Heiman G. Basic Statistics for the Behavioral Sciences

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118 CHAPTER 6 / z-Scores and the Normal Curve Model

negative z-scores make up 50% of any distribution. Thus, the negative z-scores back

in Figure 6.3 constitute 50% of their respective distributions, which corresponds to a

relative frequency of . On any normal z-distribution, the relative frequency of the

negative z-scores is .

Having determined the relative frequency of the z-scores, we work backwards

to identify the corresponding raw scores. In the statistics distribution in Figure 6.3,

students having negative z-scores have raw scores ranging between 15 and 30, so the

relative frequency of scores between 15 and 30 is . In the English distribution,

students having negative z-scores have raw scores between 10 and 40, so the relative

frequency of these scores is .

Similarly, approximately 68% of the scores always fall between the z-scores of

and . Thus, in Figure 6.3, students having z-scores between constitute approxi-

mately 68% of each distribution. Working backwards to the raw scores we see that sta-

tistics grades between 25 and 35 constitute approximately 68% of the statistics

distribution, and English grades between 30 and 50 constitute approximately 68% of

the English distribution.

In the same way, we can determine the relative frequencies for any set of scores.

Thus, in a normal distribution of IQ scores (whatever the and may be), we know

that those IQs producing negative z-scores have a relative frequency of .50, and that

about 68% of all IQ scores will fall between the scores at z-scores of . The same will

be true for a distribution of running speeds, a distribution of personality test scores, or

for any normal distribution.

We can also use z-scores to determine the relative frequency of scores in any other

portion of a distribution. To do so, we employ the standard normal curve.

The Standard Normal Curve

Because all normal z-distributions are similar, we don’t need to draw a different

z-distribution for every set of raw scores. Instead, we envision one standard curve that,

in fact, is called the standard normal curve. The standard normal curve is a perfect

normal z-distribution that serves as our model of any approximately normal z-distribu-

tion. It is used in this way: Most data produce only an approximately normal distribu-

tion, producing a roughly normal z-distribution. However, to simplify things, we

operate as if the z-distribution always fits one, perfect normal curve, which is the stan-

dard normal curve. We use this curve to first determine the relative frequency of partic-

ular z-scores. Then, as we did above, we work backwards to determine the relative

frequency of the corresponding raw scores. This is the relative frequency we would

expect, if our data formed a perfect normal distribution. Usually, this provides a reason-

ably accurate description of our data, although how accurate we are depends on how

closely the data conform to the true normal curve. Therefore, the standard normal curve

is most accurate when (1) we have a large sample (or population) of (2) interval or ratio

scores that (3) come close to forming a normal distribution.

The first step is to find the relative frequency of the z-scores and for that we look at

the area under the standard normal curve. Statisticians have already determined the pro-

portion of the area under various parts of the normal curve, as shown in Figure 6.4. The

numbers above the axis indicate the proportion of the total area between the z-scores.

The numbers below the axis indicate the proportion of the total area between the

mean and the z-score. (Don’t worry, you won’t need to memorize them.)

Each proportion is also the relative frequency of the z-scores—and raw scores—

located in that part of the curve. For example, between a z of 0 and a z of 1 is .34131

;121

.50

Using z-Scores to Determine the Relative Frequency of Raw Scores 119

Mean

.50

.0215 .1359 .3413 .3413 .1359 .0215 .0013.0013

–3 –2 –1

0+1+2+3

.4987 .4987

.4772.4772

.3413 .3413

.50

z-scores

FIGURE 6.4

Proportions of total

area under the standard

normal curve

The curve is symmetrical:

50% of the scores fall below

the mean, and 50% fall above

the mean.

(or 34.13%) of the area, so, as we’ve seen, about 34% of the scores are here. Or, scores

between a z of and a z of occur of the time, and this added to , gives

a total of 0.4772 of the scores between the mean and . Finally, with .3413 of the

scores between the mean and and of the scores between the mean and

, a total of of the area under the curve is between . (See, approxi-

mately 68% of the distribution really is between from the mean!)

We can also add together nonadjacent portions of the curve. For example, out in

the lower tail beyond is of the area under the curve (because

. Likewise, in the upper tail beyond is also

of the area under the curve. Thus, a total of .0456 (or 4.56%) of all scores fall in

the tails beyond .

Finally, notice that z-scores beyond or beyond occur only .0013 of the time,

for a total of .0026 (.26 of 1%!), which is why the range of z is essentially between .

This also explains why Chapter 5 said that the should be about one-sixth of the

range of the raw scores. The range is roughly between zs of , a distance of six times

the standard deviation. If the range is six times the standard deviation, then the stan-

dard deviation is one-sixth of the range.

REMEMBER For any approximately normal distribution, transform the raw

scores to z-scores and use the standard normal curve to find the relative fre-

quency of the scores.

We usually apply the standard normal curve in one of four ways. First, we can find

relative frequency. Most often we begin with a particular raw score in mind and then

compute its z-score (using our original z-score formula). For example, in our original

attractiveness scores, say that another man, Cubby, has a raw score of 80, which, with

and , is a z of . We can envision Cubby’s location as in Figure 6.5.

We might first ask what proportion of scores are expected to fall between the mean and

Cubby’s score. We see that of the total area falls between the mean and .

Therefore, we expect or 47.72%, of the attractiveness scores to fall between the

mean and Cubby’s score of 80. (Conversely, of the area under the curve—and

scores—are above his score.)

.0228

.4772

z 512.4772

12S

5 10X 5 60

2313

z 5 ;2

.0228z 512.0215 1 .0013 5 .02282

.0228z 522

;1S

;1.6826z 511

.3413z 521

z 512

.3413.13591211

120 CHAPTER 6 / z-Scores and the Normal Curve Model

Mean

.50

Cubby

.4772 .0228.50

.9772

.50

z-scores

Raw scores

–3 +1 +2 +3–2 –1

30 70 80

35 45 55 65 75

40 50

FIGURE 6.5

Location of Cubby’s score

on the z-distribution of

attractiveness scores

Cubby’s raw score of 80

is a z-score of .12

Second, we can find simple frequency. We might ask how many people scored

between the mean and Cubby’s score. Then we would convert the above relative fre-

quency to simple frequency by multiplying the N of the sample times the relative fre-

quency. Say that our was 1000. If we expect of the scores to fall here, then

, so we expect about 477 people to have scores between the

mean and Cubby’s score.

Third, we can find a raw score’s percentile. Recall that a percentile is the percent of

all scores below—graphed to the left of—a score. After computing the z-score for a raw

score, first see if it is above or below the mean. As in Figure 6.5, the mean is also the

median (the 50th percentile). A positive z-score is above the mean, so Cubby’s z-score

of is above the 50th percentile. In fact, his score is above the .4772 that fall between

the mean and his score. Thus, adding the .50 of the scores below the mean to the .4772

of the scores between the mean and his score gives a total of .9772 of all scores below

Cubby’s. We usually round off percentile to a whole number, so Cubby’s raw score of

80 is at the 98th percentile. Conversely, anyone scoring above the raw score of 80

would be in about the top 2% of scores.

On the other hand, say that Elvis obtained an attractiveness score of 40, producing a

z-score of . As in Figure 6.6, a total of .0228 (2.28%) of the distribution is below (to

the left of) Elvis’s score. With rounding, Elvis ranks at the 2nd percentile.

Fourth, we can find the raw score at a specific percentile. We can also work in the

opposite direction to find the raw score located at a particular percentile (or relative fre-

quency). Say that we had started by asking what attractiveness score is at the 2nd per-

centile (or we had asked below what raw score is .0228 of the distribution?). As in

Figure 6.6, a z-score of is at the 2nd percentile, or below is of the

distribution. Then to find the raw score at this z, we use a formula for transforming a

z-score into a raw score. For example, we saw that in a sample, . We’d

find that the corresponding attractiveness score is 40.

Using the z-Table

So far our examples have involved whole-number z-scores, although with real data a

z-score may contain decimals. However, fractions of z-scores do not result in propor-

tional divisions of the previous areas. Instead, find the proportion of the total area

X 5 1z21S

21 X

.0228z 52222

1.477221100025 477.2

.4772N

Using z-Scores to Determine the Relative Frequency of Raw Scores 121

Mean

.50

.0228

Elvis

.50

z-scores

Raw scores

–3

+1 +2 +3

–2 –1

70 80 90

35 45 55 65 75 85

40 50

FIGURE 6.6

Location of Elvis’s score on the z-distribution of attractiveness scores

Elvis is at approximately the 2nd percentile.

under the standard normal curve for any two-decimal z-score by looking in Table 1

of Appendix C. A portion of this “z-table” is reproduced in Table 6.1.

Say that you seek the area under the curve that is above or below a .

First, locate the z in column A, labeled “z.” Then column B, labeled “Area Between

the Mean and z,” contains the proportion of the area under the curve between the

mean and the z identified in column A. Thus, .4484 of the curve is between the mean

and the z of . Because this z is positive, we place this area between the mean

and the z on the right-hand side of the distribution, as shown in Figure 6.7. Column

C, labeled “Area Beyond z in the Tail,” contains the proportion of the area under the

curve that is in the tail beyond the z-score. Thus, .0516 of

the curve is in the right-hand tail of the distribution beyond

the z of (also shown in Figure 6.7).

Notice that the z-table contains no positive or negative

signs. You must decide whether z is positive or negative,

based on the problem you’re working. Thus, if our z had been

, columns B and C would provide the respective areas

on the left-hand side of Figure 6.7. (As shown here, always

sketch the normal curve, locate the mean, and identify the

portions of the curve you’re working with. This greatly sim-

plifies the problem.)

If you get confused when using the z-table, look at the nor-

mal distribution at the top of the table, like in Table 6.1. The

different shaded portions indicate the area under the curve

described in each column.

To work in the opposite direction to find the z-score that

corresponds to a particular proportion, read the columns in

the reverse order. First, find the proportion in column B or C,

that you seek, and then identify the corresponding z-score in

21.63

11.63

z 511.63

1.60 .4452 .0548

1.61 .4463 .0537

1.62 .4474 .0526

1.63 .4484 .0516

1.64 .4495 .0505

1.65 .4505 .0495

TABLE 6.1

Sample Portion of the

z-Table

Area Between

Mean and z

Area

Beyond z

in Tail

–z

ABC

–

122 CHAPTER 6 / z-Scores and the Normal Curve Model

If You Seek First, You Should Then You

Relative frequency of transform to z find its area in column B

scores between and

Relative frequency of transform to z find its area in column C

scores beyond in tail

that marks a given Find rel. f in column B transform its z to

rel. f between and

that marks a given find rel. f in column C transform its z to

rel. f beyond in tail

Percentile of an above transform to z find its area in column B

and add .50

Percentile of an below transform to z find its area in column C

To find the simple frequency of the scores, multiply rel. f times N.

XXX

TABLE 6.2

Summary of Steps When

Using the z-Tables

column A. For example, say that you seek the z-score corresponding to 44.84% of the

curve between the mean and z. Find .4484 in column B and then, in column A, the z-

score is 1.63.

Sometimes, you will need a proportion that is not given in the table, or you’ll need

the proportion corresponding to a three-decimal z-score. In such cases, round to the

nearest value in the z-table or, to compute the precise value, perform “linear interpola-

tion” (described in Appendix A.2).

Use the information from the z-tables as we have done previously. For example, say

that we want to examine Bucky’s raw score, which transforms into the positive z-score

of , located at the right-hand side back in Figure 6.7. If we seek the proportion of

scores above his score, then from column C we expect that of the scores are

above this score. If we seek the relative frequency of scores between his score and the

mean, from column B we expect that of the scores are between the mean up to

his raw score. Then we can also compute simple frequency or percentile as discussed

previously. Or, if we began by asking what raw score demarcates or of the

curve, we would first find these proportions in column B or C, respectively, then find

the z-score of in column A, and then use our formula to transform the z-score to

the corresponding raw score.

Table 6.2 summarizes these procedures.

11.63

.0516.4484

.4484

.0516

11.63

0.4484

0.05160.0516

Column B

Column C

z = +1.63z = –1.63

Column B

Column C

0.4484

FIGURE 6.7

Distribution showing the

area under the curve for

and z 511.63z 521.63

Statistics in Published Research: Using z-Scores 123

■

To find the relative frequency of scores above or

below a raw score, transform it into a z-score. From

the z-tables, find the proportion of the area under

the curve above or below that z.

■

To find the raw score at a specified relative

frequency, find the proportion in the z-tables and

transform the corresponding z into its raw score.

MORE EXAMPLES

With and ,

To find the relative frequency of scores above

45: Say-

ing “above” indicates in the upper tail, so from

column C the relative frequency is .

To find the percentile of the score of 41.5:

. Between this positive z

and in column B is . This score is at the 65th

percentile because

To find the proportion below : “Below”

indicates the lower tail, so from column C is .3520.

To find the score above the mean with 15% of the

scores between it and the mean (or the score at

the 65th percentile): From column B, the pro-

portion closest to is so . Then

.X 5 110.39214.021 40 5 41.56

z 510.39.1517.15

z 520.38

.1480 1 .50 5 .6480 5 .65.

.1480X

z 5 141.5 2 402>4 510.38

.1056

z 5 1X 2 X2>S

5 145 2 402>4 511.25.

5 4X 5 40

For Practice

For a sample: X

苶

 65, S

 12, and N  1000.

1. What is the relative frequency of scores

below 59?

2. What is the percentile of 75?

3. How many scores are between the mean and 70?

4. What raw score delineates the top 3%?

Answers

1. z  (59  65)/12 .50; “below” is the lower tail, so

from column C is .3085.

2. z  (75  65)/12 .83; between z and the mean, from

column B, is .2967. Then

percentile.

3. z  (70  65)/12 .42; from column B is .1628;

(.1628)(1000) gives about 163 scores.

4. The “top” is the upper tail, so from column C

the closest to .03 is .0301, with z 1.88;

so X  (1.88)(12)  65  87.56.

.7967 5 80th.2967 1 .50 5

A QUICK REVIEW

STATISTICS IN PUBLISHED RESEARCH: USING z-SCORES

A common use of z-scores is with diagnostic psychological tests such as intelligence or

personality tests. Often, however, test results are also shared with people who do not

understand z-scores; imagine someone learning that he or she has a negative personality

score! Therefore, z-scores are often transformed to more user-friendly scores. A famous

example of this is the Scholastic Aptitude Test (SAT). To eliminate negative scores and

decimals, sub-test scores are transformed so that the mean is about 500 and the standard

deviation is about 100. We’ve seen that z-scores are usually between 3, which is why

SAT scores are limited to between 200 and 800 on each part. (You may hear of higher

scores but this occurs by adding together the subtest scores.)

The normal curve and z-scores are also used when researchers create a “statistical

definition” of a psychological or sociological attribute. When debating such issues as

what a genius is or how to define “abnormal,” researchers often rely on relative standing.

For example, the term “genius” might be defined as scoring above a z of 2 on an intel-

ligence test. We’ve seen that only about 2% of any distribution is above this score, so we

have defined genius as being in the top 2% on the intelligence test. Or, “abnormal”

might be defined as having a z-score below 2 on a personality inventory. Such scores

are statistically abnormal because they are very infrequent, extremely low scores.

124 CHAPTER 6 / z-Scores and the Normal Curve Model

Finally, when instructors “curve grades” it means they are assigning grades based on

relative standing, and the curve they usually use is the normal curve and z-scores. If the

instructor defines A students as the top 2%, then students with z-scores greater than 2

receive As. If B students are the next 13%, then students having z-scores between 1

and 2 receive Bs, and so on.

USING z-SCORES TO DESCRIBE SAMPLE MEANS

Now we will discuss how to compute a z-score for a sample mean. This procedure is

very important because all inferential statistics involve computing something like a

z-score for our sample data. We’ll elaborate on this procedure in later chapters but, for

now, simply understand how to compute a z-score for a sample mean and then apply the

standard normal curve model.

To see how the procedure works, say that we give a part of the SAT to a sample of

25 students at Prunepit U. Their mean score is 520. Nationally, the mean of individual

SAT scores is 500 (and is 100), so it appears that at least some Prunepit students

scored relatively high, pulling the mean to 520. But how do we interpret the perform-

ance of the sample as a whole? The problem is the same as when we examined individ-

ual raw scores: Without a frame of reference, we don’t know whether a particular

sample mean is high, low, or in-between.

The solution is to evaluate a sample mean by computing its z-score. Previously, a

z-score compared a particular raw score to the other scores that occur in this situa-

tion. Now we’ll compare our sample mean to the other sample means that occur in

this situation. Therefore, the first step is to take a small detour and create a distribu-

tion showing these other means. This distribution is called the sampling distribution

of means.

The Sampling Distribution of Means

If the national average SAT score is 500, then, in other words, the of the population

of SAT scores is 500. Because we randomly selected a sample of 25 students and

obtained their SAT scores, we essentially drew a sample of 25 scores from this popula-

tion. To evaluate our sample mean, we first create a distribution showing all other pos-

sible means we might have obtained.

One way to do this would be to record all SAT scores from the population on slips of

paper and deposit them into a very large hat. We could then hire a statistician to sample

this population. So that we can see all possible sample means that might occur the stat-

istician would sample the population an infinite number of times: She would randomly

select a sample with the same size as ours (25), compute the sample mean, replace

the scores in the hat, draw another 25 scores, compute the mean, and so on. (She’d get

very bored, so the pay would have to be good.)

Even though the of individual SAT scores is 500, our “bored statistician” would

not obtain a sample mean equal to 500 every time. There are a variety of SAT scores in

the population and sometimes the luck of the draw would produce an unrepresentative

mix of them: sometimes a sample would contain too many high scores and not enough

low scores compared to the population, so the sample mean would be above 500 to

some degree. At other times, a sample would contain too many low scores and not

enough high scores, so the mean would be below 500 to some degree. Therefore, over



Using z-Scores to Describe Sample Means 125

XXX X X X XX X XX X X

Lower means

500

Higher means

FIGURE 6.8

Sampling distribution of

random sample means of

SAT scores

The axis is labeled to

show the different values of

we obtain when we sample

a population where the

mean in 500.

the long run, the statistician would obtain many different sample means. To see them

all, she would create a frequency polygon, which is called the sampling distribution of

means. The sampling distribution of means is the frequency distribution of all possi-

ble sample means that occur when an infinite number of samples of the same size are

randomly selected from one raw score population.

Our SAT sampling distribution of means is shown in Figure 6.8. This is similar to a

distribution of raw scores, except that here each score on the axis is a sample mean.

(Still think of the distribution as a parking lot full of people, except that now each per-

son is the captain of a sample, having the sample’s mean score.)

The sampling distribution is the population of sample means so its mean is symbol-

ized by and it stands for the average sample mean. (Yes, that’s right, it’s the mean of

the means!) Here the of the sampling distribution equals 500, which was also the

of the raw scores. To the right of are the sample means the statistician obtained that

are greater than 500, and to the left of are the sample means that were less than 500.

However, the sampling distribution forms a normal distribution. This is because most

scores in the population are close to 500, so most of the time the statistician will get a

sample containing scores that are close to 500, so the sample mean will be close to 500.

Less frequently, the statistician will obtain a strange sample containing mainly scores

that are farther below or above 500, producing means that are farther below or above

500. Once in a great while, some very unusual samples will be drawn, resulting in sam-

ple means that deviate greatly from 500. However, because the individual SAT scores

are balanced around 500, over the long run, the sample means created from those

scores will also be balanced around 500, so the average mean will equal 500.

The story about the bored statistician is useful because it helps you to understand

what a sampling distribution is. Of course, in reality, we cannot “infinitely” sample a

population. However, we know that the sampling distribution would look like Figure 6.8

because of the central limit theorem. The central limit theorem is a statistical principle

that defines the mean, the standard deviation, and the shape of a sampling distribution.

From the central limit theorem, we know that the sampling distribution of means always

(1) forms an approximately normal distribution, (2) has a equal to the of the

underlying raw score population from which the sampling distribution was created, and

(3), as you’ll see shortly, has a standard deviation that is mathematically related to the

standard deviation of the raw score population.

The importance of the central limit theorem is that with it we can describe the sam-

pling distribution from any variable without actually having to infinitely sample the

population of raw scores. All we need to know is (1) that the raw score population



12







126 CHAPTER 6 / z-Scores and the Normal Curve Model

forms a normal distribution of ratio or interval scores (so that computing the mean is

appropriate) and (2) what the and of the raw score population is. Then we’ll know

the important characteristics of the sampling distribution of means.

REMEMBER The central limit theorem allows us to envision the sampling

distribution of means, which shows all means that occur through exhaustive

sampling of a raw score population.

Why do we want to see the sampling distribution? Remember that we took a small

detour, but the original problem was to evaluate our Prunepit mean of 520. Once we

envision the distribution back in Figure 6.8, we have a model of the frequency distri-

bution of all sample means that occur when measuring SAT scores. Then we can

use it to determine the relative standing of our sample mean. To do so, we simply

determine where a mean of 520 falls on the axis of the sampling distribution in

Figure 6.8 and then interpret the curve accordingly. If 520 lies close to 500, then it is

a frequent, common mean when sampling SAT scores (the bored statistician fre-

quently obtained this result). But if 520 lies toward the tail of the distribution, far

from 500, then it is a more infrequent and unusual sample mean (the statistician

seldom found such a mean).

The sampling distribution is a normal distribution, and you already know how to

determine the location of any “score” on a normal distribution: We use—you guessed

it—z-scores. That is, we determine how far the sample mean is from the mean of the

sampling distribution when measured using the standard deviation of the distribution.

This will tell us the sample mean’s relative standing among all possible means that

occur in this situation.

To calculate the z-score for a sample mean, we need one more piece of information:

the standard deviation of the sampling distribution.

The Standard Error of the Mean

The standard deviation of the sampling distribution of means is called the standard

error of the mean. (The term standard deviation was already taken.) Like a standard

deviation, the standard error of the mean can be thought of as the “average” amount

that the sample means deviate from the of the sampling distribution. That is, in some

sampling distributions, the sample means may be very different from one another and,

“on average,” deviate greatly from the average sample mean. In other distributions, the

may be very similar and deviate little from .

For the moment, we’ll discuss the true standard error of the mean, as if we had

actually computed it using the entire sampling distribution. Its symbol is . The

indicates that we are describing a population, but the subscript indicates

that we are describing a population of sample means—what we call the sampling dis-

tribution of means. The central limit theorem tells us that can be found using the

following formula:

σσ

Xs



The formula for the true standard error of the mean is

Using z-Scores to Describe Sample Means 127

Notice that the formula involves , the true standard deviation of the underlying raw

score population, and , our sample size.

The size of depends first on the size of . This is because with more variable raw

scores the statistician often gets a very different set of scores from one sample to the

next, so the sample means will be very different (and will be larger). But, if the raw

scores are not so variable, then different samples will tend to contain the same scores,

and so the means will be similar (and will be smaller). Second, the size of

depends on the size of . With a very small (say 2), it is easy for each sample to be

different from the next, so the sample means will differ (and will be larger). How-

ever, with a large , each sample will be more like the population, so all sample means

will be closer to the population mean (and will be smaller).

To compute for the sampling distribution of SAT means, we know that is 100

and our is 25. Thus, using the above formula we have

The square root of 25 is 5, so

and thus

A of 20 indicates that in our SAT sampling distribution, our individual sample

means differ from the of 500 by something like an “average” of 20 points.

Notice that although the individual SAT scores differ by an “average” of 100, their

sample means differ by only an “average” of 20. This is because the bored statisti-

cian will often encounter a variety of high and low scores in each sample, but they will

usually balance out to produce means at or close to 500. Therefore, the sample means will

not be as spread out around 500 as the individual scores are. Likewise, every sampling dis-

tribution is less spread out than the underlying raw score population used to create it.

Now, at last, we can calculate a z-score for our sample mean.

Computing a z-Score for a Sample Mean

We use this formula to compute a z-score for a sample mean:

1σ



5 20

100

125

The formula for the transforming a sample mean into a

z-score is

z 5

2 

In the formula, stands for our sample mean, stands for the mean of the sampling

distribution (which equals the mean of the underlying raw score population) and

stands for the standard error of the mean. As we did with individual scores, this

formula measures how far a score is from the mean of a distribution, measured using

X