 Approximately how many emails had fewer than 1 thousand characters? Based just on this histogram, we cannot know the exact answer to this question. We only know that 19 emails had between 0 and 5 thousand characters. If the number of
emails is evenly distribution on this interval, then we can estimate that approximately 19/5 ≈ 4 emails fell in the range between 0 and 1 thousand. EXAMPLE 2.15 What percent of the emails had 10 thousand or more characters? From the ﬁrst example, we know that 31 emails had fewer than 10 thousand characters. Since there are 50 emails in total, there must be 19 emails that have 10 thousand or more characters. To ﬁnd the percent, compute 19/50 = 0.38 = 38%. 8Character counts for individual emails. 66 CHAPTER 2. SUMMARIZING DATA Sometimes questions such as the ones above can be answered more easily with a cumulative frequency histogram. This type of histogram shows cumulative, or total, frequency achieved by each bin, rather than the frequency in that particular bin. Characters (in thousands) Cumulative Frequency 0-5 5-10 10-15 15-20 20-25 25-30 30-35 · · · 55-60 60-65 19 31 37 39 42 47 47 · · · 49 50 Figure 2.9: The cumulative frequencies for the binned num char data. Figure 2.10: A cumulative frequency histogram of num char. This histogram uses bins or class intervals of width 5. Compare frequency, relative frequency, cumula. tive frequency, and cumulative relative frequency histograms on Tableau Public EXAMPLE 2.16 How many of the emails had fewer than 20 thousand characters? By tracing the height of the 15-20 thousand bin over to the vertical axis, we can see that it has a height just under 40 on the cumulative frequency scale. Therefore, we estimate that ≈39 of the emails had fewer than 30 thousand characters. Note that, unlike with a regular frequency histogram, we do not add up the height of the bars in a cumulative frequency histogram because each bar already represents a cumulative sum. EXAMPLE 2.17 Using the cumulative frequency table and histogram, how many of the emails had 10-15 thousand characters? To answer this question, we do a subtraction. 37 emails had less than or equal to 10-15 thousand characters and 31 emails had less than or equal to 5-10 thousand characters, so 6 emails must have had 10-15 thousand characters. Number of Characters (in thousands)Cumulative Frequency01020304050600102030405005101520253035404550556065 2.1. EXAM
INING NUMERICAL DATA 67 EXAMPLE 2.18 Approximately 25 of the emails had fewer than how many characters? This time we are given a cumulative frequency, so we start at 25 on the vertical axis and trace it across to see which bin it hits. It hits the 5-10 thousand bin, so 25 of the emails had fewer than a value somewhere between 5 and 10 thousand characters. Knowing that 25 of the emails had fewer than a value between 5 and 10 thousand characters is useful information, but it is even more useful if we know what percent of the total 25 represents. Knowing that there were 50 total emails tells us that 25/50 = 0.5 = 50% of the emails had fewer than a value between 5 and 10 thousand characters. When we want to know what fraction or percent of the data meet a certain criteria, we use relative frequency instead of frequency. Relative frequency is a fancy term for percent or proportion. It tells us how large a number is relative to the total. Just as we constructed a frequency table, frequency histogram, and cumulative frequency histogram, we can construct a relative frequency table, relative frequency histogram, and cumulative relative frequency histogram. GUIDED PRACTICE 2.19 How will the shape of the relative frequency histograms diﬀer from the frequency histograms?9 PAY CLOSE ATTENTION TO THE VERTICAL AXIS OF A HISTOGRAM We can misinterpret a histogram if we forget to check whether the vertical axis represents frequency, relative frequency, cumulative frequency, or cumulative relative frequency. 2.1.4 Describing Shape Frequency and relative frequency histograms are especially convenient for describing the shape of the data distribution. Figure 2.8 shows that most emails have a relatively small number of characters, while fewer emails have a very large number of characters. When data trail oﬀ to the right in this way and have a longer right tail, the shape is said to be right skewed.10 Data sets with the reverse characteristic – a long, thin tail to the left – are said to be left skewed. We also say that such a distribution has a long left tail. Data sets that show roughly equal trailing oﬀ in both directions are called symmetric. LONG TAILS TO IDENTIFY SKEW When data trail oﬀ in one direction, the distribution has a long tail. If a distribution has a long left tail, it is left skewed. If a distribution has a long right tail, it is right
skewed. GUIDED PRACTICE 2.20 Take a look at the dot plot in Figure 2.6. Can you see the skew in the data? Is it easier to see the skew in the frequency histogram, the dot plot, or the stem-and-leaf plot?11 9The shape will remain exactly the same. Changing from frequency to relative frequency involves dividing all the frequencies by the same number, so only the vertical scale (the numbers on the y-axis) change. 10Other ways to describe data that are right skewed: skewed to the right, skewed to the high end, or skewed to the positive end. 11The skew is visible in all three plots. However, it is not easily visible in the cumulative frequency histogram. 68 CHAPTER 2. SUMMARIZING DATA GUIDED PRACTICE 2.21 Would you expect the distribution of number of pets per household to be right skewed, left skewed, or approximately symmetric? Explain.12 In addition to looking at whether a distribution is skewed or symmetric, histograms, stem-andleaf plots, and dot plots can be used to identify modes. A mode is represented by a prominent peak in the distribution.13 There is only one prominent peak in the histogram of num char. Figure 2.11 shows histograms that have one, two, or three prominent peaks. Such distributions are called unimodal, bimodal, and multimodal, respectively. Any distribution with more than 2 prominent peaks is called multimodal. Notice that in Figure 2.8 there was one prominent peak in the unimodal distribution with a second less prominent peak that was not counted since it only diﬀers from its neighboring bins by a few observations. Figure 2.11: Counting only prominent peaks, the distributions are (left to right) unimodal, bimodal, and multimodal. GUIDED PRACTICE 2.22 Height measurements of young students and adult teachers at a K-3 elementary school were taken. How many modes would you anticipate in this height data set?14 LOOKING FOR MODES Looking for modes isn’t about ﬁnding a clear and correct answer about the number of modes in a distribution, which is why prominent is not rigorously deﬁned in this book. The important part of this examination is to better understand your data and how it might be structured. 12We suspect most households would have 0, 1, or
2 pets but that a smaller number of households will have 3, 4, 5, or more pets, so there will be greater density over the small numbers, suggesting the distribution will have a long right tail and be right skewed. 13Another deﬁnition of mode, which is not typically used in statistics, is the value with the most occurrences. It is common to have no observations with the same value in a data set, which makes this other deﬁnition useless for many real data sets. 14There might be two height groups visible in the data set: one of the students and one of the adults. That is, the data are probably bimodal. 048121605101504812162005101504812162005101520 2.1. EXAMINING NUMERICAL DATA 69 2.1.5 Descriptive versus inferential statistics Finally, we note that the graphical summaries of this section and the numerical summaries of the next section fall into the realm of descriptive statistics. Descriptive statistics is about describing or summarizing data; it does not attribute properties of the data to a larger population. Inferential statistics, on the other hand, uses samples to generalize or to infer something about a larger population. We will have to wait until Chapter 5 to enter the exciting world of inferential statistics. Section summary • A scatterplot is a bivariate display illustrating the relationship between two numerical variables. The observations must be paired, which is to say that they correspond to the same case or individual. The linear association between two variables can be positive or negative, or there can be no association. Positive association means that larger values of the ﬁrst variable are associated with larger values of the second variable. Negative association means that larger values of the ﬁrst variable are associated with smaller values of the second variable. Additionally, the association can follow a linear trend or a curved (nonlinear) trend. • When looking at a univariate display, researchers want to understand the distribution of the variable. The term distribution refers to the values that a variable takes and the frequency of those values. When looking at a distribution, note the presence of clusters, gaps, and outliers. • Distributions may be symmetric or they may have a long tail. If a distribution has a long left tail (with greater density over the higher numbers), it is left skewed. If a distribution has a long right tail (with greater density over
the smaller numbers), it is right skewed. • Distributions may be unimodal, bimodal, or multimodal. • Two graphs that are useful for showing the distribution of a small number of observations are the stem-and-leaf plot and dot plot. These graphs are ideal for displaying data from small samples because they show the exact values of the observations and how frequently they occur. However, they are impractical for larger data sets. • For larger data sets it is common to use a frequency histogram or a relative frequency histogram to display the distribution of a variable. This requires choosing bins of an appropriate width. • To see cumulative amounts, use a cumulative frequency histogram. A cumulative rel- ative frequency histogram is ideal for showing percentile. • Descriptive statistics describes or summarizes data, while inferential statistics uses sam- ples to generalize or infer something about a larger population. 70 CHAPTER 2. SUMMARIZING DATA Exercises 2.1 ACS, Part I. Each year, the US Census Bureau surveys about 3.5 million households with The American Community Survey (ACS). Data collected from the ACS have been crucial in government and policy decisions, helping to determine the allocation of federal and state funds each year. Some of the questions asked on the survey are about their income, age (in years), and gender. The table below contains this information for a random sample of 20 respondents to the 2012 ACS.15 female Income Age Gender 53,000 1600 70,000 12,800 1,200 30,000 4,500 20,000 25,000 42,000 28 male 18 54 male 22 male 18 34 male 21 male 28 29 33 male female female female 1 2 3 4 5 6 7 8 9 10 Income Age Gender female 34 female 55 33 female 41 male 47 30 male 61 male 50 24 19 male 670 29,000 44,000 48,000 30,000 60,000 108,000 5,800 50,000 11,000 female female female 11 12 13 14 15 16 17 18 19 20 (a) Create a scatterplot of income vs. age, and describe the relationship between these two variables. (b) Now create two scatterplots: one for income vs. age for males and another for females. (c) How, if at all, do the relationships between income and age diﬀer for males and females? 2.2 MLB stats. A baseball team’s success in a season is usually measured by their
number of wins. In order to win, the team has to have scored more points (runs) than their opponent in any given game. As such, number of runs is often a good proxy for the success of the team. The table below shows number of runs, home runs, and batting averages for a random sample of 10 teams in the 2014 Major League Baseball season.16 Team 1 Baltimore 2 Boston 3 Cincinnati 4 Cleveland 5 Detroit 6 Houston 7 Minnesota 8 NY Yankees 9 Pittsburgh 10 San Francisco Runs Home runs Batting avg. 0.256 211 0.244 123 0.238 131 0.253 142 0.277 155 0.242 163 0.254 128 0.245 147 0.259 156 0.255 132 705 634 595 669 757 629 715 633 682 665 (a) Draw a scatterplot of runs vs. home runs. (b) Draw a scatterplot of runs vs. batting averages. (c) Are home runs or batting averages more strongly associated with number of runs? Explain your rea- soning. 15United States Census Bureau. Summary File. 2012 American Community Survey. U.S. Census Bureau’s American Community Survey Oﬃce, 2013. Web. 16ESPN: MLB Team Stats - 2014. 2.1. EXAMINING NUMERICAL DATA 71 2.3 Fiber in your cereal. The Cereal FACTS report provides information on nutrition content of cereals as well as who they are targeted for (adults, children, families). We have selected a random sample of 20 cereals from the data provided in this report. Shown below are the ﬁber contents (percentage of ﬁber per gram of cereal) for these cereals.17 Brand 1 Pebbles Fruity 2 Rice Krispies Treats 3 Pebbles Cocoa 4 Pebbles Marshmallow 5 Frosted Rice Krispies 6 Rice Krispies 7 Trix 8 Honey Comb 9 Rice Krispies Gluten Free 10 Frosted Flakes Fiber % 0.0% 0.0% 0.0% 0.0% 0.0% 3.0% 3.1% 3.1% 3.3% 3.3% Brand 11 Cinnamon Toast Crunch 12 Reese’s Puﬀs 13 Cheerios Honey Nut 14 Lucky Charms 15 Pebbles Boulders Chocolate PB 16 Corn Pops 17 Frosted Flakes Reduced Sugar 18 Cliﬀord Crunch
19 Apple Jacks 20 Dora the Explorer Fiber % 3.3% 3.4% 7.1% 7.4% 7.4% 9.4% 10.0% 10.0% 10.7% 11.1% (a) Create a stem and leaf plot of the distribution of the ﬁber content of these cereals. (b) Create a dot plot of the ﬁber content of these cereals. (c) Create a histogram and a relative frequency histogram of the ﬁber content of these cereals. (d) What percent of cereals contain more than 7% ﬁber? 2.4 Sugar in your cereal. The Cereal FACTS report from Exercise 2.3 also provides information on sugar content of cereals. We have selected a random sample of 20 cereals from the data provided in this report. Shown below are the sugar contents (percentage of sugar per gram of cereal) for these cereals. Brand 1 Rice Krispies Gluten Free 2 Rice Krispies 3 Dora the Explorer 4 Frosted Flakes Red. Sugar 5 Cliﬀord Crunch 6 Rice Krispies Treats 7 Pebbles Boulders Choc. PB 8 Cinnamon Toast Crunch 9 Trix 10 Honey Comb Sugar % 3% 12% 22% 27% 27% 30% 30% 30% 31% 31% Brand 11 Corn Pops 12 Cheerios Honey Nut 13 Reese’s Puﬀs 14 Pebbles Fruity 15 Pebbles Cocoa 16 Lucky Charms 17 Frosted Flakes 18 Pebbles Marshmallow 19 Frosted Rice Krispies 20 Apple Jacks Sugar % 31% 32% 34% 37% 37% 37% 37% 37% 40% 43% (a) Create a stem and leaf plot of the distribution of the sugar content of these cereals. (b) Create a dot plot of the sugar content of these cereals. (c) Create a histogram and a relative frequency histogram of the sugar content of these cereals. (d) What percent of cereals contain more than 30% sugar? 17JL Harris et al. “Cereal FACTS 2012: Limited progress in the nutrition quality and marketing of children’s cereals”. In: Rudd Center for Food Policy & Obesity. 12 (2012). 72 CHAPTER 2. SUMMARIZING DATA 2.5 Mammal life spans. Data were
collected on life spans (in years) and gestation lengths (in days) for 62 mammals. A scatterplot of life span versus length of gestation is shown below.18 (a) What type of an association is apparent between life span and length of gestation? (b) What type of an association would you expect to see if the axes of the plot were reversed, i.e. if we plotted length of gestation versus life span? (c) Are life span and length of gestation inde- pendent? Explain your reasoning. 2.6 Associations. Indicate which of the plots show (a) a positive association, (b) a negative association, or (c) no association. Also determine if the positive and negative associations are linear or nonlinear. Each part may refer to more than one plot. 18T. Allison and D.V. Cicchetti. “Sleep in mammals: ecological and constitutional correlates”. In: Arch. Hydrobiol 75 (1975), p. 442. lllllllllllllllllllllllllllllllllllllllllllllllllllllllGestation (days)Life Span (years)02004006000255075100(1)(2)(3)(4) 2.2. NUMERICAL SUMMARIES AND BOX PLOTS 73 2.2 Numerical summaries and box plots What are the diﬀerent ways to measure the center of a distribution, and why is there more than one way to measure the center? How do you know if a value is “far” from the center? What does it mean to an outlier? We will continue with the email50 data set and investigate multiple quantitative summarizes for numerical data. Learning objectives 1. Calculate, interpret, and compare the two measures of center (mean and median) and the three measures of spread (standard deviation, interquartile range, and range). 2. Understand how the shape of a distribution aﬀects the relationship between the mean and the median. 3. Identify and apply the two rules of thumb for identify outliers (one involving standard deviation and mean and the other involving Q1 and Q3). 4. Describe the distribution a numerical variable with respect to center, spread, and shape, noting the presence of outliers. 5. Find the 5 number summary and IQR, and draw a box plot with outliers shown. 6. Understand the e�
�ect changing units has on each of the summary quantities. 7. Use quartiles, percentiles, and Z-scores to measure the relative position of a data point within the data set. 8. Compare the distribution of a numerical variable using dot plots / histograms with the same scale, back-to-back stem-and-leaf plots, or parallel box plots. Compare the distributions with respect to center, spread, shape, and outliers. 2.2.1 Measures of center In the previous section, we saw that modes can occur anywhere in a data set. Therefore, mode is not a measure of center. We understand the term center intuitively, but quantifying what is the center can be a little more challenging. This is because there are diﬀerent deﬁnitions of center. Here we will focus on the two most common: the mean and median. The mean, sometimes called the average, is a common way to measure the center of a distribution of data. To ﬁnd the mean number of characters in the 50 emails, we add up all the character counts and divide by the number of emails. For computational convenience, the number of characters is listed in the thousands and rounded to the ﬁrst decimal. ¯x = 21.7 + 7.0 + · · · + 15.8 50 = 11.6 The sample mean is often labeled ¯x. The letter x is being used as a generic placeholder for the variable of interest, num char, and the bar on the x communicates that the average number of characters in the 50 emails was 11,600. 74 CHAPTER 2. SUMMARIZING DATA MEAN The sample mean of a numerical variable is computed as the sum of all of the observations divided by the number of observations: ¯x = 1 n xi = xi n = x1 + x2 + · · · + xn n where is the capital Greek letter sigma and xi means take the sum of all the individual x values. x1, x2,..., xn represent the n observed values. GUIDED PRACTICE 2.23 Examine Equations (2.23) and (2.23) above. What does x1 correspond to? And x2? What does xi represent?19 GUIDED PRACTICE 2.24 What was n in this sample of emails?20 The email50 data set represents a sample from a larger population of emails
that were received in January and March. We could compute a mean for this population in the same way as the sample mean, however, the population mean has a special label: µ. The symbol µ is the Greek letter mu and represents the average of all observations in the population. Sometimes a subscript, such as x, is used to represent which variable the population mean refers to, e.g. µx. EXAMPLE 2.25 The average number of characters across all emails can be estimated using the sample data. Based on the sample of 50 emails, what would be a reasonable estimate of µx, the mean number of characters in all emails in the email data set? (Recall that email50 is a sample from email.) The sample mean, 11,600, may provide a reasonable estimate of µx. While this number will not be perfect, it provides a point estimate of the population mean. In Chapter 5 and beyond, we will develop tools to characterize the reliability of point estimates, and we will ﬁnd that point estimates based on larger samples tend to be more reliable than those based on smaller samples. EXAMPLE 2.26 We might like to compute the average income per person in the US. To do so, we might ﬁrst think to take the mean of the per capita incomes across the 3,142 counties in the county data set. What would be a better approach? The county data set is special in that each county actually represents many individual people. If we were to simply average across the income variable, we would be treating counties with 5,000 and 5,000,000 residents equally in the calculations. Instead, we should compute the total income for each county, add up all the counties’ totals, and then divide by the number of people in all the counties. If we completed these steps with the county data, we would ﬁnd that the per capita income for the US is $27,348.43. Had we computed the simple mean of per capita income across counties, the result would have been just $22,504.70! Example 2.26 used what is called a weighted mean, which will not be a key topic in this textbook. However, we have provided an online supplement on weighted means for interested readers: www.openintro.org/go?id=stat extra weighted mean 19x1 corresponds to the number of characters in the ﬁrst email in the sample (21.7, in thousands), x
2 to the number of characters in the second email (7.0, in thousands), and xi corresponds to the number of characters in the ith email in the data set. 20The sample size was n = 50. 2.2. NUMERICAL SUMMARIES AND BOX PLOTS 75 The median provides another measure of center. The median splits an ordered data set in half. There are 50 character counts in the email50 data set (an even number) so the data are perfectly split into two groups of 25. We take the median in this case to be the average of the two middle observations: (6,768 + 7,012)/2 = 6,890. When there are an odd number of observations, there will be exactly one observation that splits the data into two halves, and in this case that observation is the median (no average needed). MEDIAN: THE NUMBER IN THE MIDDLE In an ordered data set, the median is the observation right in the middle. If there are an even number of observations, the median is the average of the two middle values. Graphically, we can think of the mean as the balancing point. The median is the value such that 50% of the area is to the left of it and 50% of the area is to the right of it. Figure 2.12: A histogram of num char with its mean and median shown. EXAMPLE 2.27 Based on the data, why is the mean greater than the median in this data set? Consider the three largest values of 42 thousand, 43 thousand, and 64 thousand. These values drag up the mean because they substantially increase the sum (the total). However, they do not drag up the median because their magnitude does not change the location of the middle value. THE MEAN FOLLOWS THE TAIL In a right skewed distribution, the mean is greater than the median. In a left skewed distribution, the mean is less than the median. In a symmetric distribution, the mean and median are approximately equal. GUIDED PRACTICE 2.28 Consider the distribution of individual income in the United States. Which is greater: the mean or median? Why?21 21Because a small percent of individuals earn extremely large amounts of money while the majority earn a modest amount, the distribution is skewed to the right. Therefore, the mean is greater than the median. Number of Characters (in thousands)Frequency01020304050607005101520051015202530
3540455055606570MeanMedian 76 CHAPTER 2. SUMMARIZING DATA 2.2.2 Standard deviation as a measure of spread The U.S. Census Bureau reported that in 2019, the median family income was $80,944 and the mean family income was $108,587. Is a family income of $60,000 far from the mean or somewhat close to the mean? In order to answer this question, it is not enough to know the center of the data set and its range (maximum value - minimum value). We must know about the variability of the data set within that range. Low variability or small spread means that the values tend to be more clustered together. High variability or large spread means that the values tend to be far apart. EXAMPLE 2.29 Is it possible for two data sets to have the same range but diﬀerent spread? If so, give an example. If not, explain why not. Yes. An example is: 1, 1, 1, 1, 1, 9, 9, 9, 9, 9 and 1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 9. The ﬁrst data set has a larger spread because values tend to be farther away from each other while in the second data set values are clustered together at the mean. Here, we introduce the standard deviation as a measure of spread. Though its formula is a bit tedious to calculate by hand, the standard deviation is very useful in data analysis and roughly describes how far away, on average, the observations are from the mean. We call the distance of an observation from its mean its deviation. Below are the deviations for the 1st, 2nd, 3rd, and 50th observations in the num char variable. For computational convenience, the number of characters is listed in the thousands and rounded to the ﬁrst decimal. x1 − ¯x = 21.7 − 11.6 = 10.1 x2 − ¯x = 7.0 − 11.6 = −4.6 x3 − ¯x = 0.6 − 11.6 = −11.0... x50 − ¯x = 15.8 − 11.6 = 4.2 If we square these deviations and then take an average, the result is about equal to the sample variance, denoted by s2: s2 = = 10.12 + (−4.6)2 +
(−11.0)2 + · · · + 4.22 50 − 1 102.01 + 21.16 + 121.00 + · · · + 17.64 49 = 172.44 We divide by n − 1, rather than dividing by n, when computing the variance; you need not worry about this mathematical nuance for the material in this textbook. Notice that squaring the deviations does two things. First, it makes large values much larger, seen by comparing 10.12, (−4.6)2, (−11.0)2, and 4.22. Second, it gets rid of any negative signs. The standard deviation is deﬁned as the square root of the variance: √ s = 172.44 = 13.13 The standard deviation of the number of characters in an email is about 13.13 thousand. A subscript of x may be added to the variance and standard deviation, i.e. s2 x and sx, as a reminder that these are the variance and standard deviation of the observations represented by x1, x2,..., xn. The x subscript is usually omitted when it is clear which data the variance or standard deviation is referencing. 2.2. NUMERICAL SUMMARIES AND BOX PLOTS 77 CALCULATING THE STANDARD DEVIATION The standard deviation is the square root of the variance. It is roughly the “typical” distance of the observations from the mean. sX = 1 n − 1 (xi − ¯x)2 The variance is useful for mathematical reasons, but the standard deviation is easier to interpret because it has the same units as the data set. The units for variance will be the units squared (e.g. meters2). Formulas and methods used to compute the variance and standard deviation for a population are similar to those used for a sample.22 However, like the mean, the population values have special symbols: σ2 for the variance and σ for the standard deviation. The symbol σ is the Greek letter sigma. THINKING ABOUT THE STANDARD DEVIATION It is useful to think of the standard deviation as the “typical” or “average” distance that observations fall from the mean. EXAMPLE 2.30 Earlier, we reported that the mean family income in the U.S. in 2019 was $108,587. Estimating the standard deviation of income as approximately $50,000, is a family
income of $60,000 far from the mean or relatively close to the mean? Because $60,000 is less that one standard deviation from the mean, it is relatively close to the mean. If the value were more than 2 standard deviations away from the mean, we would consider it far from the mean. In the next section, we encounter a bell-shaped distribution known as the normal distribution. The empirical rule tells us that for nearly normal distributions, about 68% of the data will be within one standard deviation of the mean, about 95% will be within two standard deviations of the mean, and about 99.7% will be within three standard deviations of the mean. However, as seen in Figures 2.13 and 2.14, these percentages generally do not hold if the distribution is not bell-shaped. Figure 2.13: In the num char data, 40 of the 50 emails (80%) are within 1 standard deviation of the mean, and 47 of the 50 emails (9 4%) are within 2 standard deviations. The empirical rule does not hold well for skewed data, as shown in this example. 22The only diﬀerence is that the population variance has a division by n instead of n − 1. Number of Characters (in thousands)llllllllllllllllllllllllllllllllllllllllllllllllll010203040506070 78 CHAPTER 2. SUMMARIZING DATA Figure 2.14: Three very diﬀerent population distributions with the same mean µ = 0 and standard deviation σ = 1. GUIDED PRACTICE 2.31 On page 67, the concept of shape of a distribution was introduced. A good description of the shape of a distribution should include modality and whether the distribution is symmetric or skewed to one side. Using Figure 2.14 as an example, explain why such a description is important.23 When describing any distribution, comment on the three important characteristics of center, spread, and shape. Also note any especially unusual cases. EXAMPLE 2.32 In the data’s context (the number of characters in emails), describe the distribution of the num char variable shown in the histogram below. The distribution of email character counts is unimodal and very strongly skewed to the right. Many of the counts fall near the mean at 11,600, and most fall within one standard deviation (13,130) of the mean. There is one exceptionally long email
with about 65,000 characters. In this chapter we use standard deviation as a descriptive statistic to describe the variability in a given data set. In Chapter 4 we will use standard deviation to assess how close a sample mean is likely to be to the population mean. 23Figure 2.14 shows three distributions that look quite diﬀerent, but all have the same mean, variance, and standard deviation. Using modality, we can distinguish between the ﬁrst plot (bimodal) and the last two (unimodal). Using skewness, we can distinguish between the last plot (right skewed) and the ﬁrst two. While a graph tells a more complete story, we can use modality and shape (symmetry/skew) to characterize basic information about a distribution. −3−2−10123−3−2−10123−3−2−10123Number of Characters (in thousands)Frequency01020304050600510152005101520253035404550556065 2.2. NUMERICAL SUMMARIES AND BOX PLOTS 79 2.2.3 Z-scores Knowing how many standard deviations a value is from the mean is often more useful than simply knowing how far a value is from the mean. EXAMPLE 2.33 Previously, we saw that the mean family income in the U.S. in 2019 was $108,587. Let’s round this to $100,000 and estimate the standard deviation of income as $50,000. Using these estimates, how many standard deviations above the mean is a family income of $200,000? The value $200,000 is $100,000 above the mean. $100,000 is 2 standard deviations above the mean. This can be found by doing 200, 000 − 100, 000 50, 000 = 2 The number of standard deviations a value is above or below the mean is known as the Z-score. A Z-score has no units, and therefore is sometimes also called standard units. THE Z-SCORE The Z-score of an observation is the number of standard deviations it falls above or below the mean. We compute the Z-score for an observation x that follows a distribution with mean µ and standard deviation σ using Z = x − µ σ Observations above the mean always have positive Z-scores, while those below the mean always have negative Z-scores
. If an observation is equal to the mean, then the Z-score is 0. EXAMPLE 2.34 Head lengths of brushtail possums have a mean of 92.6 mm and standard deviation 3.6 mm. Compute the Z-scores for possums with head lengths of 95.4 mm and 85.8 mm. For x1 = 95.4 mm: For x2 = 85.8 mm: Z1 = x1 − µ σ = 95.4 − 92.6 3.6 = 0.78 Z2 = 85.8 − 92.6 3.6 = −1.89 We can use Z-scores to roughly identify which observations are more unusual than others. An observation x1 is said to be more unusual than another observation x2 if the absolute value of its Zscore is larger than the absolute value of the other observation’s Z-score: \|Z1\| > \|Z2\|. This technique is especially insightful when a distribution is symmetric. 80 CHAPTER 2. SUMMARIZING DATA GUIDED PRACTICE 2.35 Which of the observations in Example 2.34 is more unusual?24 GUIDED PRACTICE 2.36 Let X represent a random variable from a distribution with µ = 3 and σ = 2, and suppose we observe x = 5.19. (a) Find the Z-score of x. (b) Interpret the Z-score.25 Because Z-scores have no units, they are useful for comparing distance to the mean for distri- butions that have diﬀerent standard deviations or diﬀerent units. EXAMPLE 2.37 The average daily high temperature in June in LA is 77◦F with a standard deviation of 5◦F. The average daily high temperature in June in Iceland is 13◦C with a standard deviation of 3◦C. Which would be considered more unusual: an 83◦F day in June in LA or a 19◦C day in June in Iceland? Both values are 6◦ above the mean. However, they are not the same number of standard deviations above the mean. 83 is (83−77)/5 = 1.2 standard deviations above the mean, while 19 is (19−13)/3 = 2 standard deviations above the mean. Therefore, a 19◦C day in June in Iceland would be more unusual than an 83◦F day in
June in LA. 2.2.4 Box plots and quartiles A box plot summarizes a data set using ﬁve summary statistics while also plotting unusual observations, called outliers. Figure 2.15 provides a box plot of the num char variable from the email50 data set. The ﬁve summary statistics used in a box plot are known as the ﬁve-number summary, which consists of the minimum, the maximum, and the three quartiles (Q1, Q2, Q3) of the data set being studied. Q2 represents the second quartile, which is equivalent to the 50th percentile (i.e. the median). Previously, we saw that Q2 (the median) for the email50 data set was the average of the two middle values: 6,768+7,012 = 6,890. Q1 represents the ﬁrst quartile, which is the 25th percentile, and is the median of the smaller half of the data set. There are 25 values in the lower half of the data set, so Q1 is the middle value: 2,454 characters. Q3 represents the third quartile, or 75th percentile, and is the median of the larger half of the data set: 15,829 characters. 2 We calculate the variability in the data using the range of the middle 50% of the data: Q3−Q1 = 13,375. This quantity is called the interquartile range (IQR, for short). It, like the standard deviation, is a measure of variability or spread in data. The more variable the data, the larger the standard deviation and IQR tend to be. 24Because the absolute value of Z-score for the second observation (x2 = 85.8 mm → Z2 = −1.89) is larger than that of the ﬁrst (x1 = 95.4 mm → Z1 = 0.78), the second observation has a more unusual head length. 25(a) Its Z-score is given by Z = x−µ σ = 5.19−3 2 = 2.19/2 = 1.095. (b) The observation x is 1.095 standard deviations above the mean. We know it must be above the mean since Z is positive. 2.2. NUMERICAL SUMMARIES AND BOX PLOTS 81 Figure 2.15: A labeled box plot for the number of characters
in 50 emails. The median (6,890) splits the data into the bottom 50% and the top 50%. Explore dozens of boxplots with histograms using American Community Survey data on Tableau Public. INTERQUARTILE RANGE (IQR) The IQR is the length of the box in a box plot. It is computed as where Q1 and Q3 are the 25th and 75th percentiles. IQR = Q3 − Q1 OUTLIERS IN THE CONTEXT OF A BOX PLOT When in the context of a box plot, deﬁne an outlier as an observation that is more than 1.5 × IQR above Q3 or 1.5 × IQR below Q1. Such points are marked using a dot or asterisk in a box plot. To build a box plot, draw an axis (vertical or horizontal) and draw a scale. Draw a dark line denoting Q2, the median. Next, draw a line at Q1 and at Q3. Connect the Q1 and Q3 lines to form a rectangle. The width of the rectangle corresponds to the IQR and the middle 50% of the data is in this interval. Extending out from the rectangle, the whiskers attempt to capture all of the data remaining outside of the box, except outliers. In Figure 2.15, the upper whisker does not extend to the last three points, which are beyond Q3 + 1.5 × IQR and are outliers, so it extends only to the last point below this limit.26 The lower whisker stops at the lowest value, 33, since there are no additional data to reach. Outliers are each marked with a dot or asterisk. In a sense, the box is like the body of the box plot and the whiskers are like its arms trying to reach the rest of the data. 26You might wonder, isn’t the choice of 1.5×IQR for deﬁning an outlier arbitrary? It is! In practical data analyses, we tend to avoid a strict deﬁnition since what is an unusual observation is highly dependent on the context of the data. Number of Characters (in thousands)010203040506070lower whiskerQ1 (first quartile)Q2 (median)Q3 (third quartile)upper whiskermax whisker reachoutliers 82 CHAPTER 2. SUMMARIZING
DATA EXAMPLE 2.38 Compare the box plot to the graphs previously discussed: stem-and-leaf plot, dot plot, frequency and relative frequency histogram. What can we learn more easily from a box plot? What can we learn more easily from the other graphs? It is easier to immediately identify the quartiles from a box plot. The box plot also more prominently highlights outliers. However, a box plot, unlike the other graphs, does not show the distribution of the data. For example, we cannot generally identify modes using a box plot. EXAMPLE 2.39 Is it possible to identify skew from the box plot? Yes. Looking at the lower and upper whiskers of this box plot, we see that the lower 25% of the data is squished into a shorter distance than the upper 25% of the data, implying that there is greater density in the low values and a tail trailing to the upper values. This box plot is right skewed. GUIDED PRACTICE 2.40 True or false: there is more data between the median and Q3 than between Q1 and the median.27 EXAMPLE 2.41 Consider the following ordered data set. 5 5 9 10 15 16 30 40 80 Find the 5 number summary and identify how small or large a value would need to be to be considered an outlier. Are there any outliers in this data set? There are nine numbers in this data set. Because n is odd, the median is the middle number: 15. When ﬁnding Q1, we ﬁnd the median of the lower half of the data, which in this case includes 4 numbers (we do not include the 15 as belonging to either half of the data set). Q1 then is the average of 5 and 9, which is Q1 = 7, and Q3 is the average of 30 and 40, so Q3 = 35. The min is 5 and the max is 80. To see how small a number needs to be to be an outlier on the low end we do: On the high end we need: Q1 − 1.5 × IQR = Q1 − 1.5 × (Q3 − Q1) = 7 − 1.5 × (35 − 7) = −35 Q3 + 1.5 × IQR = Q3 + 1.5 × (Q3 − Q1) = 35 + 1.5 × (35 − 7) = 77 There are no numbers less
than -41, so there are no outliers on the low end. The observation at 80 is greater than 77, so 80 is an outlier on the high end. 27False. Since Q1 is the 25th percentile and the median is the 50th percentile, 25% of the data fall between Q1 and the median. Similarly, 25% of the data fall between Q2 and the median. The distance between the median and Q3 is larger because that 25% of the data is more spread out. 2.2. NUMERICAL SUMMARIES AND BOX PLOTS 83 2.2.5 Technology: summarizing 1-variable statistics Online calculators such as Desmos or a handheld calculator can be used to calculate summary statistics. More advanced statistical software packages include R (which was used for most of the graphs in this text), Python, SAS, and STATA. Get started quickly with this Desmos 1-VarStats Calculator (available at openintro.org/ahss/desmos). Calculator instructions TI-83/84: ENTERING DATA The ﬁrst step in summarizing data or making a graph is to enter the data set into a list. Use STAT, Edit. 1. Press STAT. 2. Choose 1:Edit. 3. Enter data into L1 or another list. CASIO FX-9750GII: ENTERING DATA 1. Navigate to STAT (MENU button, then hit the 2 button or select STAT). 2. Optional: use the left or right arrows to select a particular list. 3. Enter each numerical value and hit EXE. 84 CHAPTER 2. SUMMARIZING DATA TI-84: CALCULATING SUMMARY STATISTICS Use the STAT, CALC, 1-Var Stats command to ﬁnd summary statistics such as mean, standard deviation, and quartiles. 1. Enter the data as described previously. 2. Press STAT. 3. Right arrow to CALC. 4. Choose 1:1-Var Stats. 5. Enter L1 (i.e. 2ND 1) for List. If the data is in a list other than L1, type the name of that list. 6. Leave FreqList blank. 7. Choose Calculate and hit ENTER. TI-83: Do steps 1-4, then type L1 (i.e. 2nd 1) or the list’s name and hit ENTER. Calculating the summary statistics will return the
following information. down arrow to see all of the summary statistics. It will be necessary to hit the ¯x Σx Σx2 Sx σx Mean Sum of all the data values Sum of all the squared data values Sample standard deviation Population standard deviation Sample size or # of data points n minX Minimum Q1 Med Median maxX Maximum First quartile TI-83/84: DRAWING A BOX PLOT 1. Enter the data to be graphed as described previously. 2. Hit 2ND Y= (i.e. STAT PLOT). 3. Hit ENTER (to choose the ﬁrst plot). 4. Hit ENTER to choose ON. 5. Down arrow and then right arrow three times to select box plot with outliers. 6. Down arrow again and make Xlist: L1 and Freq: 1. 7. Choose ZOOM and then 9:ZoomStat to get a good viewing window. TI-83/84: WHAT TO DO IF YOU CANNOT FIND L1 OR ANOTHER LIST Restore lists L1-L6 using the following steps: 1. Press STAT. 2. Choose 5:SetUpEditor. 3. Hit ENTER. CASIO FX-9750GII: DRAWING A BOX PLOT AND 1-VARIABLE STATISTICS 1. Navigate to STAT (MENU, then hit 2) and enter the data into a list. 2. Go to GRPH (F1). 3. Next go to SET (F6) to set the graphing parameters. 4. To use the 2nd or 3rd graph instead of GPH1, select F2 or F3. 5. Move down to Graph Type and select the (F6) option to see more graphing options, then select Box (F2). 6. If XList does not show the list where you entered the data, hit LIST (F1) and enter the correct list number. 7. Leave Frequency at 1. 8. For Outliers, choose On (F1). 9. Hit EXE and then choose the graph where you set the parameters F1 (most common), F2, or F3. 10. If desired, explore 1-variable statistics by selecting 1-Var (F1). 2.2. NUMERICAL SUMMARIES AND BOX PLOTS 85 CASIO FX-9750GII: DELETING A DATA LIST 1. Navigate to STAT (MENU
, then hit 2). 2. Use the arrow buttons to navigate to the list you would like to delete. 3. Select (F6) to see more options. 4. Select DEL-A (F4) and then F1 to conﬁrm. GUIDED PRACTICE 2.42 Enter the following 10 data points into a calculator. Find the summary statistics and make a box plot of the data.28 5, 8, 1, 19, 3, 1, 11, 18, 20, 5 GUIDED PRACTICE 2.43 Use the email50 data set at openintro.org/data and Screen 2 of this Desmos 1-Var Stats Calculator to summarize the num char variable (number of characters in an email).29 28The summary statistics should be ¯x = 9.1, Sx = 7.48, Q1 = 3, etc. Using a TI, the boxplot looks like this: 29Remember, the Desmos Calculators and Activities in this book can be found at openintro.org/ahss/desmos. Down the email50 CSV ﬁle and open it. Copy and paste the num char column into the Desmos calculator, replacing the data currently in x1. Adjust window as needed and you should get the following: 86 CHAPTER 2. SUMMARIZING DATA 2.2.6 Outliers and robust statistics RULES OF THUMB FOR IDENTIFYING OUTLIERS There are two rules of thumb for identifying outliers: • More than 1.5× IQR below Q1 or above Q3 • More than 2 standard deviations above or below the mean. Both are important for the AP exam. In practice, consider these to be only rough guidelines. GUIDED PRACTICE 2.44 For the email50 data set,Q1 = 2,536 and Q3 = 15, 411. ¯x = 11,600 and s = 13,130. What values would be considered an outlier on the low end using each rule?30 GUIDED PRACTICE 2.45 Because there are no negative values in this data set, there can be no outliers on the low end. What does the fact that there are outliers on the high end but not on the low end suggestion?31 How are the sample statistics of the num char data set aﬀected by the observation, 64,401? What would have happened if this email wasn’t
observed? What would happen to these summary statistics if the observation at 64,401 had been even larger, say 150,000? These scenarios are plotted alongside the original data in Figure 2.16, and sample statistics are computed under each scenario in Figure 2.17. Figure 2.16: Dot plots of the original character count data and two modiﬁed data sets. 30 Q1 − 1.5 × IQR = 2536 − 1.5 × (15411 − 2536) = −16, 749.5, so values less than -16,749.5 would be considered an outlier using the ﬁrst rule of thumb. Using the second rule of thumb, a value less than ¯x−2×s = 11, 600−2×13, 130 = −14, 660 would be considered an outlier. Note tht these are just rules of thumb and yield diﬀerent values. 31It suggests that the distribution has a right hand tail, that is, that it is right skewed. llllllllllllllllllllllllllllllllllllllllllllllllll Number of Characters (in thousands)050100150OriginallllllllllllllllllllllllllllllllllllllllllllllllllDrop 64,401llllllllllllllllllllllllllllllllllllllllllllllllll64,401 to 150,000 2.2. NUMERICAL SUMMARIES AND BOX PLOTS 87 scenario original num char data drop 64,401 observation move 64,401 to 150,000 robust median 6,890 6,768 6,890 IQR 12,875 11,702 12,875 not robust s ¯x 13,130 11,600 10,798 10,521 22,434 13,310 Figure 2.17: A comparison of how the median, IQR, mean (¯x), and standard deviation (s) change when extreme observations are present. GUIDED PRACTICE 2.46 (a) Which is more aﬀected by extreme observations, the mean or median? Figure 2.17 may be helpful. (b) Is the standard deviation or IQR more aﬀected by extreme observations?32 The median and IQR are called robust estimates because extreme observations have little eﬀect on their
values. The mean and standard deviation are much more aﬀected by changes in extreme observations. EXAMPLE 2.47 The median and IQR do not change much under the three scenarios in Figure 2.17. Why might this be the case? Since there are no large gaps between observations around the three quartiles, adding, deleting, or changing one value, no matter how extreme that value, will have little eﬀect on their values. GUIDED PRACTICE 2.48 The distribution of vehicle prices tends to be right skewed, with a few luxury and sports cars lingering out into the right tail. If you were searching for a new car and cared about price, should you be more interested in the mean or median price of vehicles sold, assuming you are in the market for a regular car?33 32(a) Mean is aﬀected more. (b) Standard deviation is aﬀected more. Complete explanations are provided in the material following Guided Practice 2.46. 33Buyers of a “regular car” should be concerned about the median price. High-end car sales can drastically inﬂate the mean price while the median will be more robust to the inﬂuence of those sales. 88 CHAPTER 2. SUMMARIZING DATA 2.2.7 Linear transformations of data EXAMPLE 2.49 Begin with the following list: 1, 1, 5, 5. Multiply all of the numbers by 10. What happens to the mean? What happens to the standard deviation? How do these compare to the mean and the standard deviation of the original list? The original list has a mean of 3 and a standard deviation of 2. The new list: 10, 10, 50, 50 has a mean of 30 with a standard deviation of 20. Because all of the values were multiplied by 10, both the mean and the standard deviation were multiplied by 10. 34 EXAMPLE 2.50 Start with the following list: 1, 1, 5, 5. Multiply all of the numbers by -0.5. What happens to the mean? What happens to the standard deviation? How do these compare to the mean and the standard deviation of the original list? The new list: -0.5, -0.5, -2.5, -2.5 has a mean of -1.5 with a standard deviation of 1. Because all of the values were multiplied by -
0.5, the mean was multiplied by -0.5. Multiplying all of the values by a negative ﬂipped the sign of numbers, which aﬀects the location of the center, but not the spread. Multiplying all of the values by -0.5 multiplied the standard deviation by +0.5 since the standard deviation cannot be negative. EXAMPLE 2.51 Again, start with the following list: 1, 1, 5, 5. Add 100 to every entry. How do the new mean and standard deviation compare to the original mean and standard deviation? The new list is: 101, 101, 105, 105. The new mean of 103 is 100 greater than the original mean of 3. The new standard deviation of 2 is the same as the original standard deviation of 2. Adding a constant to every entry shifted the values, but did not stretch them. Suppose that a researcher is looking at a list of 500 temperatures recorded in Celsius (C). The mean of the temperatures listed is given as 27◦C with a standard deviation of 3◦C. Because she is not familiar with the Celsius scale, she would like to convert these summary statistics into Fahrenheit (F). To convert from Celsius to Fahrenheit, we use the following conversion: xF = 9 5 xC + 32 Fortunately, she does not need to convert each of the 500 temperatures to Fahrenheit and then recalculate the mean and the standard deviation. The unit conversion above is a linear transformation of the following form, where a = 9/5 and b = 32: aX + b Using the examples as a guide, we can solve this temperature-conversion problem. The mean was 27◦C and the standard deviation was 3◦C. To convert to Fahrenheit, we multiply all of the values by 9/5, which multiplies both the mean and the standard deviation by 9/5. Then we add 32 to all of the values which adds 32 to the mean but does not change the standard deviation further. ¯xF = ¯xC + 32 (27) + 32 9 5 5 9 = 80.6 = σF = σC 9 5 9 5 = 5.4 = (3) 34Here, the population standard deviation was used in the calculation. These properties can be proven mathemat- ically using properties of sigma (summation). 2.2. NUMERICAL SUMMARIES AND BOX PLOTS 89 Figure 2.18: 500
temperatures shown in both Celsius and Fahrenheit. ADDING SHIFTS THE VALUES, MULTIPLYING STRETCHES OR CONTRACTS THEM Adding a constant to every value in a data set shifts the mean but does not aﬀect the standard deviation. Multiplying the values in a data set by a constant will change the mean and the standard deviation by the same multiple, except that the standard deviation will always remain positive. EXAMPLE 2.52 Consider the temperature example. How would converting from Celsuis to Fahrenheit aﬀect the median? The IQR? The median is aﬀected in the same way as the mean and the IQR is aﬀected in the same way as the standard deviation. To get the new median, multiply the old median by 9/5 and add 32. The IQR is computed by subtracting Q1 from Q3. While Q1 and Q3 are each aﬀected in the same way as the median, the additional 32 added to each will cancel when we take Q3 − Q1. That is, the IQR will be increase by a factor of 9/5 but will be unaﬀected by the addition of 32. For a more mathematical explanation of the IQR calculation, see the footnote.35 2.2.8 Comparing numerical data across groups Some of the more interesting investigations can be considered by examining numerical data across groups. The methods required here aren’t really new. All that is required is to make a numerical plot for each group. To make a direct comparison between two groups, create a pair of dot plots or a pair of histograms drawn using the same scales. It is also common to use back-to-back stem-and-leaf plots, parallel box plots, and hollow histograms, the three of which are explored here. We will take a look again at the county data set and compare the median household income for counties that gained population from 2010 to 2017 versus counties that had no gain. While we might like to make a causal connection here, remember that these are observational data and so such an interpretation would be, at best, half-baked. There were 1,454 counties where the population increased from 2010 to 2017, and there were 1,672 counties with no gain (all but one were a loss). A random sample of 100 counties from the ﬁrst group and 50 from the second group are shown in
Figure 2.19 to give a better sense of some of the raw median income data. 35New IQR = 9 5 Q3 + 32 − 9 5 Q1 + 32 = 9 5 (Q3 − Q1) = 9 5 × (old IQR). 204060801000204060CelsiusFahrenheitFrequencyTemperature 90 CHAPTER 2. SUMMARIZING DATA Median Income for 150 Counties, in $1000s Population Gain 43.6 51.8 63.3 34.1 46.3 40.6 34.7 51.4 57.6 54.6 49.1 46.5 63 47.6 49 44.7 55.5 42.2 40.7 52.1 45.5 82.2 46.3 54 56.5 69.2 55 57.2 38.9 49.6 55.9 45.6 71.7 38.6 61.5 48.1 60.3 52.8 43.6 62.4 42.9 62 48.4 46.4 44.1 50.9 53.7 39.1 39 35.3 52.7 51.1 56.4 49.8 49.1 39.7 44.1 52.2 46 40.5 39.9 50 56 77.5 57.8 38.8 100.2 63 38.2 44.6 40.6 51.1 80.8 75.2 51.9 61 53.8 53.1 63 46.6 74.2 63.2 50.4 57.2 42.6 45.7 41.9 51.7 51 49.4 51.3 45.1 46.4 48.6 56.7 38.9 34.6 60 42.6 37.1 No Population Gain 50.7 60.3 48.3 40.3 40.4 39.3 45.9 47.2 57 46.1 41.5 42.3 46.4 51.7 44.9 50.5 51.8 29.1 34.9 30.9 27 61.1 51.8 40.7 45.7 34.7 43.4 37 21 33.1 45.7 31.9 52 45.7 38.7 56.3 21.7 43 56.2 48.9 34.5 44.5 43.2 42.1 50.9 41.3 35.4 33.6 56.5 43.4 Figure 2.19