# sum of the scores divided by the number of scores.

33

Central Tendency and Variability

Chapter Outline

✪ Central Tendency 34

✪ Variability 43

✪ Controversy: The Tyranny of the Mean 52

✪ Central Tendency and Variability in Research Articles 55

As we noted in Chapter 1, the purpose of descriptive statistics is to make agroup of scores understandable. We looked at some ways of getting that un-derstanding through tables and graphs. In this chapter, we consider the main statistical techniques for describing a group of scores with numbers. First, you can describe a group of scores in terms of a representative (or typical) value, such as an average. A representative value gives the central tendency of a group of scores. A representative value is a simple way, with a single number, to describe a group of scores (and there may be hundreds or even thousands of scores). The main represen- tative value we consider is the mean. Next, we focus on ways of describing how spread out the numbers are in a group of scores. In other words, we consider the amount of variation, or variability, among the scores. The two measures of variabil- ity you will learn about are called the variance and standard deviation.

In this chapter, for the first time in this book, you will use statistical formulas. Such formulas are not here to confuse you. Hopefully, you will come to see that they actually simplify things and provide a very straightforward, concise way of describ- ing statistical procedures. To help you grasp what such formulas mean in words, whenever we present formulas in this book we always also give the “translation” in ordinary English.

✪ Summary 57

✪ Key Terms 57

✪ Example Worked-Out Problems 57

✪ Practice Problems 59

✪ Using SPSS 62

✪ Chapter Notes 65

CHAPTER 2

T I P F O R S U C C E S S Before beginning this chapter, you should be sure you are comfort- able with the key terms of variable, score, and value that we consid- ered in Chapter 1.

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34 Chapter 2

Central Tendency The central tendency of a group of scores (a distribution) refers to the middle of the group of scores. You will learn about three measures of central tendency: mean, mode, and median. Each measure of central tendency uses its own method to come up with a single number describing the middle of a group of scores. We start with the mean, the most commonly used measure of central tendency. Understanding the mean is also an important foundation for much of what you learn in later chapters.

The Mean Usually the best measure of central tendency is the ordinary average, the sum of all the scores divided by the number of scores. In statistics, this is called the mean. The average, or mean, of a group of scores is a representative value.

Suppose 10 students, as part of a research study, record the total number of dreams they had during the last week. The numbers of dreams were as follows:

7, 8, 8, 7, 3, 1, 6, 9, 3, 8

The mean of these 10 scores is 6 (the sum of 60 dreams divided by 10 students). That is, on the average, each student had 6 dreams in the past week. The information for the 10 students is thus summarized by the single number 6.

You can think of the mean as a kind of balancing point for the distribution of scores. Try it by visualizing a board balanced over a log, like a rudimentary teeter- totter. Imagine piles of blocks set along the board according to their values, one for each score in the distribution (like a histogram made of blocks). The mean is the point on the board where the weight of the blocks on one side balances exactly with the weight on the other side. Figure 2–1 shows this for the number of dreams for the 10 students.

Mathematically, you can think of the mean as the point at which the total distance to all the scores above that point equals the total distance to all the scores below that point. Let’s first figure the total distance from the mean to all the scores above the mean for the dreams example shown in Figure 2–1. There are two scores of 7, each of which is 1 unit above 6 (the mean). There are three scores of 8, each of which is 2 units above 6. And, there is one score of 9, which is 3 units above 6. This gives a total distance of 11 units from the mean to all the scores above the mean. Now, let’s look at the scores below the mean. There are two scores of 3, each of which is 3 units below 6 (the mean). And there is one score of 1, which is 5 units below 6. This gives a total distance of 11 units from the mean to all of the scores below the mean. Thus, you can see that the total distance from the mean to the scores above the mean is the same as the total distance from the mean to the scores below the mean. The scores above the mean balance out the scores below the mean (and vice-versa).

(3 + 3 + 5)

(1 + 1 + 2 + 2 + 2 + 3)

mean arithmetic average of a group of scores; sum of the scores divided by the number of scores.

5 6 7 8 91 2 3 4

M = 6

Figure 2–1 Mean of the distribution of the number of dreams during a week for 10 students, illustrated using blocks on a board balanced on a log.

central tendency typical or most representative value of a group of scores.

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Central Tendency and Variability 35

Some other examples are shown in Figure 2–2. Notice that there doesn’t have to be a block right at the balance point. That is, the mean doesn’t have to be a score ac- tually in the distribution. The mean is the average of the scores, the balance point. The mean can be a decimal number, even if all the scores in the distribution have to be whole numbers (a mean of 2.30 children, for example). For each distribution in Figure 2–2, the total distance from the mean to the scores above the mean is the same as the total distance from the mean to the scores below the mean. (By the way, this analogy to blocks on a board, in reality, works out precisely only if the board has no weight of its own.)

Formula for the Mean and Statistical Symbols The rule for figuring the mean is to add up all the scores and divide by the number of scores. Here is how this rule is written as a formula:

(2–1)

M is a symbol for the mean. An alternative symbol, (“X-bar”), is sometimes used. However, M is almost always used in research articles in psychology, as rec- ommended by the style guidelines of the American Psychological Association (2001). You will see used mostly in advanced statistics books and in articles about statistics. In fact, there is not a general agreement for many of the symbols used in statistics. (In this book we generally use the symbols most widely found in psychol- ogy research articles.) S, the capital Greek letter sigma, is the symbol for “sum of.” It means “add up

all the numbers for whatever follows.” It is the most common special arithmetic symbol used in statistics.

X stands for the scores in the distribution of the variable X. We could have picked any letter. However, if there is only one variable, it is usually called X. In later chapters we use formulas with more than one variable. In those formulas, we use a second letter along with X (usually Y ) or subscripts (such as and ).

is “the sum of X.” This tells you to add up all the scores in the distribution of the variable X. Suppose X is the number of dreams of our 10 students: X is

, which is 60.7 + 8 + 8 + 7 + 3 + 1 + 6 + 9 + 3 + 8 ©

X

X

M = gX

N

M mean.

5 6 7 8 91 2 3 4

5 6 7 8 91 2 3 4

5 6 7 8 91 2 3 4

5 6 7 8 91 2 3 4

M = 6

M = 3.60

M = 6

M = 6

Figure 2–2 Means of various distributions illustrated with blocks on a board balanced on a log.

The mean is the sum of the scores divided by the number of scores.

S sum of; add up all the scores follow- ing this symbol.

X scores in the distribution of the variable X.

T I P F O R S U C C E S S Think of each formula as a statisti- cal recipe, with statistical symbols as ingredients. Before you use each formula, be sure you know what each symbol stands for. Then carefully follow the formula to come up with the end result.

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36 Chapter 2

N stands for number—the number of scores in a distribution. In our example, there are 10 scores. Thus, N equals 10.1

Overall, the formula says to divide the sum of all the scores in the distribution of the variable X by the total number of scores, N. In the dreams example, this means you divide 60 by 10. Put in terms of the formula,

Additional Examples of Figuring the Mean Consider the examples from Chapter 1. The stress ratings of the 30 students in the first week of their statistics class (based on Aron et al., 1995) were:

8, 7, 4, 10, 8, 6, 8, 9, 9, 7, 3, 7, 6, 5, 0, 9, 10, 7, 7, 3, 6, 7, 5, 2, 1, 6, 7, 10, 8, 8

In Chapter 1 we summarized all these numbers into a frequency table (Table 1–3). You can now summarize all this information as a single number by figuring the mean. Figure the mean by adding up all the stress ratings and dividing by the num- ber of stress ratings. That is, you add up the 30 stress ratings:

, for a total of 193. Then you divide this total by the number of scores, 30. In terms of the formula,

This tells you that the average stress rating was 6.43 (after rounding off). This is clearly higher than the middle of the 0–10 scale. You can also see this on a graph. Think again of the histogram as a pile of blocks on a board and the mean of 6.43 as the point where the board balances on the fulcrum (see Figure 2–3). This single rep- resentative value simplifies the information in the 30 stress scores.

M = gX

N =

193

30 = 6.43

7 + 5 + 2 + 1 + 6 + 7 + 10 + 8 + 8 8 + 6 + 8 + 9 + 9 + 7 + 3 + 7 + 6 + 5 + 0 + 9 + 10 + 7 + 7 + 3 + 6 +

8 + 7 + 4 + 10 +

M = gX

N =

60

10 = 6

N number of scores in a distribution.

T I P F O R S U C C E S S When an answer is not a whole number, we suggest that you use two more decimal places in the an- swer than for the original numbers. In this example, the original num- bers did not use decimals, so we rounded the answer to two deci- mal places.

7

6

5

4

3

2

1

0 0 1 2 3 4 5 6 7 8 9 10

Balance Point

Fr eq

ue nc

y

6.43Stress Rating

Figure 2–3 Analogy of blocks on a board balanced on a fulcrum showing the means for 30 statistics students’ ratings of their stress level. (Data based on Aron et al., 1995.)

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Central Tendency and Variability 37

Similarly, consider the Chapter 1 example of students’ social interactions (McLaughlin-Volpe et al., 2001). The actual number of interactions over a week for the 94 students are listed on page 8. In Chapter 1, we organized the original scores into a frequency table (see Table 1–5). We can now take those same 94 scores, add them up, and divide by 94 to figure the mean:

This tells us that during this week these students had an average of 17.39 social in- teractions. Figure 2–4 shows the mean of 17.39 as the balance point for the 94 social interaction scores.

Steps for Figuring the Mean Figure the mean in two steps.

❶ Add up all the scores. That is, figure ΣX. ❷ Divide this sum by the number of scores. That is, divide ΣX by N.

The Mode The mode is another measure of central tendency. The mode is the most common single value in a distribution. In our dreams example, the mode is 8. This is because there are three students with 8 dreams and no other number of dreams with as many students. Another way to think of the mode is that it is the value with the largest frequency in a frequency table, the high point or peak of a distribution’s histogram (as shown in Figure 2–5).

M = gX

N =

1,635

94 = 17.39

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0 2.5 7.5 12.5 22.5 27.5 42.5 47.5

Number of Social Interactions in a Week

Fr eq

ue nc

y

32.5 37.517.5

Balance Point

17.39

Figure 2–4 Analogy of blocks on a board balanced on a fulcrum illustrating the mean for number of social interactions during a week for 94 college students. (Data from McLaughlin-Volpe et al., 2001.)

mode value with the greatest frequency in a distribution.

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In a perfectly symmetrical unimodal distribution, the mode is the same as the mean. However, what happens when the mean and the mode are not the same? In that situation, the mode is usually not a very good way of describing the central ten- dency of the scores in the distribution. In fact, sometimes researchers compare the mode to the mean to show that the distribution is not perfectly symmetrical. Also, the mode can be a particularly poor representative value because it does not reflect many aspects of the distribution. For example, you can change some of the scores in a distribution without affecting the mode—but this is not true of the mean, which is affected by any change in the distribution (see Figure 2–6).

5 6 7 8 91 2 3 4

Mode = 8

Figure 2–5 Mode as the high point in a distribution’s histogram, using the example of the number of dreams during a week for 10 students.

115 6 7 8 9 102 3 4

5 6 7 8 9 102 3 4

5 6 7 8 9 102 3 4

Mean = 8.30

Mode = 8

Mean = 5.10

Mode = 8

Mode = 8

Mean = 7

Figure 2–6 Effect on the mean and on the mode of changing some scores, using the example of the number of dreams during a week for 10 students.

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Central Tendency and Variability 39

7 7633 98881

Median

Figure 2–7 The median is the middle score when scores are lined up from lowest to highest, using the example of the number of dreams during a week for 10 students.

T I P F O R S U C C E S S When figuring the median, remem- ber that the first step is to line up the scores from lowest to highest. Forgetting to do this is the most common mistake students make when figuring the median.

median middle score when all the scores in a distribution are arranged from lowest to highest.

outlier score with an extreme value (very high or very low) in relation to the other scores in the distribution.

On the other hand, the mode is the usual way of describing the central tendency for a nominal variable. For example, if you know the religions of a particular group of people, the mode tells you which religion is the most frequent. However, when it comes to the numerical variables that are most common in psychology research, the mode is rarely used.

The Median Another alternative to the mean is the median. If you line up all the scores from low- est to highest, the middle score is the median. Figure 2–7 shows the scores for the number of dreams lined up from lowest to highest. In this example, the fifth and sixth scores (the two middle ones) are both 7s. Either way, the median is 7.

When you have an even number of scores, the median is between two different scores. In that situation, the median is the average (the mean) of the two scores.

Steps for Finding the Median Finding the median takes three steps.

❶ Line up all the scores from lowest to highest. ❷ Figure how many scores there are to the middle score by adding 1 to the num-

ber of scores and dividing by 2. For example, with 29 scores, adding 1 and divid- ing by 2 gives you 15. The 15th score is the middle score. If there are 50 scores, adding 1 and dividing by 2 gives you 251⁄2. Because there are no half scores, the 25th and 26th scores (the scores on either side of 251⁄2) are the middle scores.

❸ Count up to the middle score or scores. If you have one middle score, this is the median. If you have two middle scores, the median is the average (the mean) of these two scores.

Comparing the Mean, Mode, and Median Sometimes, the median is better than the mean (and mode) as a representative value for a group of scores. This happens when a few extreme scores would strongly affect the mean but would not affect the median. Reaction time scores are a common example in psychology research. Suppose you are asked to press a key as quickly as possible when a green circle is shown on the computer screen. On five showings of the green circle, your times (in seconds) to respond are .74, .86, 2.32, .79, and .81. The mean of these five scores is 1.1040: that is, . However, this mean is very much influenced by the one very long time (2.32 seconds). (Perhaps you were dis- tracted just when the green circle was shown.) The median is much less affected by the extreme score. The median of these five scores is .81—a value that is much more rep- resentative of most of the scores. Thus, using the median deemphasizes the one ex- treme time, which is probably appropriate. An extreme score like this is called an outlier. In this example, the outlier was much higher than the other scores, but in other cases an outlier may be much lower than the other scores in the distribution.

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The importance of whether you use the mean, mode, or median can be seen in a controversy among psychologists studying the evolutionary basis of human mate choice. One set of theorists (e.g., Buss & Schmitt, 1993) argue that over their lives, men should prefer to have many partners, but women should prefer to have just one reliable partner. This is because a woman can have only a small number of children in a lifetime and her genes are most likely to survive if those few children are well taken care of. Men, however, can have a great many children in a lifetime. Therefore, according to the theory, a shotgun approach is best for men, because their genes are most likely to survive if they have a great many partners. Consistent with this assumption, evolutionary psycholo- gists have found that men report wanting far more partners than do women.

Other theorists (e.g., Miller & Fishkin, 1997), however, have questioned this view. They argue that women and men should prefer about the same number of partners. This is because individuals with a basic predisposition to seek a strong intimate bond are most likely to survive infancy. This desire for strong bonds, they argue, remains in adulthood. These theorists also asked women and men how many partners they wanted. They found the same result as the previous researchers when using the mean: men wanted an average of 64.3 partners, women an average of 2.8 partners. However, the picture looks drastically different if you look at the median or mode (see Table 2–1). Figure 2–8, taken directly from their article, shows why. Most women and most men want just one partner. A few want more, some many more. The big difference is that

Pe rc

en ta

ge o

f M

en a

nd W

om en

– – – –

60 – – – – –

50 – – – – –

40 – – – – –

30 – – – – –

20 – – – – –

10 – – – – –

0 – – 0 – 1 – 2 – 3 – 4 – 5 – 6 – 7 – 8 – 9 – 10 –

11 –2

0 – 21

–3 0 –

31 –4

0 – 41

–5 0 –

51 –6

0 – 61

–7 0 –

71 –8

0 – 81

–9 0 –

91 –1

00 –

10 0–

10 00

– 10

01 –1

00 00

Women %

Men %

Number of Partners Desired in the Next 30 Years

Women 2.8 1

Men 64.3 1

Mean Median

Measures of central tendency

Figure 2–8 Distributions for men and women for the ideal number of partners desired over 30 years. Note: To include all the data, we collapsed across categories farther out on the tail of these distributions. If every category represented a single number, it would be more apparent that the tail is very flat and that distributions are even more skewed than is apparent here.

Source: Miller, L. C., & Fishkin, S. A. (1997). On the dynamics of human bonding and reproductive suc- cess: Seeking windows on the adapted-for-human-environmental interface. In J. Simpson & D. T. Kenrick (Eds.), Evolutionary social psychology (pp. 197–235). Mahwah, NJ: Erlbaum.

Table 2–1 Responses of 106 Men and 160 Women to the Question, “How many partners would you ideally desire in the next 30 years?”

Mean Median Mode

Women 2.8 1 1

Men 64.3 1 1

Source: Data from Miller & Fishkin (1997).

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Central Tendency and Variability 41

there are a lot more men in the small group that want many more than one partner. These results were also replicated in a more recent study (Pedersen et al., 2002).

So which theory is right? You could argue either way from these results. The point is that focusing just on the mean can clearly misrepresent the reality of the distribution. As this example shows, the median is most likely to be used when a few extreme scores would make the mean unrepresentative of the main body of scores. Figure 2–9 illustrates this point, by showing the relative location of the mean, mode, and median for three types of distribution that you learned about in Chapter 1. The distribution in Figure 2–9a is skewed to the left (negatively skewed); the long tail of the distribution points to the left. The mode in this distribution is the highest point of the distribution, which is on the far right hand side of the distribution. The median is the point at which half of the scores are above that point and half are below. As you can see, for that to happen, the median must be a lower value than the mode. Finally, the mean is strongly influenced by the very low scores in the long tail of the distribution and is thus a lower value than the median. Figure 2–9b shows the location of the mean, mode, and median for a distribution that is skewed to the right (positively skewed). In this case, the mean is a higher value than either the mode or median because the mean is strongly influ- enced by the very high scores in the long tail of the distribution. Again, the mode is the highest point of the distribution, and the median is between the mode and the mean. In Figures 2–9a and 2–9b, the mean is not a good representative value of the scores, because it is unduly influenced by the extreme scores.

Mean Median Mode

MeanMedianMode

Mean Mode

Median

(a)

(b)

(c)

Figure 2–9 Locations of the mean, mode, and median on (a) a distribution skewed to the left, (b) a distribution skewed to the right, and (c) a normal curve.

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Figure 2–9c shows a normal curve. As for any distribution, the mode is the high- est point in the distribution. For a normal curve, the highest point falls exactly at the midpoint of the distribution. This midpoint is the median value, since half of the scores in the distribution are below that point and half are above it. The mean also falls at the same point because the normal curve is symmetrical about the midpoint, and every score in the left hand side of the curve has a matching score on the right hand side. So, for a normal curve, the mean, mode, and median are always the same value.

In some situations psychologists use the median as part of more complex statis- tical methods. Also, the median is the usual way of describing the central tendency for a rank-order variable. Otherwise, unless there are extreme scores, psychologists almost always use the mean as the representative value of a group of scores. In fact, as you will learn, the mean is a fundamental building block for most other statistical techniques.

A summary of the mean, mode, and median as measures of central tendency is shown in Table 2–2.

Table 2–2 Summary of Measures of Central Tendency

Measure Definition When Used

How are you doing?

1. Name and define three measures of central tendency. 2. Write the formula for the mean and define each of the symbols. 3. Figure the mean of the following scores: 2, 8, 3, 6, and 6. 4. For the following scores find (a) the mean, (b) the mode, and (c) the median: 5,

3, 2, 13, 2. (d) Why is the mean different from the median?

1.The mean is the ordinary average, the sum of the scores divided by the num- ber of scores. The mode is the most frequent score in a distribution. The me- dian is the middle score; that is, if you line the scores up from lowest to highest, it is the halfway score.

2.The formula for the mean is is the mean; is the symbol for “sum of”—add up all the scores that follow; Xis the variable whose scores you are adding up; Nis the number of scores.

3.. 4.(a) The mean is 5; (b) the mode is 2; (c) the median is 3; (d) The mean is differ-

ent from the median because the extreme score (13) makes the mean higher than the median.

Mean Sum of the scores divided by the number of scores

• With equal-interval variables • Very commonly used in psychology

research

Mode Value with the greatest frequency in a distribution

• With nominal variables • Rarely used in psychology research

Median Middle score when all the scores in a distribution are arranged from lowest to highest

• With rank-ordered variables • When a distribution has one or more

outliers • Rarely used in psychology research

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Central Tendency and Variability 43

(a) (b)

1.70 Mean

^ 3.20

Mean

^

3.20 Mean

^

3.20 Mean

^ 2.50

Mean

^

3.20 Mean

^

Figure 2–10 Examples of distributions with (a) the same mean but different amounts of spread, and (b) different means but the same amount of spread.

Variability Researchers also want to know how spread out the scores are in a distribution. This shows the amount of variability in the distribution. For example, suppose you were asked, “How old are the students in your statistics class?” At a city-based university with many returning and part-time students, the mean age might be 29. You could answer, “The average age of the students in my class is 29.” However, this would not tell the whole story. You could have a mean of 29 because every student in the class was exactly 29 years old. If this is the case, the scores in the distribution are not spread out at all. In other words, there is no variation, or variability, among the scores. You could also have a mean of 29 because exactly half the class members were 19 and the other half 39. In this situation, the distribution is much more spread out; there is considerable variability among the scores in the distribution.

You can think of the variability of a distribution as the amount of spread of the scores around the mean. Distributions with the same mean can have very different amounts of spread around the mean; Figure 2–10a shows histograms for three differ- ent frequency distributions with the same mean but different amounts of spread around the mean. A real-life example of this is shown in Figure 2–11, which shows the distributions of the housing prices in two neighborhoods: one with diverse hous- ing types and the other with a consistent type of housing. As with Figure 2–10a, the mean housing price is the same in each neighborhood. However, the distribution for

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MeanHousing Prices

Neighborhood with Consistent Type of Housing

MeanHousing Prices

Neighborhood with Diverse Types of Housing

Figure 2–11 Example of two distributions with the same mean but different amounts of spread: housing prices for a neighborhood with diverse types of housing and for a neigh- borhood with a consistent type of housing.

variance measure of how spread out a set of scores are; average of the squared deviations from the mean.

deviation score score minus the mean.

squared deviation score square of the difference between a score and the mean.

the neighborhood with diverse housing types is much more spread out around the mean than the distribution for the neighborhood that has a consistent type of housing. This tells you that there is much greater variability in the prices of housing in the neighborhood with diverse types of housing than in the neighborhood with a consis- tent housing type. Also, distributions with different means can have the same amount of spread around the mean. Figure 2–10b shows three different distributions with dif- ferent means but the same amount of spread. So, while the mean provides a represen- tative value of a group of scores, it doesn’t tell you about the variability of the scores. You will now learn about two measures of the variability of a group of scores: the variance and standard deviation.2

The Variance The variance of a group of scores tells you how spread out the scores are around the mean. To be precise, the variance is the average of each score’s squared difference from the mean.

Here are the four steps to figure the variance: 