Heiman G. Basic Statistics for the Behavioral Sciences

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68 CHAPTER 4 / Measures of Central Tendency: The Mean, Median, and Mode

skewed distribution, and the mean gets pulled away from where most of your grades are,

toward that low grade. What hurts is then telling someone your average because it’s mis-

leading: It sounds as if all of your grades are relatively low, even though you have only

that one zinger.

The mean is always pulled toward the tail of any skewed distribution because it must

balance the entire distribution. You can see this starting with the symmetrical distribu-

tion containing the scores 1, 2, 2, 2, and 3. The mean is 2, and this accurately describes

most scores. However, adding the score of 20 skews the sample. Now the mean is

pulled up to 5. But! Most of these scores are not at or near 5. As this illustrates,

although the mean is always at the mathematical center, in a skewed distribution, that

center is not where most of the scores are located.

The solution is to use the median to summarize a skewed distribution. Figure 4.6

shows the relative positions of the mean, median, and mode in skewed distributions. In

both graphs, the mean is pulled toward the extreme tail and is not where most scores

are located. The mode is toward the side away from the extreme tail and so the distri-

bution is not centered here either. Thus, of the three measures, the median most accu-

rately reflects the central tendency—the overall address—of a skewed distribution.

REMEMBER Use the mean to summarize normal distributions of interval or

ratio scores; use the median to summarize skewed distributions.

It is for the above reasons that the government uses the median to summarize such

skewed distributions as yearly income or the price of houses. For example, recent U.S.

census data indicated that the median income is slightly above $50,000. But a relatively

small number of corporate executives, movie stars, professional athletes, and the like

make millions! If we include these salaries and compute the average income, it is about

$68,000. However, this is misleading because most people do not earn at or near this

higher figure. Instead, the median is a better summary of this distribution.

Believe it or not, we’ve now covered the basic measures of central tendency. In sum,

the first step in summarizing data is to compute a measure of central tendency to

describe the score around which the distribution tends to be located.

■

Compute the mode with nominal data or with a distinctly bimodal distribution of

any type of scores.

■

Compute the median with ordinal scores or with a skewed distribution of interval or

ratio scores.

■

Compute the mean with a normal distribution of interval or ratio scores.

FIGURE 4.6

Measures of central

tendency for skewed

distributions

The vertical lines show the

relative positions of the

mean, median, and mode.

Negative skew

scores

High

. . . . . .

Middle Low

Positive skew

scores

Low . . . Middle . . . High

Mode

Median

Mean

Median

Most often the data in behavioral research are summarized using the mean. This is

because most often we measure variables using interval or ratio scores and most often

with such scores, “mother nature” produces a reasonably normal distribution. Because

the mean is used so extensively, we will delve further into its characteristics and uses in

the following sections.

TRANSFORMATIONS AND THE MEAN

Recall that we perform transformations to make scores easier to work with or to make

scores from different variables comparable. The simplest transformation is to add, sub-

tract, multiply, or divide each score by a constant. This brings up the burning question:

How do transformations affect the mean?

If we add a constant to each raw score in a sample, the new mean of the trans-

formed scores will equal the old mean of the raw scores plus . For example, the scores

7, 8, and 9 have a mean of 8. Adding 5 to each score produces 12, 13, and 14. The new

mean is 13. The old mean of 8 plus the constant 5 also equals 13. Thus, the rule is that

new . The same logic applies for other mathematical operations.

When subtracting K from each score, new . When multiplying each

score by ,new . When dividing each score by ,new .

The above rules also apply to the median and to the mode. In essence, using a con-

stant merely changes the location of each score on the variable by points, so we also

move the “address” of the distribution by the same amount.

DEVIATIONS AROUND THE MEAN

To understand why the mean is used so extensively in statistics and research, you must

first understand why the mean is the center of a distribution. The mean is the center

score because it is just as far from the scores above it as it is from the scores below it.

That is, the total distance that some scores lie above the mean equals the total distance

that other scores lie below the mean.

The distance separating a score from the mean is called the score’s deviation, indi-

cating the amount the score “deviates” from the mean. A score’s deviation is equal to

the score minus the mean, or in symbols:

X 5 1old X2>KKX 5 1old X2KK

X 5 old X – K

X 5 old X 1 K

1K2

Deviations Around the Mean 69

The formula for computing a score’s deviation is

X – X

Thus, if the sample mean is 47, a score of 50 deviates by , because .

A score of 40 deviates from the mean of 47 by , because .

REMEMBER Always subtract the mean from the raw score when computing

a score’s deviation.

Do not, however, think of deviations as simply positive or negative numbers. Think

of a deviation as having two components: The number, which indicates distance from

40 2 47 52727

50 2 47 is 1 313

70 CHAPTER 4 / Measures of Central Tendency: The Mean, Median, and Mode

the mean (which is always positive), and the sign, which indicates direction from the

mean. For example, the table in Figure 4.7 shows that we computed the deviation

scores for our previous creativity scores by subtracting the mean of 4 from each score.

In the polygon in Figure 4.7 the axis is labeled twice, once using each creativity score

and, underneath, using its corresponding deviation score. Thus, you can see that a posi-

tive deviation indicates that the raw score is larger than the mean and graphed to the

right of the mean. A negative deviation indicates that the score is less than the mean

and graphed to the left of the mean. The size of the deviation (regardless of its sign) in-

dicates the distance the raw score lies from the mean: the larger the deviation, the

farther into the tail the score is from the mean. A deviation of 0 indicates that an indi-

vidual’s raw score equals the mean.

Now you can see why the mean is the mathematical center of any distribution. If we

add together all of the positive deviations we have the total distance that some scores are

above the mean. Adding all of the negative deviations, we have the total distance that

other scores are below the mean. If we add all of the positive and negative deviations

together, we have what is called the sum of the deviations around the mean. As in the

table in Figure 4.7, the total of the positive deviations will always equal the total of the

negative deviations. Therefore:

The sum of the deviations around the mean always equals zero.

Thus, the mean is the center of a distribution because, in total, it is an equal distance

form the scores above and below it. Therefore, the half of the distribution that is below

the mean balances with the half of the distribution that is above the mean.

The sum of the deviations around the mean always equals zero, regardless of the

shape of the distribution. For example, in the skewed sample of 4, 5, 7 and 20, the mean

is 9, which produces deviations of , , , and , respectively. Their sum is

again zero.

Some of the formulas you will see in later chapters involve something similar

to computing the sum of the deviations around the mean. The statistical code for

finding the sum of the deviations around the mean is . Al-

ways work inside parentheses first, so this says to first find the devia-

tion for each score, The indicates to then find the sum

of the deviations. Thus, say we have the scores 3, 4, 6, and 7. With

, then , which equals

zero.

REMEMBER is the symbol for the sum of the devia-

tions around the mean.

Σ1 X 2 X

1 1 1 2Σ1X 2 X2 5 22 1 21X 5 5

Σ1X 2 X2.

Σ1X 2 X2

111222425

Score Minus Score Equals Deviation

Sum 05

1352

1252

1152

052

2152

2252

2352

FIGURE 4.7

Deviations Around the Mean

The mean is subtracted from each score,

resulting in the score’s deviation.

Mean

Creativity raw scores

Deviations

12345678

–3

–2

–1 0 +1 +2 +3

The importance of the sum of the deviations equaling zero is that this makes the

mean the score that, literally, everyone in the sample scored around, with scores above

or below it to the same extent. Therefore, we think of the mean as the typical score be-

cause it more or less describes everyone’s score, with the same amounts of more and

less. This is why the mean is such a useful tool for summarizing a distribution. It is also

why we use the mean when we are predicting any individual scores.

Using the Mean to Predict Scores

Recall that a goal of behavioral science is to predict a behavior in a particular situation.

This translates into predicting the scores found in that situation. When we don’t know

anything else, the mean is our best prediction about the score that any individual ob-

tains. Because it is the central, typical score, we act as if all the scores were the mean

score, and so we predict that score every time. This is why, if the class average on an

exam is 80, you would predict that each student’s grade is 80. Further, for any students

who were absent, you’d predict that they will score an 80 as well. Likewise, if your

friend has a B average in college, you would predict that he or she received a B in every

course. For any future course, you’d also predict a B.

However, not every score in a sample will equal the mean, so our predictions will

sometimes be wrong. To measure the amount of our error when predicting unknown

scores, we measure how well we can predict the known scores in our data. The amount

of error in any single prediction is the difference between what someone actually gets

and what we predict he or she gets . In symbols, this difference is . We’ve

seen that this is called a deviation, but alter your perspective here: In this context, a de-

viation is the amount of error we have when we predict the mean as someone’s score.

REMEMBER When we use the mean to predict scores, a deviation

indicates our error: the difference between the we predict for someone and

the that he or she actually gets.

If we determine the amount of error in every prediction, our total error is equal to the

sum of the deviations. As we’ve seen, in any data the sum of the deviations is always

zero. Thus, by predicting the mean score every time, the errors in our predictions will,

over the long run, cancel out to equal zero. For example, the test scores 70, 75, 85, and

90 have a of 80. One student scored the 70, but we would predict he scored 80, so we

would be wrong by . But, another student scored the 90; by predicting an 80 for

her, we would be off by In the same way, our errors for the sample will cancel out

so that the total error is zero. Likewise, we assume that other participants will behave

similarly to those in our sample, so that using the mean to predict any unknown scores

should also result in a total error of zero.

If we predict any score other than the mean, the total error will be greater than zero.

A total error of zero means that, over the long run, we overestimate by the same amount

that we underestimate. A basic rule of statistics is that if we can’t perfectly describe

every score, then the next best thing is to have our errors balance out. There is an old

joke about two statisticians shooting at a target. One hits 1 foot to the left of the target,

and the other hits 1 foot to the right. “Congratulations,” one says. “We got it!” Like-

wise, if we cannot perfectly describe every score, then we want our errors—our over-

and underestimates—to balance out to zero. Only the mean provides this capability.

Of course, although our total error will equal zero, any individual prediction may be

very wrong. Later chapters will discuss how to reduce these errors. For now, however,

110.

210

1X 2 X2

X 2 X1X21X2

Deviations Around the Mean 71

72 CHAPTER 4 / Measures of Central Tendency: The Mean, Median, and Mode

DESCRIBING THE POPULATION MEAN

Recall that ultimately we seek to describe the population of scores we would find in a

given situation. Populations are unwieldy, so we must also summarize them. Usually

we have interval or ratio scores that form at least an approximately normal distribution,

so we usually describe the population using the mean.

The symbol for the mean of a population is the Greek letter (pronounced “mew”).

Thus, to indicate that the population mean is 143, we’d say that . The symbol

simply shows that we’re talking about a population instead of a sample, but a mean

is a mean, so a population mean has the same characteristics as a sample mean: is the

average score in the population, it is the center of the distribution, and the sum of the

deviations around equals zero. Thus, is the score around which everyone in the

population scored, it is the typical score, and it is the score that we predict for any indi-

vidual in the population.

REMEMBER The symbol for a population mean is .␮

␮␮

␮

␮ 5 143

␮

A QUICK REVIEW

■

A deviation is the difference between a score and

the mean (or ) and indicates a score’s loca-

tion relative to the mean.

■

When using the mean to predict a score, the score’s

deviation indicates the amount of error in the

prediction. The total error over all such predictions

is , the sum of the deviations around the

mean, which equals zero.

MORE EXAMPLES

In some data, . Therefore, scores above 30

will produce positive deviations which will cancel out

with the negative deviations from the scores below the

mean, so that the sum of the deviations equals zero.

For every participant, we’d predict a score of 30. Over

the long run, our over- and under-estimates will can-

cel out. In symbols, each error is the difference be-

tween the actual and the that we predict, so theX

X 5 30

Σ1X 2 X2

1X 2 X2

total error is the sum of the deviations, ,

which is zero.

For Practice

1. By performing , you are computing a ____.

2. By performing , you are computing ____.

3. By saying that , you are saying

that the mean is located ____ relative to the scores

in a sample.

4. By saying that , you are saying

that when predicting someone’s score is the mean,

our errors ____.

Answers

1. deviation

2. the sum of the deviations around the mean

3. in the center of the distribution

4. balance out to equal a total of zero

Σ1X 2 X2 5 0

Σ1X 2 X2

X 2 X

Σ1X 2 X2

remember that unless we have additional information about the scores, the mean is our

best prediction of any score.

REMEMBER We use the mean as everyone’s predicted score because, over

the long run, the over- and underestimates will cancel out so that the total er-

ror in our predictions equals zero.

How do we determine the value of If all the scores in the population are known to

us, we compute using the same formula that we used to compute :

Usually, however, a population is infinitely large, so we cannot directly compute .

Instead, we estimate based on the mean of a random sample. If, for example, a sam-

ple’s mean in a particular situation is 99, then, assuming the sample accurately repre-

sents the population, we would estimate that in this situation is also 99. We make

such an inference because it is a population with a mean of 99 that is most likely to pro-

duce a sample with a mean of 99. That is, a of 99 indicates a sample with mostly

scores around 99 in it. What population would be most likely to provide these scores?

The population containing mostly scores around 99—where the population mean is 99.

Thus, we assume that most scores in a sample are located where most scores in the pop-

ulation are located, so a sample mean is a good estimate of .

SUMMARIZING RESEARCH

Now you can understand how measures of central tendency are used in research. Usu-

ally we compute the mean because we have normally distributed interval or ratio

scores. Thus, we might compute the mean number of times our participants exhibit a

particular behavior, or compute their mean response to a survey question. In a correla-

tional study we compute their mean score on the variable and the mean score on the

variable. Using such means, we can describe the typical score and predict the scores

of other individuals, including those of the entire population.

Instead we might compute the median or the mode if we have other types of scores

or distributions. When considering the shape of the distribution, we are usually con-

cerned with the shape of distribution for the population, because ultimately that is what

we want to describe. How do we know its shape? The first step in conducting a study is

to read relevant published research reports. From these you will learn many things, in-

cluding what other researchers say about the population and how they compute central

tendency.

Summarizing an Experiment

We perform similar steps when summarizing the results of an experiment. Remember

though that the results that need summarizing are the scores from the dependent vari-

able. Therefore, it is the characteristics of the dependent scores that determine whether

we compute the mean, median, or mode.

REMEMBER The measure of central tendency to compute in an experiment

is determined by the type of scale used to measure the dependent variable.

Usually it is appropriate to compute the mean, and we do so for each condition of

the independent variable. For example, here is a very simple study. Say that we think

people will make more mistakes when recalling a long list of words than when recall-

ing a short list. We create three conditions of the independent variable of list length. In

one condition, participants read a list containing 5 words and then recall it. In another

condition, participants read a 10-item list and recall it, and in a third condition, they

␮

␮ 5

ΣX

X␮

␮?

Summarizing Research 73

74 CHAPTER 4 / Measures of Central Tendency: The Mean, Median, and Mode

read a 15-item list and recall it. For each participant, we measure the dependent vari-

able of number of errors made in recalling the list.

First, look at the individual scores shown in Table 4.1. A relationship is present here

because a different and higher set of error scores occurs in each condition. Most experi-

ments involve a much larger , however, so to see the relationship buried in the raw

scores, we compute a measure of central tendency.

In our memory experiment, the variable of recall errors is a ratio variable that is as-

sumed to form an approximately normal distribution. Therefore, we compute the mean

score in each condition by computing the mean of the scores in each column. These

are shown under each column in Table 4.1.

Normally in a research report you will not see the individual raw scores. There-

fore, to interpret the mean in any study, simply envision the scores that would typi-

cally produce such a mean. For example, when , envision a normal distribution

of scores above and below 3, with most scores close to 3. Likewise, for each mean,

essentially envision the kinds of raw scores shown in our columns. Thus, the means

show that recalling a 5-item list resulted in one distribution located around three er-

rors, but recalling a 10-item list produced a different distribution at around six errors,

and recalling a 15-item list produced still another distribution at around nine errors.

Further, we use the mean score to describe the individual scores in each condition.

In Condition 1, for example, we’d predict that any participant would make about

three errors.

Most important is the fact that, by looking at the means alone, we see that a rela-

tionship is present here: as the conditions change (from 5 to 10 to 15 items in a list),

the scores on the dependent variable also change (from around 3, to around 6, to

around 9 errors, respectively).

Note, however, that not all means must change for a relationship to be present. For

example, we might find that only the mean in the 5-item condition is different from

the mean in the 15-item condition. We still have a relationship if, at least sometimes,

as the conditions of the independent variable change, the dependent scores also

change.

REMEMBER We summarize an experiment usually by computing the mean

of the dependent scores in each condition. A relationship is present if the

means from two or more conditions are different.

The above logic also applies to the median or mode. For example, say that we study

political party affiliation as a function of a person’s year in college. Our dependent

variable is political party, a nominal variable, so the mode is the appropriate measure

of central tendency. We might see that freshmen most often claim to be Republican,

but the mode for sophomores is Democrat; for juniors, Socialist; and for seniors,

Communist. These data reflect a relationship because they indicate that as college

level changes, political affiliation tends to change. Likewise, say that the median

income for freshmen is lower than

the median income for sophomores,

which is lower than for upperclass-

men. This tells us that the location

of the distribution of incomes is dif-

ferent for each class, so we know

that the income “scores” of individ-

uals are changing as their year in

college changes.

5 3

Condition 1: Condition 2: Condition 3:

5-Item List 10-Item List 15-Item List

25 7

36 9

4711

5 9X 5 6X 5 3

TABLE 4.1

Errors Made by

Participants Recalling a

5-, 10-, or 15-Item List

The mean of each condition

is under each column.

Graphing the Results of an Experiment

Recall that the independent variable involves the conditions “given” to participants so

it is plotted on the axis. The dependent variable is plotted on the axis. However, be-

cause we want to summarize the data, usually we do not plot the individual scores.

Rather, we plot either the mean, median, or mode of the dependent scores from each

condition.

REMEMBER Always plot the conditions of the independent variable on the

axis and the mean, median, or mode of the dependent scores on the axis.

Note: Do not be confused by the fact that we use to represent the scores when

computing the means. We still plot them on the axis.

We complete the graph by creating either a line graph or a bar graph. The type of

graph to select is determined by the characteristics of the independent variable.

Line Graphs Create a line graph when the independent variable is an interval or a

ratio variable. In a line graph adjacent data points are connected with straight lines.

For example, our independent variable of list length involves a ratio scale. Therefore,

we create the line graph shown on the left in Figure 4.8. Note that the label on the

axis is “mean recall errors.” Then we place a data point above the 5-item condition

opposite the mean of 3 errors, another data point above the 10-item condition at the

mean of 6 errors, and so on. We use straight lines to connect the data points here for the

same reason we did when producing polygons: Anytime the variable on the axis in-

volves an interval or ratio scale, we assume that it is a continuous variable and there-

fore we draw lines. The lines show that the relationship continues between the points

shown on the axis. For example, we assume that if there had been a 6-item list, the

mean error score would fall on the line connecting the means for the 5- and 10-item lists.

(The APA publication guidelines do not recommend using histograms to summarize

experiments.)

Summarizing Research 75

FIGURE 4.8

Line graphs showing (A) the relationship for mean errors in recall as a function of list length and (B) the data points we

envision around each mean

Length of list

Mean recall errors

51015

Length of list

Recall errors

51015

is envisioned as

76 CHAPTER 4 / Measures of Central Tendency: The Mean, Median, and Mode

The previous Figure 4.8 conveys the same information as the means did back in

Table 4.3, where a different mean indicates a different distribution of scores. We envi-

sion these distributions as shown in the right-hand graph. Each mean implies a sample

of scores and their corresponding data points are around—above and below—the

mean’s data point. Because the vertical positions of the means change as the conditions

change, we know that the raw scores also change, so a relationship is present.

Notice that you can easily spot such a relationship because the different means pro-

duce a line graph that is not horizontal. On any graph, if the summary data points form

a line that is not horizontal, it indicates that the individual Y scores are changing as

the X scores change, so a relationship is present. On the other hand, say that each con-

dition had produced a mean of 5 errors. As shown below on the left in Figure 4.9, this

results in a horizontal (flat) line, indicating that as list length changes, the mean error

score stays the same. This implies that (as in the figure on the right) the individual

scores stay the same regardless of the condition, so no relationship is present. Thus, on

any graph, if the summary data points form a horizontal line, it indicates that the indi-

vidual Y scores do not change as the X scores change, and so a relationship is not

present.

Bar Graphs Create a bar graph when the independent variable is a nominal or an

ordinal variable. Notice that the rule here is the same as it was in Chapter 3: Create a

bar graph whenever the X scores are nominal or ordinal scores. With experiments, we

place a bar above each condition on the axis to the height on the axis that corre-

sponds to the mean for that condition. As usual, adjacent bars do not touch: Recall that

nominal or ordinal scales are discrete, meaning that you can have one score or the

other, but nothing in-between. The spaces between the bars communicate this.

For example, say that we compared the recall errors made by psychology majors,

English majors, and physics majors. The independent variable of college major is a

nominal variable, so we have the bar graph shown in Figure 4.10. Because the tops of

the bars do not form a horizontal line, we know that different means and thus different

scores occur in each condition. We can again envision that we would see individual er-

ror scores at around 8 for physics majors, around 4 for psychology majors, and around

12 for English majors. Thus, the scores change as a function of college major, so a rela-

tionship is present.

FIGURE 4.9

Line graphs showing (A) no relationship for mean errors in recall as a function of list length and (B) the data points we

envision around each mean

Length of list

Mean recall errors

51015 0

Length of list

Recall errors

51015

is envisioned as

Summarizing Research 77

FIGURE 4.10

Bar graph showing mean

errors in recall as a func-

tion of college major

The height of each bar corre-

sponds to the mean score for

the condition.

College major

PsychologyPhysics

English

Mean recall errors

A QUICK REVIEW

■

Graph the independent variable on the axis and

the mean, median, or mode of the dependent scores

on the Y axis.

■

Create a line graph when the independent variable

is interval or ratio; produce a bar graph when it is

nominal or ordinal.

MORE EXAMPLES

We ask males and females to rate their satisfaction

with an instructor and the mean scores are 20 and 30,

respectively. To graph this, gender is a nominal inde-

pendent variable, so plot a bar graph, with the labels

“male” and “female” on and the means for each on

. Instead, say we measure the satisfaction scores of

students tested with either a 10-, 40-, or 80-question

final exam and, because the scores form skewed dis-

tributions, compute the median scores. Test length is a

ratio independent variable, so plot a line graph, with

the labels 10, 40, and 80 on and the median of each

condition on .Y

For Practice

1. The independent variable is plotted on the ____

axis, and the dependent variable is plotted on

the ____ axis.

2. A ____ shows a data point above each , with

adjacent data points connected with straight lines.

3. A ____ shows a discrete bar above each .

4. The characteristics of the ____ variable determine

whether to plot a line or bar graph.

5. Create a bar graph with ____ or ____ variables.

6. Create a line graph with ____ or ____ variables.

Answers

2. line graph

3. bar graph

4. independent

5. nominal; ordinal

6. interval; ratio

X; Y

Note: In a different experiment we might have measured a nominal or an ordinal

dependent variable. In that case we would plot the mode or median on the axis for

each condition. Then, again depending on the characteristics of the independent vari-

able, we would create either a line or bar graph.

REMEMBER The scale of measurement of the dependent variable determines

the measure of central tendency to calculate. The scale of the independent

variable determines the type of graph to create.