Basic Statistical Concepts and Methods for Earth 

Scientists 

By Ricardo A. Olea 

Open-File Report 2008–1017 

U.S. Department of the Interior 

U.S. Geological Survey 
U.S. Department of the Interior 

DIRK KEMPTHORNE, Secretary 

U.S. Geological Survey 

Mark D. Myers, Director 

U.S. Geological Survey, Reston, Virginia 2008 

Revised and reprinted: 2008 

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Telephone: 1-888-ASK-USGS 

Suggested citation: 

Olea, R.A., 2008, Basic statistical concepts and methods for earth scientists: U.S. Geological Survey OpenFile Report 2008–1017, 191 p. 

Any use of trade, product, or firm names is for descriptive purposes only and does not imply 

endorsement by the U.S. Government. 

Although this report is in the public domain, permission must be secured from the individual 

copyright owners to reproduce any copyrighted material contained within this report. 
Basic statistical concepts and 

methods for earth scientists
 

Ricardo A. Olea
 

Open-File Report 2008–1017
 

U.S. Department of the Interior 

U.S. Geological Survey 
CONTENTS
 

1. Introduction 32. Data collection issues 113. Univariate statistics 354. Bivariate statistics 785. Nondetect statistics 1006. Statistical testing 1117. Multivariate statistics 153

These notes have been prepared for teaching one-day course intended to refresh and upgrathe statistical background of the participants. 
1. INTRODUCTION
 
STATISTICS
 

Statistics is the science of collecting, analyzing,interpreting, modeling, and displaying masses ofnumerical data primarily for the characterization understanding of incompletely known systems. 

Over the years, these objectives have lead to a 

amount of analytical work to achieve, substantiaand guide descriptions and inferences. 
WHY STATISTICS?
 

• Given any district, time and economics ordinari

preclude acquiring perfect knowledge of a singattribute of interest, let alone a collection of theresulting in uncertainty and sometimes into bulkdatasets. 

• Note that uncertainty is not an intrinsic property 

geological systems; it is the result of incomplete knowledge by the observer. 

• To a large extent, earth sciences aim to inferrin

processes and predicting future events based orelationships among attributes, preferably quanuncertainty. 

Statistics is an important component in the eme

fields of data mining and geoinformatics. 
WHEN NOT TO USE STATISTIC

•	 There are no measurements for the attribute(s) 

interest. 

•	 There are very few observations, say 3 or 4. •	 The attribute is perfectly known and there is no 

interest in having associated summary informapreparing any generalizations, or making any tyquantitative comparisons to other attributes. 
CALCULATIONS
 

As shown in this course, most of the statistical calcfor the case of one attribute are fairly trivial, not reqmore than a calculator or reading a table of values

Multivariate methods can be computationally inte

suited for computer assistance. Computer programto be cumbersome to utilize, some requiring the mof special computer languages. 

Modern computer programs are easy to employ 

are driven by friendly graphical user interfaces.Calculations and graphical display are performed tdirect manipulation of graphical icons. 
EXAMPLE OF MODERN PROGR
COURSE OBJECTIVES
 

This short course is intended to refresh basic conand present various tools available for optimal exof information from data. 

At the end of the course, the participants: 

•	 should have increased their ability to read the 

statistical literature, particularly those publicatiolisted in the bibliography, thus ending better quto independently apply statistics; 

•	 may have learned some more theoretically sou

convincing ways to prepare results; 

•	 might be more aware both of uncertainties com

associated with geological modeling and of the multiple ways that statistics offers for quantifyinuncertainties. 
© The New Yorker Collection 1977 Joseph Mirachi

from cartoonbank.com. All rights reserved. 

“That’s the gist of what I want to say. Now get 

me some statistics to base it on.” 
2. DATA COLLECTION 

ISSUES
 
ACCURACY AND PRECISION
ACCURACY
 

Accuracy is a property of measured and calcula

quantities that indicates the quality of being closthe actual, true value. •	 For example, the value 3.2 is a more accurat

representation of the constant π (3.14159265

than 3.2564. 

•	 Accuracy is related to, but not synonymous w

precision. 
PRECISION
 

In mathematics, precision is the number of signdigits in a numerical value, which may arise from measurement or as the result of a calculation. 

The number 3.14 is less precise representatio

the constant π than the number 3.55745. 

For experimentally derived values, precision 

related to instrumental resolution, thus to the abreplicating measurements. 
ACCURACY AND PRECISION

Mathematically, a calculationor a measurement can be: 

•	 Accurate and precise. For example 3.141592653 is

both an accurate and precise value of 

π. 

• Precise but not accurate, 

like 

π = 3.5893627002.
 

•	 Imprecise and inaccurate, 

such as 

π = 4. 

•	 Accurate but imprecise, such as 

π = 3.1.
 Accuracy 

Precision 

Experimental co

While precision is obvious to assess, accuracy 

not. To a large extent, what is behind statistics ieffort to evaluate accuracy. 
SIGNIFICANT DIGITS
 
SIGNIFICANT DIGITS
 

Significant digits are the numerals in a number thasupported by the precision of a measurement or calculation. 

•	 The precision of a measurement is related to th

discriminating capability of the instrument. 

•	 The precision of a calculation is related to the n

involved in the arithmetic processing and is deca few simple rules. 
SIGNIFICANT DIGITS IN A NUMB

•	 Digits from 1-9 are always significant. 

•	 Zeros between two other significant digits are al

significant. 

•	 Zeros used solely for spacing the decimal point (placeholders) are not significant. For example, 

value 0.00021 has two significant digits. 

•	 Trailing zeros, both to the right of the decimal pl

of another significant digit, are significant. For e2.30 has three significant digits. 

•	 For real numbers, zeros between the rightmost significant digit and the decimal place are not sig

Hence, 4000. is automatically understood to havsignificant digit. If, for example, indeed there are use the notation 4.0.10

3

. 

•	 Whole numbers have unlimited number of signif

digits. So, 4000 has infinite significant digits. 
SIGNIFICANT DIGITS IN CALCULAT

Reporting results often requires some manual rouespecially if using calculators or computers. 

In all four basic operations of addition, substrac

multiplication, and division, the number with the lesignificant digits in decimal places determines thesignificant decimal digits to report in the answer.Examples: 

• 2.1 + 2.08 = 4.2 • (8.1.2.08)/4 = 4.2 

Easy way corollary: 

If all the operands have the same number of si

decimal places, the significant decimal places iresult are the same as those of the operands. E0.38.27.90 - 4.28/10.25 = 10.18 
DETECTION LIMIT
 
LIMITS IN CHEMICAL ANALYSE

In analytical chemistry, the detection limit is the 

quantity of a substance at which it can be decidewhether an analyte is present or not, a problem tarises from the fact that instruments produce reaeven in the complete absence of the analyte. 

The limit actually measured, 

x

L, is:
 

x 

= x 

+ k ⋅ s L bl bl 

where 

xbl is the mean of blank measurements, 

their standard deviation, and k a reliability factor commonly taken equal to 3. If S is the sensitivity the calibration curve, then the detection limit, LO

LOD 

= k ⋅ sbl ⋅S 
EXAMPLE OF COPPER DETECTIO

In this case the interest is in the detection limit for Cu in Fe. From the measurements in Fe with actually any Cu: 

xbl 

= 2.594 cps 

sbl 

= 0.366 cps 

So, taking k 

=

3: 

xL 

= 2.594 

+ 3 ⋅0.366 

= 3.692 

Measured limit of detec
8 

ond ce s per6 sunto ced, 4 uraseM

2 

0 0.1 0.2 

Standard, perce

EXAMPLE OF COPPER DETECTIO

The sensitivity of the calibration curve, 

S, is the 

inverse of the rate of change of instrumental reading with changes in the concentration of the analyte above the detection limit, LOD. In our example, 

S 

= 0.074 %/cps. 

Thus 

LOD 

= 3 ⋅0.366 ⋅0.074 

= 0.08 %Cu 
INSTRUMENTAL DETECTION LIM

The Limit of Detection presumes a matrix clean interfering substances. In such a case, the Limit Detection is a special case of Instrumental DetecLimit, IDL: 

LOD 

= IDL .
 
METHOD DETECTION LIMIT 

In the presence of complex matrices that can complicate the calculation of the background properties, it is common to use the Method DetecLimit, MDL, instead of simply the Instrumental Detection Limit, IDL. 

In the detection of copper, iron is a clean matrix 

stainless steel would be a complex matrix with sealloys. 

The Method Detection Limit can be anywhere f

2-5 times higher than the Instrumental Detection 

MDL 

=

(2 to 5

)⋅IDL 
LIMIT OF QUANTITATION
 

A common misconception is that the LOD is the minimum concentration that can be measured. Instead, LOD is the concentration at which one creasonably certain that the concentration is not z

Just because one can tell something from nois

does not mean that one can necessarily know homuch of the analyte there actually is. Quantitatiogenerally agreed to start at 5 times the Method Detection Limit. This higher concentration is callPractical Quantitation Limit, PQL. Thus 

PQL 

= 5 ⋅MDL
 
SUMMARY
 

In summary: 

LOD : IDL : MDL : PQL = 1:1: (2to5): (10 to
LEGACY DATA
 
LEGACY DATA
 

Legacy data are information in the development which an organization has invested significantresources in its preparation and that has retaineimportance, but that has been created or stored software and/or hardware that is perceived outmor obsolete by current standards. •	 Working with legacy data is ordinarily a difficu

frustrating task. 

•	 Customarily, legacy data are too valuable to b

ignored. 

• The nature of problems is almost boundless, y

possible to group them in some general categ
COMMON PROBLEMS WITH LEGAC

• Data quality challenges • Database design problems
 • Data architecture problems
 • Processing difficulties 
TYPICAL DATA QUALITY PROBLE

•	 Different technologies in acquiring the data. 

•	 Obsolete methods and conventions were used t

prepare the data. 

•	 The purpose of a column in a tabulation is deter

the value of one or several other columns. 

•	 Inconsistent data formatting. •	 Frequently missing data •	 Multiple codes have the same meaning. 

•	 Varying default values 
COMMON DATABASE DESIGN PROB

• Inadequate documentation • Ineffective or no naming conventions. • Text and numbers appear in the same column. 
COMMON ARCHITECTURE PROBL

• Different hardware platforms • Different storage devices • Redundant data sources • Inaccessible data in obsolete media. 
LEGACY DATA EVALUATION 

Issues to consider include: • Are the data needed to achieve an established • What will be lost if this information is eliminate• Are the data consistent? • Are the data accurate and up-to-date? • How much data are missing? • What is the format of the data? • Can the new system support the same format? • Is the information redundant? • Is this information stored in multiple ways or m

times?
 

We are all generating legacy data. Be visiona
3. UNIVARIATE 

STATISTICS
 
EVERYTHING AND A PIECE
 

In statistics, population is the collection of all possoutcomes or individuals comprising the complete of our interest, e.g. all people in USA. 

Populations may be hard or impossible to analy

exhaustively. In statistics, a limited collection of measurements is called sample, e.g. a Gallup Po

Unfortunately, the term “sample” is employed w

different meanings in geology and statistics. 

Geology Statistics 

collection sample sample observation 

The statistical usage of the term sample is obse

what follows. 
RANDOM VARIABLE 

Frequency 

Variate 

• A random variable 

variate is a quantimay take any of thvalues within a givwith specified relafrequencies. 

• 

The concept is heutilized in statisticcharacterize a poor convey the unkvalue that an attrimay take. 
DESCRIPTIVE ANALYSIS
 

•	 A sample of an attribute ordinarily comprises se

measurements, which are best understood wheorganized in some way, which is an important aof statistics. 

•	 The number of measurements in a sample is th

sample size. 

•	 There are multiple options to make the data mo

intelligible, some more convenient than others,depending on factors such as the sample size the ultimate objectives of the study. 
SAMPLE VISUALIZATION
 
FREQUENCY TABLE
 

If the range of the data is divided into class intecustomarily of the same length and all observaare assigned to the corresponding class, the reis a count of relative frequency of the classes. 
UNCF EXAMPLE FREQUENCY TA

Class Count Frequency, % 

1 7680-7710 1 1.43 

2 7710-7740 1 1.43 

3 774 -7770 0 1 1.43 

4 7770-7800 2 2.86 


 
 
 
 
 
 
 5 7800 783 - 0 5 7.14 


 
 
 
 
 

6 0-783 7860 6 8.57 7 7860-7890 10 14.298 7890-7920 11 15.719 7920-7950 13 18.57

10 7950-7980 7 10.00

11 7980-8010 7 10.00


 
 
 
 
 
 
 
 
 
 
 
 
 
 12 8010-8040 4 5.71 13 8040-8070 2 1.43 

Total 70 100.00 


 
 
 
 
 
 
 
 
 
 
 
 


This example relates to depth to an unconformitan undisclosed location by a major oil company

It will be employed as a common reference to 


 
 
 
 
 
 
 
 
 
 
 
 


graphically illustrate other definitions in this cha
HISTOGRAM
 

0 4 8 

12 16 20 

Frequency, percent 

7695 7755 7815 7875 7935 7995 8055
 

Depth, ft
 

A histogram is a graphical representation of afrequency table. 
CUMULATIVE FREQUENCY
 

Summaries based on frequency tables depend on selection of the class interval and origin. 

A display of pro

data not larger thavalue versus such eliminates this dra

The data 

zi are r

and the ordinate dinto classes of wid

The ordinate of an

center of the class same rank as 

zi. 

Customarily, the vertical axis is scaled so that da

a normal distribution displays as a straight line. 
SUMMARY STATISTICS
 
SUMMARY STATISTICS
 

Summary statistics are another alternative tohistograms and cumulative distributions. 

A statistic is a synoptic value calculated from 

sample of observations, which is usually but nonecessarily an estimator of some populationparameter. 

Generally, summary statistics are subdivided

into three categories: • Measures of location or centrality • Measures of dispersion • Measures of shape 
MEASURES OF LOCATION
 

Measures of location give an idea about the cetendency of the data. They are: • the mean; • the median; • the mode. 
MEANS 

The arithmetic mean or simply the mean, 

mˆ , of a 

sample of size n is the 

additive average of all theobservations, 

zi: 

1 n 

mˆ 
 

=

∑

zi . n i = 
 1 

The less frequently used

geometric mean, 

mˆ 

g , is the 

nth root of the product:

⎛ 1/ 

n n 

⎞

mˆ g 


 

= ⎜ ⎟ = ⋅ ⋅ ⋅ ⎜∏

n

zi ⎟

⎠

z1 

z2

K

⎝ 
 i =1 


 

z n . The arithmetic m

Always: UNCF sample is 79

and its geometric m


 mˆ 

≥

mˆ 

g . 7911.8 ft. 
THE MEDIAN 

The median, Q2, of a sample is the value thatevenly splits the numberof observations zi into a 

lower half of smallest observations and the upper half of largest measurements. 

If zi is sorted by

increasing values, then 

⎧z(n 1)/ 2, if n is odd, ⎪ + Q2 = ⎨⎪0.5 ⋅(z + z ), if n is even. ⎩ n / 2 (n / 2)+1 

The median of the UNCF sample is 7918 ft. 
THE MODE
 

Frequency, percent 

20
 16
 12
 

8
 4
 0 

7695 7755 7815 7875 7935 7995 8055
 

Depth, ft
 

The mode of a samthe most probable frequent value. 

For the UNCF sa

the center of the clthe most observati7935 ft. 
ROBUSTNESS
 

Robustness denotes the ability of statistical meto work well not only under ideal conditions, bupresence of data problems, mild to moderate departures from assumptions, or both. 

For example, in the presence of blunders, the 

median is a more robust statistic than the mean
MEASURES OF SPREAD
 

Measures of spread provide an idea of the dispersion of the data. The most common meaare: • the variance; • the standard deviation; • extreme values; • quantiles; • the interquartile range. 
VARIANCES
 

The variance, σˆ 2, is the average squared dispe

around the mean: 

) 2 1 n σ = ∑(z − 2 i m) n i =1 

expression that is commonly restricted to estimvariances of finite populations. 

On dealing with samples, the denominator is 

changed to n - 1. 

Because this is a quadratic measure, it is less

robust than other measures of spread. 

The variance of the UNCF sample is 5474 sq 
STANDARD DEVIATION 

The standard deviation is the positive square rootof the variance. 

It has the advantage of

being in the same unitsas the attribute. 

The UNCF sample

standard deviation is 74.5 ft. 

σ

σ

σ

According to Chebyshev’s

theorem, for any sample and t 

>

1, the proportion 

that deviates from the mean 

mˆ at least t ⋅σˆ is at m

1 Prop

( X 

−

mˆ 

≥ t ⋅σˆ 

)

≤

2t 
EXTREME VALUES
 

The extreme values are the minimum and the maximum. 

For the UNCF 

sample, the minimumis 7696 ft and the maximum is 8059 ft. 
QUANTILES
 

The idea of the median splitting the ranked saminto two equal size halves can be generalized tnumber of partitions with equal number ofobservations, called quantiles or fractiles. The names for the boundaries for the most common quantiles are: • Median, for 2 partitions • Quartiles, for 4 partitions. • Deciles, for 10 partitions. • Percentiles, for 100 partitions. There is always one less boundary than the nuof partitions. 
UNCF QUARTILES
 

Q1 = 7871.75 ft 

Q2 = 7918 ft Q3 = 7965.75 ft 
INTERQUARTILE RANGE
 

The interquartile range,iqr, is the difference between the upper andthe lower quartiles 

iqr 

=

Q3 

−

Q1, 

thus measuring the datacentral spread. 

For the UNCF sample,

the interquartile range is94 ft. iqr 
OUTLIER
 

Atypical extreme values are called outliers.
 •	 Outliers may be genuine 

values or sampling blunders. 

7695 

7755 

7815 

7875 

7935 

7995 

8055 

8085 

81

20 

mild 

16 

Frequency, percent 12 8 

4 0 

Depth, ft 

• A practical rule of thumb is 

to regard as a mild outlierany value deviating fromthe median more than 1.5 times the interquartile range and an extreme outlier if it departs more 3times. 

For the UNCF sample, all mild outliers seem to b

legitimate values while the extreme outlier of 8210 blunder. 
BOX-AND-WHISKER PLOT

The box-and whisker plot isa simple graphical way to summarize several of the statistics: 

Depth, ft 

UNCF
 

8100 8000 7900 7800 7700 7600 

Max. 

Min. 

Mean 

+


 

• Minimum • Quartiles • Maximum • Mean Variants abound. Extremes may exclude outliers, in which case the outliers are individually posted as open circles. Extremes sometimes are replaced by the 5 and 95 percentiles. 
MEASURES OF SHAPE
 

The most commonly used measures of shapthe distribution of values are: • The coefficient of skewness • The quartile skew coefficient • The coefficient of kurtosis. 
COEFFICIENT OF SKEWNESS

The coefficient of skewness is a measure of asymof the histogram. It is given by: 

1 

∑

n 

(

zi −

m

)3 n 20 

7695 7755 7815 7875 793

Depth, ft 

B1 = -0.38 

16 

Frequency, percent 12 8 

40 

i =1 

B1 

=

σ3 

If : 

B1 

<

0, the left tail is 

longer;
 

B 

=

0, the distribution is 1 

symmetric; 

B1 

>

0, the right tail is 

longer. 

The UNCF coefficient of skewness is -0.38. 
QUARTILE SKEW COEFFICIEN

It serves the same purpose as the coefficient of sbut it is more robust, yet more sensitive to the cenof the distribution. Its definition is: 

20 

qs 

=

(Q 

−Q 

)

−

(Q 

−Q 

) 3 2 2 1 

16 

qs=0.02 

7695 7755 7815 7875 

Depth, ft 

iqr
 

Frequency, percent 12 If : qs 

<

0, the left tail is longer;
 

8

qs 

= 0 4 , the distribution is 

symmetric; 0

qs 

> 0 , the right tail is longer. 

The UNCF quartile skew coefficient is 0.02. 
COEFFICIENT OF KURTOSIS 

This statistic measures the concentration of valuethe mean. Its definition is: 

Gaussian dist

n 0.5 1 

4

∑

(

zi −

m

) 0.4 

n i=1 B

= 0.3 2 

σ4 

0.2 

If : 

B2 

<

3, the distribution is 

more peaked than 

0.1 

Gaussian; -5 -2.5 0 

0
 

B2 

=

3, it is as peaked as 

x
 

f(x) 

Gaussian; 

B2 

> 3 , it is less peaked 

than Gaussian. 

The UNCF coefficient of kurtosis 

0 4 8 

12 16 20 

Frequency, percent 

B2=3.08 

7695 7755 7815 7875 

Depth, ft
 is 3.08. 
MODELS
 
PROBABILITY
 

Probability is a measure of the likelihood that an uevent, A, will occur. It is commonly denoted by P

•	 The probability of an impossible event is zero, P

It is the lowest possible probability. 

•	 The maximum probability is Pr[A] = 1, which den

certainty. 

• When two events A and B cannot take place 

simultaneously, Pr[A or B] = Pr[A] + Pr[B].
 

•	 Frequentists claim that Pr[A] = NA/N, where N is 

number of outcomes and NA the number of outc

A. They can be counted theoretically or experim

•	 Subjective probabilities are a degree of belief in 
BAYES’S THEOREM 

This is one of the most widely used probability relaIf event B already occurred, the conditional probabevent A, Pr[

A|B], is: 

[ Pr

[B | A

] Pr

A | B 

]

= Pr [ ] A Pr[ ] B 

Example
 Suppose we have two boxes. A blue ball is drawn (event B). What is the probability it came from box #1 (event 

A)?

• In the absence of additional information, Pr[

A] = 

Number o

Box Blue Red 

#1 20 5 

#2 12 18 

23 

• Pr[B|A] = 20/25 = 0.8. 32 • Pr[B] = 32/55 = 0.59. Hence 

• Pr

[ 0.8 A | B

]

= 0.5 

= 0.69

0.59 
PROBABILITY FUNCTIONS
 

Analytical functions approximating experimentafluctuations are the alternative to numerical desand measures. They provide powerful approximof general conditions. The drawback is loss of fdetails in favor of simpler models. 

Models approximating histograms are called

probability density functions. 

Variations in the parameters of a probability d

function allow the generation of a family of distrisometimes with radically different shapes. 

Main subdivision is into discrete and continuo

distributions, of which the binomial and normal distribution are two typical and common exampl
BINOMIAL DISTRIBUTION

This is the discrete probability density function,.f 

(

x;p,n

), of the number of 

successes in a series of 

independent (Bernoulli) trials, such as head or tails in coin flipping. If the probability of success at every trial is 

p, the 

probability of x successes in n independent trials is 

0 

y
 0.3 ilit


 

bab0.2
 orP0.1
 

0 1 2

3

4

5

6

7

Number of suc

0.4


 

y tilobabirP 0.3 


 0.2
 0.1
 0
 

0 1 2

3

4

5 6 7

Number of suc

0.4 

f 

(

n! x;p,n

)

= p 

x

(

1

−

p

n

−

x

) , x 

=

0

x

,1, 2,

K, n !(n 

−

x

)! 
NORMAL DISTRIBUTION
 

The most congenial of allcontinuous models is the distribution, also known aGaussian distribution. Its parameters are 

μ and 

σ, w

coincide with the mean an

0 standard deviation. 

0.1 0.2 0.3 0.4 0.5 

f(x; 0,1) 

-5 -2.5 0 2.5 5 

(x

−

μ ) x 

2

1 

−

2

2

σ f 

(

x;μ,σ

)

= e ,− ∞

< x < ∞

σ

2

π

If X = log(

Y) is normally distributed, Y is said to 

a lognormal distribution. Lognormal distributions positively defined and positively skewed. 
PROBABILITY FROM MODELS x

Prob 

[X 

≤ x 

]

1 

1 

=

∫f x( )dx Prob

[X 

≤

x1]

−∞

x

f(x)

Gaussian distribution 0.5 

1 

0.4 0.88

0.8 

0.88 

-5 -2.5 0 2.5 5
 -5 -2.5 0 

F (x ;0,1) 0.6
 0.4
 

f(x;0,1)0.3
 0.2
 

0.2 

0 

0.1 

0 1.

1.17 

x x 

Examples: Gaussian distr

[

1.17 

Prob X 

≤ 1.17] =

∫Normal

(

x;0,1

)dx Prob[X 

≤ 1.17] =

−∞

= 

= 0.88 
EXPECTED VALUE
 

Let X be a random variable having a probability distribution f x ( ), and let 

u

(

x

) be a function of 

expected value of x is denoted by the operator 

E

and it is its average value. 

Continuous case, such as temperature: 

E

[

∞

u x ( )

]

=

∫ u x f −∞ ( ) ( ) x dx 

Discrete case, like coin flipping: 

E

[

u

(

x

)

]

=

∑

u

(x f ) (

x

)

x 
EXAMPLE 

For the trivial case 

u(x) = x 

in the discrete case, if all values are equally probabexpected value turns into 

E[ ] = 1 x x ∑n x 

which is exactly the definition of the mean. 
MOMENT
 

Moment is the name given to the expected valuwhen the function of the random variable, if it extakes the form (x − a)k, where k is an integer la

than zero, called the order. 

If a is the mean, then the moment is a central 

moment. 

The central moment of order 2 is the variance

an equally probable discrete case, 

) 2 1 2 M2 = σ = ∑n (xi − m) n i =1 
SIMULATION
 
1.1

Cumulative probability

1 

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 

0 

xmin 

At

SIMULATION
 

Reality is often too complex to be able to analyticaprobabilities as in flipping coins. Modern approaccomputers to model a system and observe it repewith each repetition being the equivalent of flippin

• Relevant attributes take the form of random varia

instead of being deterministic variables. 

• Generators of random sequences of numbers be

0-1 play an important part in simulation. 

• The other important component is 

the Monte Carlo method, which allows drawing values from the distributions. 

• The final result is the histogram(s) 

for the output variable(s). 
BOOTSTRAP 

• This is a resampling form of the 

Monte Carlo method. 

• It works with the sample directly 

instead of a distribution. 

• In its simplest form, the general 

steps are: 

1. Randomly pick as many

measurements as the samplesize. Some values may betaken more than once, while others not at all. 

2. Calculate and save as many

statistics as interested. 

3. Go back to step 1 and repeat the

process at least 1,000 times.
 

4. BIVARIATE STATISTIC
TOOLS
 

Frequently there is interest in comparing two or mmeasurements made for the same object or site. the most common alternatives, we have: • Scatterplot • Correlation coefficient • Regression • Quantile-Quantile plot • Probability-Probability plot 

Some of these concepts can be generalized to 

than two variables, and all multivariate techniques Chapter 7 are valid for two variables. 
SCATTERPLOTS 

A bivariate scatterplot is a Cartesian posting in which the abscissa and the ordinate are any two variables consistently measured for a series of objects. 

Scatterplots are 

prepared for exploring or revealing form, direction, and strength of association between two attributes. 

Mecklenburg Bay seafloo
COVARIANCE
 

Given two random variables X and Y with means μ 

their covariance is the expected value: 

Cov X,Y = E[(X − μ X )(Y − μY )]

The covariance estimator when using a sample o

measurements is: 

1 n n n Covˆ X,Y = ∑ 1 xi ⋅ yi − ∑xi ⋅ n ∑y − i 1 i =1 n ⋅(n −1) i =1 i =1 

The covariance is a measure of joint variation. 
CORRELATION COEFFICIENT 

This coefficient is the number most commonly used to summarize bivariate comparisons. If 

σ X and 

σ

Y are the standard 

deviations for two variables, their correlation coefficient, 

ρ, is given by: 

Cov 

ρ

=

X ,Y

σ X ⋅σ Y 

•	 It only makes sense to employ 

ρ for 

assessing linear associations. 

•	 

ρ varies continuously from: 

1, perfect direct linear correlation 0, no linear correlation -1, perfectly inverse correlation 
SENSITIVITY TO OUTLIERS
 Moore and Notz, 2006 
REGRESSION
 

Regression is a method for establishing analytidependency of one or more variables on anothmainly to determine the degree of their dependto estimate values not included in the samplingto summarize the sampling. •	 The variable in the abscissa is called the 

regressor, independent, or explanatory varia

•	 The variable in the ordinate is the regressed

dependent, or response variable. 

•	 In many studies, which variable goes into wh

axis is an arbitrary decision, but the result is different. Causality or the physics of the procmay help in solving the indetermination. 
REGRESSION MODEL
 

• The model is: 

Response variable (y ) 

εi 

( ;θ) i xf 

yi = f 

(

xi 

;θ

)

+

ε 

• f 

(

xi 

;θ

) is any contin

function of x that is j

selected by the user

unknown parameter

• Term 

ε is a random 

accounting for the e

•	 Having selected f 

(

x

parameters 

θ are ca

by minimizing total e

Explanatory variable (

x) which there are seve

methods. 

Avoid using the model outside the sample range oexplanatory variable. 
LINEAR REGRESSION
 

ρ = 0.94 

The simplest case is linear Mecklenburg Bay seafloregression with parametersobtained minimizing the mean square error, MS

E: 

n

1

2 MSE 

=

∑

ε i n i =1 

In this special situation, 

ρ2 

accounts for the proportionof variation accounted bythe model. 

In the example, 

ρ = 0.94. 

Hence, in this case, the 

linear regression explains 88% 

(100 ⋅ ρ2 

)

of the variation. 
NONLINEAR REGRESSION
 

Linear Quadratic polynomial 

MSE = 25.47 MSE = 20.56 

Cubic polynomial Sixth degree polynomial 

MSE MSE = 20.33 = 19.98 

• In theory, t

higher thepolynomial the better t

• In practice

higher thepolynomialless robust solution. 

• Overfitting 

capture nonot systemvariation. 
IMPLICATIONS
 

Life expectancy, year 

Countries with population over
 

20 million in 1990
 

ρ = -0.81
 80 

75 70 65 60 55 50 

1 10 100 1000 

Persons per television set 

High to good correla

•	 allows prediction of 

variable when only is known, but the inmay be inaccurate,particularly if the cocoefficient is low; 

•	 means the variable

related, but the assmay be caused by common link to a thlurking variable, marelationship meanin

•	 does not necessari

cause-and-effect. 
“In a new effort to increase third-world life expectancy, aid organizations today began delivery 

of 100,000 television sets.” 

After Moore and Notz, 2006 
QUANTILE-QUANTILE PLOT 

•	 A quantile-quantile or QQ-plot is a scatterplot based on ranked data. 

•	 The paring is independent from the object or site where the

observations were taken. The first pair of coordinates has the minimumvalue for each attribute, the secondpair is made of the second smallestreadings, and so on until finishingwith the two maximum values. Interpolations are necessary fordifferent size samples. 

•	 Identical distributions generate a straight line along the main

diagonal. 

•	 QQ-plots are sensitive to a shift and scaling of the distributions. 
STANDARDIZED VARIATE
 

If X is a random variable with mean 

μ and 

standard deviation 

σ, 

the standardized variate, 

Z, is the 

transformation. 

X 

−

μ Z 

=

σ

A standardized variate always has a mean of zand variance of one. 
PROBABILITY-PROBABILITY PL

V

A PP-plot is another scatterplot prepared extractinformation from the cumulative distributions of tvariates. 

If the variates are in different units, preliminary

standardization is necessary. 

For given thresholds, the axes show the cumu

probabilities for the two distributions being comp
SPECIAL TOPICS
 
PROPAGATION OF ERRORS

Error propagation is a rather Analytically:archaic term referring to the N(30,5) = N(12,3) influence that uncertainty in variables have on a function Monte Carlo simula

of them. 0.5 

0 

0.1 0.2 0.3 0.4 0.5 

f(x) 

N(12,3) 

• Under certain simple 0.4 0.3 

conditions, it is possible 

f(x) 

0.2 0.1 to analytically express 0 

0 3 6 9 12 15 18 21 24 0 

the impact of variable 

x 

uncertainty on the function uncertainty. 

•	 Today the situation is 

resolved through numerical simulation. 
BIAS
 

Bias in sampling denotes preference in taking thmeasurements. 

Bias in an estimator implies that the calculatio

preferentially loaded in one direction relative to tvalue, either systematically too high or too low. • When undetected or not compensated, bias in

to erroneous results. 

• The opposite of the quality of being bias is unb
SEMIVARIOGRAM
 

Let 

si be the geographical location of a sampling s

The semivariogram is a

squared average comparisonof the degree of dissimilaritywith geographical distance,h, for pairs of measurementsof Z at sites s and s + 

h: 

1

2

γ ( ) h 

=

E

[{Z(s 

+ h 

)

−

Z( ) s 

}

] 2 

If 

Nh is the number of pairs 

a distance h apart, the most 

popular unbiased estimate is: Nh 

ˆ( ) = − 1 

∑

[Z(s 

+

h

)

−

Z( ) si ]

2

γ h i 2

Nh i =1
 
LEARNING FROM THE SEMIVARIO

•	 The autocovariance is a 

function that results from grouping covariance valuesby distance class. 

•	 The semivariogram and the 

autocovariance are related through:

γ ( ) h 

=

σ

− Cov

(

) 2 h . 

•	 The sill is equal to the 

population variance,

σ

2

. 

•	 The range, 

a, gives a

quantitative indication to the Lag 

a

notion of radius of influence of a measurement.
 

Second order moment 

σ2 
COMPOSITIONS 

In some studies, an object can bedecomposed in constituent parts. Bydefinition, these parts add to the 25 whole object. Compositional data are 

0 

20 

measurements of these components.
 

Ash, percent weight 15 10 5 Compositional data only contain relative information.
 

Compositional data are common in 75 80 85 

Fixed carbon, 

geochemical analysis, taking unitssuch as parts per million and percent. 

Only positive measurements are possible, betw

zero and the value denoting the whole object, saythe parts are percentages of the whole. 

The fact that compositions are constraint data b

correlation to the components. 
CAUTION
 

In general, straight application of statistics to data aa constant produces suboptimal or inconsistent resMost successful ad hoc developments still employ cstatistics, but after taking special precautions. 

The most common approach is to transform the d

possibility is the isometric log-ratio transformation, for two components, X1 and X2, is: 

1 X 1 X Y = log 1 or Y = 2 2 log , with X + X 1 X2 

2 2 X1 

where c is a known closure constant. This transform

• brings the range of variation to (−∞,∞); 

• eliminates the correlation due to closure; • properly handles distances in attribute space.Results require backtransformation.


 
5. NONDETECT STATIST
STATISTICS OF THE UNDETECT

•	 Undetected values have peculiar characterist

put them apart from those actually measured

•	 Values below detection limit require different 

treatment than precise measurements. 

• There have been several attempts to analyze 

samples with values below detection limit.
 
STATISTICAL METHODS 

The main approaches for the calculation of summstatistics are: •	 Replacement of the values below detection by 

arbitrary number. 

•	 Maximum likelihood method 

•	 Kaplan-Meier method 

•	 Bootstrap 
REPLACEMENT 

•	 This is the prevailing practice. 

•	 This is not really a method. It has poor to no 

justification. 

•	 It can lead to completely wrong results. •	 The most common practice is to replace the v

below detection by half that value. 
f(x)

0.40.5 0.3 0.2 0.

0

f(x|θ ) 

MAXIMUM LIKELIHOOD METHO

THE GENERAL CASE
 

This is a method for the inference of parameters, 

θ

parent probability density function f 

(x | θ

) based on 

multiple outcomes, 

xi. 

The function 

L

(

θ

)

= f 

(x ,K,x |θ ,K,θ

)

is called the

1 n 1 m 

likelihood function. 

Given outcomes 

xi, the optimal parameters 

θ are 

that maximize 

L

(

θ

), which come from solving 

∂

L

(

θ

)

For example, for the normal distribution, n 2 

( n (x

−

θ )

1

x

−

θ 2 

∑ i 1 ) 1 

( )

−

2

θ22 ( )

⎛ 1 

⎞ 2 

−

i =1 

f x | θ

=

e

L

θ

2

θ2

2

= ⎜

⎜ ⎟ 2 ⎟ e 

2πθ 2 

⎝

2πθ 2

⎠

the solution is: 

1 n 

2 1 

θ

2 2 

1 

=

∑

n 

xi =

μˆ 

θ 2 

=

∑

(

x

n n i −

μˆ 

)

=

σˆ 

i =1 i =1 
MAXIMUM LIKELIHOOD
 

OF THE UNDETECTED
 

•	 The method is parametric: it requires assuming 

distribution that the data would have followed in all values would have been accurately measure

•	 Essential for the success of the method is the 

assumption of the correct distribution. 

•	 Most commonly selected models are the norma

the lognormal distributions. 

•	 It is commonly used to estimate mean, variance

quartiles. 

•	 The method tends to outperform the others for s

of size larger than 50, or when 50 to 80% of the are below detection limit. 
ARSENIC CONCENTRATION
 

OAHU, HAWAII
 

Mean: 0.945 St. dev.: 0.656 
KAPLAN-MEIER METHOD
 

•	 The Kaplan-Meier method works in terms of a s

function, a name derived from clinical applicatiowhich the interest is on survival time after startimedical treatment. 

•	 The Kaplan-Meier method was developed for r

censored data—exact measurements are missabove certain values. Left censored data must flipped using a convenient but arbitrary thresholarger than the maximum. 

•	 Calculated parameters must be backtransforme

•	 The Kaplan-Meier method tends to work better 

there are more exact measurements than data detection limit. 
OAHU DATA SURVIVAL FUNCTI

0 

0.2 0.4 0.6 0.8 

1 

Survival probability Mean: 0.949 St. dev.: 0.807 

0 1 2 3 4 5 

Flipped concentration, micro g/L 

Flipping constant: 5 micro g/L 
BOOTSTRAPPING FOR NONDETE

Remember that the bootstrap is aresampling form of the MonteCarlo method. The adjusted stepsfor handling nondetects are: 1. Randomly pick as many values 

as the sample size. 

2. If a picked value is a nondetect, 

replace it with a number randomly selected between zero and the detection limit. 

3. Make a log-ratio transformation. 4. Calculate all statistics of interest. 5. Save the statistics. 6. Go back to step 1 and repeat 

the process at least 1,000 times. 
COMPARISON OF THE FOUR MET

EXAMPLE FROM OAHU, HAWA

 Mean St. dev. 

Q1 Median 

μg/L 

Substituting with zero 0.567 0.895 0.0 0.0 

Subs. half detection lim. 1.002 0.699 0.5 0.95 Subs. with detection lim. 1.438 0.761 0.75 1.25 

Lognormal Max. Likelihd. 0.945 0.656 0.509 0.777 

Kaplan-Meier, nonparm. 0.949 0.807 0.5 0.7 

Bootstrap average 1.002 0.741 0.501 0.752 Log-ratio bootstrap aver. 0.722* 0.492 0.738 

*geometric mean 
6. STATISTICAL TESTIN
STATISTICAL TESTING
 

The classical way to make statistical comparisoto prepare a statement about a fact for which it possible to calculate its probability of occurrenceThis statement is the null hypothesis and its 

counterpart is the alternative hypothesis. 

The null hypothesis is traditionally written as 

H

and the alternative hypothesis as 

H1. 

A statistical test measures the experimental 

strength of evidence against the null hypothesis

Curiously, depending on the risks at stake, the 

hypothesis is often the reverse of what the experimenter actually believes for tactical reasothat we will examine. 
EXAMPLES OF HYPOTHESES 

Let 

μ1 and 

μ2 be the means of two samples. If on

wants to investigate the likelihood that their meanthe same, then the null hypothesis is: 

H0 : 

μ1 

=

μ2 

and the alternative hypothesis is: 

H1 : 

μ1 

≠

μ2 

but it could also be: 

H1 : 

μ1 

>

μ2 

The first example of 

H1 is said to be two-sided o

tailed because includes both 

μ1 

>

μ2 and

μ1 

<

μ2 . 

second is said to be one-sided or one-tailed. 

The number of sides has implications on how to 

formulate the test. 
POSSIBLE OUTCOMES
 

H0 is correct 

Correct decision 

H0	 is accepted Probability: 

1

−

α

Type I error 

H0	 is rejected Probabilty: 

α

H0 is incor

Type II er

Probability

Correct dec

Probability:

1

•	 The probability α of committing a Type I error 

called the level of significance. 

•	 

α is set before performing the test. 

•	 In a two-sided test, α is split between the two • Often, H0 and 

α are designed with the intentio

rejecting 

H0, thus risking a Type I error and av

the unbound Type II error. The more likely ththe more power the test has. Power is 

1

−

β . 
IMPORTANCE OF CHOOSING 

H

Selection of what is null and what is alternative hypohas consequences in the decision making. Customoperate on the left column of the contingency table harder to analyze right column remains unchecked. environmental remedial action: 

Selection A 

Ho: Site is clean 

True False 

Test 

action 

Accept Correct Reject Wrong 

A. Wrong rejection msite is declared contwhen it is actually clwhich should lead to unnecessary cleanin

B. Now, the wrong d

Selection B 

Ho: Site is contaminated 

True False

Test 

action 

Accept Correct Reject Wrong 

declares a contaminsite clean. No actionprolongs a health ha

In both cases, Pr[Type I error] 

≤

α . 
PARTITION 

The level of significance is employed to partition the of possible values of the statistic into two classes. •	 One interval, usually the longest one, contains th

values that, although not necessarily satisfying thhypothesis exactly, are quite possibly the result ovariation. If the statistic falls in this interval—the interval in our cartoon—the null hypothesis is acc

accept	 reject 

•	 The red interval comprises those values that, alth

possible, are highly unlikely to occur. In this situanull hypothesis is rejected. The departure from thhypothesis most likely is real, significant. 

•	 When the test is two-sided, there are two rejectio

reject	 accept reject 
STATISTIC 

A key step in the feasibility of being able to run a is the ability of finding an analytical expression fostatistic such that: •	 It is sensitive to all parameters involved in the 

null hypothesis. 

•	 It has an associated probability distribution. 

Given some data, the p-value is the probability t

the statistic takes values beyond the value calculausing the data. The p-value allows to convert the statistic to probability units. For a one-sided test: • The p-value is also the smallest level of signific

leading to rejection of the test. 

• The lower the p-value, the stronger is the evide

against the null hypothesis provided by the dat
SAMPLING DISTRIBUTION 

The sampling distribution of a statistic is the distribution of values taken by the statistic for apossible random samples of the same size from the same population. Examples of such sampling distributions are: 

•	 Standard normal and the t-distributions for th

comparison of two means. 

•	 The F-distribution for the comparison of two 

variances. 

•

χ

2

	 The distribution for the comparison of two 

distributions. 
PROBABILITY REMINDER AND P-VA

∫

1 

Prob 

[

x 

∞

X 

≤ x 

]

= f x( )dx Prob 

[X 

> x 1]

=

1 

∫

−∞

x

f(x)

0.5 0.4 0.3 

0.9 

1

f(x;0,1) 0.2
 

0.1 

0 

-5 -2.5 0 

x 

Gaussian dis

Example: 

[

1.28 

Prob X 

≤ 1.28

]

=

∫Normal

(

x;0,1

)dx

−∞

= 0.90

Prob

[X 

> 1.28

]

∞

=

∫Normal

(

x;0,1

)dx 

1.28 

= 0.10 

In this example, assuming a one-sided test, whe

statistic is 1.28, the 

p-value would be 0.1 (10%). 
TESTING PROCEDURE
 

z

zs 

f(z) 

1. Select the null hypothesis 

H0 and 

the alternative hypothesis 

H1. 

2. Choose the appropriate statistic. 

3. Set the level of significance 

α. 

4. Evaluate the statistic for the case 

of interest, 

zs. 

5. Use the distribution for the statistic in combinati

the level of significance to define the acceptancrejection intervals. Find out either the correspon

• 

p-value of the statistic in the probability spa

• level of significance in the statistic space, 

z

6. Accept the null hypothesis if 

zs < 

zα. Otherwise

H0 because its chances to be true are less than 
PARAMETER TESTING
 
TESTING TWO MEANS
 
INDEPENDENCE
 

The random outcomes A and B are 

statistically independent if knowing the result of one does not change the probability for the unknown outcome of the other. 

Prob[A | B] 

= Prob[

A] 

For example: 

• Head and Tail in successive flips of a fair are independent events. • Being dealt a King from a deck of cards ahaving two Kings on the table are not independent events. 
TESTING PARENT POPULATION

Given a sample, the interest is on whether the vaare a partial realization of a population with knowand variance. 

A one-sided test would be: 

H0 : m

n

≤

μ p 

H1 : m

n

>

μ p 

where 

mn is the sample mean, 

μp is the population mean, 

σ

p

2 is the population variance. 
CENTRAL LIMIT THEOREM
 

Let X , X ,K, X be n random variables that: 1 2 n •	 share the same probability distribution 

D; 

•	 are independent; 

•	 the mean of the probability distribution is 

μ;
 

•	 the standard deviation of the distribution σ

exists and is finite; and 

•	 let m 

=

(X 

+ X 

+

K

+ X 

)/ 

n

.

n 1 2	 n 

Then, for any 

D: 

lim 

F(

m

)

= Normal 

(μ,σ

2 / 

n

)

n→∞

n 
CENTRAL LIMIT THEOREM IN ACT

Examples 

n 

=

2
 

n 

=

4
 

n 

= 25
 
STANDARD NORMAL DISTRIBUT

As mentioned before, standardization is the transformation obtained by subtracting the mean from each observation and dividing the result by the standard deviation. 

In the Central Limit case: 

m 

n

−

μ

z 

= p 

σ p 1/ n 

Normal

0.5 0.4 0.3 0.2 0.1 

0 

-4 -3 -2 -1 0 

z 

f(z) 

Thus, in this case, the appropriate statistic is th

sample mean standardized by employing the poparameters. By the Central Limit theorem, this sfollows a standard normal distribution, 

N(0,1). 
EXAMPLE
 

 Size Mean Stand. dev. 

Population 16.5 6.0 Sample 25 18.0 

1. H 0 : m n p ≤ μ 	 0.5 

H 1 : m n p > μ 	 0.4 

2. α = 0.05 (5%) 0.3 

3. The statistic is z-score. 0.2 

18 −16.5	 0.1 4. zs = = 1.25 6 1/ 25 0 

-4 -3 -2 -1 0 1 2 3 

z 5. For a Normal(0,1) 1.25 1.65

distribution, the

cumulative probability is 0.05 above zα = 1.65. 

6. zs<zα, therefore, H0 is accepted. 

f(z) 
INFLUENCE OF SAMPLE SIZE 

The table shows sensitivity of the results to sampwhen the experimental mean remains fixed at 18.the previous example. 

Sample 

size 

Statistic 

zs 

P-value, 

percent 

Ho

(α=5%) 

10 0.79 21.48 Accepted 

25 1.25 10.16 Accepted 

100 2.50 0.62 Rejected 

250 3.95 0.01 Rejected 

The larger the sample size, the more likely a reje•	 Specification of sample size adds context to a •	 Specificity of a test is poor for small sample si

•	 For large sample sizes, findings can be statist

significant without being important. 
DEGREES OF FREEDOM
 

Degrees of freedom is a measure of the independeof information on which a parameter estimate is bas

The number of degrees of freedom for any estima

number of observations minus all relationships prevestablished among the data. The number of degrefreedom is at least equal to the number of other panecessary to compute for reaching to the calculatioparameter of interest. 

For example, for a sample of size 

n, the number o

of freedom for estimating the variance is n – 1 becaneed first to estimate the mean, after which one obcan be estimated from the others. E. g., for the firs

x 

+ x 

+

K

+

x 1 2 

n x 

= n ⋅ −

(x 

+ x 

+

K

+ x 

1 2 3 n 

1 44 2n 4 34

mean 
STUDENT’S T-DISTRIBUTION

We have seen that the mean of any independenidentically distributed random variables is normaprovided that: 

•	 One knows the means and variance of the 

population. 

•	 The sample size is large. The rule of thumb i

a size above 30 is large. 

The Student’s t distribution operates analogou

the standard normal distribution, N ( ) 0,1 , and shoused instead when any of the requirements abonot met. 
EXAMPLES OF T-DISTRIBUTION

Probability density function Cumulative distributio

k is the degrees of freedom. 
TESTING TWO VARIANCES
 
F-DISTRIBUTION
 

The F-distribution is another distribution particudeveloped for statistical testing. 

Its parameters are two degrees of freedom th

vary independently from each other. 

It is employed to test variances and it operate

completely analogously to the other distributionprovides a reference value based on which the hypothesis is accepted or rejected according to level of significance. 

The main limitation is that the samples must 

from normal distributions. 
EXAMPLES OF F-DISTRIBUTION

Probability density function Cumulative distribution 

d1 and d2 are the degrees of freedom. 
EXAMPLE OF VARIANCE TESTI

 Size Variance 

Sample 1 11 22.4 

Sample 2 26 10.0 

H 0 1 2 :σ

=

σ

H 1 1 2 :σ

>

σ

The test statistic is the ratio of largest to smallest variance 

22.4 F 

=

10 

= 2.24 

F

Above F 

= 2.24, the 

p-value is 5%. Thus 

H0 is 10,25 

rejected if 

α

> 5.0%, when 

zα will be less than 2.

Otherwise, it is accepted. 
TESTING MORE THAN TWO MEA
ANALYSIS OF VARIANCE
 

•	 Analysis of variance is an application of the F-te

purpose of testing the equality of several means

•	 Strategies and types of variances abound, but it 

revolves around calculating some variances and proving or disproving their equality. 

•	 The interest is on deciding whether the discrepa

measurements are just noise or part of a significvariation. 

•	 The simplest form is the one-way ANOVA, in wh

same attribute is measured repeatedly for more specimens: 

H0 : 

μ1 

=

μ2 

L

=

μk ,k 

≥ 3 

H1 : at least one mean is different. 

•	 In this case, the variances going into the statistic 

variance for the mean among localities and the vwithin specimens. 
2 m ⎛ ⎞ ⎜ n SS ⎟ E = ∑ ∑xij − X j ⎜ ⎟

j =1 ⎝ i =1 ⎠

N = n ⋅ m

n 1 X j = ∑xij n i =1 

m n 1 X = ∑∑xij N j =1 i =1 

m 2

SS A = ∑⎜⎛ X j − X ⎟⎞ ⎝ ⎠ j =1 

m n 2 ⎛ ⎞ SST = ∑∑⎜ xij − X ⎟ ⎝ ⎠ j =1 i =1 

MS 

MS

ONE-WAY ANOVA TABLE
 

Source of 

variation Sum of squares 

Degrees of 

freedom Variance F te

Among localities SSA m - 1 MSA 

M

Within specimens (“Error”) SSE N - m MSE Total Variation SST N - 1 

m : number of localities 

n : number of specimens 

xij : specimen i for locality j 
EXAMPLE

Shell 

Specimen Locality 1 Locality 2 

1 16.5 14.3

2 17.3 16.1

3 18.7 17.6

width, mm 

Locality 3 Locality 4 Locality 5 

8.7 15.9 17.6

9.5 17.6 18.4

11.4 18.4 19.1

4 19.2 18.7

5 21.3 19.3

6 22.4 20.2

25

12.5 19.0 19.9

14.3 20.3 20.2

16.5 22.5 24.3

Width, mm20 + + +

+

15 + 10

5

1 1 2 3 4 5

Specimens 

Source of Degrees of 

variation Sum of squares freedom Variances F test statistic 

Among localities 237.42 4 59.35 

Within specimens 10.14 146.37 25 5.85 (“Error”) 

Total Variation 383.79 29

When F4,25 = 10.14, p = 0.00005, so for α > 0.00005, at

least one mean is significantly different from the others. 

140
DISTRIBUTIONAL TESTING
 
THE χ2DISTRIBUTION 

Let X1, X2,K, Xk be k random variables that: 

• are independent; • are all normally distributed; • the means μ

• each distribution has a standard deviation σ 

then, the sum of the standardized squares 

k 

2 

k ⎛ x − μ ⎞2 ∑z i = ∑⎜⎜ i i ⎟⎟

i =1 i =1 ⎝ σ i ⎠

i are not necessarily equal; 

follows a χk2 distribution with k degrees of

freedom. Like the t-distribution, lim χ2k = N 0,1 k→∞ ( 

. 
EXAMPLES OFχ2 DISTRIBUTIO

Probability density function Cumulative distributio

k is the degrees of freedom. 
χ2 TEST OF FITNESS 

χ k2 can be used to test whether the relative frequen

an observed event follow a specified frequency distThe test statistic is 

n (Oi −Ei )2 2 χ = ∑

i =1 Ei 

Oi is an observed frequency for a given class, 

Ei is the frequency corresponding to the target dist

As usual, if the statistic lands beyond the value of 

corresponding to the degrees of freedom k and leve

significance α , the null hypothesis is rejected. 

Weaknesses of the test are: 

• result depends on the number of classes, • no class can be empty, • there must be at least 5 observations per class 

least 80% of the classes. 
1 2 3 4 5 

WEST LYONS FIELD, KANSAS 

1. The testing will be for 

normality, 6 classes, and a level of significance of 5%. 

2. In the normal distribution, 

7.2 11.92 19.23 23.99 class width should be 

100/6=16.66% probability. Boundaries for those equiprobable classes are 7.20, 11.92, 15.64, 19.23, and 23.99. Each class contains Ei = 76/6=12.66 

measurements exactly. Values for Oi must be 

counted. 
WEST LYONS FIELD, KANSAS (

4. In this case, the statistic is: 2

2

2 

2 

(11

−12.66

)

(17 

−12.66

)

(12 

−12.66

)

(10 

−1

χs

=

+

+

+

12.66 12.66 12.66 12.

(15 

−12.66

)2 

(11

−12.66

)2 

= 2.94 

5. We have calcula

parameters (mestandard deviatithe there are 6 - degrees of freed

2 6. For a level of significance of 5%, 

χ α

(4,0.05

)

= 9

. 

2 2 So, because 

χs 

<

χ α , there is no evidence to s

that the thickness values are not normally distr

f(x) 

x 

2.94 9.49 

(4) 2 χ

66.12 66.12

+

+
KOLMOGOROV-SMIRNOV TES

This is a much simpler test than the chi-square test because: 

•	 The result does not 

depend on the number of classes. 

•	 It is nonparametric. 

The statistic is the 

maximum discrepancy, D, between the two
 

cumulative distributions 

Fn 

(

x

)and F x ( ) under com

D 

= max

(

Fn 

(

x

)

−

F( ) x 

)
Ds 

WEST LYONS FIELD, KANSAS

1. We want to test that 

the data are N(15.64,8.75) for a level of significance of 5%. 

2. Ds = 0.090 (9%). 

3. Critical value for 

Dα(76,0.05) = 0.102. 

4. There is no evidence 

to postulate that thethickness values are not normally distributed beDs<Dα. The test, however, presumes the sampl

from independent and identically distributed obs
Q-Q AND P-P PLOTS
 

Scatterplots of the quantiles and West the cumulative probabilities of the twodistributions are the ultimate tests in terms of simplicity. •	 If the distributions are the same, the 

points align along the main diagonal. 

•	 There is no statistic or level of 

significance for evaluating theresults. 

•	 PP-plots are insensitive to shifting

and scaling, and the vertical scale isin the same units as in KolmogorovSmirnov test. 

•	 The QQ-plot is good here at calling 

the user’s attention about the normal distribution being able to take negative values. 
FINAL REMARKS
 
SPATIOTEMPORAL DEPENDEN

Many natural phenomena exhibit fluctuations thashow continuity in space and time. Continuity dethe common experience that in proximity of any observation, another measurement will be approximately the same. •	 Given a spatiotemporally continuous phenome

and the location of an observation, close proxdoes tell something about the outcome of a seobservation; they are not necessarily indepen

•	 The degree and extent of this dependence ca

estimated through the semivariogram. 

•	 Modeling spatiotemporally dependent data, of

the form of maps, is the realm of geostatistics
SEMIVARIOGRAM
 

1000 random numbers Gravimetric anomaly, 

Spatial independence Spatial depe
7. MULTIVARIATE 

STATISTICS
 
METHODS
 

Multivariate statistics deals with the analysis and of objects with two or more attributes consistentlymeasured for several specimens. The main motivare better understanding of the data and introducsome simplifications.

The main families of methods available are: 

• cluster analysis; • discriminant analysis; • principal component analysis; • factor analysis. 

While missing values are not an insurmountabl

problem, they are a situation to avoid. Often one 

value in a record requires dropping the entire rec

All methods are complex enough to require a c

for performing the calculations. 
MATRICES 

A matrix is a rectangular array of numbers, such as 

A. 

When n = 

m, 

A is a square 

matrix of order n. 

Matrices are a convenient 

notation heavily used in dealing with large systems of linear equations, which notably reduce in size to just A X = 

B. 

Main diagonal of a square 

matrix is the sequence of element 

a11, 

a22, ..., 

ann from 

upper left to lower right. 

⎡

a11 a 12 

L

⎢

a a 

⎢ 21 22 

L

A 

=

⎢L

⎢

L L 

⎣a 

n1 a 

n2 

L

B 

=

[

b1 

b2 

L

X 

=

[

x1 

x2 

L
CLUSTER ANALYSIS
 
AIM
 

The general idea is to group objects in attribute space into clusters as internally homogeneous as possibleand as different form the other clusters as possible. 

Different types of distances,

proximity criteria, and approaches for preparing the clusters haveresulted in several methods. 

Some methods render results as 

dendrograms, which allow displayingthe data in two dimensions. 

Large distance increments provide

the best criteria for deciding on thenatural number of clusters. 

10 

8 6 420 

Attribute 2 

0 2 4 

Attr

1 

2 

3 

4 

5 

A 

B 

A 
158

DISSIMILARITIES

If Σ is the covariance matrix for a multivariate sample of m

attributes, the following distances are the most common 

measurements for dissimilarity between vectors p and q:

m

• Euclidean, ∑

m

(p q − )

2

i i • Manhattan, ∑ pi − qi

= i =1 i 1

m

Mahalanobis, ∑(p −q)

'

Σ−1

(p− • q). i =1

Distances can be in original data space or standardized.

Mahalanobis distances account for distance relative to 

direction and global variability through the covariance matrix.

Euclidean distances can be regarded as a special case of 

Mahalanobis distance for a covariance matrix with ones 

along the main diagonal and zeros anywhere else.
PROXIMITY AND METHODS 

Proximity between clusters is commonly decided based on the average inter-cluster distance. The most common methods are:
 

• Agglomerative hierarchical • Divisive hierarchical • K-means 

Choice of dissimilarity measure may have great

influence in the results than the selection of the m
DATA SET TO BE CLUSTERED

Physical properties of carbonates 

Mineral 

Spec. gravity 

g/cc 

Refractive index

 Smallest Largest 

Aragonite (ar) 2.94 1.530 1.685 

Azurite (az) 3.77 1.730 1.838 Calcite (ca) 2.72 1.486 1.658 

Cerusite (ce) 6.57 1.803 2.076 

Dolomite (do) 2.86 1.500 1.679 Magnesite (mg) 2.98 1.508 1.700 

Rhodochrosite (rh) 3.70 1.597 1.816 

Smithsonite (sm) 4.43 1.625 1.850 Siderite (si) 3.96 1.635 1.875 Strontianite (st) 3.72 1.518 1.667 

Witherite (wi) 4.30 1.529 1.677 

Often all attributes are standardized to avoid thdominance of results by those with the largest numerical values. 
AGGLOMERATIVE CLUSTERIN

• Initially there are as many 

clusters as records. 

• At any stage, the two 

closest clusters are merged together, reducing the number of clusters by one. 

• The procedure ends with 

the last two clusters merging into a single cluster. 

Euclidean distance is the most common choice (Wmethod). 
DIVISIVE CLUSTERING
 

1.	 Initially all records

are in one cluster. 

2.	 At any stage, all

distances inside each cluster are calculated. 

3.	 The cluster with the 

largest specimen-tospecimen distanceis broken apart,increasing thenumber of clusters by one. 
DIVISIVE CLUSTERING
 

4. These specimens with 

the largest distance become the seed of the two new clusters. All other specimens in the breaking apart cluster are assigned to the closest seed. 

5. The procedure ends 

with the last two true clusters breaking apart into individual specimens. 
K-MEANS CLUSTERING
 

The final number, 

k, of k = 5

clusters is here decided at the outset. The algorithm is: 1. Select the location of 

k
 

centroids at random. 

2. All objects are assigned 

to the closest centroid.
 

3. Recalculate the location 

of the k centroids. 

4. Repeat steps 2 and 3 until reaching convergence

The method is fast, but the solution may be sensitithe selection of the starting centroids. 

ca 

do 

ar 

ma 

rh 

st 

wi 

az 
CLUSTERING REMARKS
 

•	 The method is primarily a classification tool de

a statistical background. 

• The more complex the dataset, the more likely 

different methods will generate different results

•	 The k-means methods is an extreme case, as 

different runs for the same number of clusters produce different results. 

•	 Often solutions are suboptimal, failing to perfe

honor the intended objective of the method. 

•	 In the absence of clear cut selection criteria, convenience in the eye of the user remains as 

ultimate consideration on choosing the numbefinal clusters and clustering method. 

•	 If totally lost, go for the Ward’s method followe

the k-means method starting from the clustergenerated by Ward’s method. 
DISCRIMINANT ANALYSIS
 
BASIC IDEA
 

Discriminant analysis is a mixture of classification aprediction method under different availability of datcluster analysis. 

10

86 4 2 0 

0 2 4 

Attribute 

Attribute 2 

•	 The classes are known for all 

objects in a training set. 

•	 The training set is a data set 

intended for a second stage classification of objects without class assignments. 

•	 The problem is solved by 

minimizing misclassification, which starts by finding class geometric boundaries for the training set in the data space. 
ASSUMPTIONS
 

Discriminant analysis is a true statistical method bon multivariate distribution modeling. Importantassumptions for making the analysis possible are•	 The data are a sample from a multivariate norm

distribution. Thus, all attributes are normallydistributed within each class. 

•	 Any specimen has the same probability of belo

to any of the classes.

•	 None of the attributes is a linear combination o

others. 

•	 The means for attributes across groups are not 

correlated with the variances. 

Although in practice all assumptions are neversimultaneously satisfied, the formulations are robenough to tolerate departures. 
VARIANTS
 

There are different approaches to discriminant abut the main difference is in the type of surfaces employed to establishing the class boundaries. •	 Linear methods, in which the surfaces are 

hyperplanes in an m dimensional space, wher

the number of attributes considered in the anaLinear discriminant analysis results from assuall classes have the same covariance matrix. 

•	 Quadratic methods, in which the boundaries a

polynomials of order up to 2, with each class ha different covariance matrix. 
TESTS
 

The following tests are integral parts of the proc• Homogeneity of covariances • Equality of means • Normality 
TEXAS GULF COAST

Median grain size, mm 

0.355 

0.35 

0.345 

0.34 

0.335 

0.33 

0.325 

0.32 

0.315 

Offs

+ 

+ 

+ 

Beach sands

1.05	 1.1 1.15 1.2 1

Sorting coefficie

 SAND EXAM

Test for homogeneity of covariances 

p = 0.85 

Test for equality of means 

p = 0.00

Classification table 

 Beach Offshore

Beach 31 

3

Offshore 3 44

Kolmogorov-Smirnov test for normality 

 Statistic Probability

Beach 0.056 0.96390Offshore 0.034 0.99999

Assignments (

+) 

Sorting Med. size Pr[Beach

1.22 0.327 0.97 

1.17 0.334 0.85 1.12 0.342 0.40 

1.27 0.331 0.12 
PRINCIPAL COMPONENT ANALY
BASIC IDEA
 

The main objective of principal component analysis is to reduce the dimensionality of the sample. •	 The new variables are linear 

combinations of the old ones. 

•	 The new axes are orthogonal 

and oriented in directions of maximum variation. 

•	 The new variables account for 

the same total variance, but in decreasing proportions. 

•	 If explaining less than 100% of 

the total variation is acceptable, one can drop the less relevant new variables. 
MATRIX TERMINOLOGY 

The determinant of a square matrix, |

A|, of order m 

expression having additive and subtractive product

coefficients. The exact rules are complicated, but 2, for example, A 

=

a11 ⋅a22

−

a12 ⋅a21 . 

Given a symmetric square 

matrix, its eigenvalues are 

the solution, 

Λ, to the 

equation that results from subtracting the unknown value 

λ from the main 

diagonal of its determinant. 

An eigenvector 

Xi is the 

solution to equation system obtained by subtracting an eigenvalue from all diagonal terms of A when B = 

0. 

a 11

−

λ a 12

L 

a 21 a 22

−

λ

L 

L

L

L

a 

m1 a 

m

2

L a 

Λ

= [λ1 

λ2 

L 

λ

−

−

λ

⎡

−

λ

⎢⎢⎢⎢⎣

a a 

L a 

11 i 12 1m 

a a 21 22 i L

a

2

m

L L L L 

a a m1 

m2 

L

amm 
METHOD
 

Solution to the problem comes from finding theorthogonal combination of attributes with maximvariance. 

Given n records containing measurements for 

attributes, one can calculate an m by m symmet

covariance matrix. The actual multivariate distribis irrelevant. 

The directions of maximum variance are provid

the eigenvectors in the form of directional cosineaxes in the new system of reference remain orth

The length of each axis is taken as twice the

eigenvalue. 

Borrowing from the properties of matrices, the 

axes are oriented along the directions of maximuvariance and ranked by decreasing length. 
BOX EXAMPLE
 

Variables 

x1 

= long edge 

x2 

= middle edge 

x3 

= short edge 

x4 

= longest diagonal 

radius of smallest circums

x 5 

=

radius of largest inscrib

long edge 

+ intermediate e

x 6 

=

short edge
 

surface area
 x 7 

=

volume 
PRINCIPAL COMPONENTS EXAM

Coefficient matrix 

⎡
 5.400 ⎤ ⎢ 3.260 ⎢ 5.846
 

⎥⎥ ⎢ 0.779 1.465 2.774 ⎥

A
 

= ⎢ 6.391 ⎢


 6.083 2.204 9.107
 

⎥⎥ ⎢ 2.155 ⎢ 3.035 ⎢ 1.312 2.877


 
 

− 3.839 − 5.167 

1.611 

2.783 

10.71414.774 


 

20.776
 

⎥⎥⎥ ⎢

⎣ −1.996

− 2.370

−1.740

− 3.283 2.252 2.622 2.594⎥

⎦


 

Eigenvectors 

⎡ 0.164 

− 0.422 

− 0.645 0.225 0.415 0.385 ⎤ ⎢ 0.142 ⎢

 

− 0.447 0.713 0.395 ⎥ 0.329 ⎥ ⎢− 0.173 

− 0.257 0.130

− 0.629

− 0.607 0.280 0.211 ⎥ ⎢ Χ

= 0.170 ⎢

 

− 0.650 

− 0.146

− 0.212

− 0.403 ⎥ 

− 0.565 ⎥ ⎢ 0.546 ⎢

⎢ 0.768 

 0.135 0.133 

− 0.105

.1490

− 0.164

− 0.161

− 0.207 

− 0.596 

0.465

0.514 ⎥ ⎥ − 0.327 ⎥ ⎢

⎣ 0.313 

− 0.719 0.596 0.107 ⎥

⎦

Orientation 

of second 

new axis 

Λ

= [34.491

d variance 120 10080aine 60lve exp 40mulati 20Cu

0

Eigenvalues 

18.999 2.539 0.806 0.


 
 
 
 
 
 

1 2 3 4 

Principal compo
FACTOR ANALYSIS
 
INTENDED PURPOSE
 

Factor analysis was originally formulated in psychology to assess intangible attributes, like intelligence, in terms of abilities amenable of testing, such as reading comprehension. 

Principal component analysis can 

be used to run a factor analysis, which sometimes contributes to confusion of the two methods. 

Principal component factor analysis 

employs a correlation coefficient matrix obtained after standardizing the data. 
GENERAL ASSUMPTIONS
 

Factor analysis, unlike principal component analysthe canons of statistical modeling, thus • opening possibilities in terms of testing, yet • creating restrictions in utilization: data must comsome assumption for the method to be applicable. 

Analogously to regression, the model is: 

p 

xi =

∑

aik ⋅fk 

+

ε i , i 

= 1,2,

K,m 

k 

=1 

where 

xi is the 

ith observed attribute. 

aik and 

fk are the loadings and factors to com

the analysis. 

εi is a random error. 

It is also assumed that all variables are multinorm

distributed. 
VARIANTS
 

The two main approaches to factor analysis are: 

•	 principal components 

•	 maximum likelihood 

In addition to the general assumptions, maximum likelihood factor analysis assumes that: •	 Factors follow a standard normal distribution, 

•	 All of the factors are independent. 

•	 The correlation coefficient matrix prepared wit

data is a good approximation to the population correlation coefficient matrix . 
BOX EXAMPLE REVISITED
 

Variables 

x1 

= long edge 

x2 

= middle edge 

x3 

= short edge 

x4 

= longest diagonal 

radius of smallest circumscribe sphere x 5 

=

radius of largest inscribe sphere
 

long edge 

+ intermediate edge x 6 

=

short edge
 

surface area
 x 7 

=

volume 
PRINCIPAL COMPONENT FACTO

ANALYSIS FOR THE BOX EXAMPL

Coefficient matrix 

⎡ 1.000⎢


 


 

⎤⎥ 0.580 1.000 ⎢ ⎥ ⎢ 0.201 0.364 1.000 ⎥ ⎢ ⎥ A
 

= ⎢


 0.911 0.834 0.439 1.000

⎢


 

⎥

0.283 0.166
 

− 0.704 0.163 1.000 ⎥ ⎢ 
 

⎥ ⎢ 0.287 0.261
 

−

0.681 0.202 0.990 1.000
 

⎥ ⎢

⎣ −

0.533

−

0.609

− 0 649 ⎥ . −

0.676 0.427 0.357 1.000⎦


 

Eigenvectors (Factors) 

⎡− 0.405 0.293

−

0.667

− 0.227

− 0.410 0.278 ⎤ ⎢ ⎥ −

0.432 0.222 0.698

− 0.437

− 0.144 0.254 ⎢

⎢−

⎥

⎢ 0

 .385 

− 0.356 0.148 0.628 0.512

− 0.188 0.108 ⎥

⎥ Χ

=

−

−

− ⎢ 0.494 0.232 0.119 0.210 0.105 0.588

− 0.536 ⎥ ⎢

⎢ 0.128 0.575 0.111 0.389 0.423 0.556 ⎥

⎥ ⎢ 0.580 0.174 0.355

− 0.500

− 0.497 ⎥ ⎢ ⎥ ⎣ 0.481 0.130 0.735

− 0.455 ⎦

Orientation 

of second 

new axis 

70

Principal compon

60


 
 
 
 
 Factor analysis 50
 403020

e percent Total varianc , 

10
 0

1 2 3 4 5 

New axes 

Λ

Eigenvalues 


 

=
 

[3.395 2.805 0.437 0.278 0.08
ACKNOWLEDGMENTS
I am indebted for the valuable suggestions received as part of the review process of these n

Emil Attanasi (USGS), Geoffrey Bohling (Kansas Geological Survey), and Vera Pawlowsky-

(Universitat de Girona, Spain). I am also grateful to early participants of my lectures who cowith their remarks to make corrections and reformulate some topics. 
BIBLIOGRAPHY
 
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Chernick, M. R., 2008, Bootstrap Methods: A Guide for Practitioners and Researchers: Wiley Inte

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Davis, J. C., 2002, Statistics and Data Analysis in Geology: John Wiley and Sons, New York, thir

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Good, P. I. and J. W. Hardin, 2006, Common Errors in Statistics (and How to Avoid Them): Wiley 

Hoboken, NJ, 254 p. 

Hastie, T., R. Tibshirani, and J. Friedman, 2001, The Elements of Statistical Learning—Data Min

and Prediction: Springer, New York, 533 p. 

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Holmgren, E. B., 1995, The P-P Plot as a Method for Comparing Treatment Effects : Journal of 

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561 p. 

Pawlowsky-Glahn, V., editor, 2005, Special issue on advances in compositional data: Mathemati

vol. 37, no. 7, p 671–850. 

Salkind, N. J., 2007, Statistics for People Who (Think They) Hate Statistics: Sage Publications, T

CA, 402 p. 

Thomsen, V., D. Schatzlein, and D. Mercuro, 2003, Limits of detection in spectroscopy: 

www.spectroscopyonline.com, vol. 18, no. 12, p. 112-114, 

Venables, W. N. and B. D. Ripley, 2003, Modern Applied Statistics with S: Springer, New York, 4Walpole, R. E., R. H. Myers, S. L. Myers, and K. Ye, 2006, Probability and Statistics for Enginee

Scientists: Prentice Hall, eighth edition, 848 p. 
INDEX
 
accuracy 13, 15 

agglomerative hierarchical method 

159, 161 

alternative hypothesis 112–113, 

115, 120 

analysis 

cluster 156–165, 167 descriptive 38 discriminant analysis 167-171 

factor 179–183 principal component 173–177, 

179–181, 183 

variance 138-140 

arithmetic mean 47 

assignment 167, 171 

autocovariance 97 Bayes’s theorem 66 bias 95 binomial distribution 68 bootstrap method 76, 109–110 

box-and-whisker plot 59 causality 88 central limit theorem 125–127 Chebyshev’s theorem 53 chi-square 

distribution 142–143 test 144–147 

classification 167, 169 

cluster analysis 156–165, 167 coefficient 

kurtosis 63 

quartile skew 62 skewness 61, 69 

coin flipping 68, 71, 75, 123 compositional data 98–99 

Conditional probability 66 correlation 98 

coefficient 82, 86, 88, 179 

covariance 81–82, 97, 158, 

169–171 

matrix 158, 169, 175 

cumulative distribution 43 decile 56 degrees of freedom 130, 132, 

134–135, 142–144, 146 

dendrogram 157 dependent variable 84 detection limit 21–27, 101–110 

instrumental 24method 25


 

quantitatio


 

n 26

determinant 174 


 

discriminant analysis 167–171 dissimilarity 96, 159 distance 157–158 

Euclidean 158, 161 Mahalanobis 158 Manhattan 158 

distribution 67, 125 

binomial 68 chi-square 142–143 continuous 67 

discrete 67 

F 134–135 Gaussian 43

104–105, 1

142, 145, 1

lognormal 69normal 43, 63

104–105, 1142, 145, 1

sampling 119 student’s t 13testing 142–1

divisive hierarchic

162–163 

eigenvalue 174–1eigenvector 174–error 85 

mean square propagation 9

Type I 114Type II 114 


 

Euclidean distanc

expected value 7explanatory variaextreme value 54factor 180–181 

analysis 179–

F-distribution 134fractile 56 frequency table 4
Gaussian distribution 43, 63, 

69–70, 104–105, 125–128, 131, 134, 142, 145, 148, 168, 170, 180 

geometric mean 47, 110 geostatistics 151 histogram 42, 67, 75 hypothesis 120 

alternative 112–113, 115, 120 null 112–117, 120, 128–129, 

134, 144 

independence 123, 148, 151, 152, 

181 

independent variable 84, 142 instrumental detection limit 24 interquartile range 57–58 k-means method 159, 164–165 Kaplan-Meier method 107, 110 Kolmogorov-Smirnov test 147–149, 

171 

kurtosis 63 legacy data 29–34 level of significance 114, 116–117, 

120, 134, 144, 146, 149 

limit of quantitation 26 linear 

correlation 82 method 169 regression 86–87 

lognormal distribution 69, 105 log-ratio transformation 99, 109 

lurking variable 88 Mahalanobis distance 158 main diagonal 155, 174 Manhattan distance 158 matrix 155, 174–175, 177, 183 

correlation coefficient 179, 181 covariance 158, 169, 175 order 155, 174 square 155 

maximum 54 

likelihood method 104–105, 

110, 181 

mean 47, 50, 59, 69, 72, 81, 91, 

105, 124–125, 131, 142, 146, 168, 170–171 

arithmetic mean 47 geometric mean 47, 110 testing 124–128, 138–140 

measure 

centrality 46–49 dispersion 51–58 shape 60–63 

median 48, 50, 56 method 

agglomerative hierarchical 159, 

161 

bootstrap 76, 109–110 detection limit 25 divisive hierarchical 159, 

162–163 

k-means 159, 164–165 

linear 169 Kaplan-Meier maximum like

110, 181 

Monte Carlo quadratic 169 replacement Ward’s 161, 

minimum 54 mode 49 moment 73 Monte Carlo meth

109 

noise 87, 152 nonlinear regress

normal distributio

104–105, 125–142, 145, 148, 

null hypothesis 1

128–129, 134, 

outlier 58 order 155, 174 P-P plot plot 92, 1p-value 117, 119–percentile 56 plot 

P-P 92, 149 Q-Q 90, 149 

population 36, 12power 114 precision 14–15, 
principal component analysis 

173–177, 179–181, 183 

probability 65, 70, 117–118 

conditional 66 density function 67 

probability-probability plot 92, 149 

proximity 159 Q-Q plot 90, 149 quadratic method 169 quantile 56 quantile-quantile plot 90, 149 quartile 56–57, 59, 105 

radius of influence 97 random variable 37, 75 

range 97 regressed variable 84 

regression 84–88, 180 regressor 84 replacement method 103, 110 response variable 84 robustness 50, 87, 168 

sample 36, 95 

size 38, 105, 129–131 

sampling distribution 118 

scatterplot 80, 149 semivariogram 96–97, 151–152 

significant digit 17–19 sill 97 simulation 75 skewness 61, 69 

standard deviation 53, 69, 91, 125, 

142, 146 

standardization 91–92, 127, 142, 

158, 160, 179 

statistic 45, 76, 109, 116–120, 144, 

146, 147, 149 

statistics 4–8, 37, 127, 136 student’s t distribution 131–132 test 

chi-square 144–146 distribution 142–149 Kolmogorov-Smirnov 147–148, 

171 

one-sided 113, 116, 123 

two-sided 113, 116, 140 mean 124–128, 138–140 

variance 134–136 

theorem 

Bayes’s 66 central limit 125–127 Chebyshev’s theorem 53 

unbiasedness 95–96 uncertainty 5, 94 

variable 

dependent 84explanatory 84


 

–85 

independent 84 

lurking 88random 37


 

, 75

regressed 84


 

response 84


 
 

variance 52–53, 

134, 136, 138

variate 37, 75 

testing 134–

Ward’s method