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Statistics For Psychology

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

A s we noted in Chapter 1, the purpose of descriptive statistics is to make a group 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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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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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 X2X1

M = gX

M mean.

5 6 7 8 91 2 3 4

M = 6

M = 3.60

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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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 =

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.

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

Balance Point

F re

qu en

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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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

0 2.5 7.5 12.5 22.5 27.5 42.5 47.5

Number of Social Interactions in a Week

F re

qu en

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

Mean = 8.30

Mode = 8

Mean = 5.10

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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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.

(©X)>N = 5.52> 5 = 1.1040

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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

P er

ce nt

ag e

of 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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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?

Answers

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.

M=(©X)>N=(2+8+3+6+6)>5=5

g M=(©X)>N. M

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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(a) (b)

1.70 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:

❶ Subtract the mean from each score. This gives each score’s deviation score, which is how far away the score is from the mean.

❷ Square each of these deviation scores (multiply each by itself). This gives each score’s squared deviation score.

❸ Add up the squared deviation scores. This total is called the sum of squared deviations.

❹ Divide the sum of squared deviations by the number of scores. This gives the average (the mean) of the squared deviations, called the variance.

Suppose one distribution is more spread out than another. The more spread-out distribution has a larger variance because being spread out makes the deviation scores bigger. If the deviation scores are bigger, the squared deviation scores and the average of the squared deviation scores (the variance) are also bigger.

sum of squared deviations total of all the scores of each score’s squared dif- ference from the mean.

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In the example of the class in which everyone was exactly 29 years old, the variance would be exactly 0. That is, there would be no variance (which makes sense, because there is no variability among the ages). (In terms of the numbers, each person’s deviation score would be ; 0 squared is 0. The average of a bunch of zeros is 0.) By contrast, the class of half 19-year-olds and half 39- year-olds would have a rather large variance of 100. (The 19-year-olds would each have deviation scores of . The 39-year-olds would have deviation scores of . All the squared deviation scores, which are either squared or 10 squared, come out to 100. The average of all 100s is 100.)

The variance is extremely important in many statistical procedures you will learn about later. However, the variance is rarely used as a descriptive statistic. This is be- cause the variance is based on squared deviation scores, which do not give a very easy-to-understand sense of how spread out the actual, nonsquared scores are. For ex- ample, a class with a variance of 400 clearly has a more spread-out distribution than one whose variance is 10. However, the number 400 does not give an obvious insight into the actual variation among the ages, none of which is anywhere near 400.3

The Standard Deviation The most widely used way of describing the spread of a group of scores is the standard deviation. The standard deviation is directly related to the variance and is figured by taking the square root of the variance. There are two steps in figuring the standard deviation.

❶ Figure the variance. ❷ Take the square root. The standard deviation is the positive square root of the

variance. (Any number has both a positive and a negative square root. For ex- ample, the square root of 9 is both and .)

If the variance of a distribution is 400, the standard deviation is 20. If the vari- ance is 9, the standard deviation is 3.

The variance is about squared deviations from the mean. Therefore, its square root, the standard deviation, is about direct, ordinary, not-squared deviations from the mean. Roughly speaking, the standard deviation is the average amount that scores differ from the mean. For example, consider a class where the ages have a standard deviation of 20 years. This tells you that the ages are spread out, on the average, about 20 years in each direction from the mean. Knowing the standard deviation gives you a general sense of the degree of spread.4

The standard deviation does not, however, perfectly describe the shape of the distribution. For example, suppose the distribution of the number of children in fam- ilies in a particular country has a mean of 4 and standard deviation of 1. Figure 2–12 shows several possibilities of the distribution of number of children, all with a mean of 4 and a standard deviation of 1.

Formulas for the Variance and the Standard Deviation We have seen that the variance is the average squared deviation from the mean. Here is the formula for the variance.

(2–2)SD2 = g(X – M)2

– 3+ 3

– 1039 – 29 = 10 19 – 29 = – 10

29 – 29 = 0

standard deviation square root of the average of the squared deviations from the mean; the most common descriptive statistic for variation; approximately the average amount that scores in a distribu- tion vary from the mean.

The variance is the sum of the squared deviations of the scores from the mean, divided by the number of scores.

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SD2 variance.

SD standard deviation. SD2 is the symbol for the variance. This may seem surprising. SD is short for

standard deviation. The symbol SD2 emphasizes that the variance is the standard devi- ation squared. (Later, you will learn other symbols for the variance, S 2 and —the lowercase Greek letter sigma squared. The different symbols are for different situa- tions in which the variance is used. In some cases, it is figured slightly differently.)

The top part of the formula is the sum of squared deviations. X is for each score and M is the mean. Thus, X – M is the score minus the mean, the deviation score. The superscript number (2) tells you to square each deviation score. Finally, the sum sign (Σ) tells you to add up all these squared deviation scores.

The sum of squared deviations of the scores from the mean, which is called the sum of squares for short, has its own symbol, SS. Thus, the variance formula can be written using SS instead of Σ(X – M)2:

(2–3)

Whether you use the simplified symbol SS or the full description of the sum of squared deviations, the bottom part of the formula is just N, the number of scores.

SD2 = SS

�2sum of squares (SS) sum of squared deviations.

T I P F O R S U C C E S S The sum of squared deviations is an important part of many of the procedures you learn in later chap- ters; so be sure you fully under- stand it, as well as how it is figured.

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

Figure 2–12 Some possible distributions for family size in a country where the mean is 4 and the standard deviation is 1.

The variance is the sum of squares divided by the number of scores. ISB

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Central Tendency and Variability 47

That is, the formula says to divide the sum of the squared deviation scores by the number of scores in the distribution.

The standard deviation is the square root of the variance. So, if you already know the variance, the formula is

(2–4)

The formula for the standard deviation, starting from scratch, is the square root of what you figure for the variance:

(2–5)

(2–6)

Examples of Figuring the Variance and Standard Deviation Table 2–3 shows the figuring for the variance and standard deviation for the number of dreams example. (The table assumes you have already figured the mean to be 6 dreams.) Usually, it is easiest to do your figuring using a calculator, especially one with a square root key. The standard deviation of 2.57 tells you that roughly speak- ing, on the average, the number of dreams vary by about 21⁄2 from the mean of 6.

Table 2–4 shows the figuring for the variance and standard deviation for the example of students’ number of social interactions during a week (McLaughlin- Volpe et al., 2001). (To save space, the table shows only the first few and last few scores.) Roughly speaking, this result tells you that a student’s number of social in- teractions in a week varies from the mean (of 17.39) by an average of 11.49. This can also be shown on a histogram (see Figure 2–13).

Measures of variability, such as the variance and standard deviation, are heavily influenced by the presence of one or more outliers (extreme values) in a distribution.

SD = A SS

SD = B ©(X – M)2

SD = 2SD2

Table 2–3 Figuring the Variance and Standard Deviation in the Number of Dreams Example

Score (Number of

Dreams) �

Mean Score (Mean Number

of Dreams) � Deviation

Score

Squared Deviation

Score

7 6 1 1

8 6 2 4

7 6 1 1

3 6 9

1 6 25

6 6 0 0

9 6 3 9

3 6 9

8 6 2 4

Standard deviation = SD = 2SD 2 = 26.60 = 2.57 Variance = SD 2 =

g(X – M )2

N =

SS N

= 66 10

= 6.60

©: 0

– 3

– 5 – 3

The standard deviation is the square root of the variance.

The standard deviation is the square root of the result of taking the sum of the squared deviations of the scores from the mean divided by the number of scores.

The standard deviation is the square root of the result of taking the sum of squares divided by the number of scores.

T I P F O R S U C C E S S When figuring the variance and standard deviation, lay your work out as in Tables 2–3 and 2–4. This helps you follow all the steps and end up with the correct answers.

T I P F O R S U C C E S S Always check that your answers make intuitive sense. For example, looking at the scores for the dreams example, a standard deviation— which, roughly speaking, repre- sents the average amount that the scores vary from the mean—of 2.57 makes sense. If your answer had been 21.23, however, it would mean that, on average, the number of dreams varied by more than 20 from the mean of 6. Looking at the group of scores, that just couldn’t be true.

T I P F O R S U C C E S S Notice in Table 2–3 that the devia- tion scores (shown in the third col- umn) add up to 0. The sum of the deviation scores is always 0 (or very close to 0, allowing for round- ing error). So, to check your figur- ing, always sum the deviation scores. If they do not add up to 0, do your figuring again!

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Table 2–4 Figuring the Variance and Standard Deviation for Number of Social Interactions During a Week for 94 College Students

Number of Interactions �

Mean Number of Interactions �

Deviation Score

Squared Deviation

Score

48 17.39 30.61 936.97

15 17.39 5.71

33 17.39 15.61 243.67

3 17.39 207.07

21 17.39 3.61 13.03

– – – –

35 17.39 17.61 310.11 9 17.39 70.39

30 17.39 12.61 159.01

8 17.39 88.17

26 17.39 8.61 74.13

12,406.44

Source: Data from McLaughlin-Volpe et al. (2001).

Standard deviation = 2SD 2 = 2131.98 = 11.49 Variance = SD 2 =

g(X – M )2

N =

12,406.44 94

= 131.98

©: 0.00 – 9.39

– 8.39

– 14.39

– 2.39

2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5

Number of Social Interactions in a Week

F re

qu en

^ ^ ^ ^ ^ ^ ^ ^ ^ ^

M 1 SD

FREQUENCY 12 16 16 16 10 11 4 3 3 3

INTERVAL 0 – 4 5 – 9

10 – 14 15 – 19 20 – 24 25 – 29 30 – 34 35 – 39 40 – 44 45 – 49

1 SD

Figure 2–13 The standard deviation as the distance along the base of a histogram, using the example of number of social interactions in a week. (Data from McLaughlin-Volpe et al., 2001.)

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T I P F O R S U C C E S S A common mistake when figuring the standard deviation is to jump straight from the sum of squared deviations to the standard devia- tion (by taking the square root of the sum of squared deviations). Remember, before finding the standard deviation, first figure the variance (by dividing the sum of squared deviations by the number of scores, N). Then take the square root of the variance to find the standard deviation.

The scores in the number of dreams example were 7, 8, 8, 7, 3, 1, 6, 9, 3, 8, and we figured the standard deviation of the scores to be 2.57. Now imagine that one addi- tional person is added to the study and that the person reports having 21 dreams in the past week. The standard deviation of the scores would now be 4.96, which is almost double the size of the standard deviation without this additional single score.

Computational and Definitional Formulas In actual research situations, psychologists must often figure the variance and the standard deviation for distributions with many scores, often involving decimals or large numbers. In the days before computers, this could make the whole process quite time-consuming, even with a calculator. To deal with this problem, in the old days researchers developed various shortcuts to simplify the figuring. A shortcut for- mula of this type is called a computational formula.

The traditional computational formula for the variance of the kind we are dis- cussing in this chapter is as follows:

(2–7)

means that you square each score and then take the sum of the squared scores. However, means that you first add up all the scores and then take the square of this sum. Although this sounds complicated, this formula was actually eas- ier to use than the one you learned before if a researcher was figuring the variance for a lot of numbers by hand or even with an old-fashioned handheld calculator, be- cause the researcher did not have to first find the deviation score for each score.

However, these days computational formulas are mainly of historical interest. They are used by researchers only on rare occasions when computers with statistics software are not readily available to do the figuring. In fact, today, even many hand- held calculators are set up so that you need only enter the scores and press a button or two to get the variance and the standard deviation.

In this book we give a few computational formulas just so that you have them if you someday do a research project with a lot of numbers and you don’t have access to statistical software. However, we very definitely recommend not using the computa- tional formulas when you are learning statistics, even if they might save you a few min- utes of figuring a practice problem. The computational formulas usually make it much harder to understand the meaning of what you are figuring. The only reason for figuring problems at all by hand when you are learning statistics is to reinforce the underlying principles. Thus, you would be undermining the whole point of the practice problems if you use a formula that had a complex relation to the basic logic. The formulas we give you for the practice problems and for all the examples in the book are designed to help strengthen your understanding of what the figuring means. Thus, the usual formula we give for each procedure is what statisticians call a definitional formula.

The Importance of Variability in Psychology Research Variability is an important topic in psychology research because much of the re- search focuses on explaining variability. We will use a couple of examples to show what we mean by “explaining variability.” As you might imagine, different students experience different levels of stress with regard to learning statistics: Some experi- ence little stress; for other students, learning statistics can be a source of great stress. So, in this example, explaining variability means identifying the factors that explain why students differ in the amount of stress they experience. Perhaps how much experience students have had with math explains some of the variability. That is,

(©X)2 ©X2

SD2 = gX2 – A AgX B 2 > N B

computational formula equation mathematically equivalent to the defini- tional formula. Easier to use for figuring by hand, it does not directly show the meaning of the procedure.

definitional formula equation for a statistical procedure directly showing the meaning of the procedure.

The variance is the sum of the squared scores minus the result of taking the sum of all the scores, squaring this sum and dividing by the number of scores, then taking this whole difference and dividing it by the number of scores.

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according to this explanation, the differences (the variability) among students in amount of stress are partially due to the differences (the variability) among students in the amount of experience they have had with math. Thus, the variation in math experience partially explains, or accounts for, the variation in stress. What factors might explain the variation in students’ number of weekly social interactions? Per- haps a factor is variation in the extraversion of students, with more extraverted stu- dents tending to have more interactions. Or perhaps it is variation in gender, with one gender having consistently more interactions than the other. Much of the rest of this book focuses on procedures for evaluating and testing whether variation in some specific factor (or factors) explains the variability in some variable of interest.

The Variance as the Sum of Squared Deviations Divided by Researchers often use a slightly different kind of variance. We have defined the vari- ance as the average of the squared deviation scores. Using that definition, you divide the sum of the squared deviation scores by the number of scores (that is, the variance is SS�N). But you will learn in Chapter 7 that for many purposes it is better to define the variance as the sum of squared deviation scores divided by 1 less than the number of scores. In other words, for those purposes the variance is the sum of squared deviations divided by (that is, variance is SS�[ ]). (As you will learn in Chapter 7, you use this dividing by approach when you have scores from a particular group of people and you want to estimate what the variance would be for the larger group of people whom these individuals represent.)

The variances and standard deviations given in research articles are usually fig- ured using SS�( ). Also, when calculators or computers give the variance or the standard deviation automatically, they are usually figured in this way (for exam- ple, see the Using SPSS section at the end of this chapter). But don’t worry. The ap- proach you are learning in this chapter of dividing by N (that is, figuring variance as SS�N) is entirely correct for our purpose here, which is to use descriptive statistics to describe the variation in a particular group of scores. It is also entirely correct for the material you learn in Chapters 3 through 6. We mention this other approach (vari- ance as SS�[ ]) now only so that you will not be confused if you read about variance or standard deviation in other places or if your calculator or a computer pro- gram gives a surprising result. To keep things simple, we wait to discuss the dividing by approach until it is needed, starting in Chapter 7. N – 1

N – 1

N – 1 N – 1N – 1

N � 1

How are you doing?

1. (a) Define the variance and (b) indicate what it tells you about a distribution and how this is different from what the mean tells you.

2. (a) Define the standard deviation; (b) describe its relation to the variance; and (c) explain what it tells you approximately about a group of scores.

3. Give the full formula for the variance and indicate what each of the symbols means.

4. Figure the (a) variance and (b) standard deviation for the following scores: 2, 4, 3, and 7 ( ).

5. Explain the difference between a definitional and a computational formula. 6. What is the difference between the formula for the variance you learned in this

chapter and the formula that is typically used to figure the variance in re- search articles?

M = 4

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Answers

1.(a) The variance is the average of the squared deviation of each score from the mean. (b) The variance tells you about how spread out the scores are (that is, their variability), while the mean tells you the central tendency of the distribution.

2.(a) The standard deviation is the square root of the average of the squared de- viations from the mean. (b) The standard deviation is the square root of the variance. (c) The standard deviation tells you approximately the average amount that scores differ from the mean.

3.is the variance.means the sum of what follows. Xis for the scores for the variable being studied. Mis the mean of the scores. Nis the number of scores.

4. (a)Variance:

(b) Standard deviation: 5.A definitional formula is the standard formula in the straightforward form that

shows the meaning of what the formula is figuring. A computational formula is a mathematically equivalent variation of the definitional formula, but the com- putational formula tends not to show the underlying meaning. Computational formulas were often used before computers were available and researchers had to do their figuring by hand with a lot of scores.

6.The formula for the variance in this chapter divides the sum of squares by the number of scores (that is, SS�N). The variance in research articles is usually figured by dividing the sum of squares by one less than the number of scores (that is, SS�[]). N-1

SD=2SD 2

=23.50=1.87. 14>4=3.50. (7-4)

2 ]>4=

SD 2 =[©(X-M)

2 ]>N=[(2-4)

2 +(4-4)

2 +(3-4)

2 +

=[©1X-M2 2 ]>N. SD

You are learning statistics for the fun of it, right? No? Or maybe so, after all. If you become a psychologist, at some time or other you will form a hypothesis, gather data, and analyze them. (Even if you plan a career as a psychother- apist or other mental health practitioner, you will proba- bly eventually wish to test an idea about the nature of your patients and their difficulties.) That hypothesis— your own original idea—and the data you gather to test it are going to be very important to you. Your heart may well be pounding with excitement as you analyze the data.

Consider some of the comments of social psycholo- gists we interviewed for our book The Heart of Social Psychology (Aron & Aron, 1989). Deborah Richardson, who studies interpersonal relationships, confided that her favorite part of being a social psychologist is looking at the statistical output of the computer analyses:

It’s like putting together a puzzle. . . . It’s a highly arous- ing, positive experience for me. I often go through periods

of euphoria. Even when the data don’t do what I want them to do . . . [there’s a] physiological response. . . . It’s exciting to see the numbers come off—Is it actually the way I thought it would be?—then thinking about the alternatives.

Harry Reis, former editor of the Journal of Personality and Social Psychology, sees his profession the same way:

By far the most rewarding part is when you get a new data set and start analyzing it and things pop out, partly a confirmation of what led you into the study in the first place, but then also other things. . . . “Why is that?” Trying to make sense of it. The kind of ideas that come from data. . . . I love analyzing data.

Bibb Latane, an eminent psychologist known for, among other things, his work on why people don’t al- ways intervene to help others who are in trouble, reports eagerly awaiting

B O X 2 – 1 The Sheer Joy (Yes, Joy) of Statistical Analysis

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Controversy: The Tyranny of the Mean Looking in the behavioral and social science research journals, you would think that statistical methods are their sole tool and language, but there have also been rebel- lions against the reign of statistics. We are most familiar with this issue in psychology, where the most unexpected opposition came from the leader of behaviorism, the school of psychology most dedicated to keeping the field strictly scientific.

Behaviorism opposed the study of inner states because inner events are impossi- ble to observe objectively. (Today most research psychologists claim to measure inner events indirectly but objectively.) Behaviorism’s most famous advocate, B. F. Skinner, was quite opposed to statistics. Skinner even said, “I would much rather see a graduate student in psychology taking a course in physical chemistry than in statistics. And I would include [before statistics] other sciences, even poetry, music, and art” (Evans, 1976, p. 93).

Skinner was constantly pointing to the information lost by averaging the results of a number of cases. For instance, Skinner (1956) cited the example of three overeating mice—one naturally obese, one poisoned with gold, and one whose hy- pothalamus had been altered. Each had a different curve for learning to press a bar for food. If these learning curves had been summed or merged statistically, the result would have represented no actual eating habits of any real mouse. As Skinner said, “These three individual curves contain more information than could probably ever be generated with measures requiring statistical treatment, yet they will be viewed with suspicion by many psychologists because they are single cases” (p. 232).

In clinical psychology and the study of personality, voices have always been raised in favor of the in-depth study of one person instead of or as well as the aver- aging of persons. The philosophical underpinnings of the in-depth study of individ- uals can be found in phenomenology, which began in Europe after World War I (Husserl, 1970). This viewpoint has been important throughout the social sciences, not just in psychology.

Today, the rebellion in psychology is led by qualitative research methodologists (e.g., McCracken, 1988), an approach that is much more prominent in other behav- ioral and social sciences. The qualitative research methods, developed mainly in an- thropology, can involve long interviews or observations of a few individuals. The highly skilled researcher decides, as the event is taking place, what is important to remember, record, and pursue through more questions or observations. The mind of the researcher is the main tool because, according to this approach, only that mind can find the important relationships among the many categories of events arising in the respondent’s speech.

Many who favor qualitative methods argue for a blend: First, discover the im- portant categories through a qualitative approach. Then, determine their incidence in

. . . the first glimmerings of what came out . . . [and] using them to shape what the next question should be . . . You need to use everything you’ve got, . . . every bit of your experience and intuition. It’s where you have the biggest effect, it’s the least routine. You’re in the room with the tiger, face to face with the core of what you are doing, at the moment of truth.

Bill Graziano, whose work integrates developmental, personality, and social psychology, calls the analysis of his data “great fun, just great fun.” And in the same vein, Margaret Clark, who studies emotion and cognition, de- clares that “the most fun of all is getting the data and looking at them.”

So you see? Statistics in the service of your own cre- ative ideas can be a pleasure indeed.

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the larger population through quantitative methods. Too often, these advocates argue, quantitative researchers jump to conclusions about a phenomenon without first exploring the human experience of it through free-response interviews or observations.

Finally, Carl Jung, founder of Jungian psychology, sometimes spoke of the “sta- tistical mood” and its effect on a person’s feeling of uniqueness. Jung had no prob- lem with statistics—he used them in his own research. He was concerned about the cultural impact of this “statistical mood”—much like the impact of being on a jammed subway and observing the hundreds of blank faces and feeling diminished, “one of a crowd.” He held that the important contributions to culture tend to come from people who feel unique and not ordinary. As we increasingly describe our- selves statistically—“90% of men under thirty think . . .”—we tend to do just that, think like 90% of men under thirty. To counteract this mood, Jungian analyst Marie Louise von Franz (1979) wrote, “An act of loyalty is required towards one’s own feelings” (pp. IV-18). Feeling “makes your life and your relationships and deeds feel unique and gives them a definite value” (pp. IV-18–IV-19). Your beloved is like no one else. Your own death is a face behind a door. And the meaning of ‘civilian deaths this month due to the war were 20,964’ is unfathomable horror—not a number.

In short, there have been many who have questioned an exclusively statistical view of our subject matter, and their voices should be considered too as you proceed with your study of what has become the predominant, but not exclusive, means of doing psychology research.

B O X 2 – 2 Gender, Ethnicity, and Math Performance From time to time, someone tries to argue that because some groups of people score better on math tests and make careers out of mathematics, these groups have a genetic advantage in math (or statistics). Other groups are said or implied to be innately inferior at math. The issue comes up about gender, about racial and ethnic groups, and of course in arguments about overall intelli- gence as well as math. There’s little evidence for such genetic differences (a must-see article is Block, 1995), but the stereotypes persist.

The impact of these stereotypes has been well estab- lished in research by Steele and his colleagues (1997), who have done numerous studies on what they call “stereo- type threat.” This phenomenon occurs when a negative stereotype about a group you belong to becomes relevant to you because of the situation you are in, like taking a math test, and provides an explanation for how you will behave. A typical experiment creating stereotype threat (Spencer et al., 1999) involved women taking a difficult math test. Half were told that men generally do better on the test, and the other half that women generally do equally well. Those who were told that women do worse did indeed score substantially lower than the other group. In the other condition, there was no difference. (In fact, in

two separate studies, men performed a little worse when they were told there was no gender difference, as if they had lost some of their confidence.)

The same results occur when African Americans are given parts of the graduate record examination. They do fine on the test when they are told no racial differences in the scores have been found, and they do worse when they are told that such differences have been found (Steele, 1997).

Stereotype threat has also been found to occur in the United States for Latinos (Gonzales et al., 2002) and the poor (Croizet & Claire, 1998). Many lines of research in- dicate that prejudices, not genetics, are the probable cause of differences in test scores between groups. Al- though some researchers (Rushton & Jensen, 2005) con- tinue to argue for genetic differences, the evidence is still substantial that stereotype threat plays the main role in lower test scores (Suzuki & Aronson, 2005). For exam- ple, the same difference of 15 IQ points between a domi- nant and minority group has been found all over the world, even when there is no genetic difference between the groups, and in cases where opportunities for a group have changed, such as when they emigrate, differences have rapidly disappeared (Block, 1995).

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If groups such as women and African Americans are not inherently inferior but perform worse on tests, what might be the reasons? The usual explanation is that they have internalized the “superior” group’s prejudices. Steele thinks the problem might not be so internal but may have to do with the situation. The stigmatized groups perform worse when they know that’s what is expected—when they experience the threat of being stereotyped. They either become too anxious or give up and avoid the subject.

What Can You Do for Yourself? So, do you feel you belong to a group that is expected to do poorly at math? (Perhaps the group of “math dumb- bells” in the class?) What can you do to get out from under the shadow of stereotype threat as you take this course?

First, care about learning statistics. Don’t discount it to save your self-esteem. Fight for your right to know this subject. Consider these words from the former presi- dent of the Mathematics Association of America:

The paradox of our times is that as mathematics becomes increasingly powerful, only the powerful seem to benefit from it. The ability to think mathematically—broadly interpreted—is absolutely crucial to advancement in vir- tually every career. Confidence in dealing with data, skepticism in analyzing arguments, persistence in pene- trating complex problems, and literacy in communicating about technical matters are the enabling arts offered by the new mathematical sciences. (Steen, 1987, p. xviii)

Second, once you care about succeeding at statistics, realize you are going to be affected by stereotype threat. Think of it as a stereotype-induced form of test anxiety and work on it that way (see Box 1–2).

Third, root out the effects of that stereotype in your- self as much as you can. It takes some effort. That’s why we are spending time on it here. Research on stereotypes shows that they can be activated without our awareness (Fiske, 1998), even when we are otherwise low in preju- dice or a member of the stereotyped group.

Some Points to Think About For women, yes, the very top performers tend to be male, but the differences are slight, and the lowest performers are not more likely to be female. Indeed, gender differ- ences on test performance have been declining (National Center for Education Statistics, 2001). Tobias (1982) cites numerous studies for why women might not make it to the very top in math. For example, in a study of students identified by a math talent search, it was found that few

parents arranged for their daughters to be coached before the talent exams. Sons were almost invariably coached. In another study, parents of mathematically gifted girls were not even aware of their daughters’ abilities, whereas par- ents of boys invariably were. In general, girls tend to avoid higher math classes, according to Tobias, because parents, peers, and even teachers often advise them against pursuing too much math. Indeed, mothers’ views of their child’s math abilities are strong predictors of their later performance (Bleeker & Jacobs, 2004). Girls fre- quently outperform boys in math, yet still greatly under- estimate their abilities (Heller & Ziegler, 1996). So, even though women are earning more PhDs in math than ever before, it is not surprising that math is the field with the highest dropout rate for women.

We checked the grades in our own introductory statis- tics classes and found no reliable difference for gender. More generally, Schram (1996) analyzed results of 13 in- dependent studies of performance in college statistics and found an overall average difference of almost exactly zero (the slight direction of difference favored females). Steele (1997) also found that the grades of African Americans, for example, rose substantially when they were enrolled in a transition-to-college program emphasizing that they were the cream of the crop and much was expected of them. Meanwhile, African American students at the same school who were enrolled in a remedial program for mi- norities received considerable attention, but their grades improved very little and many more of them dropped out of school than in the other group. Steele argues that the very idea of a remedial program exposed those students to a subtle stereotype threat.

Cognitive research on stereotype threat has demon- strated that it most affects math problems relying on long- term memory and spills over into subsequent tasks not normally affected by stereotype threat (Beilock et al., 2007).

Another point to ponder is a study cited by Tobias (1995) comparing students in Asia and the United States on an international mathematics test. The U.S. stu- dents were thoroughly outperformed, but more important was why: Interviews revealed that Asian students saw math as an ability fairly equally distributed among people and thought that differences in performance were due to hard work. Contrarily, U.S. students thought some people are just born better at math; so hard work matters little.

In short, our culture’s belief that “math just comes naturally to some people” could be holding you back. But then, doing well in this course may even be more sat- isfying for you than for others.

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Central Tendency and Variability in Research Articles The mean and the standard deviation are very commonly reported in research arti- cles. However, the mode, median, and variance are only occasionally reported. Sometimes the mean and standard deviation are included in the text of an article. For our dreams example, the researcher might write, “The mean number of dreams in the last week for the 10 students was 6.00 ( ).” Means and standard deviations are also often listed in tables, especially if a study includes several groups or several different variables. For example, Selwyn (2007) conducted a study of gender-related perceptions of information and communication technologies (such as games ma- chines, DVD players, and cell phones). The researcher asked 406 college students in Wales to rate 8 technologies in terms of their level of masculinity or femininity. The students rated each technology using a 7-point response scale, from for very fem- inine to for very masculine, with a midpoint of 0 for neither masculine or femi- nine. Table 2–5 (reproduced from Selwyn’s article) shows the mean, standard deviation, and variance of the students’ ratings of each technology. As the table shows, games machines were rated as being more masculine than feminine, and land- line telephones were rated as being slightly more feminine than masculine. Notice that Table 2–5 is one of those rare examples where the variance is shown (usually just the standard deviation is given). Overall, the table provides a useful summary of the descriptive results of the study. In another part of the study, Selwyn compared women’s and men’s perceptions of the masculinity or femininity of different aspects of computers and computing. We will describe those results in Chapter 8; so be sure to look out for them!

Another interesting example is shown in Table 2–6 (reproduced from Norcross et al., 2005). The table shows the application and enrollment statistics for psychology doctoral programs in the United States, broken down by area of psychology and by year (1973, 1979, 1992, and 2003). The table does not give standard deviations, but it does give both means and medians. For example, in 2003 the mean number of ap- plicants to doctoral counseling psychology programs was 71.0, but the median was only 59. This suggests that some programs had very high numbers of applicants that

+ 3 – 3

SD = 2.57

Table 2–5 Mean Scores for Each Technology

N Mean S.D. Variance

Games machine (e.g., Playstation) 403 1.92 1.00 .98

DVD Player 406 .44 .85 .73

Personal Computer (PC) 400 .36 .82 .68

Digital radio (DAB) 399 .34 .99 .98

Television set 406 .26 .78 .62

Radio 404 .81 .65

Mobile phone 399 .88 .77

Landline telephone 404 1.03 1.07

Note: Mean scores range from �3 (very feminine) to +3 (very masculine). The midpoint score of .0 denotes “neither masculine nor feminine.” Source: Selwyn, N. (2007). Hi-tech = guy-tech? An exploration of undergraduate students’ gendered perceptions of information and communication technologies. Sex Roles, 56, 525–536. Copyright © 2007. Reprinted by permission of Springer Science and Business Media.

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skewed the distribution. In fact, you can see from the table that for almost every kind of program and for both applications and enrollments, the means are typically higher than the medians. You may also be struck by just how competitive it is to get into doctoral programs in many areas of psychology. It is our experience that one of the factors that makes a lot of difference is doing well in statistics courses!

1. The mean is the most commonly used measure of central tendency of a distrib- ution of scores. The mean is the ordinary average—the sum of the scores divided by the number of scores. In symbols, .

2. Other, less commonly used ways of describing the central tendency of a distribu- tion of scores are the mode (the most common single value) and the median (the value of the middle score when all the scores are lined up from lowest to highest).

3. The variability of a group of scores can be described by the variance and the standard deviation.

4. The variance is the average of the squared deviation of each score from the mean. In symbols, . The sum of squared deviations,

, is also symbolized as SS. Thus . 5. The standard deviation is the square root of the variance. In symbols,

It is approximately the average amount that scores differ from the mean.

6. There have always been a few psychologists who have warned against statistical methodology because in the process of creating averages, knowledge about the individual case is lost.

7. Means and standard deviations are often given in research articles in the text or in tables.

M = (©X)>N

Summary

Figuring the Mean Find the mean for the following scores: 8, 6, 6, 9, 6, 5, 6, 2.

Answer You can figure the mean using the formula or the steps.

Using the formula: . Using the steps:

❶ Add up all the scores. . ❷ Divide this sum by the number of scores. .48> 8 = 6

8 + 6 + 6 + 9 + 6 + 5 + 6 + 2 = 48

Example Worked-Out Problems

Key Terms

central tendency (p. 34) mean (M) (pp. 34, 35) Σ (sum of ) (p. 35) X (p. 35) N (number of scores) (p. 36) mode (p. 37) median (p. 39)

outlier (p. 39) variance (SD2 ) (p. 44) deviation score (p. 44) squared deviation score (p. 44) sum of squared deviations (sum of

squares) (SS ) (pp. 44, 46) standard deviation (SD) (p. 45)

SD2 (p. 46) SD (p. 46) computational formula (p. 49) definitional formula (p. 49)

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Finding the Median Find the median for the following scores: 1, 7, 4, 2, 3, 6, 2, 9, 7.

Answer ❶ Line up all the scores from lowest to highest. 1, 2, 2, 3, 4, 6, 7, 7, 9. ❷ Figure how many scores there are to the middle score by adding 1 to the

number of scores and dividing by 2. There are 9 scores; so the middle score is the result of adding 1 to 9 and then dividing by 2, which is 5. The middle score is the fifth score.

❸ Count up to the middle score or scores. The fifth score from the bottom is 4; so the median is 4.

Figuring the Sum of Squares and the Variance Find the sum of squares and the variance for the following scores: 8, 6, 6, 9, 6, 5, 6, 2. (These are the same scores used in the previous example for the mean: .)

Answer You can figure the sum of squares and the variance using the formulas or the steps.

Using the formulas:

Table 2–7 shows the figuring, using the following steps:

❶ Subtract the mean from each score. ❷ Square each of these deviation scores. ❸ Add up the squared deviation scores. This gives the sum of squares (SS). ❹ Divide the sum of squared deviations by the number of scores. This gives the

variance (SD2).

SD2 = SS>N = 30> 8 = 3.75. = 30 = 4 + 0 + 0 + 9 + 0 + 1 + 0 + 16

= 22 + 02 + 02 + 32 + 02 + – 12 + 02 + – 42 + (9 – 6)2 + (6 – 6)2 + (5 – 6)2 + (6 – 6)2 + (2 – 6)2

M = 6

Table 2–7 Figuring for Example Worked-Out Problem for the Sum of Squares and Variance Using Steps

❶ ❷

Score Mean Deviation Squared Deviation

8 6 2 4

6 6 0 0

9 6 3 9

6 6 0 0

5 6 1

6 6 0 0

2 6 16

❸ ❹ Variance = 30/ 8 = 3.75

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Figuring the Standard Deviation Find the standard deviation for the following scores: 8, 6, 6, 9, 6, 5, 6, 2. (These are the same scores used above for the mean, sum of squares, and variance. .)

Answer You can figure the standard deviation using the formula or the steps.

Using the formula: . Using the steps:

❶ Figure the variance. The variance (from above) is 3.75. ❷ Take the square root. The square root of 3.75 is 1.94.

Outline for Writing Essays on Finding the Mean, Variance, and Standard Deviation

1. Explain that the mean is a measure of the central tendency of a group of scores. Mention that the mean is the ordinary average, that is, the sum of the scores divided by the number of scores.

2. Explain that the variance and standard deviation both measure the amount of variability (or spread) among a group of scores.

3. The variance is the average of each score’s squared difference from the mean. Describe the steps for figuring the variance.

4. Roughly speaking, the standard deviation is the average amount that scores dif- fer from the mean. Explain that the standard deviation is directly related to the variance and is figured by taking the square root of the variance.

SD = 2SD2 = 23.75 = 1.94

SD2 = 3.75

These problems involve figuring. Most real-life statistics problems are done on a computer with special statistical software. Even if you have such software, do these problems by hand to ingrain the method in your mind. To learn how to use a comput- er to solve statistics problems like those in this chapter, refer to the Using SPSS sec- tion at the end of this chapter and the Study Guide and Computer Workbook that accompanies this text.

All data are fictional unless an actual citation is given.

Set I (for Answers to Set I Problems, see pp. 674–675) 1. For the following scores, find the (a) mean, (b) median, (c) sum of squared

deviations, (d) variance, and (e) standard deviation:

32, 28, 24, 28, 28, 31, 35, 29, 26

2. For the following scores, find the (a) mean, (b) median, (c) sum of squared deviations, (d) variance, and (e) standard deviation:

6, 1, 4, 2, 3, 4, 6, 6

3. For the following scores, find the (a) mean, (b) median, (c) sum of squared deviations, (d) variance, and (e) standard deviation:

2.13, 6.01, 3.33, 5.78

Practice Problems

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4. Here are the noon temperatures (in degrees Celsius) in a particular Canadian city on Boxing Day (usually December 26) for the 10 years from 1998 through 2007: , , , 0, , , , , and . Describe the typical tem- perature and the amount of variation to a person who has never had a course in statistics. Give three ways of describing the representative temperature and two ways of describing its variation, explaining the differences and how you figured each. (You will learn more if you try to write your own answer first, before read- ing our answer at the back of the book.)

5. A researcher is studying the amygdala (a part of the brain involved in emotion). Six participants in a particular fMRI (brain scan) study are measured for the increase in activation of their amygdala while they are viewing pictures of violent scenes. The activation increases are .43, .32, .64, .21, .29, and .51. Figure the (a) mean and (b) standard deviation for these six activation increases. (c) Explain what you have done and what the results mean to a person who has never had a course in statistics.

6. Describe and explain the location of the mean, mode, and median for a normal curve.

7. A researcher studied the number of anxiety attacks recounted over a two-week period by 30 people in psychotherapy for an anxiety disorder. In an article de- scribing the results of the study, the researcher reports: “The mean number of anxiety attacks was 6.84 ( ).” Explain these results to a person who has never had a course in statistics.

8. In a study by Gonzaga et al. (2001), romantic couples answered questions about how much they loved their partner and also were videotaped while revealing something about themselves to their partner. The videotapes were later rated by trained judges for various signs of affiliation. Table 2–8 (reproduced from their Table 2) shows some of the results. Explain to a person who has never had a course in statistics the results for self-reported love for the partner and for the number of seconds “leaning toward the partner.”

Set II 9. (a) Describe and explain the difference between the mean, median, and mode. (b)

Make up an example (not in the book or in your lectures) in which the median would be the preferred measure of central tendency.

SD = 3.18

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Table 2–8 Mean Levels of Emotions and Cue Display in Study 1

Women ( ) Men ( )

Indicator M SD M SD

Emotion reports

Self-reported love 5.02 2.16 5.11 2.08

Partner-estimated love 4.85 2.13 4.58 2.20

Affiliation-cue display

Affirmative head nods 1.28 2.89 1.21 1.91

Duchenne smiles 4.45 5.24 5.78 5.59

Leaning toward partner 32.27 20.36 31.36 21.08

Gesticulation 0.13 0.40 0.25 0.77

Note: Emotions are rated on a scale of 0 (none) to 8 (extreme). Cue displays are shown as mean seconds displayed per 60 s. Source: Gonzaga, G. C., Keltner, D., Londahl, E. A., & Smith, M. D. (2001). Love and the commitment problem in romantic relationships and friendship. Journal of Personality and Social Psychology, 81, 247–262. Published by the American Psychological Association. Reprinted with permission.

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10. (a) Describe the variance and standard deviation. (b) Explain why the standard deviation is more often used as a descriptive statistic than the variance.

11. For the following scores, find the (a) mean, (b) median, (c) sum of squared deviations, (d) variance, and (e) standard deviation:

2, 2, 0, 5, 1, 4, 1, 3, 0, 0, 1, 4, 4, 0, 1, 4, 3, 4, 2, 1, 0

12. For the following scores, find the (a) mean, (b) median, (c) sum of squared deviations, (d) variance, and (e) standard deviation:

1,112; 1,245; 1,361; 1,372; 1,472

13. For the following scores, find the (a) mean, (b) median, (c) sum of squared deviations, (d) variance, and (e) standard deviation:

3.0, 3.4, 2.6, 3.3, 3.5, 3.2

14. For the following scores, find the (a) mean, (b) median, (c) sum of squared deviations, (d) variance, and (e) standard deviation:

8, –5, 7, –10, 5

15. Make up three sets of scores: (a) one with the mean greater than the median, (b) one with the median and the mean the same, and (c) one with the mode greater than the median. (Each made-up set of scores should include at least five scores.)

16. A psychologist interested in political behavior measured the square footage of the desks in the official office of four U.S. governors and of four chief executive offi- cers (CEOs) of major U.S. corporations. The figures for the governors were 44, 36, 52, and 40 square feet. The figures for the CEOs were 32, 60, 48, and 36 square feet. (a) Figure the means and standard deviations for the governors and for the CEOs. (b) Explain, to a person who has never had a course in statistics, what you have done. (c) Note the ways in which the means and standard deviations differ, and speculate on the possible meaning of these differences, presuming that they are representative of U.S. governors and large corporations’ CEOs in general.

17. A developmental psychologist studies the number of words that seven infants have learned at a particular age. The numbers are 10, 12, 8, 0, 3, 40, and 18. Fig- ure the (a) mean, (b) median, and (c) standard deviation for the number of words learned by these seven infants. (d) Explain what you have done and what the re- sults mean to a person who has never had a course in statistics.

18. Describe and explain the location of the mean, mode, and median of a distribu- tion of scores that is strongly skewed to the left.

19. You figure the variance of a distribution of scores to be – 4.26. Explain why your answer cannot be correct.

20. A study involves measuring the number of days absent from work for 216 em- ployees of a large company during the preceding year. As part of the results, the researcher reports, “The number of days absent during the preceding year ( ) was . . . .” Explain what is written in parentheses to a person who has never had a course in statistics.

21. Payne (2001) gave participants a computerized task in which they first see a face and then a picture of either a gun or a tool. The task was to press one button if it was a tool and a different one if it was a gun. Unknown to the participants while they were doing the study, the faces served as a “prime” (something that starts you thinking a particular way); half the time they were of a black person and half the time of a white person. Table 2–9 shows the means and standard de- viations for reaction times (the time to decide if the picture is of a gun or a tool) after either a black or white prime. (In Experiment 2, participants were told to

M = 9.21; SD = 7.34

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decide as fast as possible.) Explain the results to a person who has never had a course in statistics. (Be sure to explain some specific numbers as well as the general principle of the mean and standard deviation.)

Table 2–9 Mean Reaction Times (in Milliseconds) in Identifying Guns and Tools in Experiments 1 and 2

Prime

Black White

Target M SD M SD

Experiment 1

Gun 423 64 441 73

Tool 454 57 446 60

Experiment 2

Gun 299 28 295 31

Tool 307 29 304 29

Source: Payne, B. K. (2001). Prejudice and perception: The role of automatic and controlled processes in misperceiving a weapon. Journal of Personality and Social Psychology, 81, 181–192. Published by the American Psychological Association. Reprinted with permission.

Using SPSS

The U in the following steps indicates a mouse click. (We used SPSS version 15.0 to carry out these analyses. The steps and output may be slightly different for other versions of SPSS.)

Finding the Mean, Mode, and Median ❶ Enter the scores from your distribution in one column of the data window. ❷ U Analyze. ❸ U Descriptive statistics. ❹ U Frequencies. ➎ U on the variable for which you want to find the mean, mode, and median, and

then U the arrow. ➏ Statistics. ❼ U Mean, U Median, U Mode, U Continue. ❽ Optional: To instruct SPSS not to produce a frequency table, U the box labeled

Display frequency tables (this unchecks the box). ❾ U OK.

Practice the steps above by finding the mean, mode, and median for the number of dreams example at the start of the chapter (the scores are 7, 8, 8, 7, 3, 1, 6, 9, 3, 8). Your output window should look like Figure 2–14. (If you instructed SPSS not to show the frequency table, your output will show only the mean, median, and mode.)

Finding the Variance and Standard Deviation As mentioned earlier in the chapter, most calculators and computer software— including SPSS—calculate the variance and standard deviation using a formula that involves dividing by N – 1 instead of N. So, if you request the variance and standard

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deviation directly from SPSS (for example, by clicking Variance and Std. deviation in Step ❼), the answers provided by SPSS will be different from the answers in this chapter.5 The following steps show you how to use SPSS to figure the variance and standard deviation using the dividing-by-N method you learned in this chapter. It is easier to learn these steps using actual numbers; so we will use the number of dreams example again.

❶ Enter the scores from your distribution in one column of the data window (the scores are 7, 8, 8, 7, 3, 1, 6, 9, 3, 8). We will call this variable “dreams.”

❷ Find the mean of the scores by following the preceding steps for Finding the Mean, Mode, and Median. The mean of the dreams variable is 6.

❸ You are now going to create a new variable that shows each score’s squared deviation from the mean. U Transform, U Compute Variable. You could call the new variable any name you want, but we will call it “sqdev” (for “squared

Figure 2–14 Using SPSS to find the mean, median, and mode for the number of dreams example.

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deviation”). So, write sqdev in the box labeled Target Variable. You are now going to tell SPSS how to figure sqdev. In the box labeled Numeric Expression, write (dreams ) * (dreams ). (The asterisk is how you show multiplication in SPSS.) As you can see, this formula takes each score’s deviation score and multiplies it by itself to give the squared deviation score. Your Compute Variable window should look like Figure 2–15. U OK. You will see that a new variable called sqdev has been added to the data window (see Figure 2–16). The scores are the squared deviations of each score from the mean.

❹ As you learned in this chapter, the variance is figured by dividing the sum of the squared deviations by the number of scores. This is the same as taking the mean of the squared deviation scores. So, to find the variance of the dreams scores, follow the steps shown earlier to find the mean of the sqdev variable. This comes out to 6.60; so the variance of the dreams scores is 6.60.

➎ To find the standard deviation, use a calculator to find the square root of 6.60, which is 2.57.

If you were conducting an actual research study, you would most likely request the variance and standard deviation directly from SPSS. However, for our purpose in this chapter (describing the variation in a group of scores), the steps we just outlined are entirely appropriate.

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Figure 2–15 SPSS compute variable window for Step ❸ of finding the variance and standard deviation for the number of dreams example.

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Figure 2–16 SPSS data window after Step ❸ for finding the variance and standard deviation for the number of dreams example.

1. In more formal, mathematical statistics writing, the symbols can be more com- plex. This complexity allows formulas to handle intricate situations without confusion. However, in books on statistics for psychologists, even fairly ad- vanced texts, the symbols are kept simple. The simplified form rarely creates ambiguities in the kinds of statistical formulas psychologists use.

2. This section focuses on the variance and standard deviation as indicators of spread, or variability. Another way to describe the spread of a group of scores is in terms of the range—the highest score minus the lowest score. Suppose that in a particular class the oldest student is 39 years of age and the youngest is 19; the range is 20 (that is, ). Psychology researchers rarely use the range because it is such an imprecise way to describe the spread; it does not take into account how clumped together the scores are within the range.

3. Why don’t statisticians use the deviation scores themselves, make all deviations positive, and just use their average? In fact, the average of the deviation scores (treating all deviations as positive) has a formal name—the average deviation or mean deviation. This procedure was actually used in the past, and some psy- chologists have noted subtle advantages of the average deviation (Catanzaro & Taylor, 1996). However, the average deviation does not work out very well as part of more complicated statistical procedures.

39 – 19 = 20

Chapter Notes

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4. It is important to remember that the standard deviation is not exactly the average amount that scores differ from the mean. To be precise, the standard deviation is the square root of the average of scores’ squared deviations from the mean. This squaring, averaging, and then taking the square root gives a slightly different re- sult from simply averaging the scores’ deviations from the mean. Still, the result of this approach has technical advantages that outweigh the slight disadvantage of giving only an approximate description of the average variation from the mean (see Chapter Note 3).

5. Note that if you request the variance from SPSS, you can convert it to the vari- ance as we figure it in this chapter by multiplying the variance from SPSS by

(that is, the number of scores minus 1) and then dividing the result by N (the number of scores). (That is the variance as we are figuring it in this chapter � SPSS’s variance multiplied by [ ]�N.) Taking the square root of the resulting value will give you the standard deviation (using the formula you learned in this chapter). We use a slightly longer approach to figuring the variance and standard deviation in order to show you how to create new variables in SPSS.

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