Standard Deviation

What do you want to do?

How many variables?

What level of data?

Central tendency

Describe

Median

Make inferences

Univariate Bivariate Multivariate

Nominal Ordinal Interval/ Ratio

Central tendency

Mode

Central tendency

Mean

Dispersion Dispersion Dispersion

Variance, Standard deviation

Average, Absolute deviation

Range, Index of

dispersion

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Learning Objectives

Understand the difference between dispersion and deviation.■■

Understand the range, variance, and standard deviation as measures of dis­■■

persion.

Identify the proper measure of dispersion for analyzing data.■■

Explain how to calculate the range, variance, and standard deviation.■■

It is important to know the central value of a distribution, but it is just one character­ istic of the distribution. It is also important to know how spread out the values are, together with the central value, because it is possible to have two data sets with the same measure of central tendency but with very different distributions. For example, two tests were given, with the following results:

Test 1: 10 80 85 90 95 100

Test 2: 75 75 75 75 75 175

Each test has a mean of 75. The scores of test 1 range from 0 to 100, whereas all of the scores for test 2 are 75. These distributions have the same mean but very differ­ ent spreads of scores. If you did not know your score and had to choose which group you were going to be in, you would want to know the spread of scores. If you are a weak student, it might be better to go with the all­75 group, whereas if you are a good student, you would definitely want to go with the spread group. This is the reason it is important to know the dispersion of the distribution together with the measure of central tendency. Measures of dispersion measure how narrow or how spread out the values are around the central value.

1 Deviation and Dispersion5- Dispersion and deviation are largely synonymous. The only practical difference is that deviation typically refers to the difference between a single value or case and the measure of central tendency, whereas dispersion is used more to refer to the overall difference between all cases or values and the measure of central tendency.

Dispersion and deviation are important to research. All variables have dispersion; if not, they would not be variables (values that change between cases), they would be

Chapter 5

Measures of Dispersion

121

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constants (a constant value no matter what the case). Take juvenile delinquency, for example. Some juveniles do not commit any delinquent acts during their teenage years (at least theoretically), many juveniles commit some delinquent acts, and some juveniles commit many delinquent acts.1 This is what makes the variable delinquency a variable: The number of delinquent acts varies among juveniles. Delinquency is not a constant because not every juvenile commits the same number of delinquent acts. This also means there is dispersion among the number of juvenile acts committed: some juveniles com­ mit a small number of acts, some a larger number, and some commit many delinquent acts. If a researcher asked how many delinquent acts a juvenile committed on average, the answer would represent the central value of juvenile delinquency. That does not tell the entire story, however. The researcher also needs to know how the juveniles differ from each other in their delinquency. This is the measure of dispersion.

Not only is dispersion an important addition to measures of central tendency but the two are closely related. As introduced in “Measures of Central Tendency” ( Chapter 4), two properties of the mean are that (1) the sum of the deviations from the mean always equals zero, and (2) the sum of the squared deviations from the mean is the smallest of any of the measures of central tendency. The first of these properties can be illustrated here. If we took the data (X) shown in Table 5-1 and subtracted

X X 2 X X 2 Me X 2 Mo

7 3.066 3 4

7 3.066 3 4

6 2.066 2 3

5 1.066 1 2

5 1.066 1 2

5 1.066 1 2

4 .066 0 1

4 .066 0 1

3 2.934 21 0

3 2.934 21 0

3 2.934 21 0

3 2.934 21 0

2 21.934 22 21

1 22.934 23 22

1 22.934 23 22

S 59 0 21 14

Table 5­1 Sum of Deviations from the Mean, Median, and Mode

122 Chapter 5 n Measures of Dispersion

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each value from the mean (X = 3.934), the sum of those scores would be zero. Here, deviations from the mean are also compared to deviations from the median and the mode. As can be seen, both of these numbers produced nonzero deviations, whereas the deviations from the mean are zero.

As also discussed in “Measures of Central Tendency” (Chapter 4), the sum of the squared deviations from the mean is the smallest value for summed deviations, smaller than if the same calculations were made for the mode or median. This prin­ ciple is shown in Table 5-2, where the second column shows the result of subtracting the mean (3.934) from each value and then squaring the result. When these values are summed, the result is 50.93. The third column shows the result of subtracting the median (4) from each value and then squaring the result. When these values are summed, the result is 51, larger than the sum of squared deviations from the mean. Finally, the fourth column shows the result of subtracting the mode (3) from each value and then squaring the result. When these values are summed, the result is 64, substantially larger than the sum of the squared deviations from the mean. This char­ acteristic of the mean becomes very important when discussing correlation and regres­ sion. The characteristics of deviations and dispersion are also important in discussing measures of dispersion as a univariate descriptive statistic.