Measures of Statistical Dispersion | Wyzant Resources Resources / Lessons / Math / Statistics and Probability / Averages / Dispersion - Deviation an... Dispersion - Deviation and Variance Dispersion measures how the various elements behave with regards to some sort of central tendency, usually the mean. Measures of dispersion include range , interquartile range , variance, standard deviation and absolute deviation. We've already looked at the first two in the Averages section, so let's move on to the other measures. Absolute Deviation Absolute deviation for a given data set is defined as the average of the absolute difference between the elements of the set and the mean (average deviation) or the median element (median absolute deviation). The average deviation is calculated as follows: which means that the average deviation is the average of the differences between each element of the data set and the mean.

The median absolute deviation is calculated as follows: Example 1 The heights of a group of 10 students randomly selected from a given school are as follows (in ft): 5.5, 3.5, 4.6, 6.1, 5.7, 5.11, 4.9, 5.0, 5.0, 5.5 a) Find the absolute deviation from the mean. b) Find the absolute deviation from the median. Solution a) To find the absolute deviation from the mean, we need to first find the mean of the heights. We know that the mean x̄ is given by: Using the above, we calculate the mean as: The mean height is 5.091 ft. The deviation from the mean for each of the elements in the data set is obtained by subtracting the mean from that element, as follows: For 5.5: We find all the deviations and then take their average (remember that we only consider their absolute values): b) To find the absolute deviation from the median, we need to first find the median height for the data set.

We know that to find the median value, we arrange the elements in the data set in ascending or descending order and the find that element that lies in the middle. Arranged in ascending order from the smallest to the largest: Finding the median: Since we had an even number of elements in the data set, it comes as no surprise that we're unable to obtain a median by canceling out corresponding elements. We're left with two elements and so we find their mean which then becomes our median. Having obtained our median as 5.25, we can proceed to find the average deviation from the median using the same steps as in the previous question. Variance and Standard Deviation Variance, as the name suggests, is a measure of how different the elements in a given population are. Variance is used to indicate how spread out these elements are from the mean of the population. There are two kinds of variance: population variance and sample variance. Population variance is the variance of the entire population and is denoted by σ2 while sample variance is the variance of a sample space of the population; and is denoted by S2 Standard deviation is the square root of variance. Standard deviation is a measure of how precise the mean of a population or sample is.

It is used to indicate trends in the elements in a given data set with respect to the mean, i.e, the spread of these elements from the mean. Just as we have a population and sample variance, we also have a population and sample standard deviation. Population standard deviation is denoted by σ while the sample standard deviation is denoted by S Although absolute deviation is also a measure of dispersion, variance and standard deviation are better measures because of the way they're calculated. Calculating variance involves squaring the differences (deviations) between the element and the mean and this makes the differences larger and thus more manageable. Making the differences larger adds a weighting factor to them making trends easier to spot. The population variance can be calculated from the following: where μ is the population mean. The sample variance is given by where x̄ is the sample mean. Standard deviation is simply the square root of variance, so we can calculate it by taking the square root of the above variance formulae: Population standard deviation Measures of Variability Measures of Variability Determine the relative variability of two distributions Compute the range Compute the variance in the population Estimate the variance from a sample Compute the standard deviation from the variance What is Variability?

Variability refers to how "spread out" a group of scores is. To see what we mean by spread out, consider graphs in Figure 1. These graphs represent the scores on two quizzes. The mean score for each quiz is 7.0. Despite the equality of means, you can see that the distributions are quite different. Specifically, the scores on Quiz 1 are more densely packed and those on Quiz 2 are more spread out. The differences among students were much greater on Quiz 2 than on Quiz 1. Quiz 1 Quiz 2 Figure 1. Bar charts of two quizzes. The terms variability, spread, and dispersion are synonyms, and refer to how spread out a distribution is. Just as in the section on central tendency where we discussed measures of the center of a distribution of scores, in this chapter we will discuss measures of the variability of a distribution. There are four frequently used measures of variability: the range, interquartile range, variance, and standard deviation. In the next few paragraphs, we will look at each of these four measures of variability in more detail. Range The range is the simplest measure of variability to calculate, and one you have probably encountered many times in your life.

The range is simply the highest score minus the lowest score. Lets take a few examples. What is the range of the following group of numbers: 10, 2, 5, 6, 7, 3, 4? Well, the highest number is 10, and the lowest number is 2, so 10 - 2 = 8. The range is 8. Lets take another example. Heres a dataset with 10 numbers: 99, 45, 23, 67, 45, 91, 82, 78, 62, 51. What is the range? The highest number is 99 and the lowest number is 23, so 99 - 23 equals 76; the range is 76. Now consider the two quizzes shown in Figure 1. On Quiz 1, the lowest score is 5 and the highest score is 9. Therefore, the range is 4. The range on Quiz 2 was larger: the lowest score was 4 and the highest score was 10. Therefore the range is 6.

Interquartile Range The interquartile range (IQR) is the range of the middle 50% of the scores in a distribution. It is computed as follows: IQR = 75th percentile - 25th percentile For Quiz 1, the 75th percentile is 8 and the 25th percentile is 6. The interquartile range is therefore 2. For Quiz 2, which has greater spread, the 75th percentile is 9, the 25th percentile is 5, and the interquartile range is 4. Recall that in the discussion of box plots , the 75th percentile was called the upper hinge and the 25th percentile was called the lower hinge. Using this terminology, the interquartile range is referred to as the H-spread. A related measure of variability is called the semi-interquartile range . The semi-interquartile range is defined simply as the interquartile range divided by 2. If a distribution is symmetric, the median plus or minus the semi-interquartile range contains half the scores in the distribution.

Variance Variability can also be defined in terms of how close the scores in the distribution are to the middle of the distribution. Using the mean as the measure of the middle of the distribution, the variance is defined as the average squared difference of the scores from the mean. The data from Quiz 1 are shown in Table 1. The mean score is 7.0. Therefore, the column "Deviation from Mean" contains the score minus 7. The column "Squared Deviation" is simply the previous column squared. Table 1. Calculation of Variance for Quiz 1 scores. Scores Statistics of dispersion - Handbook of Biological Statistics Handbook of Biological Statistics Statistics of dispersion Summary A statistic of dispersion tells you how spread out a set of measurements is. Standard deviation is the most common, but there are others. Introduction Summarizing data from a measurement variable requires a number that represents the "middle" of a set of numbers (known as a "statistic of central tendency" or "statistic of location"), along with a measure of the "spread" of the numbers (known as a "statistic of dispersion"). You use a statistic of dispersion to give a single number that describes how compact or spread out a set of observations is.

Although statistics of dispersion are usually not very interesting by themselves, they form the basis of most statistical tests used on measurement variables. Range: This is simply the difference between the largest and smallest observations. This is the statistic of dispersion that people use in everyday conversation; if you were telling your Uncle Cletus about your research on the giant deep-sea isopod Bathynomus giganteus, you wouldn't blather about means and standard deviations, you'd say they ranged from 4.4 to 36.5 cm long (Biornes-Fourzán and Lozano-Alvarez 1991). Then you'd explain that isopods are roly-polies, and 36.5 cm is about 14 American inches, and Uncle Cletus would finally be impressed, because a roly-poly that's over a foot long is pretty impressive. Range is not very informative for statistical purposes. The range depends only on the largest and smallest values, so that two sets of data with very different distributions could have the same range, or two samples from the same population could have very different ranges, purely by chance.

In addition, the range increases as the sample size increases; the more observations you make, the greater the chance that you'll sample a very large or very small value. There is no range function in spreadsheets; you can calculate the range by using =MAX(Ys)−MIN(Ys), where Ys represents a set of cells. Sum of squares: This is not really a statistic of dispersion by itself, but I mention it here because it forms the basis of the variance and standard deviation. Subtract the mean from an observation and square this "deviate". Squaring the deviates makes all of the squared deviates positive and has other statistical advantages. Do this for each observation, then sum these squared deviates. This sum of the squared deviates from the mean is known as the sum of squares. It is given by the spreadsheet function DEVSQ(Ys) (not by the function SUMSQ). You'll probably never have a reason to calculate the sum of squares, but it's an important concept. Parametric variance : If you take the sum of squares and divide it by the number of observations (n), you are computing the average squared deviation from the mean.

As observations get more and more spread out, they get farther from the mean, and the average squared deviate gets larger. This average squared deviate, or sum of squares divided by n, is the parametric variance. You can only calculate the parametric variance of a population if you have observations for every member of a population, which is almost never the case. I can't think of a good biological example where using the parametric variance would be appropriate; I only mention it because there's a spreadsheet function for it that you should never use, VARP(Ys). Sample variance: You almost always have a sample of observations that you are using to estimate a population parameter. To get an unbiased estimate of the population variance, divide the sum of squares by n−1, not by n. This sample variance, which is the one you will always use, is given by the spreadsheet function VAR(Ys). From here on, when you see "variance," it means the sample variance. You might think that if you set up an experiment where you gave 10 guinea pigs little argyle sweaters, and you measured the body temperature of all 10 of them, that you should use the parametric variance and not the sample variance.

You would, after all, have the body temperature of the entire population of guinea pigs wearing argyle sweaters in the world. However, for statistical purposes you should consider your sweater-wearing guinea pigs to be a sample of all the guinea pigs in the world who could have worn an argyle sweater, so it would be best to use the sample variance. Even if you go to Española Island and measure the length of every single tortoise (Geochelone nigra hoodensis) in the population of tortoises living there, for most purposes it would be best to consider them a sample of all the tortoises that could have been living there. Standard deviation: Variance, while it has useful statistical properties that make it the basis of many statistical tests, is in squared units. A set of lengths measured in centimeters would have a variance expressed in square centimeters, which is just weird; a set of volumes measured in cm3 would have a variance expressed in cm6, which is even weirder. Taking the square root of the variance gives a measure of dispersion that is in the original units.

The square root of the parametric variance is the parametric standard deviation, which you will never use; is given by the spreadsheet function STDEVP(Ys). The square root of the sample variance is given by the spreadsheet function STDEV(Ys). You should always use the sample standard deviation; from here on, when you see "standard deviation," it means the sample standard deviation. The square root of the sample variance actually underestimates the sample standard deviation by a little bit. Gurland and Tripathi (1971) came up with a correction factor that gives a more accurate estimate of the standard deviation, but very few people use it. Their correction factor makes the standard deviation about 3% bigger with a sample size of 9, and about 1% bigger with a sample size of 25, for example, and most people just don't need to estimate standard deviation that accurately. Neither SAS nor Excel uses the Gurland and Tripathi correction; I've included it as an option in my descriptive statistics spreadsheet. If you use the standard deviation with the Gurland and Tripathi correction, be sure to say this when you write up your results.

In addition to being more understandable than the variance as a measure of the amount of variation in the data, the standard deviation summarizes how close observations are to the mean in an understandable way. Many variables in biology fit the normal probability distribution fairly well. If a variable fits the normal distribution, 68.3% (or roughly two-thirds) of the values are within one standard deviation of the mean, 95.4% are within two standard deviations of the mean, and 99.7 (or almost all) are within 3 standard deviations of the mean. Thus if someone says the mean length of men's feet is 270 mm with a standard deviation of 13 mm, you know that about two-thirds of men's feet are between 257 and 283 mm long, and about 95% of men's feet are between 244 and 296 mm long. Here's a histogram that illustrates this: Left: The theoretical normal distribution. Right: Frequencies of 5,000 numbers randomly generated to fit the normal distribution.

The proportions of this data within 1, 2, or 3 standard deviations of the mean fit quite nicely to that expected from the theoretical normal distribution. Left: The theoretical normal distribution. Right: Frequencies of 5,000 numbers randomly generated to fit the normal distribution. The proportions of this data within 1, 2, or 3 standard deviations of the mean fit quite nicely to that expected from the theoretical normal distribution. The proportions of the data that are within 1, 2, or 3 standard deviations of the mean are different if the data do not fit the normal distribution, as shown for these two very non-normal data sets: Left: Frequencies of 5,000 numbers randomly generated to fit a distribution skewed to the right. Right: Frequencies of 5,000 numbers randomly generated to fit a bimodal distribution. Left: Frequencies of 5,000 numbers randomly generated to fit a distribution skewed to the right. Right: Frequencies of 5,000 numbers randomly generated to fit a bimodal distribution. Coefficient of variation.

Coefficient of variation is the standard deviation divided by the mean; it summarizes the amount of variation as a percentage or proportion of the total. It is useful when comparing the amount of variation for one variable among groups with different means, or among different measurement variables. For example, the United States military measured foot length and foot width in 1774 American men . The standard deviation of foot length was 13.1 mm and the standard deviation for foot width was 5.26 mm, which makes it seem as if foot length is more variable than foot width. However, feet are longer than they are wide. Dividing by the means (269.7 mm for length, 100.6 mm for width), the coefficients of variation is actually slightly smaller for length (4.9%) than for width (5.2%), which for most purposes would be a more useful measure of variation. Example Here are the statistics of dispersion for the blacknose dace data from the central tendency web page.

In reality, you would rarely have any reason to report all of these: Range 90 Variance 1029.5 Standard deviation 32.09 Coefficient of variation 45.8% How to calculate the statistics Spreadsheet I have made a spreadsheet that calculates the range, sample variance, sample standard deviation (with or without the Gurland and Tripathi correction), and coefficient of variation, for up to 1000 observations. Web pages This web page calculates standard deviation and other descriptive statistics for up to 10000 observations. This web page calculates range, variance, and standard deviation, along with other descriptive statistics. I don't know the maximum number of observations it can handle. R Salvatore Mangiafico's R Companion has a sample R program for calculating range, sample variance, standard deviation, and coefficient of variation . SAS PROC UNIVARIATE will calculate the range, variance, standard deviation (without the Gurland and Tripathi correction), and coefficient of variation. It calculates the sample variance and sample standard deviation. For examples, see the central tendency web page. Reference Briones-Fourzán, P., and E. Lozano-Alvarez.

1991. Aspects of the biology of the giant isopod Bathynomus giganteus A. Milne Edwards, 1879 (Flabellifera: Cirolanidae), off the Yucatan Peninsula. Journal of Crustacean Biology 11: 375-385. Gurland, J., and R.C. Tripathi. 1971. A simple approximation for unbiased estimation of the standard deviation. American Statistician 25: 30-32. Table of Contents This page was last revised July 20, 2015. Its address is It may be cited as: McDonald, J.H. 2014. Handbook of Biological Statistics (3rd ed.). Sparky House Publishing, Baltimore, Maryland. This web page contains the content of pages 107-110 in the printed version . ©2014 by John H. McDonald. You can probably do what you want with this content; see the permissions page for details.

Measures of Variability | Real Statistics Using Excel Real Statistics Using Excel Contact Us Measures of Variability We consider a random variable x and a data set S = {x1, x2, …, xn} of size n which contains possible values of x. The data set can represent either the population being studied or a sample drawn from the population. The mean is the statistic used most often to characterize the center of the data in S. We now consider the following commonly used measures of variability of the data around the mean, namely the standard deviation, variance, squared deviation and average absolute deviation. In addition we also explore three other measures of variability that are not linked to the mean, namely the median absolute deviation, range and inter-quartile range. Of these statistics the variance and standard deviation are most commonly employed. Excel Functions: If R is an Excel range which contains the data elements in S then the Excel function which calculates each of these statistics is shown in Figure 1. Functions marked with an asterisk are supplemental functions found in the Real Statistics Resource Pack, although equivalent formulas in standard Excel are described later. Statistic Figure 1 – Measures of Variability Observation: These functions ignore any empty or non-numeric cells.

Variance Definition 1: The variance is a measure of the dispersion of the data around the mean. Where S represents a population the population variance (symbol σ2) is calculated from the population mean µ as follows: Where S represents a sample the sample variance (symbol s2) is calculated from the sample mean x̄ as follows: The reason the expression for the population variance involves division by n while that of the sample variance involves division by n – 1 is explained in Property 3 of Estimators , where division by n – 1 is required to obtained an unbiased estimator of the population variance. Excel Function: The sample variance is calculated in Excel using the worksheet function VAR. The population variance is calculated in Excel using the function VARP. In Excel 2010/2013 the alternative forms of these functions are VAR.S and VAR.P. Example 1: If S = {2, 5, -1, 3, 4, 5, 0, 2} represents a population, then the variance = 4.25. This is calculated as follows.

First, the mean = (2+5-1+3+4+5+0+2)/8 = 2.5, and so the squared deviation SS = (2–2.5)2 + (5–2.5)2 + (-1–2.5)2 + (3–2.5)2 + (4–2.5)2 + (5–2.5)2 + (0–2.5)2 + (2–2.5)2 = 34. Thus the variance = SS/n = 34/8 = 4.25 If instead S represents a sample, then the mean is still 2.5, but the variance = SS/(n–1) = 34/7 = 4.86. These can be calculated in Excel by the formulas VARP(B3;B10) and VAR(B3:B10), as shown in Figure 2. Figure 2 – Examples of measures of variability Observation: When data is expressed in the form of frequency tables then the following properties are useful. Click here for the proofs of these properties.

Property 1 : If x̄ is the mean of the sample S = {x1, x2, …, xn}, then the sample variance can be expressed by Property 2 : If µ is the mean of the population S = {x1, x2, …, xn}, then the population variance can be expressed by Standard Deviation Definition 2: The standard deviation is the square root of the variance. Thus the population and sample standard deviations are calculated respectively as follows: Excel Function: The sample standard deviation is calculated in Excel using the worksheet function STDEV. The population standard deviation is calculated in Excel using the function STDEVP. In Excel 2010/2013 the alternative forms of these functions are STDEV.S and STDEV.P.

Example 2: If S = {2, 5, -1, 3, 4, 5, 0, 2} is a population, then the standard deviation = square root of the population variance = = 2.06 If S is a sample, then the sample standard deviation = square root of the sample variance = = 2.20 These are the results of the formulas STDEVP(B3:B10) and STDEV(B3:B10), as shown in Figure 2.

Real Statistics Functions: The Real Statistics Resource Pack furnishes the following array functions: VARCOL(R1) = a row range which contains the sample standard variances of each of the columns in R1 STDEVCOL(R1) = a row range which contains the sample standard deviations of each of the columns in R1 VARROW(R1) = a column range which contains the sample standard variances of each of the rows in R1 STDEVROW(R1) = a column range which contains the sample standard deviations of each of the rows in R1 Example 3: Use the VARCOL and STDEVCOL functions to calculate the sample variance and standard deviation of each of the columns in the range L4:N11 of Figure 3. The formula =VARCOL(J4:L11) produces the first result (in range J15:L15), while the formula =STDEVCOL(J4:L11) produces the second result (in range J16:L16). Remember that after entering either of these formulas you must press Ctrl-Shft-Enter.

Figure 3 – Sample Variance and Standard Deviation by Column Property 3 : If the population {x1, x2, …, xn} has mean µx and standard deviation σx and the population {y1, y2, …, ym} has mean µy and standard deviation σy, then the variance of the combined population is Thus if µx = µy the combined population variance would be Property 4 : If the sample {x1, x2, …, xn} has mean x̄ and standard deviation sx and the sample {y1, y2, …, ym} has mean ȳ and standard deviation sy, then the variance of the combined sample is Thus if x̄ = ȳ, the combined sample variance would be Example 4: Find the mean and variance of the sample which results from combining the two samples {3, 4, 6, 7} and {6, 1, 5}. Figure 4 – Calculation of combined mean and standard deviation The data in the two samples is given in the range B3:C7 of Figure 4.

From these, the mean, variance and standard deviation are calculated for each of the two samples (ranges B12:B15 and C12:C15). Using Property 4, we can calculate the mean and variance of the combined sample (D13 and D14). If we simply combine the two samples we obtain the data in the range F3:F10, from which we can calculate the mean, variance and standard deviation in the normal way (range D12:D18). As we can see the results are the same. Observation: In practice instead of using Property 3 and 4, we use the approach shown in the following example, especially since it can be applied to more than two samples or populations. Example 5: Find the mean and variance of the sample which results from combining the three samples shown in range A3:D6 of Figure 5. Figure 5 – Calculation of combined mean and variance We have three samples whose total sample size is 58 (cell B7), calculated via =SUM(B4:B6). The sum of the elements in each sample can be calculated from the mean as shown in range F4:F6. E.g.

the sum of all the data elements in sample 1 is 276 (cell F4), calculated via the formula =B4*C4. Thus the sum of all the elements in all three sample is 786 (cell F7), calculated via the formula =SUM(F4:F6). The mean of the combined sample is therefore 13.5517 (cell C7), calculated via the formula =F7/B7. The calculation of the combined variance is similar. The key is to first find the sum of the squares of all the elements in each sample. These are given in range I4:I6. E.g. the sum of the squares of all the elements in sample 1 is 5512 (cell I4), calculated by =G4+H4 (using Property 1), where G4 contains the formula =B4*C4^2 and H4 contains =(B4-1)*D4. Thus the sum of squares of all the elements in the combined sample is 19,832 (cell I7), calculated by =SUM(I4:I6).

Finally, the variance for the combined sample is 161.059 (cell D7), calculated by =(I7-B7*C7^2)/(B7-1), based on Property 1. The standard deviation is therefore 12.6909. Squared Deviation Definition 3: The squared deviation (symbol SS for sum of squares) is most often used in ANOVA and related tests. It is calculated as Excel Function: The squared deviation is calculated in Excel using the worksheet function DEVSQ. Example 6: If S = {2, 5, -1, 3, 4, 5, 0, 2}, the squared deviation = 34. This is the same as the result of the formula DEVSQ(B3:B10) as shown in Figure 2. Average Absolute Deviation Definition 4: The average absolute deviation (AAD) of data set S is calculated as Excel Function: The average absolute deviation is calculated in Excel using the worksheet function AVEDEV.

Example 7: If S = {2, 5, -1, 3, 4, 5, 0, 2}, the average absolute deviation = 1.75. This is the same as the result of the formula AVEDEV(B3:B10) as shown in Figure 2. Median Absolute Deviation Definition 5: The median absolute deviation (MAD) of data set S is calculated as Median {|xi – where = median of the data elements in S. Excel Formula: If R is a range which contains the data elements in S then the MAD of S can be calculated in Excel by the array formula: =MEDIAN(ABS(R-MEDIAN(R))) Even though the value is presented in a single cell it is essential that you press Ctrl-Shft-Enter to obtain the array value, otherwise the result won’t come out correctly. This function only works properly when R doesn’t contain any empty cell or cell with a non-numeric value. Alternatively, you can use the supplemental function MAD(R) which is contained in the Real Statistics Resource Pack. This function works properly even when R contains empty cells and/or cells with non-numeric values.

You don’t need to press Ctrl-Shft-Enter to use this function. Example 8: If S = {2, 5, -1, 3, 4, 5, 0, 2}, the median absolute deviation = 2 since S = {-1, 0, 2, 2, 3, 4, 5, 5}, and so the median of S = (2+3)/2 = 2.5. Thus MAD = the median of {3.5, 2.5, 0.5, 0.5, 0.5, 1.5, 2.5, 2.5} = {0.5, 0.5, 0.5, 1.5, 2.5, 2.5, 2.5, 3.5}, i.e. (1.5+2.5)/2 = 2. You can achieve the same result using the supplemental formula =MAD(E3:E10) as shown in Figure 2.

Observation: This metric is less affected by extremes in the tails because the data in the tails have less influence on the calculation of the median than they do on the mean. Range Definition 6: The range of a data set S is a crude measure of variability and consists simply of the difference between the largest and smallest values in S. Excel Formula: If R is a range which contains the data elements in S then the range of S can be calculated in Excel by the formula: =MAX(R) – MIN(R) Alternatively, you can use the supplemental function RNG(R) which is contained in the Real Statistics Resource Pack. Example 9: If S = {2, 5, -1, 3, 4, 5, 0, 2}, the range = 5 – (-1) = 6. You can achieve the same result using the supplemental formula =RNG(E3:E10) as shown in Figure 2. Inter-quartile Range Definition 7: The inter-quartile range (IQR) of a data set S is calculated as the 75% percentile of S minus the 25% percentile.

The IQR provides a rough approximation of the variability near the center of the data in S. Excel Formula: If R is a range which contains the data elements in S then the IQR of S can be calculated in Excel by the formula: =QUARTILE(R, 3) – QUARTILE(R, 1) In Excel 2010/2013 there is a new version of the quartile function called QUARTILE.EXC. An alternative version of IQR is therefore =QUARTILE.EXC(R, 3) – QUARTILE.EXC(R, 1) See Ranking Functions in Excel for further information about the QUARTILE and QUARTILE.EXC functions. Alternatively, you can calculate the inter-quartile range via the supplemental function IQR(R, b) which is contained in the Real Statistics Resource Pack. When b = FALSE (default), the first version of IQR is returned, while when b = TRUE the second version is returned.

Example 10: If S = {2, 5, -1, 3, 4, 5, 0, 2}, then the first version of IQR = 4.25 – 1.5 = 2.75, while the second version is IQR = 4.75 – 0.5 = 4.25. You can achieve the same result using the Real Statistics formulas =IQR(B3:B10) and =IQR(B3:B10,TRUE), as shown in Figure 2. Observation: The variance, standard deviation, average absolute deviation and median absolute deviation measure both the variability near the center and the variability in the tails of the distribution which represents the data. The average absolute deviation and median absolute deviation do not give undue weight to the tails. On the other hand, the range only uses the two most extreme points and the interquartile range only uses the middle portion of the data.

Coefficient of Variation Definition 8: The coefficient of variation (aka the coefficient of variability), V (or CV), of the data set S is calculated as V = s/x̄ Since s and x̄ have the same units of measurement, V has no units of measurement. This statistic only makes sense for ratio scale data. The higher the value of V the more dispersion there is. Clearly, the coefficient of variation is only defined when the mean is not zero. Excel Formula: If R is a range which contains the data elements in S then the coefficient of variation for S can be calculated in Excel by the formula: =VAR.S(R)/AVERAGE(R) Difference Between Dispersion and Skewness: Dispersion vs Skewness Home / Science & Nature / Science / Mathematics / Difference Between Dispersion and Skewness Difference Between Dispersion and Skewness Posted on Dispersion vs Skewness In statistics and probability theory, often the variation in the distributions has to be expressed in a quantitative manner for the purposes of comparison. Dispersion and Skewness are two statistical concepts where the shape of the distribution is presented in a quantitative scale. More about Dispersion In statistics, the dispersion is the variation of a random variable or its probability distribution.

It is a measure of how far the data points lie from the central value. To express this quantitatively, measures of dispersion are used in descriptive statistic. Variance, Standard Deviation, and Inter-quartile range are the most commonly used measures of dispersion. If the data values have a certain unit, due to the scale, the measures of dispersion may also have the same units. Interdecile range, Range, mean difference, median absolute deviation, average absolute deviation, and distance standard deviation are measures of dispersion with units. In contrast, there are measures of dispersion which has no units, i.e dimensionless. Variance, Coefficient of variation, Quartile coefficient of dispersion, and Relative mean difference are measures of dispersion with no units. Dispersion in a system can be originated from errors, such as instrumental and observational errors. Also, random variations in the sample itself can cause variations. It is important to have a quantitative idea about the variation in data before making other conclusions from the data set. More about Skewness In statistics, skewness is a measure of asymmetry of the probability distributions. Skewness can be positive or negative, or in some cases non-existent. It can also be considered as a measure of offset from the normal distribution.

If the skewness is positive, then the bulk of the data points is centred to the left of the curve and the right tail is longer. If the skewness is negative, the bulk of the data points is centred towards the right of the curve and the left tail is rather long. If the skewness is zero, then the population is normally distributed. In a normal distribution, that is when the curve is symmetric, the mean, median, and mode have the same value. If the skewness is not zero, this property does not hold, and the mean, mode, and median may have different values. Pearson’s first and second coefficients of skewness are commonly used for determining the skewness of the distributions. Pearson’s first skewness coffeicent = (mean – mode) / (standard deviation) Pearson’s second skewness coffeicent = 3(mean – mode) / (satndard deviation) In more sensitive cases, adjusted Fisher-Pearson standardized moment coefficient is used. G = {n / (n-1)(n-2)} ∑ni=1 ((y-ӯ)/s)3 What is the difference between Dispersion and Skewness?

Dispersion concerns about the range over which the data points are distributed, and the skewness concerns the symmetry of the distribution. Both measures of dispersion and skewness are descriptive measures and coefficient of skewness gives an indication to the shape of the distribution. Measures of dispersion are used to understand the range of the data points and offset from the mean while skewness is used for understanding the tendency for the variation of data points into a certain direction. Related posts: Variability in Data Share with Friends How to Measure Variability in a Data Set In this lesson, we discuss three measures that are used to quantify the amount of variation in a data set - the range, the variance, and the standard deviation. For example, consider a population of elements {5, 5 ,5, 5}. Here, each of the values in the data set are equal, so there is no variation. The set {3, 5, 5, 7}, on the other hand, has some variation since some some elements in the data set have different values. Notation The following notation is helpful, when we talk about variability. σ2: The variance of the population. σ: The standard deviation of the population. s2: The variance of the sample.

s: The standard deviation of the sample. μ: The population mean . N: Number of observations in the population. n: Number of observations in the sample. P: The proportion of elements in the population that has a particular attribute. p: The proportion of elements in the sample that has a particular attribute. Q: The proportion of elements in the population that does not have a specified attribute. Note that Q = 1 - P. q: The proportion of elements in the sample that does not have a specified attribute. Note that q = 1 - p. Note that capital letters refer to population parameters , and lower-case letters refer to sample statistics . The Range The range is the simplest measure of variation. It is difference between the biggest and smallest random variable. Range = Maximum value - Minimum value Therefore, the range of the four random variables (3, 5, 5, 7} would be 7 minus 3 or 4. Variance of the Mean It is important to distinguish between the variance of a population mean and the variance of a sample mean. They have different notation, and they are computed differently. The variance of a population mean is denoted by σ2; and the variance of a sample mean, by s2.

The variance of a population mean is the average squared deviation from the population mean, as defined by the following formula: σ2 = Σ ( Xi - μ )2 / N where σ2 is the population variance, μ is the population mean, Xi is the ith element from the population, and N is the number of elements in the population. The variance of a sample mean is defined by slightly different formula: s2 = Σ ( xi - x )2 / ( n - 1 ) where s2 is the sample variance, x is the sample mean, xi is the ith element from the sample, and n is the number of elements in the sample. If you are working with a simple random sample, the sample variance can be considered an unbiased estimate of the true population variance. Therefore, if you want to estimate the unknown population variance, based on known data from a simple random sample, use this formula. Example 1 A population consists of four observations: {1, 3, 5, 7}. What is the variance? Solution: First, we need to compute the population mean.

μ = ( 1 + 3 + 5 + 7 ) / 4 = 4 Then we plug all of the known values in to formula for the variance of a population, as shown below: σ2 = Σ ( Xi - μ )2 / N σ2 = [ ( 1 - 4 )2 + ( 3 - 4 )2 + ( 5 - 4 )2 + ( 7 - 4 )2 ] / 4 σ2 = [ ( -3 )2 + ( -1 )2 + ( 1 )2 + ( 3 )2 ] / 4 σ2 = [ 9 + 1 + 1 + 9 ] / 4 = 20 / 4 = 5 Example 2 A simple random sample consists of four observations: {1, 3, 5, 7}. What is the best estimate of the population variance? Solution: This problem is handled exactly like the previous problem, except that we use the formula for calculating sample variance, rather than the formula for calculating population variance.

s2 = Σ ( xi - x )2 / ( n - 1 ) s2 = [ ( 1 - 4 )2 + ( 3 - 4 )2 + ( 5 - 4 )2 + ( 7 - 4 )2 ] / ( 4 - 1 ) s2 = [ ( -3 )2 + ( -1 )2 + ( 1 )2 + ( 3 )2 ] / 3 s2 = [ 9 + 1 + 1 + 9 ] / 3 = 20 / 3 = 6.667 Standard Deviation of the Mean The standard deviation is the square root of the variance. It is important to distinguish between the standard deviation of a population and the standard deviation of a sample. They have different notation, and they are computed differently. The standard deviation of a population is denoted by σ; and the standard deviation of a sample, by s. The standard deviation of a population mean is defined by the following formula: σ - 1 where s is the sample standard deviation, x is the sample mean, xi is the ith element from the sample, and n is the number of elements in the sample.

Variance of a Proportion The variance formulas introduced in the previous section can be used with confidence for any random variable - even proportions. However, for proportions the formulas can be expressed in a form that is easier to compute. When all of the elements of the population are known, the variance of a population proportion is defined by the following formula: σ2 = PQ where P is the population proportion and Q equals 1 - P. When the population proportion is estimated from sample data, the variance of the sample proportion is estimated by slightly different formula: s2 = pq where p is the sample estimate of the true proportion, and q is equal to 1 - p. Given a simple random sample, this sample variance can be considered an unbiased estimate of the true population variance. Therefore, if you need to estimate the unknown population variance, based on known data from a simple random sample, this is the formula to use. Standard Deviation of a Proportion The standard deviation of a proportion is the square root of the variance of the proportion.

Thus, the standard deviation of a population proportion is: σ What is Standard Deviation - PdfSR.com What is Standard Deviation In statistics and probability theory, standard deviation shows how much variation or "dispersion" exists from the average (mean, or expected value). A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data points are spread out over a large range of values. The standard deviation of a random variable, statistical population, data set, or probability distribution is the square root of its variance. It is algebraical y simpler though practical y less robust than the average absolute deviation.[1][2] A useful property of standard deviation is that, unlike variance, it is expressed in the same units as the data. In addition to expressing the variability of a population, standard deviation is commonly used to measure confidence in statistical conclusions. For example, the margin of error in polling data is determined by calculating the expected standard deviation in the results if the same pol were to be conducted multiple times. The reported margin of error is typical y about twice the standard deviation - the radius of a 95 percent confidence interval.

In science, researchers commonly report the standard deviation of experimental data, and only effects that fall far outside the range of standard deviation are considered statistical y significant Know More About Math Answers Math.Tutorvista.com - normal random error or variation in the measurements is in this way distinguished from causal variation. Standard deviation is also important in finance, where the standard deviation on the rate of return on an investment is a measure of the volatility of the investment. When only a sample of data from a population is available, the population standard deviation can be estimated by a modified quantity called the sample standard deviation, explained below. This quantity is the population standard deviation; it is equal to the square root of the variance. The formula is valid only if the eight values we began with form the complete population. If they instead were a random sample, drawn from some larger, "parent" population, then we should have used 7 (which is n - 1) instead of 8 (which is n) in the denominator of the last formula, and then the quantity thus obtained would have been cal ed the sample standard deviation. See the section Estimation below for more details.

A slightly more complicated real life example, the average height for adult men in the United States is about 70", with a standard deviation of around 3". This means that most men (about 68%, assuming a normal distribution) have a height within 3" of the mean (67"-73") -- one standard deviation -- and almost all men (about 95%) have a height within 6" of the mean (64"-76") -- two standard deviations. If the standard deviation were zero, then al men would be exactly 70" tal . If the standard deviation were 20", then men would have much more variable heights, with a typical range of about 50"-90". Three standard deviations account for 99.7% of the sample population being studied, assuming the distribution is normal (bel shaped). For example, each of the three populations {0, 0, 14, 14}, {0, 6, 8, 14} and {6, 6, 8, 8} has a mean of 7. Their standard deviations are 7, 5, and 1, respectively.

The third population has a much smal er standard deviation than the other two because its values are al close to 7. It wil have the same units as the data points themselves. Learn More How To Do Fractions Math.Tutorvista.com Page No. :- 2/4 If, for instance, the data set {0, 6, 8, 14} represents the ages of a population of four siblings in years, the standard deviation is 5 years. As another example, the population {1000, 1006, 1008, 1014} may represent the distances traveled by four athletes, measured in meters. It has a mean of 1007 meters, and a standard deviation of 5 meters. Standard deviation may serve as a measure of uncertainty. In physical science, for example, the reported standard deviation of a group of repeated measurements should give the precision of those measurements. When deciding whether measurements agree with a theoretical prediction, the standard deviation of those measurements is of crucial importance: if the mean of the measurements is too far away from the prediction (with the distance measured in standard deviations), then the theory being tested probably needs to be revised.

This makes sense since they fal outside the range of values that could reasonably be expected to occur if the prediction were correct and the standard deviation appropriately quantified. See prediction interval. While the standard deviation does measure how far typical values tend to be from the mean, other measures are available. An example is the mean absolute deviation, which might be considered a more direct measure of average distance, compared to the root mean square distance inherent in the standard deviation. Math.Tutorvista.com Lecture 11- Measures of Dispersion 1 Sum of the Squared Deviations VARIANCE is simply the mean of these squared and summed deviations (i.e., the average of the squared deviations) The symbol for VARIANCE is s2 accompanied by a subscript for the corresponding variable.

Here's the formula for variance of variable X: For the data above, the variance computes as STANDARD DEVIATION, is the square root of the variance The symbol for STANDARD DEVIATION (often shortened to STD DEV) is s The formula for the standard deviation of variable X: For the data above, the standard deviation computes as This is the most important formula in statistics But read on page 24 of the SPSS Guide that "for theoretical reasons" (n-1) is used in the demoninator to compute the standard devation, not n. Indeed, SPSS routinely uses n-1 and not n when it computes the standard deviation. But for now, learn this formula as the definition of the standard deviation. We will get to the "theoretical reasons" later.

Dispersion and the Normal Distribution Occasionally in lecture I spoke about the "shape" of a distribution of values, using these terms: unimodal -- the distribution had only a single value that occurred most frequently symmetrical -- the left side of the distribution of values mirrored the right side, i.e, it was neither skewed to the left, nor was it skewed to the right bell-shaped -- the frequencies of cases declined toward the extreme values in the right and left tails, so that the distribution had the appearance of a "bell." When such a "bell-shaped" distribution demonstrates certain properties, it is said to have a normal distribution. In one sense, it is called "normal" because a unimodal, symmetrical, bell-shaped distribution "normally" develops in the long run through accidents of nature when the events are equally-likely to occur. In an infinite amount of time, a random process could ultimately generate any structured result: e.g., a group of monkeys seated at typewriters could peck out all the great works of literature. This would be an extremely rare event, but it is conceivable. The normal curve is a mathematical formula that assigns probabilities to the occurrence of rare events.

Statistically speaking, it is a probability distribution for a continuous random variable: The ordinate represents the probability density for the occurrence of a value. The baseline represents the values. The exact shape of the curve is given by a complicated formula that you do NOT need to know. The area under the curve is interpreted as representing all occurrences of the variable, X. We can consider the area as representing 100 PERCENT of the occurrences; in PROPORTIONS this is expressed as 1.0. We can then interpret AREAS under the curve as representing certain PROPORTIONS of occurrences or "probabilities". We cannot assign a probability to any point, but we can attach probabilities to INTERVALS on the baseline associated with AREAS under the curve: e.g., the mean has 50% of the cases standing to each side. Special properties of the normal distribution: Its shape is such that it Embraces 68.26% of the cases within 1 s.d. around the mean. Embraces 95.46% of the cases within 2 s.d. around the mean. Embraces 99.74% of the cases within 3 s.d. around the mean.

More roughly speaking, 68%, 95%, and 99% of the cases are embraced within 1, 2, and 3 standard deviations from the mean in a normal distribution. Determining whether a distribution is "normal" The "Eyeball" test that we discussed earlier: Is the distribution unimodal? Is the distribution symmetrical? More exacting mathematical tests: measured according to "moments" or "deviations" from the mean First, the meaning of "moment" -- in physics, it is the distance or deviation from a reference point. In statistics, a "moment" is a deviation from the mean. If deviations are raised to powers of 1, 2, 3, and 4--"moments" of 1, 2, 3, and 4 are created. FIRST MOMENT: If skewness and kurtosis values tend toward 0, then the distribution approximates a normal distribution. Suppose the distribution is not normal? No matter how the original observations are distributed, the mean plus or minus two standard deviations will include at least 75% of the observations.

No matter how the original observations are distributed, the mean plus or minus three standard deviations witll include 89% or more. Assignment for today: Compute measures of dispersion for pctwomen, pctblack, and bush2000. Which is closest to a "normal distribution"? Transforming variables to approach a normal distribution The SPSS Guide, page 27, discusses a method to transform a heavily skewed variable so that it distributes more "normally." The aim is to change its underlying scale of measurement so that large values count for less. The method described (there are others) involves taking not the absolute value of the observation but the logarithm (base 10) of the value. The chart on page illustrates how countries like China and India tend to skew the distribution of the population variable when population is measured in thousands. When population is measured as the log of the population, the variable approaches a normal distribution. The transformation reduces the extreme impact of countries with very large populations (like China and Indis) in statistical analyses that involve population as a variable. You'll understand the importance of this later. Obviously, this simple method of transformation works only for variables that are right-skewed. Fortunately, that accounts for most variables of interest to social scientists.

Fortunately also, there are other techniques for transforming left-skewed variables, if that's needed. Variability in Statistics Share with Friends How to Measure Variability in Statistics Statisticians use summary measures to describe the amount of variability or spread in a set of data. The most common measures of variability are the range, the interquartile range (IQR), variance, and standard deviation. The Range The range is the difference between the largest and smallest values in a set of values. For example, consider the following numbers: 1, 3, 4, 5, 5, 6, 7, 11. For this set of numbers, the range would be 11 - 1 or 10. The Interquartile Range (IQR) The interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles . Quartiles divide a rank-ordered data set into four equal parts. The values that divide each part are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.

Q1 is the "middle" value in the first half of the rank-ordered data set. Q2 is the median value in the set. Q3 is the "middle" value in the second half of the rank-ordered data set. The interquartile range is equal to Q3 minus Q1. For example, consider the following numbers: 1, 3, 4, 5, 5, 6, 7, 11. Q1 is the middle value in the first half of the data set. Since there are an even number of data points in the first half of the data set, the middle value is the average of the two middle values; that is, Q1 = (3 + 4)/2 or Q1 = 3.5. Q3 is the middle value in the second half of the data set. Again, since the second half of the data set has an even number of observations, the middle value is the average of the two middle values; that is, Q3 = (6 + 7)/2 or Q3 = 6.5.

The interquartile range is Q3 minus Q1, so IQR = 6.5 - 3.5 = 3. An Alternative Definition for IQR In some texts, the interquartile range is defined differently. It is defined as the difference between the largest and smallest values in the middle 50% of a set of data. To compute an interquartile range using this definition, first remove observations from the lower quartile. Then, remove observations from the upper quartile. Then, from the remaining observations, compute the difference between the largest and smallest values. For example, consider the following numbers: 1, 3, 4, 5, 5, 6, 7, 11. After we remove observations from the lower and upper quartiles, we are left with: 4, 5, 5, 6. The interquartile range (IQR) would be 6 - 4 = 2. When the data set is large, the two definitions usually produce the same (or very close) results. However, when the data set is small, the definitions can produce different results.

The Variance In a population , variance is the average squared deviation from the population mean, as defined by the following formula: σ2 = Σ ( Xi - μ )2 / N where σ2 is the population variance, μ is the population mean, Xi is the ith element from the population, and N is the number of elements in the population. Observations from a simple random sample can be used to estimate the variance of a population. For this purpose, sample variance is defined by slightly different formula, and uses a slightly different notation: s2 = Σ ( xi - x )2 / ( n - 1 ) where s2 is the sample variance, x is the sample mean, xi is the ith element from the sample, and n is the number of elements in the sample. Using this formula, the sample variance can be considered an unbiased estimate of the true population variance. Therefore, if you need to estimate an unknown population variance, based on data from a simple random sample, this is the formula to use. The Standard Deviation The standard deviation is the square root of the variance.

Thus, the standard deviation of a population is: σ - 1 where s is the sample standard deviation, x is the sample mean, xi is the ith element from the sample, and n is the number of elements in the sample. Effect of Changing Units Sometimes, researchers change units (minutes to hours, feet to meters, etc.). Here is how measures of variability are affected when we change units. If you add a constant to every value, the distance between values does not change. As a result, all of the measures of variability (range, interquartile range, standard deviation, and variance) remain the same. On the other hand, suppose you multiply every value by a constant. This has the effect of multiplying the range, interquartile range (IQR), and standard deviation by that constant. It has an even greater effect on the variance. It multiplies the variance by the square of the constant. Test Your Understanding The correct answer is (B). First, we need to compute the sample mean.

x = ( 1 + 3 + 5 + 7 ) / 4 = 4 Then we plug all of the known values into formula for the standard deviation of a sample, as shown below: s = sqrt [ Σ ( xi - x )2 / ( n - 1 ) ] s = sqrt { [ ( 1 - 4 )2 + ( 3 - 4 )2 + ( 5 - 4 )2 + ( 7 - 4 )2 ] / ( 4 - 1 ) } s = sqrt { [ ( -3 )2 + ( -1 )2 + ( 1 )2 + ( 3 )2 ] / 3 } s = sqrt { [ 9 + 1 + 1 + 9 ] / 3 } = sqrt (20 / 3) = sqrt ( 6.67 ) = 2.58 Note: This problem asked us to estimate the standard deviation of a population, based on sample data. To do this, we used (n - 1) in the denominator of the standard deviation formula.

If the problem had asked us to compute the standard deviation of a population based on population data, we would have used N in the denominator. Descriptive Statistics Descriptive Statistics 1. Understand concepts of sample vs. population 2. Describe data with measures of central tendency 3. Describe data with measures of dispersion 4. Understand positive (right) vs. negative (left) skew 5. Interpret histograms and boxplots 6. Understand the concept of outliers The following page describes the major concepts and measures in descriptive statistics that may be covered on the statistics diagnostic exam. I have highlighted the major concepts in green. Nearly all introductory statistics texts should have a chapter or two on descriptive statistics. Descriptive statistics involves summarizing and organizing the data so they can be easily understood. Descriptive statistics, unlike inferential statistic, seeks to describe the data, but do not attempt to make inferences from the sample to the whole population. We typically describe the data in a sample. A sample is the selected portion of a population, often selected through a random process (such as simple random sampling, or a more complex stratified random sampling approach). A population is composed of those entities, individuals or objects of interest.

For example, we may be concerned about measuring the diameter of loblolly pines (a coniferous tree common in North Carolina) in a forest tract in the Duke Forest–the population of interest includes loblollys in the forest tract, while the sample would be those trees selected for measurement. The data that we collect may be either qualitative (may also be called categorical or nominal) or quantitative (numeric). Gender, MEM concentration, nation of origin are all qualitative or categorical measures, whereas height, distance, number of students in a class are quantitative. There is no natural ordering in categorical data, just distinct categories in which an individual/object can be placed. Quantitative data may either be discrete (such as counts of species occurrence in a plot) or continuous (such as height). The most basic descriptive statistics include measures of central tendency (mean, median, mode) and measures of dispersion (range, IQR, variance and standard deviation). We can also discuss the skewness of a distribution. When a distribution has a right tail it is likely positively skewed. Alternatively, when a distribution has a left tail, this often suggests a negative (or left) skew. When the data are normally distributed, the mean, median and mode are equivalent.

However, when the data are right (or positively skewed), the median will be less than the mean. Alternatively, when the data are left (or negatively skewed), the mean will be less than the median. Histograms In addition to the basic summary statistics described above, we can also describe the data graphically. For continuous data, the two most common graphs are the histogram and the boxplot. The green histogram below displays the diameter and breast height (DBH) measurements of 250 randomly selected loblolly pines in (imaginary) tract A in the Duke Forest. The histogram shows no obvious skew, however, the green boxplot below suggests that the distribution of DBH values is positively skewed with four outliers. A Right (Positively) Skewed Distribution Side-by-Side Boxplots of DBH (cm) of Loblolly Pines in Tracts A and B Boxplots are another means to display the distribution of data points. In the boxplots below, one can easily see the median (the solid line in the middle of the box), and the 25th and 75th percentiles (the top/bottom edges of the box). The IQR is the distance between these edges.

The whiskers extend out from either edge of the box. In this case (it can vary by software/statistician’s preference), the end of the whisker extends to the point just within the 1.5*IQR + 75th percentile (on the high end) and 25th percentile – IQR*1.5 on the low end of the distribution (these are called the fences). In this case, outliers are identified as any point outside the fences. There are four outlier DBH measurements identified in Tract A. Because of the positive skewness of the data, we would expected the mean to be greater than the median in each of these tracts. The descriptive (summary) statistics are shown below the boxplots. As we can see in the figure above, four out of the ten distributions contain outliers according to the 1.5*IQR rule, even though the points were randomly drawn from a normal distribution. This suggests that outliers may occur through a process of random chance.

Question: In probability theory and statistics, what is a measure of the variability or dispersion of a population, a data set, or a probability distribution?
Answer: Standard deviation