standard deviation

Moonpie

unit4homework.pdf

Home >Law homework help >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

16304_CH05_Walker.indd 120 8/2/12 4:17:08 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

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

16304_CH05_Walker.indd 121 8/2/12 4:17:08 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

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

6 2.066 2 3

5 1.066 1 2

4 .066 0 1

3 2.934 21 0

2 21.934 22 21

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

16304_CH05_Walker.indd 122 8/2/12 4:17:09 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

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.

X 1X 2 X22 1X 2 Me22 1X 2 Mo22 7 9.40 9 16

7 9.40 9 16

6 4.27 4 9

5 1.14 1 4

4 0 0 1

3 .87 1 0

2 3.74 4 1

1 8.61 9 4

o 59 5.93 51 64

Table 52 Sum of Squared Deviations from the Measures of Central Tendency

12351 Deviation and Dispersion

16304_CH05_Walker.indd 123 8/2/12 4:17:09 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

2 Measures of Dispersion5- Although there are measures of dispersion that are more appropriate for certain levels of measurement, it is not as clear cut as measures of central tendency. The dispersion of nominal level data typically is best analyzed with the range or an index of disper sion. These two, together with the mean absolute deviations from the median, can be used with ordinal level data. Interval and ratio level data can be examined with the variance or standard deviation. Each of these measures of dispersion is discussed in this chapter.

Range

The simplest measure of the dispersion of a distribution is the range. The range is generally used with nominal and partially ordered ordinal variables, although it is sometimes included as an additional measure of dispersion for higher level variables. The range is simply the difference between the highest and lowest values in a distribu tion. For example, if the spread of ages in a class was from 18 to 35, the range would be 17.

Although the range is generally stated in terms of a single figure, it is sometimes stated simply as the two extreme values. In the example above, the range of scores could be said to be 18–35. This is particularly true for nominal and ordinal level data. For example, in Table 5-3 the range could be stated as 5(6 2 1). This does not make much sense, though. It is more easily interpreted if the range is stated as “less than high school to postgraduate.”

The range is useful in that it is a quick and easily calculated measure of disper sion. Although most statistical packages include the range as a measure of dispersion (see Table 53), it is generally not necessary to use this feature except when examin ing a very large data set where the number of values might make it time consuming to determine the highest and lowest value. In the SPSS printout in Table 53, the range would be simple to calculate even without knowing the value from the descriptive statistics. Here, the range of values would be 5 (6 2 1). It might be cumbersome, how ever, to determine the range simply by examining the raw data. Once the data is in a table, however, it is a simple matter.

The range does have limitations, however. First, it gives only a general indication of the variability of the values in a distribution. For example, if the range is reported as 5, you must still examine the data to understand what that means; it could mean that the data ranges from 1 to 6, or it could mean that the data ranges from 101 to 106. Also, the range does not account for intracategory variation, or the way the values are distributed between the extremes. Stating that the range is 5 provides little informa tion about how the data is arranged within the range. There could be a relatively equal number in each category, as in the freshman class in Table 5-4, or there could be only a few values in all but one category, as in the senior class in Table 54. Here, there is a

124 Chapter 5 n Measures of Dispersion

16304_CH05_Walker.indd 124 8/2/12 4:17:09 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

What is your highest level of education?

Value Label Value Frequency Percent Valid

Percent Cumulative

Percent

Less than High School 1 16 4.6 4.8 4.8

GED 2 59 17.0 17.6 22.3

High School Graduate 3 8 2.3 2.4 24.7

Some College 4 117 33.7 34.8 59.5

College Graduate 5 72 20.7 21.4 81.0

Post Graduate 6 64 18.4 19.0 100.0

Missing 11 3.2

Total 347 100.0 100.0

N Valid 336

Missing 11

Mean 4.08

Median 4

Mode 4

Std. Deviation 1.460

Variance 2.131

Skewness 2.477

Std. Error of Skewness .133

Kurtosis 2.705

Std. Error of Kurtosis .265

Range 5

Table 53 SPSS Output of Dispersion

Number of Arrests Freshman Class SeniorClass

0 25 99

1 20 0

2 18 0

3 15 0

4 12 0

5 10 1

Total 100 100

Table 54 Distribution of Arrests for Two Criminal Justice Classes

12552 Measures of Dispersion

16304_CH05_Walker.indd 125 8/2/12 4:17:09 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

relatively even proportion of freshman in each number of arrests. Most of the seniors, however, have never been arrested. The range for these two classes, nevertheless, is the same: (5 or 0 – 5).

Finally, the range is heavily influenced by extremely high or low values that may be atypical of the distribution. Take, for example, the distribution of inmate deaths in two different prisons in a state, shown in Table 5-5. In this table, prisons A and B have similar records of inmate deaths. In fact, only one year is different between the two prisons; in 1990, prison B had a riot during which 26 prisoners were killed. This will heavily influence the range for this prison. The range for prison A is 3 (3 2 0) or 0–3, but the range for prison B is 26 (26 2 0) or 0–26. This demonstrates the problems with using the range as a measure of dispersion.

Index of Dispersion

The index of dispersion, D, is another measure of dispersion for nominal and partially ordered ordinal variables. The index of dispersion is not a commonly used measure of dispersion, but there are not many measures of association that are properly used with nominal and ordinal level data, resulting in many researchers using the variance or standard deviation when it is not appropriate for the data. Since there are not many measures of dispersion for nominal level data, the index of dispersion is discussed here as a possible measure.

As shown in the following formula, the index of dispersion is a ratio of the num ber of pairs of scores:

D 5 actual numbers of pairs

maximum number of pairs

Year Prison A Prison B

1990 0 26

1991 0 0

1992 1 1

1993 3 3

1994 1 1

1995 2 2

Table 55 Number of Prison Deaths, 1990–1995

126 Chapter 5 n Measures of Dispersion

16304_CH05_Walker.indd 126 8/2/12 4:17:09 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

In this equation, the denominator is the maximum number of pairs that could be cre ated out of the scores if each set of numbers was in a different category; the numera tor is the maximum number of unique pairs that can be made with the data at hand. Completing this formula requires two calculations. To calculate the maximum number of pairs, multiply the values of each combination of scores. The example in Table 5-6 is a simplified example of the education level of probation officers. Since there are 15 total officers and 3 categories, maximum variability would occur when each category had 5 officers. Calculating the maximum number of pairs would then be accomplished by multiplying the frequencies of each of the categories as follows: 5 scores for col lege grad 3 5 scores for some college = 25; 5 scores for college grad 3 5 scores for no college = 25; and 5 scores for some college 3 5 scores for no college = 25. Adding the 3 combinations achieves a score of 75. The actual number of pairs is then calculated by multiplying the actual frequency of the combinations of each category. Here, the actual number of pairs would be 66: 2 college grad 3 8 some college, 2 college grad 3 5 no college, and 8 some college 3 5 no college. The index of dispersion for this data, then, is simply the ratio of the maximum number of possible pairs and the actual number of pairs:

D 5 actual numbers of pairs

maximum number of pairs 5

75 5 0.88

The value of the index of dispersion runs from 0 to 1. If the cases were equally distributed across all of the possible categories, the actual number of pairs (numera tor) would equal the maximum possible number of pairs, and the index of dispersion would equal 1 (see Exhibit 5-1). If all of the scores are in one category, there would be minimum variability and the index of dispersion would be zero (see Exhibit 5-2). In Table 56 and in the equation above, the value of 0.88 shows there is almost equal dispersion among the categories.

Value f

College grad 2

Some college 8

No college 5

N 15

Table 56 Education of Probation Officers

12752 Measures of Dispersion

16304_CH05_Walker.indd 127 8/2/12 4:17:09 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

For more complicated data sets, the index of dispersion can be computed using the formula

D 5 k1N 2 2 Sf 22 N 21k 2 12

where k is the possible number of categories in the data set (in the example above, 3; even categories with 0 scores would be included); N is the number of cases (in the example, N would be 15); and Sf 2 is the sum of the squared frequencies.

Using the data from Table 56, use of this formula to calculate the index of disper sion can be demonstrated. In this example, there were 3 categories and 15 total cases. The only other calculation that needs to be made is Sf 2. In this simple case, f 2 for the

Value f

College grad 5

Some college 5

No college 5

N 15

In this set of scores, the frequencies are divided evenly among the three categories. This represents the maximum variability of the numerator. Here, the maximum number of scores that could be cre ated is 75, as outlined previously. Also, because the number of scores in each category is the same as the maximum number of pairs, the actual number of pairs is the same value (75).

D 5 actual number of pairs

maximum number of pairs 5

75 5 1

Exhibit 51 Index of Dispersion at Maximum Dispersion

Value f

College grad 0

Some college 0

No college 15

N 15

In this set of scores, the frequencies are grouped entirely into one category. This represents the minimum variability of the numerator. Here, the maximum number of scores that could be created is 75, as outlined previously. However, because the number of scores in each category is 0 except for one category, all calculations for the actual number of pairs equal 0. Therefore, the calculation of the index of dispersion is

D 5 actual number of pairs

maximum number of pairs 5

75 5 0

Exhibit 52 Index of Dispersion at Minimum Dispersion

128 Chapter 5 n Measures of Dispersion

16304_CH05_Walker.indd 128 8/2/12 4:17:10 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

three categories would be 4 (22), 64 (82), and 25 (52), for a total of 93. Entering these values into the formula above would result in the following calculation:

D 5 31152 2 932 15213 2 12

5 31225 2 932

225122

5 311322

450

5 396

450

5 0.88

This is the same value that was obtained in the previous calculations using Table 56. There is no provision in SPSS to calculate the index of dispersion, so any use in actual research will require that this formula be calculated by hand.

Mean Absolute Deviation

The mean (average) absolute deviation is somewhat different from other measures of dispersion. This measure could be used with any measure of central tendency. For example, you could calculate the mean absolute deviation of scores from the mean, the mean absolute deviation of scores from the median, or even the mean absolute devia tion of scores from the mode. Why an absolute deviation? It makes sense that if we wanted to examine deviation (dispersion) for a set of data, we could simply calculate the mean and then measure how far each value is from the mean. As discussed earlier and in “Measures of Central Tendency” (Chapter 4), however, when the deviations from the mean are summed, the result is zero. That value does not tell anything about the dispersion. What is needed is a way to remove the signs of the values so they will sum to a positive value. That can be done by taking the absolute value of each devia tion, that is, simply removing the sign.

The mean absolute deviation from the median has advantages over other mea sures because the mean absolute deviation from the median is a smaller number than for either the mean or the mode. The mean absolute deviation from the median is not often used as a measure of dispersion. There is no standard measure of dispersion for ordinal level data, however, and because the median is the most appropriate measure of central tendency for use with ordinal and skewed interval and ratio level data, it can be argued that the mean absolute deviation from the median is a valid measure of dispersion for this type of data.

The mean absolute deviation from the median is determined by calculating the median, subtracting that value from each score, taking the absolute value (removing

12952 Measures of Dispersion

16304_CH05_Walker.indd 129 8/2/12 4:17:10 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

the sign), and dividing by N. A simple example is shown in Table 5-7. Notice that this is the same prisoner escape data that was used in “Measures of Central Tendency” (Chapter 4) and Table 51. Here, the median is 4 escapes, the mean number of escapes is 3.93, and the modal number of escapes is 3. Subtracting each score from the median is shown in column d, and the same calculations for the mean and mode are shown in columns b and f. Note here that as discussed above, when the mean is subtracted from the values, the sum of these deviations is zero and is smaller than the deviations from the median (column d) or the mode (column f). When the absolute value of the deviations is calculated, however, a different distribution emerges. The absolute val ues of the scores from the median are shown in column e, and the absolute values of the mean and mode are shown in columns c and g. The total of the absolute deviations from the median is 23. This is the smallest value for the three measures, with the sum of absolute deviations from the mean being 23.066 and the sum of absolute deviations from the mode being 24. Dividing by N (15) gives a mean absolute deviation from the median of 1.53.

Using the mean absolute deviation from the median has an additional advantage. The result is a standardized measure, meaning it can be compared across groups. Like

a X

b X 2 X

c 0 X 2 X 0

d X 2 Me

e 0 X 2 Me 0

f X 2 Mo

g 0 X 2 Mo 0

7 3.066 3.066 3 3 4 4

6 2.066 2.066 2 2 3 3

5 1.066 1.066 1 1 2 2

4 .066 .066 0 0 1 1

3 2.934 .934 21 1 0 0

2 21.934 1.934 22 2 21 1

1 22.934 2.934 23 3 22 2

S 0 23.066 21 23 14 24

Table 57 Comparison of Absolute Values of Mean, Median, and Mode

130 Chapter 5 n Measures of Dispersion

16304_CH05_Walker.indd 130 8/2/12 4:17:10 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

the index of dispersion, there is no direct mechanism in SPSS for calculating the mean absolute deviation from the median. It can be calculated with macros, however, and makes a good addition to the other measures of dispersion.

Variance

At the opposite end of the scale of sophistication are the variance and standard devia tion. The standard deviation is simply the square root of the variance, so these measures are often spoken of together. These two measures of dispersion are most appropriate for interval and ratio level data, although they are often used with ordinal (especially fully ordered ordinal) level data because of their power over other measures of disper sion. The variance and standard deviation do not work well with nominal and partially ordered ordinal level data because they measure the deviation about the mean; if the mean is not appropriate for nominal and ordinal level data, neither would be measures of dispersion based on the mean.

The variance was developed by Ronald A. Fisher in 1918 in connection with his work on the analysis of variance (see Chapter 14). Fisher used the variance in studying human traits and inheritibility because it was mathematically superior to the standard deviation, which was developed by J. F. Enke (1832).

The variance and standard deviation are based on the mean. In fact, the variance, and therefore the standard deviation, are nothing more than a special kind of mean—the average of the squared deviations from the mean. Squaring the deviations is desirable because, as discussed above, the sum of the differences of each value in a dis tri bution from the mean is zero and because the sum of the squared differences of each value in a distribution from the mean yields the minimum summed value. These two properties make the variance and standard deviation ideal for mathematical calculations in other statistical procedures and play a big part in calculating the “bestfitting line,” which is a goal when comparing two or more variables.

The variance (represented by s2 for samples or s2 for populations) measures the average of squared deviations of scores around the mean. This was the process that was undertaken in Tables 51 and 57. The formula for calculating the variance is

s2 5 S1X 2 X22

If the differences are squared and then summed, the result will be the smallest distance from any given value to the mean (as in Table 52). Dividing the figure by N results in a formula that is the mean of the squared deviations, just as when calculating the mean of any data set.

Recall that in this section of the text we are dealing with descriptive statistics. That means we are assuming that the data being analyzed represents a population.

13152 Measures of Dispersion

16304_CH05_Walker.indd 131 8/2/12 4:17:10 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

Again, X is used rather than m because of convention, even though m would be more appropriate because we are examining a population. If the data were considered a sample of a population, the formula for the variance would become

s2 5 S1X 2 X22

N 2 1

Here the denominator of the formula is N 2 1 rather than N. Dividing by N 2 1 rather than N is a procedure used to account for bias in small samples. This is discussed more thoroughly in the section of the book on inferential analyses. In this portion of the text, the formula for dealing with populations is used because descriptive statistics assumes a population is being analyzed.

Calculating the variance requires calculating the mean, calculating the difference between the mean and each value (X or midpoint), squaring the differences, and divid ing by N. In Table 5-8, the mean is 4 (92/23). The third column represents calcula tions that subtract the mean from each value of X. For example, the first prison had 7 escapes in 10 years. Since the mean is 4, this prison had 3 more escapes than the mean. This can be repeated for each score, where each prison’s score can be seen as higher than, lower than, or the same as the mean. Notice also that the sum of this column is zero (as shown in Table 51).

The fourth column takes each of these values and squares them (the 3 escapes higher than the first prison above the mean is squared, resulting in a value of 9). The sum of these calculations equals 68. It is then simply a matter of dividing this value by N (23), which is the same as calculating the mean (average) of the squared deviations. The result is 2.96, so the mean of the deviations is 2.96 (this is the variance).

To interpret the variance, if all the values in a distribution are the same (the value of the mean), the value of the variance will be zero. In this case, if all the values of X had been 4, the mean would have been 4 and the variance would have been 0. The variance will be at a maximum when all the scores are grouped in the extremes of the distribution (the highest and lowest values in the distribution). For example, if the data from Table 58 were altered so each value was in the extremes of the distribution (each value was either a 7 or a 1), it would be represented by the distribution shown in Table 5-9. Here, although N remains 23, the mean will be different because the SX has changed to 95. Additionally, although the sum of differences from the mean remains 0 (actually, 0.1 because of rounding), the square of this value is much larger (206.61 rather than 68). This would be the first indicator that the variance is going to be much larger. Indeed, calculation of the variance for this data set produces a variance of 8.98, almost three times as large as the variance in Table 58.

Table 5-10 shows the output that has been used with all of the other measures of central tendency and with the range. Here the variance is 2.131. This is substantially

132 Chapter 5 n Measures of Dispersion

16304_CH05_Walker.indd 132 8/2/12 4:17:10 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

lower than either of the values in the two earlier examples. Part of the reason for this is the nature of the data; the data in Tables 58 and 59 ranges from 1 to 7, while the data in Table 510 ranges from 1 to 6. It is also evident that there is less dispersion in the data in Table 510 than in Table 58 or 59.

The variance is important to statistical analysis for several reasons. First, as dis cussed above, the variance is the mean of the squared deviations, which makes it useful in higherorder statistical procedures such as regression (discussed later in the text). The squared deviations of the variance are also important because they give weight to

X f X 2 X 1X 2 X22 7 1 3 9

7 1 3 9

6 1 2 4

5 1 1 1

4 1 0 0

3 1 21 1

2 1 22 4

1 1 23 9

o 92 23 0 68

Table 58 Calculation of the Variance

X 5 4

s2 5 S1 X 2 X22

5 68

5 2.96

13352 Measures of Dispersion

16304_CH05_Walker.indd 133 8/2/12 4:17:10 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

extreme values. This represents an increased measure of spread as the deviation rises. For example, a deviation of 2 from the mean has more weight than two deviations of 1 each (22 = 4 > 12 + 12 = 2). Because of this characteristic, data sets with many small deviations will show smaller variance than data sets with fewer but larger deviations. Since we are attempting to measure deviation from the measure of central tendency, this characteristic is important. Finally, the expected value of the variance of a sample (using N 2 1 as the denominator of the formula) is equal to the population variance, whereas the standard deviation is not. Differences in statistical procedures for samples

X f X 2 X 1X 2 X22 7 1 2.87 8.2369

7 1 2.87 8.2369

1 1 23.13 9.7969

o 95 23 .1 202.610

Table 59 Calculation of the Maximum Variance

X 5 4.13

s2 5 S1X 2 X22

5 206.61

5 8.98

134 Chapter 5 n Measures of Dispersion

16304_CH05_Walker.indd 134 8/2/12 4:17:11 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

and for populations are discussed in the section of the text on inferential analyses, but it can be stated here that statistical procedures that can cross over between samples and populations are important for some analyses.

The squaring of deviations presents a problem for the variance, however. Since all values are squared, the units of measurement no longer fit the original values, which makes the variance difficult to interpret. For example, it is difficult or impossible to interpret a value of 8.98 (from Table 59) when the values range only from 1 to 7. This is the reason the standard deviation has become important.

A final note about the variance (which also applies to the standard deviation) is that it is difficult to calculate a variance for grouped data. There are three reasons for this. First,

What is your highest level of education?

Value Label Value Frequency Percent Valid

Percent Cumulative

Percent

Less than High School 1 16 4.6 4.8 4.8

GED 2 59 17.0 17.6 22.3

High School Graduate 3 8 2.3 2.4 24.7

Some College 4 117 33.7 34.8 59.5

College Graduate 5 72 20.7 21.4 81.0

Post Graduate 6 64 18.4 19.0 100.0

Missing 11 3.2

Total 347 100.0 100.0

N Valid 336

Missing 11

Mean 4.08

Median 4

Mode 4

Std. Deviation 1.460

Variance 2.131

Skewness 2.477

Std. Error of Skewness .133

Kurtosis 2.705

Std. Error of Kurtosis .265

Range 5

Table 510 SPSS Output of Dispersion

13552 Measures of Dispersion

16304_CH05_Walker.indd 135 8/2/12 4:17:11 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

the variance assumes the data is interval or ratio level. Even if a variable is interval level, when it is grouped it generally becomes ordinal level, thereby violating the assumption of the variance. Further, when a variable is grouped, the midpoint is used rather than the numbers themselves (as in Table 33). The midpoint is an approximation of the values in a particular category and is thus not a true representation of the data. This means the variance will be slightly off (at best)—you will not get a value of zero if you add all the scores. Finally, even if you do want to use the variance with grouped data, you would have to ungroup the data. For example, if you had a midpoint of 18 and a frequency of

5, as in Table 33, you could not simply calculate X 2 X using the midpoint (18). You would have to list out 18 2 the mean, five times and repeat this for each value.

Standard Deviation

As discussed above, the variance is often difficult to interpret. You can see from the example that when the range of values is from 1 to 7, a variance of 8.98 makes little intuitive sense. The standard deviation resolves this problem by putting the disper sion in the same units as the distribution. The standard deviation is calculated by tak ing the square root of the variance. Since the differences from the mean were squared to account for the sign (thus putting the values on a scale different from that of the data), it is a simple mathematical procedure to take the square root of the sum of those values and put the sum back on the same scale as that of the original data.

The standard deviation (represented by s or s) was developed by J. F. Enke (who was a student of Gauss) in 1832. The term standard deviation was formally coined by Karl Pearson in 1894. Calculation of the standard deviation is both simple and complex. The simple part is that if the variance is known, you can simply take the square root of that value. If the variance is not known, however, that value must be determined prior to calculating the standard deviation. The formula for calculating the standard deviation without knowing the variance beforehand is

σ = −( )Σ X X N

This is, of course, the square root of the variance. In the example from Table 59, the standard deviation would be

s 5 "s2 5 "8.98 5 2.99

This value is much easier to interpret in terms of the range of scores in this distribu tion. It is also important to note that the standard deviation can be interpreted in terms

136 Chapter 5 n Measures of Dispersion

16304_CH05_Walker.indd 136 8/2/12 4:17:11 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

of the number of standard deviations from the mean (Z scores; Chapter 6). This is important in discussions of the normal curve and inferential statistics.

Uses for the Variance and Standard Deviation

Using the variance and standard deviation is not unlike using other commonly used standard measurements. This is because they can be used to compare two distribu tions. In the example given at the beginning of Chapter 4, you cannot tell much about the distributions from the measure of central tendency; you also need to know how spread out the values are. When comparing two distributions, both pieces of informa tion are needed. For example, assume that you take tests in criminal law and in police administration and score 80 on one test and 75 on the other. You really need to know not only this information but also what other students scored. You need the average score and how spread out the scores were. An 85 when the dispersion is zero is prob ably not as important to your grades as a 75 where the dispersion is very large, because in the first instance, you know you are going to get an 85 on the test, whereas in the second example, you may get any score from an A to a C.

The variance is also important in other statistical procedures. Aside from the fact that the variance, along with the mean, is the foundation of many of the statistical formulas discussed later in the text, the variance is a fundamental component of some tests, such as analysis of variance (ANOVA).

Also, the most important use for the standard deviation is yet to come. When we begin to examine the normal curve and to apply the use of the normal curve in inferential analysis, the standard deviation will become very important because this is how the normal curve is divided and discussed, as well as how to determine where a particular value stands in a distribution.

3 Selecting the Most Appropriate Measure of 5- Dispersion It is more difficult to determine the appropriate measure of dispersion than to select a measure of central tendency because the measures of dispersion are not as tied to the level of measurement. The range can always be used regardless of the level of data or form of the distribution, although it is limited as to the information provided. For nominal level data and for some ordinal level data, the index of dispersion is a good choice. For interval/ratio level data and some ordinal level data, the variance and stan dard deviation are preferred. There is a caveat, however: if the median is the preferred measure of central tendency because the data is ordinal or the distribution is skewed, the mean absolute deviation may be preferred over the variance or standard deviation.

13753 Selecting the Most Appropriate Measure of Dispersion

16304_CH05_Walker.indd 137 8/2/12 4:17:11 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

The reason for this is simple but often overlooked. If you are using the median rather than the mean, it should be because the data is such that the mean is not an appropri ate measure of central tendency, either because the data is ordinal level or because of skew. Why would you want to use a measure of dispersion based on the measure of central tendency that is not appropriate for the data? In this instance, it is more appro priate to use the mean absolute deviation because the mean absolute deviation about the median is a complementary measure of dispersion for the median.

4 Conclusion5- This chapter added to your ability to describe data in terms of the central or most frequently occurring value by outlining procedures to assess how values differ from each other within a single variable. Although there are no widely accepted measures of variability for categorical (nominal and ordinal level) data, because this level of data is not expected to vary much or in a uniform fashion, some potentially useful measures of dispersion were offered. The discussion of the variance and standard deviation has set the stage for their use in later chapters as well as for their use in measuring devia tion or dispersion within a variable.

The next chapter will conclude the description of one variable by describing the form or shape of the variable. Topics will include how spread out the overall distribu tion is (kurtosis), whether the data is grouped in one end of the values (skewness), and if there is more than one mode for the distribution.

5 Key Terms5- dispersion standard deviation index of dispersion sum of squared deviations mean absolute deviation variance range

6 Summary of Equations5- Index of Dispersion

D 5 actual number of pairs

maximum number of pairs

Index of Dispersion (complicated data sets)

D 5 k1N 2 2 Sf 22 N 21k 2 12

138 Chapter 5 n Measures of Dispersion

16304_CH05_Walker.indd 138 8/2/12 4:17:11 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

Variance (samples)

s2 5 S1X 2 X22

N Variance (population)

s2 5 S1X 2 X22

N 2 1 Standard Deviation

σ = −( )Σ X X N

7 Exercises5- 1. For the following set of data, calculate:

a. The range b. The variance c. The standard deviation

6, 7, 8, 10, 10, 10, 12, 14

2. For the following set of data, calculate: a. The range b. The variance c. The standard deviation

7, 4, 2, 3, 4, 5, 8, 1, 9, 4

3. For the following data set, calculate: a. The range b. The variance c. The standard deviation

Interval Midpoint Frequency

90–99 6

80–89 8

70–79 4

60–69 3

50–59 2

13957 Exercises

16304_CH05_Walker.indd 139 8/2/12 4:17:11 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

4. For the following set of data, calculate: a. The range b. The variance c. The standard deviation

Interval f

90–100 5

80–89 7

70–79 9

60–69 4

5. Using the frequency tables that follow, discuss the measures of dispersion that would be appropriate for each.

HOME: What type of house do you live in?

Value Label Value Frequency Percent Valid

Percent Cumulative

Percent

House 1 280 81.6 82.4 82.4

Duplex 2 3 .9 .9 83.2

Trailer 3 34 9.9 10.0 93.2

Apartment 4 21 6.1 6.2 99.4

Other 5 2 .6 .6 100.0

Missing 3 .9

Total 343 100.0 100.0

N Valid 340

Missing 3

Mean 1.41

Std. Error of Mean .051

Median 1.00

Mode 1

Std. Deviation 0.945

Variance .892

Skewness 2.001

Std. Error of Skewness .132

Kurtosis 2.613

Std. Error of Kurtosis .264

Range 5

140 Chapter 5 n Measures of Dispersion

16304_CH05_Walker.indd 140 8/2/12 4:17:11 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

ARREST: How many times have you been arrested?

Value Frequency Percent Valid

Percent Cumulative

Percent

0 243 70.8 86.2 86.2

1 23 6.7 8.2 94.3

2 10 2.9 3.5 97.9

3 3 .9 1.1 98.9

5 2 .6 .7 99.6

24 1 .3 .4 100.0

Missing 61 17.8

Total 343 100.0 100.0

N Valid 282

Missing 61

Mean .30

Std. Error of Mean .093

Median 0

Mode 0

Std. Deviation 1.567

Variance 2.455

Skewness 12.692

Std. Error of Skewness .145

Kurtosis 187.898

Std. Error of Kurtosis .289

Range 24

TENURE: How long have you lived at your current address (months)?

Value Frequency Percent Valid

Percent Cumulative

Percent

1 14 4.1 4.3 4.3

2 6 1.7 1.8 6.1

3 4 1.2 1.2 7.3

4 4 1.2 1.2 8.6

5 6 1.7 1.8 10.4

6 6 1.7 1.8 12.2

14157 Exercises

16304_CH05_Walker.indd 141 8/2/12 4:17:12 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

TENURE: How long have you lived at your current address (months)?

Value Frequency Percent Valid

Percent Cumulative

Percent

7 1 .3 .3 12.5

8 3 .9 .9 13.5

9 2 .6 .6 14.1

10 1 .3 .3 14.4

11 1 .3 .3 14.7

12 11 3.2 3.4 18.0

14 1 .3 .3 18.3

18 5 1.5 1.5 19.9

21 1 .3 .3 20.2

24 30 8.7 9.2 29.4

30 1 .3 .3 29.7

31 1 .3 .3 30.0

32 1 .3 .3 30.3

36 22 6.4 6.7 37.0

42 1 .3 .3 37.3

48 12 3.5 3.7 41.0

60 24 7.0 7.3 48.3

72 14 4.1 4.3 52.6

76 1 .3 .3 52.9

84 8 2.3 2.4 55.4

96 18 5.2 5.5 60.9

108 4 1.2 1.2 62.1

120 9 2.6 2.8 64.8

132 11 3.2 3.4 68.2

144 21 6.1 6.4 74.6

156 13 3.8 4.0 78.6

168 11 3.2 3.4 82.0

170 5 1.5 1.5 83.5

180 7 2.0 2.1 85.6

182 2 .6 .6 86.2

186 1 .3 .3 86.5

142 Chapter 5 n Measures of Dispersion

16304_CH05_Walker.indd 142 8/2/12 4:17:12 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

TENURE: How long have you lived at your current address (months)?

Value Frequency Percent Valid

Percent Cumulative

Percent

192 14 4.1 4.3 90.8

198 1 .3 .3 91.1

204 24 7.0 7.3 98.5

216 3 .9 .9 99.4

240 2 .6 .6 100.0

Missing 16 4.7

Total 343 100.0 100.0

N Valid 327

Missing 16

Mean 88.77

Std. Error of Mean 3.880

Median 72

Mode 24

Std. Deviation 70.164

Variance 4923.055

Skewness .365

Std. Error of Skewness .135

Kurtosis 21.284

Std. Error of Kurtosis .269

Range 239

SIBS: How many brothers and sisters do you have?

Value Frequency Percent Valid Percent

Cumulative Percent

0 39 11.4 11.5 11.5

1 137 39.9 40.5 52.1

2 79 23.0 23.4 75.4

3 39 11.4 11.5 87.0

4 17 5.0 5.0 92.0

5 13 3.8 3.8 95.9

6 6 1.7 1.8 97.6

14357 Exercises

16304_CH05_Walker.indd 143 8/2/12 4:17:12 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

SIBS: How many brothers and sisters do you have?

Value Frequency Percent Valid Percent

Cumulative Percent

7 4 1.2 1.2 98.8

9 1 .3 .3 99.1

10 1 .3 .3 99.4

12 1 .3 .3 99.7

15 1 .3 .3 100.0

Missing 5 1.5

Total 343 100.0

N Valid 338

Missing 5

Mean 1.94

Std. Error of Mean .098

Median 1

Mode 1

Std. Deviation 1.801

Variance 3.245

Skewness 2.664

Std. Error of Skewness .133

Kurtosis 12.027

Std. Error of Kurtosis .265

Range 15

8 References5- Enke, J. F. (1832). Über die methode der kleinsten quadrate. Berliner Astronom isches

Jahrbuch für 1834, pp. 249–312. Translated 1841 in R. Taylor (ed.), Scientific Memoirs (Vol. 2). pp. 317–369.

Fisher, R. A. (1918). The correlation between relatives on the supposition of Mende lian inheritance. Transcripts of the Royal Society of Edinburgh, 52, 399–433.

Pearson, K. (1894). On the dissection of asymmetrical frequencycurves: General theory. Philosophical Transactions of the Royal Society of London (Series A, Vol. 185). London, England: Cambridge University Press.

Zeller, R. A. (2000). On teaching about descriptive statistics in criminal justice. Jour- nal of Criminal Justice Education, 10(2), 349–360.

144 Chapter 5 n Measures of Dispersion

16304_CH05_Walker.indd 144 8/2/12 4:17:12 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U

9 For Further Reading5- Galton, F. (1883). Inquiries into Human Faculty and Its Development. London, England:

Macmillan. MacGillivray, H. L. (1981). The mean, median, mode inequality and skewness for a

class of densities. Australian Journal of Statistics, 23, 247. Pearson, K. (1895). Classification of asymmetrical frequency curves in general: Types

actually occurring. Philosophical Transactions of the Royal Society of London (Series A, Vol. 186). London: Cambridge University Press.

10 Note5- 1. An expanded version of this example can be found in Zeller (2000).

Criminal Justice on the Web Visit http://criminaljustice.jbpub.com/Stats4e to make full use of today’s teaching and tech nology! Our interactive Companion Website has been designed to specifically complement Statistics in Criminology and Criminal Justice: Analysis and Interpretation, 4th Edition. The resources available include a Glossary, Flashcards, Crossword Puzzles, Practice Quizzes, Web links, and Student Data Sets. Test yourself today!

145510 Note

16304_CH05_Walker.indd 145 8/2/12 4:17:12 PM

R O D D Y , A N T H O N Y I S A A C 3 7 2 7 B U