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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.

CHAPTER 2

Central Tendency and Variability

Chapter Outline

C Central Tendency 34

C Variability 43

C Controversy: The Tyranny of the Mean 52

Central Tendency and Variability in Research Articles 55

O Summary 57

C Key Terms 57

C Example Worked-Out Problems 57 C Practice Problems 59

C Using SPSS 62

C Chapter Notes 65

As 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.

33

M = 6

1 2 3 4 5 6 7 8 9

Chapter 2

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

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

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 (1 + 1 + 2 + 2 + 2 + 3) 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 (3 + 3 + 5) 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).

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 and Variability 35

M = 6 M = 6

1 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7 8 9

M = 3.60 M = 6

1 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7

8 9

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

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:

M = EX (2-1) 4 The mean is the sum of the scores divided by the number of scores.

M is a symbol for the mean. An alternative symbol, X ("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 X 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.)

E, 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 X 1 and X 2 ).

EX 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: E X is 7 + 8 + 8 + 7 + 3 + 1 +6+9+3+ 8, which is 60.

Eh vit fii.aiiir44 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.

M mean.

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

X scores in the distribution of the variable X.

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,

Ex 60 M = = — = 6

N 10

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

X193 M– = = 6.43

N 30

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.

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.

N number of scores in a distribution.

0

6.43

Balance Point

0 1 2 3

Stress Rating

10

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.)

Central Tendency and Variability 37

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3

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Balance Point

Number of Social Interactions

in a Week

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.)

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:

EX 1,635

M = — = N 94

= 17.39

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.

O Add up all the scores. That is, figure F.X. • Divide this sum by the number of scores. That is, divide EX 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).

mode value with the greatest frequency in a distribution.

Mode = 8

1 2 3 4 5 6 7 8

38 Chapter 2

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.

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).

Mode = 8

Mean = 8.30

2 3 4 5 6 7 8 9 10 11

Mode = 8

Mean = 5.10

2 3 4 5 6 7 8 9 10

Mode = 8

Mean = 7

2 3 4 5 6 7 8 9 10

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.

Central Tendency and Variability 39

1 3 3 6 7 7 8 8 8 9

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.

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.

O Line up all the scores from lowest to highest. @ 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 25 11/2. Because there are no half scores, the 25th and 26th scores (the scores on either side of 251/2) are the middle scores.

• 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, (IX)/N = 5.52/5 = 1.1040. 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.

. _TIP FOR SUCCESS 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.

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

Men

Source: Data from Miller & Fishkin (1997).

0 Men % ■ Women %

Measures of central tendency Mean Median

Men

64.3 1 Women 2.8 1

40 Chapter 2

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

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Number of Partners Desired in the Next 30 Years

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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.

Mean Mode

Median

Central Tendency and Variability 41

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.

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.

Chapter 2

Table 2-2 Summary of Measures of Central Tendency

Measure Definition When Used

Mean

Sum of the scores divided by the number of scores

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

research

Mode

Median

Value with the greatest frequency in a distribution

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

• With nominal variables • Rarely used in psychology research

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

outliers • Rarely used in psychology research

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.

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?

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Central Tendency and Variability 43

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

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.

Neighborhood with Diverse Types of Housing

Housing Prices

Mean

Neighborhood with Consistent Type of Housing

Housing Mean Prices

Chapter 2

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.

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:

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

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

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

0 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.

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.

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

Central Tendency and Variability 45

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 29 — 29 = 0; 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 19 — 29 = —10. The 39-year-olds would have deviation scores of 39 — 29 = 10. All the squared deviation scores, which are either —10 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.

0 Figure the variance. (9 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 +3 and —3.)

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.

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.

SD 2 = 1(X M)2 (2-2) 4 The variance is the sum of the squared deviations of the scores from the mean, divided by the number of scores.

46 Chapter 2

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1 7

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1 2

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

SD 2 is the symbol for the variance. This may seem surprising. SD is short for standard deviation. The symbol SD 2 emphasizes that the variance is the standard devi- ation squared. (Later, you will learn other symbols for the variance, S 2 and o-2—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 (1) 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 E(X - M) 2 :

SD2 variance.

SD standard deviation.

sum of squares (SS) sum of squared deviations.

vim lor-o I 411s-M1 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.

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

SD 2 = SS N

(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.

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.

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.

11111 IF • MP 1 ./T 4 ir -g -SIM

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.

MI II -fog Awl rota -1 -C411. 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!

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

SD = VSD 2 (2-4)

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

SD = E(X - M) 2

N

or

SS SD = V ITT

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.

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

8 6 2 4

7 6 1 1

3 6 —3 9

1 6 —5 25

6 6 0 0

9 6 3 9

3 6 —3 9

8 6 2 4

2,:0 66

E (X —M) 2 SS 66 Variance = SD' = — — —

N 10 = 6.60

N

Standard deviation = SD = \/SD 2 = 16.60 = 2.57

(2-5)

(2-6)

16

15

1 4

13

12

11

10

9

8

7

6

5

4

3

2

2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5

48 Chapter 2

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 -2.39 5.71

33 17.39 15.61 243.67

3 17.39 -14.39 207.07

21 17.39 3.61 13.03

35 17.39 17.61 310.11

9 17.39 -8.39 70.39

30 17.39 12.61 159.01

8 17.39 -9.39 88.17

26 17.39 8.61 74.13

E: 0.00 12,406.44

SD 2 E (X - M) 2 12,406.44

Variance = SD` = N

= 94

- 131.98

Standard deviation = VSD 2 = V131.98 = 11.49

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

INTERVAL FREQUENCY 0 - 4 12 5 - 9 16

10 - 14 16 15 - 19 16 20 - 24 10 25 - 29 11 30 - 34 4 35 - 39 3 40 - 44 3 45 - 49 3

4 1 SD )• 1

1SD M

Number of Social Interactions in a Week

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.)

1111111711,111TINTITIMITIIIIIPP

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 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.

Central Tendency and Variability 49

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:

— ((EX) 2/N) SD2 = (2- 7)

EX 2 means that you square each score and then take the sum of the squared scores. However, ( X ) 2 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,

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.

Chapter 2

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.