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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

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Learning Objectives

Understand the mode, median, and mean as measures of central tendency.■■

Identify the proper measure of central tendency to use for each level of mea-■■

surement.

Explain how to calculate the mode, median, and mean.■■

1 Univariate Descriptive Statistics4- Using frequency distributions and graphical representation, as in Chapter 3, helps researchers determine how the data is arranged and summarize it. Frequency distribu- tions and graphs, however, cannot always tell the entire story. It is usually necessary to summarize the data further. Instead of summarizing entire distributions, it is more often efficient to compare only certain characteristics of the data. To conduct this comparison, it is helpful to know certain information, such as the form of the distribu- tion, the average of the values, and how spread out they are within the distribution.

This is where univariate descriptive statistics come into play. Univariate descrip- tive statistics are used to describe and interpret the meaning of a distribution. They are called univariate because they pertain to only one variable at a time and do not attempt to measure relationships between variables. Univariate descriptive statistics make compact characterizations of distributions in terms of three properties of the data. First is the central tendency, which translates to the average, middle point, or most common value of the distribution. The second property is the dispersion of the data. This relates to how spread out the values are around the central measure. Finally, there is the form of the distribution. The form of a distribution relates to what the distribution would look like if displayed graphically. Included in the form of a dis- tribution is the number of peaks, skewness, and kurtosis. In this chapter we address the first univariate descriptive procedure: measures of central tendency. Measures of dispersion and measures of the form of a distribution are covered in Chapters 5 and 6, respectively.

Chapter 4

Measures of Central Tendency

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2 Measures of Central Tendency4- Measures of central tendency examine where the central value is in a distribution or the distribution’s most typical value. There are three common measures of central tendency, one for each level of measurement (interval and ratio are combined). These are the mode for nominal level data, the median for ordinal level data, and the mean for interval and ratio level data.

Mode

At the lowest level of sophistication is the mode (symbolized by Mo). The mode is used primarily for nominal data to identify the category with the greatest number of cases. The mode is the most frequently occurring value, or case, in a distribution. It is the tallest column on a histogram or the peak on a polygon or line chart. The mode has the advantage of being spotted easily in a distribution, and is often used as a first indicator of the central tendency of a distribution.

The mode is the only measure of central tendency appropriate for nominal vari- ables because it is simply a count of the values. Unlike other measures of central ten- dency, the mode explains nothing about the ordering of variables or variation within the variables. In fact, the mode ignores information about ordering and interval size even if it is available. So it is generally not advised to use the mode for ordinal or interval level data (unless it is used in addition to the median or mean) because too much information is lost.

There is no formula or calculation for the mode for either grouped or ungrouped data. The procedure is just to count the scores and determine the most frequently occurring value. Consider the data set in Table 4-1, which is the number of prisoner escapes from 15 prisons over a 10-year period. Here, there are 15 total escapes. There are two 7’s, one 6, three 5’s, two 4’s, four 3’s, one 2, and two 1’s. The mode in this data set would be 3 escapes, because there are more 3’s than any other value.

7 5 4 3 2

7 5 4 3 1

6 5 3 3 1

Table 4-1 Ungrouped Data

For grouped data, determining the mode is often even easier because the numbers are already counted. The data from Table 4-1 has been grouped in Table 4-2. What is the mode of this data set? Here you simply determine the category that has the highest value. In this case it would be the 3–4 category because it has a frequency of 6. If the data were plotted on a bar chart or polygon, the distribution would look like that in Figure 4-1. Here, you can see that the category 3–4 has the highest bar on the bar chart

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How do you do that?

Obtaining Univariate Statistics in SPSS

Univariate statistics include measures of central tendency, measures of dispersion, and form. You can obtain all of these statistics in the same procedure in SPSS, and it is just an extension of the same procedure you used in Chapter 3 to obtain a fre- quency distribution. The steps to follow are:

Open a data set.1.

Start SPSS.a.

Select b. File, then Open, then Data.

Select the file you want to open, then select c. Open.

Once the data is visible, select 2. Analyze, then Descriptive Statistics, then Fre- quencies.

Make sure the 3. Display Frequency Tables is checked.

Select the variables you wish to include in your distribution and press the 4. c between the two windows.

Select the 5. Statistics button at the bottom of the window.

Check the boxes of any of the univariate measures you want to include in your 6. research.

For measures of central tendency (this chapter), check the boxes in the frame a. Central Tendency, typically the mode, median, and mean.

For measures of dispersionb. (Chapter 5), check the boxes in the frame Disper- sion, typically the standard deviation, variance, and range.

For measures of form c. (Chapter 6), check the boxes in the frame Distribution, specifically skewness and kurtosis.

Select 7. Continue, then ok.

An output window should appear containing a distribution similar in format to 8. Table 4-3.

X f

7–8 2

5–6 4

3–4 6

1–2 3

Table 4-2 Modal Value for Grouped Data

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1–2 3–4 5–6 7–8

Figure 4-1 Bar Chart and Polygon of Grouped Data from Table 4-2

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 347 100.0 100.00

N Valid 336

Missing 11

Mean 4.08

Median 4.00

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 4-3 Combination Table for Education from the 1993 Little Rock Community Policing Survey

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and it forms a hump in the polygon. This highest bar or hump indicates the mode for that variable.

One caution when discussing the mode. The mode is not the frequency of the number that occurs most often but rather, the category (or class) itself. It is easy to want to state that the mode in Table 4-2 is 6 because that is the frequency that is high- est. This is not the mode, however; the mode is the category of the value that has the highest frequency: in this case, 3–4.

Data that is in a frequency table also makes calculating the mode easy. What is the mode in the frequency table in Table 4-3? The mode in this case is 4, or some college. Note that in this case, the mode can be written as either 4 or some college. When using nominal or ordinal data where value labels are assigned to the values, the mode can be expressed as either the value (number) or the value label.

The histogram with a polygon overlay for the data in Table 4-3 is shown in Figure 4-2. As shown in the figure, the highest bar on the histogram or the hump in the poly- gon is at the 4 or some college level. The mode as calculated here is what is obtained from SPSS. In the output in Table 4-3, the mode is identified as some college (4), with a frequency of 117. Notice also that the median, mean, and other measures are also included in this table. This is typical univariate output from SPSS. It provides most of the univariate descriptive statistics discussed in this chapter and the two that follow. Table 4-3 may look somewhat daunting right now, but by the time you finish Chapter 6, a frequency table and univariate output such as this should be shorthand for every- thing you need to know about a distribution.

A distribution is not confined to having only one mode. There are often situations where a distribution will have several categories that have the same or similar frequen- cies. In these cases, the distribution can be said to be bimodal or even multimodal. It is also possible for a distribution to have no mode if the frequencies are the same for each category. If the data in Table 4-1 is modified, a bimodal, multimodal, and a data set with no mode can be created, as shown in Figures 4-3 to 4-5. In Figure 4-3, categories

Less Than High School

GED

Education Type

High School Graduate

Some College

College Graduate

Post Graduate

140

120

100

F re

q u

e n

c y

Figure 4-2 Histogram and Polygon of Education Responses

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1 2 3 4 5 6 7 0

Figure 4-3 Bimodal Distribution

1 2 3 4 5 6 7 0

Figure 4-4 Multimodal Distribution

2 3 4 5 7 0

Figure 4-5 No-Modal Distribution

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3 and 4 both have the same frequency: 3. In this case, both the 3 and the 4 would be the modes because each has the same (highest) value.

In Figure 4-4, the 3, 4, and 5 categories all have frequencies of 3. This means that all three categories would be the mode for this distribution. When almost half of the categories in the distribution represent the mode, its use as a measure of central tendency is reduced.

In Figure 4-5, all of the categories have the same frequency. This does not hap- pen very often, but it is possible, especially in survey research or with other data that have a limited range of categories. The mode as a measure of central tendency in this case is practically useless, although it would be beneficial as a way of stating that all of the values have the same frequency. Although each of the modes in Figures 4-3 (3 and 4), 4-4 (3, 4, and 5), and 4-5 (2 through 7) have the same frequency, that does not always have to be the rule. There is some debate as to what constitutes a bimodal or multimodal distribution. Some propose that the frequencies have to be the same for a distribution to be multimodal. Others argue that practically any peaks in a distribution can represent modes. For example, in Figure 4-1, some would argue that both the 1–2 category and the 3–4 category represent a mode. These people argue that any peak in a polygon, or any spike in the frequency, may represent a mode. In this text, only the category or categories with the highest frequencies will be designated the mode.

Measures of central tendency are among the oldest of all descriptive statistics. The mean, for example, can be traced back to Pythagoras in the 6th century BC, although its development is surely much earlier. Galton (1883) coined the term median during his work on percentiles, but the procedure was used before this by Fechner for arriving at a value of the “middlemost ordinate.” Finally, Karl Pearson reduced the concept of the “abscissa corresponding to the ordinate of maximum frequency” to the mode in his 1895 work.

As can be seen from these distributions, the mode quickly becomes ineffective when there are multiple modes and is worthless (except for an understanding of the nature of the distribution) when each category has a modal value. This is why the mode is not widely used as a measure of central tendency in statistics except for nomi- nal level data.

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Median

If the data is at least ordinal level, the median (symbolized by Me) may be a bet- ter choice for examining the central tendency of the distribution. The median is the point of the 50th percentile of the distribution. This means that the median is the exact midpoint of a distribution or the value that cuts the distribution into two equal parts. For the simple distribution 1, 2, 3, the median would be 2 because it cuts this distribution in half. Note that 2 is not the most frequently occurring or the product of some formula but simply the value in the middle. The median will always be the middle value, but sometimes it will be necessary to resort to math to determine the exact middle value.

The median is used with ordinal level data because it does not imply distance between intervals, only direction: above the median or below it. Recall from “Variables and Measurement” (Chapter 2) that the nature of ordinal level data is that you can determine which category is greater than or less than another category, but there are not equal intervals so there is no way to determine how much greater or lesser the category is. The median also works on this principle, determining the midpoint of a distribution such that a category can be said to be less than or greater than the median, but there is no way to tell by how much. For example, take the following two distributions:

1, 2, 3, 3, 4, 4, 5 1, 1, 1, 3, 10, 50, 100

Each has the same number of values, 7, although each has very different numbers. In this case, the modes would be different: 3 and 4 in the first; 1 in the second. Also, the means would be different: 3.14 in the first, 23.71 in the second. The median for both of these distributions, however, would be 3, the middle value in the distribution. In both distributions there are three values below the median and three values above the median.

The median may be used instead of the mean in a special circumstance where the distribution takes on the quality of being skewed. The mean (discussed in the next section) is often highly influenced by extreme scores. For example, if you were to calculate the mean, or average, age of four people who are 2, 3, 4, and 50 years old, the mean would be 14.75 years old. Obviously, 14.75 is not a good measure of the central value in this distribution, but because of the way the mean is calculated, that is the value that would be obtained. The median of that distribution would be 3.5, which is much more like the central value. Even in the example above, the mean of the second distribution is 23.71, which is not really representative of the distribution. Note, however, that if the variable is interval and the distribution is not skewed, some of the explanatory power of the data is lost using the median as the measure of central tendency.

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Median for Ungrouped Data

Calculation of the median for ungrouped data is relatively simple. All that is needed is the N for the distribution. If the N is not given, simply count the number of scores (remember—do not add the scores, count them). The N is then placed in the formula 1N 1 12

2 . If the data from Table 4-1 were expanded, the median can be calculated as

shown in Exhibit 4-1. There are 23 values here. Adding 1 to this number and then dividing by 2 obtains

the exact middle of the distribution, in this case the 12th value. Once this value is calculated, if the numbers are not arranged in order, you should do so. This ensures that the middle value of the distribution is actually in the middle and that the numbers are arranged in order. Then, beginning with the lowest value, simply count up the ungrouped data until the value obtained in the formula is reached (the 12th value in Exhibit 4-1). This is the median. In this case, counting to the 12th value would produce a score of 3. So the median number of escapes in this distribution is 3.

There are several issues to note about the median. These are important to under- stand when interpreting the median. First, be careful when calculating the median because two different numbers must be dealt with. The value that is obtained from the formula is not the median but simply the number of values to count up in the distribu- tion to find the median (or median class for grouped data). The median is the score or class that contains the number from the formula. In the example above, the median is not 12. Twelve is only the number to count up from the beginning of the distribution to find the median, which is 3.

Also, if there is more than one of the same score in the median class (there are three 3’s in Figure 4-5), the median is still that score even though it occurs more than once. The key value to look for is the middle value, regardless of how many there are of that particular category. This will be brought up again in terms of calculating the median for grouped data where the class interval is greater than 1.

Finally, unlike the mode, the median does not have to be a value in the distribution. For an odd number of scores (as in Exhibit 4-1) the median will be one of the scores

7 5 4 2 2 6 5 3 2 1 6 5 3 2 1 6 5 3 2 6 4 2 2

Exhibit 4-1 Ungrouped Data

N 1 1

2 5

23 1 1

5 24

2 5 12

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because it is the point that cuts the distribution in half. If there are an even number of scores, however, the median will fall in between two of the scores. For example,

in the distribution 3, 4, 5, 6, 7, 8, 9, 10, the formula 1N 1 12

2 would give a value of

4.5. This would put the median between the 6 and 7. When this occurs, the median is a value halfway between the two scores. In this case, the median would be 6.5. This holds true even if the two numbers do not have an interval of 1. For example, in the distribution 5, 6, 8, 10, 11, 12, the number from the formula puts the median between the score of 8 and the score of 10; therefore, the median would be 9.

Median for Grouped Data

For grouped data where the class interval is 1 or where the entire class can be used as the median, the process for finding the median is essentially the same as that for ungrouped data. The first step is to find the value to count up in the distribution using

the formula 1N 1 12

2 . Then, simply count up the frequency of each class to find the

median class. If the data from Exhibit 4-1 is grouped into a frequency distribution it would look like Exhibit 4-2.

The first step is to determine the midpoint using the formula. Since the data in Exhibit 4-2 has not changed, there are still 23 values (escapes). Plugging this value into the formula would, as before, result in a value of 12. Since the values are in a frequency distribution, they are probably already ordered. Although it is possible to find the median beginning from either the lowest or highest category, it is best for consistency to begin with the lowest value. In this case, you would begin with the class of 1 and count the frequencies until the 12th value is reached, which is 3. Note that it is possible here to count from 10 to 12 and still be in the 3 class. This is fine, as long as the value we are looking for, 12, is one of the numbers in that class. If the middle value from the calculation had been a 10, 11, or 12, the 3 class would still have been the median class.

X f 7 1 6 4 5 4 4 2 3 3 2 7 1 2 N 23

Exhibit 4-2 Grouped Data with an Interval Class of 1

N 1 1

2 5

23 1 1

5 24

5 12

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The same procedure is used if the data is grouped with a class interval greater than 1 but where a median class is sufficient. The process for calculating the median where only the median class is desired is shown in Exhibit 4-3. This frequency distri- bution has the same N as the previous distributions, so the first step will be the same: calculating the value to count up to. Here again, the value is 12. Beginning with the lowest class and counting up to 12 will put you in the 16–20 class. This class contains between the 10th and 14th cases, but because it contains the value from the calcula- tion, it is the median class.

Looking again at Table 4-3, the data in this distribution could be either nominal or ordinal. It could be argued, for example, that including a GED in the distribution disrupts the ordering of the categories such that the data should properly be called nominal. It could also be argued, however, that the categories are sufficiently ordered to be called ordinal. For that reason, and to ensure some consistency of examples, the same frequency distribution used to discuss the mode is used here to discuss output for the median.

The median in Table 4-3 is the same as the mode, some college (4). This was obtained in the same manner as described above:

347 1 1

2 5 174

Counting up in the distribution (beginning at the 1 category, which is on the top in this example) puts the median in the 4th category (16 + 59 + 8 + 117). Since this category contains between the 83rd and 200th cases, it contains the 174th value. Also, since the category containing the median is sufficient in this instance, the median is said to be some college, or 4.

Calculating an exact median for grouped data with a class interval greater than 1 is somewhat more complicated. Using the data set in Exhibit 4-3, the procedure to

calculate an exact median begins as all others, with the formula 1N 1 12

2 . This produces

the value 12, the same as in previous examples. This means that the 16–20 class is the

X f 31–35 2 26–30 3 21–25 4 16–20 5 11–15 4 6–10 3 1–5 2 N 23

Exhibit 4-3 Calculating Median Class for Grouped Data

Step 1. Find the median interval N 1 1

2 5 12.

Step 2. Count up in the frequency for that class.

Step 3. That is the median class for this distribution (16–20).

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median class. As stated above, we could count to 14 in this class, beyond the 12 needed to establish a median value. The question then becomes: Where in this class does the median lie? To find out requires interpolation within the class. Assuming the scores are evenly distributed within the class,1 the formula for calculating the exact median is

Me 5 Lm 1 a 0.5N 2 cfbm

fm bi

where Lm is the lower limit of the median class, c fbm the cumulative frequency of the interval below the median class, fm the frequency of the median class, and i the width of the interval of the median class. Using this formula with the data from Exhibit 4-3, the median is calculated as follows:

Me 5 15.5 1 a0.5 1232 2 9

5 b5

5 15.5 1 a11.5 2 9 5

5 15.5 1 a2.5 5 b5

5 15.5 1 10.525 5 15.5 1 2.5

5 18

The value of 15.5 is the lower limit of the 16–20 class. N is 23, as in all other examples. The cumulative frequency is determined by adding up all the frequencies below the class containing the median. In this case, there are three classes below the median class: 1–5, 6–10, and 11–15. The frequencies of these classes (2, 3, and 4, respectively) equal 9 (c fbm). The frequency of the class containing the median in this case (16–20 class) is 5 (fm). Finally, the interval is calculated by subtracting the lower limit of the median class from the upper limit (20.5 2 15.5 = 5). In this case, the result of the cal- culations shows that the exact midpoint of the distribution is 18.2

Calculating the exact median in actual research is less often necessary. Most sta- tistical programs report the exact median from the ungrouped data, or the researchers report only the median class or report the midpoint of the median category as the median. There are times, however, when it is necessary to determine the exact median from information in journal articles. For example, you may wish to know the exact median from Table 4-4. This table shows categories that are not only greater than 1 but are unequal. The procedure would be the same as discussed above, however. Here, the median category would be 51 to 75% [(40 + 1)/2 = 20.5]. Interpolating where in that

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class the exact median would be involves using the formula given above. Application of that formula for the data in Table 4-4 is shown below.

Me 5 Lm 1 a 0.5N 2 cfbm

fm bi

5 50.5 1 a0.5 1402 2 19

14 b25

5 50.5 1 a20 2 19 14

b25

5 50.5 1 a 1 14 b25

5 50.5 1 0.071252

5 50.5 1 1.75

5 52.25

As you would expect, the median does not go very far into the median class in this example. This is evident because the frequency below the median class is 19, and the exact median is only 20.5.

This process is complicated somewhat when the median class is open-ended. For example, what is the midpoint in a distribution where the upper category for annual income is $30,000 and greater? There are several methods of dealing with this issue. Probably the best is to attempt to determine what a reasonable midpoint might be. This is also shown in the example in Table 4-4, which has two open-ended categories: less than high school and postgraduate. This would make it difficult to determine, for example, where the midpoint of a postgraduate degree would lie (some graduate

What portion of your professional research focuses on racial or ethnic issues?

Portion Number Percentage

0–10% 6 15

11–25% 1 2

26–50% 12 30

51–75% 14 35

Over 75% 7 18

Source: Edwards, White, and Pezzella (1998).

Table 4-4 Research on Racial or Ethnic Issues

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work, master’s degree, law degree, etc.). This would have to be a judgment call by the re searcher based on theory and an understanding of the data.

Mean

A statistician is a person who stands in a bucket of ice water, sticks his head in an oven, and says, “On average, I feel fine.”

—Unknown

The most popular measure of central tendency, both among statisticians and the general population, is the mean. The mean is used primarily for interval and ratio level data. Because it assumes equality of intervals, the mean is generally not used with nominal or ordinal level data. The mean is very important to statistical analysis because it is the basis, along with the variance (see the discussion of measures of dispersion in Chapter 5), of many of the formulas for higher-order statistical procedures. The mean also serves as a check on the integrity of the data. As discussed above, the mean is often heavily influenced by extreme scores. So if a 17 has been mistyped as 177, the mean will be much larger than expected. Mean scores outside what would be expected for the data should be a signal to recheck the data.

There are actually several different versions of the mean. The mean discussed in this chapter is the arithmetic mean (from here on, called the mean). There are varia- tions of the mean that are less utilized in social science research and are not discussed here. These include the weighted mean, harmonic mean, and geometric mean.

The symbolic notation for the mean is different than symbols that have been used

to this point. The mean is symbolized either by m or X, depending on whether the data is a population or sample estimate (this distinction is used most often in Chapter 15 and beyond). It is interesting that descriptive statistics deals with a population, but it has become convention that the mean most commonly used in descriptive statistics is

actually the symbol for the sample mean (X). Since most texts use this notation for the mean in descriptive analyses, it will also be used here, even though the more proper notation would be the population mean (m).

The mean is simply the average of all the values in a distribution. To obtain the mean, add up the scores in a distribution and divide by N (just as in calculating an average). In statistical terms, the mean is calculated as

X 5 Sfx

where fx is calculated by multiplying X times the frequency for each value. In the example used in Exhibit 4-2, the mean would be calculated as in Exhibit 4-4. Here, each X is multiplied by the frequency for that category (7 3 1, 6 3 4, etc.). That cre-

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ates an fx column in the table, which is then summed to obtain Sfx (84). That value is then divided by the N for the distribution (23) to obtain the mean for the distribution. In this case, there were 23 prisons that had a total of 84 escapes, so the mean (average) number of escapes for these 23 prisons was 3.65 escapes.

The procedure for calculating the mean for grouped and ungrouped data is the same. The only difference is that for grouped data where the class interval is greater than 1, the midpoint of the class is used as X.3 For example, in the frequency dis- tribution in Exhibit 4-3, the midpoints of the classes would be 2.5 (5.5 2 0.5 = 5; 5/2 = 2.5), 8.5, 13.5, and so on. These are the values that would be used for X in the formula for the mean.

The mean can be estimated from the example output that was used for the mode and median (as shown in Table 4-3). Note that this data is not interval or ratio level and is used here only to show the similarities and differences among the mean, median, and mode. Even though this is nominal/ordinal level data, SPSS treats it as interval level and uses the formula above for calculating the mean. In this example, each of the category values (1 through 6) are multiplied by the frequency for that category (1 3 16, 2 3 59, etc.). This fx value is summed to achieve a total of 1370. This is then divided by N minus the 11 missing values for a total of 336. The result is 4.077, which is what SPSS reported (rounded to 4.08).

In most cases in real research, the mean may not be accompanied by a frequency distribution, or the frequency distribution will be more for presentation than for analy- sis. In such cases, the mean may be reported alone, or it could be reported as part of a discussion or table of univariate statistics associated with the research.

The mean has several advantages over other measures of central tendency. From a practical standpoint, the mean is preferred because it is standardized. This means it can be compared across distributions. This is very beneficial when comparing similar data from different sources, such as the mean number of prisoners per institution in several states, because the two values can be directly compared. The mean is also important because the sum of the deviations of the scores from the mean is always

X f fx

7 1 7 6 4 24 5 4 20 4 2 8 3 3 9 2 7 14 1 2 2 N 23 ofx 84

Exhibit 4-4 Calculating the Mean

X 5 Sfx

5 84

23 5 3.65

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zero. That is, if each value in a distribution were subtracted from the mean, the sum of those scores would be zero. This is discussed in detail in Chapter 5. A final important characteristic of the mean is that 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 principle of sum of squares is very important to our discussions in Chapter 5 of the variance and sum of squares as they relate to regression lines.

As discussed above, the greatest problem with the mean is that it is greatly influ- enced by extreme scores in the distribution. The example in the section on the median, where a mean age of 15 was obtained when all but one of the values was less than 5, shows how much the mean can be influenced by extreme scores. That is why the median is used in cases where the data is skewed.

3 Selecting the Most Appropriate Measure of 4- Central Tendency The goal of many statistical analyses is to be able to develop summary statements, often about a large amount of data. Proper summarization depends on several factors, including the level of data, the nature of the data, the purpose of the summarization, and the interpretation.

The level of data has a substantial influence on which measure of central tendency should be used. As stated earlier, one measure is most appropriate for a particular level of data. The mode is most appropriate for nominal level data, and its use with ordinal and interval level data would result in a loss of power in terms of the information that could be gained from the data. The median is most appropriate with ordinal level data. Although it can be used with interval level data (especially skewed distributions), it should not be used with nominal level data because the rankings assumed in the median cannot be achieved with nominal level data. Finally, the mean should be used only with interval or ratio level data because it assumes equal intervals of the data that cannot be achieved by nominal and partially ordered ordinal level data. The exception here is that the mean can be used with dichotomized nominal level data because this type of data approximates interval level characteristics.

Selection of the most appropriate measure of central tendency is also sometimes based on the nature of the distribution. As discussed above, if a distribution is highly skewed, or if it can be determined that there are some extreme values (outliers) in the distribution that would make the mean inaccurate as a measure of central tendency, the median should be used rather than the mean.

The second criterion for choosing a measure of central tendency is the purpose of summarization, typically in terms of what you are trying to predict. Imagine that you were asked to state one measure that would best capture the nature of a distribution.

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How would you go about that? To put it another way, you might bet $100 to guess a number drawn at random from a distribution. Which number would you choose? One way to address these questions would be to find the score that would be at the “heart” of the distribution: the most common score, the one that cut the distribution in half, or the average score. That is the goal and the role of measures of central tendency. There are several ways to go about this.

If you knew all the values in the distribution, you could calculate the mode easily and quickly. If you are interested in predicting an exact value, you should probably use the mode because it has the highest probability of occurring in any given distribution. Both the median and the mean may produce values that are not in the distribution, so if you must guess and be absolutely right as to the number, use the mode. For example, say you are taking a multiple-choice test and have no idea which answer to a certain question is correct. If you had the distribution of correct answers for that professor for that test, you would want to choose the modal answer rather than the median or mean. This is because you must get the answer correct or it does not count. As another example, consider a prediction based on driving a car around an obstruction placed in front of it. If tests occur over a number of drivers, the distribution would be bimodal: some steering to the left and some to the right. A suggested course of action would not be the median or mean, however, as that would have the vehicle crashing into the obstacle even though it minimized the error in steering.

If, on the other hand, you want to maximize your prediction by getting closest to the number over several tries, thereby minimizing your error, the median might be a better choice. Here, whether you miss high or low is irrelevant; what is important is the size of the error. In a popular game show, contestants are given $7 and required to guess the exact numbers included in the price of a car. For each number they are off, they lose $1. If they have money left over after making all the guesses, they win the car; if they run out of money, they lose. The probability of response plays a big part in the first two or three numbers. You would not want to guess 9 for the first number, for example. If contestants are at the fourth or fifth number, however, and still have money left, they may want to choose the median value (probably a 5) to minimize the error (loss of dol- lars). Being high or low does not matter here, only deviation from the number.

Finally, if you have the opportunity to average your misses over several guesses and the signs do matter (high guesses can offset low guesses), the mean is the best choice. The mean is good in that if you do not know a value, it is often best to choose the average. For example, if you had to guess the weight of a woman whom you had never seen, you should probably choose the mean weight for women because this would minimize the error. The mean is also practically the only choice when using estimates in higher-order analyses because the mathematical properties of both the mode and the median are such that they do not lend themselves to inclusion in other formulas. The mean is less efficient, however, with highly skewed distributions.

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The final criterion for selecting a particular measure of central tendency is the interpretation. If you chose the wrong level of measurement and base your measure of central tendency on that choice, your interpretation may very well not make sense. For example, for a nominal level variable such as paint color, the mode makes sense (more people chose red than any other color). The median does not make much sense, however. For example, if you say half or fewer of the respondents chose red, what does that mean? There is no reference point because there is no order. The same holds true for the mean. How could you interpret an average of 1.8 on paint color; that the aver- age color chosen was slightly different than red? It is easier to use lower measures of central tendency with higher levels of measurement, but you lose some of the power of your interpretation. For example, it is technically correct to say the modal age in a class is 20, but it is not as precise as saying the average age is 22.4.

4 Conclusion4- In this chapter, we introduced univariate analyses by discussing the first of the

univariate descriptive statistics, measures of central tendency. Measures of central ten- dency are one of the most used descriptive statistics and provide the most information. For example, if you were to ask someone about a group of people, you might provide an answer in terms of an average age or average income.

The measures of central tendency provide the information that their name implies: a measure of the central value. Think of a seesaw. For a seesaw to work properly, it must have a balance point in the middle so the weight is distributed generally equally on each side (as in Figure 4-6). Here, the measure of central tendency is at the balance point of the distribution. The picture of a seesaw, however, could easily be replaced with a histogram of a frequency distribution. If only the X axis were retained, the seesaw could look like the bar chart in Figure 4-7. This distribution is actually unique in that, mathematically, the mean equals 4. Since 4 is the most frequently occurring value, it is also the mode; and because 4 is the middlemost point in the distribution, it is also the median. If the values of the distribution were changed some, the balance point would have to shift to keep the balance of the distribution. For example, in Figure 4-8, the mean, median, and mode are at different points. This is because of the spread and alignment of the values in the distribution.

You can see that just knowing the measure of central tendency is not always enough. Sometimes it is also important to know how spread out the values are or how they are arranged in the distribution. This is the reason that more than measures of cen- tral tendency are needed for a proper description of data. In Chapter 5, we address how

Figure 4-6 Balancing a Distribution on the Measure of Central Tendency

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spread out the values are in the distribution, and in Chapter 6 we discuss the arrange- ment of the data within the distribution. Together, these three pieces of information make up the complete analysis of a single variable (univariate analysis).

1 2 3 5 6 74

Figure 4-7 Histogram of Balanced Frequency Distribution

1 2 3 5 6 74

Mo Mean Me

Figure 4-8 Histogram of Unbalanced Distribution

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5 Key Terms4- central tendency median dispersion mode form sum of squares mean

6 Summary of Equations4- Median (Me) for ungrouped data

N 1 1

Median (Me) for grouped data

Me 5 Lm 1 a 0.5N 2 cfbm

fm bi

Mean (X)

X 5 a fx

7 Exercises4- The exercises for this chapter and Chapters 5 and 6 use the same examples. This will allow you to work through problems using all three types of univariate descriptive sta- tistics. 1. For the set of data below, calculate:

a. The mode b The median c. The mean

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

2. For the set of data below, calculate: a. The mode b. The median c. The mean

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

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3. For the set of data below, calculate: a. The mode b. The median c. The mean

Interval Midpoint Frequency

90–100 6

80–89 8

70–79 4

60–69 3

50–59 2

4. For the set of data below, calculate: a. The mode b. The median c. The mean

Interval f

90–100 5

80–89 7

70–79 9

60–69 4

5. For each of the variables in the frequency tables that follow (from the gang database), describe the level of measurement for each variable and how you determined your answer.

6. Using the frequency tables that follow (from the gang database), discuss the three measures of central tendency.

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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 343 100.0 100.00

N Valid 340

Missing 3

Mean 1.41

Std. Error of Mean .051

Median 1

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

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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

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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

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

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

Value Frequency Percent Valid

Percent Cumulative

Percent

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

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

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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

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 Total

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

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8 References4- Edwards, W. J., White, N., Bennett, I., & Pezzella, F. (1998). Who has come out of the

pipeline? African Americans in criminology and criminal justice. Journal of Crimi- nal Justice Education, 9(2), 249–266.

Galton, F. (1883). Inquiries into Human Faculty and Its Development. London, England: Macmillan.

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, England: Cambridge University Press.

9 Notes4- 1. This may not be a valid assumption, and it is possible, for example, that all the

scores could be 14, but it would be impossible to calculate the median with- out deconstructing the values, so an assumption is made that all values in the median class are equally distributed.

2. For future reference, this formula is the same (except for the 0.5) as the one used for computing percentiles because the median is the 50th percentile of the distribu- tion.

3. This procedure assumes closed intervals for each class. If you have a situation, say, where the oldest category of an age distribution is “6 and above,” it is more difficult to determine the midpoint. It is sometimes necessary to make an estimate of where the central value of the class might be.

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!

1194-9 Notes

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