Heiman G. Basic Statistics for the Behavioral Sciences

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48 CHAPTER 3 / Frequency Distributions and Percentiles

This says that to compute relative frequency, divide the frequency ( ) by the total num-

ber of scores ( ). For example, if a score occurred four times ( ) in a sample of 10

scores ( ), then filling in the formula gives

The score has a relative frequency of .40, meaning that the score occurred .40 of the

time in the sample.

As you can see here, one reason that we compute relative frequency is simply be-

cause it can be easier to interpret than simple frequency. Interpreting that a score has a

frequency of 4 is difficult because we have no frame of reference—is this often or not?

However, we can easily interpret the relative frequency of .40 because it means that the

score occurred .40 of the time.

REMEMBER Relative frequency indicates the proportion of the time (out of )

that a score occurred.

We can also begin with and compute the corresponding simple frequency. To

transform relative frequency into simple frequency, multiply the relative frequency

times . Thus, if is .4 and is 10, multiply .4 times 10 and the answer is 4; the

score occurs four times in this sample.

Finally, sometimes we transform relative frequency to percent. Converting relative

frequency to percent gives the percent of the time that a score occurred. To transform

relative frequency to percent, multiply the times 100. Above, the was .40, so

Thus, 40% of the sample had this score. Conversely, to transform

percent into relative frequency, divide the percent by 100. Thus 40% / 100 .40.

Presenting Relative Frequency in a Table or Graph

A distribution showing the relative frequency of all scores is called a relative frequency

distribution. To create a relative frequency table, first create a simple frequency table,

as we did previously. Then add a third column labeled “ ”

For example, look at Table 3.3. To compute , we need , which here is 20. Then

the score of 1, for example, has so its relative frequency is 4/20, or .20. And so on.

We can also determine the combined relative frequency of several scores by adding

their frequencies together: In Table 3.3, a score of 1 has a relative frequency of .20, and

a score of 2 has a relative frequency of .50, so together, their relative frequency is .20 1

.50, or .70; participants having a 1 or 2 compose .70 of our sample. (To check your

table, remember that, except for rounding error, the sum of all relative frequencies in a

distribution should equal 1: All scores together should constitute 100% of the sample.)

We graph a relative frequency distribution using the same rules as with simple fre-

quency: Create a bar graph if the scores involve a nominal or ordinal scale, and create a

f 5 4,

Nrel. f

rel. f.

1.402110025 40%.

rel. frel. f

Nrel. fN

rel. f

rel. f 5

5 .40

Score f rel. f

6 1 .05

5 0 .00

4 2 .10

3 3 .15

2 10 .50

1 4 .20

Total: 20 1.00 100%5

TABLE 3.3

Relative Frequency

Distribution

The left-hand column identi-

fies the scores, the middle

column shows each score’s

frequency, and the right-hand

column shows each score’s

relative frequency.

histogram or polygon if the scores involve an interval or ratio scale. Figure 3.9 presents

examples using the data from Table 3.3. The only novelty here is that the axis reflects

relative frequency, so it is labeled in increments between 0 and 1.0.

Finding Relative Frequency Using the Normal Curve

Although relative frequency is an important component of statistics, we will not emphasize

the previous formula. (You’re welcome.) Instead, most of the time our data will be nor-

mally distributed, and then we will use the normal curve to determine relative frequency.

To understand this approach, think about a normal curve in a different way. Imagine

that you are flying in a helicopter over a parking lot. The and axes are laid out on

the ground, and the people who received a particular score are standing in line in front

of the marker for their score. The lines of people are packed so tightly together that,

from the air, you only see the tops of many heads in a “sea of humanity.” If you painted

a line that went behind the last person in line at each score, you would have the normal

curve shown in Figure 3.10.

From this perspective, the height of the curve above any score reflects the number of

people standing in line at that score. Thus, in Figure 3.10, the score of 30 has the highest

frequency because the longest line of people is standing at this score in the parking lot.

Relative Frequency and the Normal Curve 49

FIGURE 3.10

Parking lot view of the

ideal normal curve.

The height of the curve above

any score reflects the number

of people standing in line at

that score.

Scores

0101520253035404550555

. . . . . .

50%

FIGURE 3.9

Examples of relative frequency distributions using the data in Table 3.3

Scores

rel.

1.0

Bar graph Histogram Polygon

234 65

1234 651234657

Scores Scores

50 CHAPTER 3 / Frequency Distributions and Percentiles

The reason for using this “parking lot view” is so that you think of the normal curve as a

picture of something solid: The space under the curve between the curve and the

axis has area that represents individuals and their scores. The entire parking lot contains

everyone in the sample and 100% of the scores. Therefore, any portion of the parking lot—

any portion of the space under the curve—corresponds to that portion of the sample.

For example, notice that in Figure 3.10 a vertical line is drawn through the middle

score of 30, and so .50 of the parking lot is to the left of the line. Because the complete

parking lot contains all participants, a part that is .50 of it contains 50% of the partici-

pants. (We can ignore those relatively few people who are straddling the line.) Partici-

pants are standing to the left of the line because they received scores of 29, 28, and so

on. So, in total, 50% of the participants received scores below 30. Now turn this around:

If 50% of the participants obtained scores below 30, then the scores below 30 occurred

50% of the time. Thus, the scores below 30 have a combined relative frequency of .50.

This logic is so simple it almost sounds tricky: if you have one-half of the parking

lot, then you have one-half of the participants and thus one-half of the scores, so those

scores occur .50 of the time. Or, if you have 25% of the parking lot, then you have 25%

of the participants and 25% of the scores, so those scores occur .25 of the time.

This is how we describe what we have done using statistical terminology: The total

space occupied by the everyone in the parking lot is called the total area under the nor-

mal curve. We identify some particular scores and determine the area of the correspon-

ding portion of the polygon above those scores. We then compare the area of this

portion to the total area to determine the proportion of the total area under the curve

that we have selected. Then, as we’ve seen,

The proportion of the total area under the normal curve that is occupied by a

group of scores corresponds to the combined relative frequency of those scores.

Of course, statisticians don’t fly around in helicopters, eyeballing parking lots, so here’s

a different example: Say that by using a ruler and protractor, we determine that in Figure

3.11 the entire polygon occupies an area of 6 square inches on the page. This total area cor-

responds to all scores, which is N. Say that the area under the curve between the scores of

30 and 35 covers 2 square inches. This area is due to the number of times these scores oc-

cur. Therefore, the scores between 30 and 35 occupy 2 out of the 6 square inches created

by all scores, so these scores constitute , or .33, of the entire distribution. Thus, the

scores between 30 and 35 constitute .33 of our N, so they have a relative frequency of .33.

We could obtain this answer by using the formula for relative frequency if, using

and each score’s , we computed the for each score between 30 and 35 and then

added them together. However, the advantage of using the area under the curve is that

we can get the answer without knowing or the simple frequencies of these scores.N

rel. ff

2>6

FIGURE 3.11

Finding the proportion of

the total area under the

curve

The complete curve occupies

6 square inches, with scores

between 30 and 35 occupying

2 square inches.

Scores

0 10152025303540455055

5 . . .. . .

2 square inches

6 square inches

COMPUTING CUMULATIVE FREQUENCY AND PERCENTILE

Researchers have one other approach for organizing the scores in addition to comput-

ing simple and relative frequency. Sometimes we want to know a score’s standing rela-

tive to the other scores. For example, it may be most informative to know that 30 people

scored above 80 or that 60 people scored below 80. When we seek such information,

the convention in statistics is to count the number of scores below the score, computing

either cumulative frequency or percentile.

Computing Cumulative Frequency and Percentile 51

A QUICK REVIEW

■

Relative frequency is the proportion of the time that

a score occurs.

■

The area under the normal curve corresponds to

100% of a sample, so a proportion of the curve will

contain that proportion of the scores, which is their

relative frequency.

MORE EXAMPLES

Below, the shaded area is .15 of the total curve (so

15% of people in the parking lot are standing at these

scores). Thus, scores between 55 and 60 occur .15 of

the time, so their combined relative frequency is .15.

Above the score of 70 is .50 of the curve, so scores

above 70 have a combined relative frequency of .50.

For Practice

1. If a score occurs 23% of the time, its relative fre-

quency is .

2. If a score’s relative frequency is .34, it occurs

percent of the time.

3. If scores occupy .20 of the area under the curve,

they have a relative frequency of .

4. Say that the scores between 15 and 20 have a rela-

tive frequency of .40. They make up of the

area under the normal curve.

Answers

1. 23%/100 5 .23

2. (.34)(100) 5 34%

3. .20

4. .40

Scores

50 55 60 65 70 75 80 85 9045

.50

.15

In fact, whatever the variable might be, whatever the might be, and whatever the ac-

tual frequency of each score is, we know that the area these scores occupy is 33% of

the total area, and that’s all we need to know to determine their relative frequency.

This is especially useful because, as you’ll see in Chapter 6, statisticians have created

a system for easily finding the area under any part of the normal curve. Therefore, we

can easily determine the relative frequency for scores in any part of a normal distribu-

tion. (No, you won’t need a ruler and a protractor.) Until then, simply remember this:

REMEMBER The total area under the normal curve corresponds to the times

that all scores occur, so a proportion of the total area under the curve corre-

sponds to the proportion of time some of the scores occur, which is their rel-

ative frequency.

52 CHAPTER 3 / Frequency Distributions and Percentiles

Computing Cumulative Frequency

Cumulative frequency is the frequency of all scores at or below a particular score. The

symbol for cumulative frequency is To compute a score’s cumulative frequency, we

add the simple frequencies for all scores below the score to the frequency for the score,

to get the frequency of scores at or below the score.

For example, Table 3.4 shows a “cumulative frequency distribution” created from a

simple frequency table to which we add a column. Begin with the lowest score. Here,

no one scored below 10, and one person scored 10, so we have a of 1 (one person

scored 10 or below 10). Next, there were two scores of 11. We add this to the previous

for 10, so the for 11 is 3 (three people scored at 11 or below 11). Next, no one

scored at 12, but three people scored below 12, so the for 12 is also 3. And so on,

each time adding the frequency for a score to the cumulative frequency for the score

immediately below it.

As a check, verify that the for the highest score equals : Here all 20 participants

obtained either the highest score or a score below it.

Nowadays, researchers seldom create graphs showing cumulative frequency, so we

won’t. In fact, cumulative frequency is not the most common way to summarize scores.

But cumulative frequency is the first step in computing percentiles, which are very

common.

Computing Percentiles

We’ve seen that the proportion of time a score occurs provides a frame of reference

that is easier to interpret than the number of times a score occurs. Therefore, our final

procedure is to transform cumulative frequency into a percent of the total. A score’s

percentile is the percent of all scores in the data that are at or below the score. Thus,

for example, if the score of 80 is at the 75th percentile, this means that 75% of the

sample scored at or below 80.

Usually, we will already know a score’s cf. Then

Ncf

cfcf

cf.

Score f cf

17 1 20

16 2 19

15 4 17

14 6 13

13 4 7

12 0 3

11 2 3

10 1 1

TABLE 3.4

Cumulative Frequency

Distribution

The left-hand column identi-

fies the scores, the center col-

umn contains the simple

frequency of each score, and

the right-hand column con-

tains the cumulative

frequency of each score.

The formula for finding the percentile for a score with a

known cf is

Score’s Percentile 5 a

b11002

This says to first divide the score’s by , which transforms the into a proportion of

the total sample. Then we multiply this times 100, which converts it into a percent of

the total. Thus, if a score has a of 5 and is 10, then so the score

is at the 50th percentile.

Table 3.5 shows the previous cumulative frequency table (where ) trans-

formed to percentiles. With one person scoring 10 or below, (1/20)(100) equals 5, so 10

is at the 5th percentile. The three people scoring 11 or below are at the 15th percentile

and so on. The highest score is, within rounding error, the 100th percentile, because

100% of the sample has the highest score or below.

N 5 20

15>102110025 50,Ncf

cfNcf

Score f cf Percentile

17 1 20 100

16 2 19 95

15 4 17 85

14 6 13 65

13 4 7 35

12 0 3 15

11 2 3 15

10 1 1 5

TABLE 3.5

Percentiles

The right-hand column

contains the percentile of

each score.

Computing Cumulative Frequency and Percentile 53

FIGURE 3.12

Normal distribution

showing the area under

the curve to the left of

selected scores

Scores

0 101520253035404550555 . . .. . .

.50 of curve

.15 of curve

.85 of curve

For finding the score at a percentile not shown in our table, the precise way is with a

computer program (such as SPSS). However, a quick way to find an approximate per-

centile is to use the area under the normal curve.

Finding Percentile Using the Area Under the Normal Curve

Percentile describes the scores that are lower than a particular score, and on the normal

curve, lower scores are to the left of a particular score. Therefore, the percentile for a

given score corresponds to the percent of the total area under the curve that is to the left

of the score. For example, on the distribution in Figure 3.12, 50% of the curve is to the

left of the middle score of 30. Because scores to the left of 30 are below it, 50% of the

distribution is below 30 (in the parking lot, 50% of the people are standing to the left of

the line and all of their scores are less than 30). Thus, the score of 30 is at the 50th per-

centile. Likewise, to find the percentile for the score of 20 in Figure 3.12, we would

find the percent of the total area that is to the left of 20. Say that we find that 15% of

the curve is to the left of 20; then 20 is at the 15th percentile.

We can also work the other way to find the score at a given percentile. Say that we

seek the score at the 85th percentile. We would measure over until 85% of the area

under the curve is to the left of a certain point. If, as shown in Figure 3.12, the score of

45 is at that point, then 45 is at the 85th percentile.

Notice that we make a slight change in our definition of percentile when we use the

normal curve. Technically, a percentile is the percent of scores at or below a score.

However, in everyday use, the at may be dropped. Then percentile becomes the percent

of scores below a particular score. This is acceptable if we are describing a large sam-

ple or a population because those participants at the score are a negligible portion of

the total (remember that we ignored those relatively few people who were straddling

the line). Thus, in Figure 3.12, the score of 30 is at the 50th percentile, so we say that

50% of the scores are below 30 and 50% are above it.

However, if we are describing a small sample, we should not ignore those at the

score, because those participants may actually constitute a sizable portion of the sam-

ple. Say that 10% have this score. If we conclude that 50% are above and 50% are be-

low, with 10% at the score, we have the impossible total of 110%! Therefore, with

small samples, percentile is calculated and defined as the percent of scores at or below

a particular score. Because of this distinction, you should use the area under the normal

curve to compute percentile when you have a large sample or a population that also fits

the normal curve.

54 CHAPTER 3 / Frequency Distributions and Percentiles

STATISTICS IN PUBLISHED RESEARCH: APA PUBLICATION RULES

The rules that we’ve used for creating tables and graphs are part of the procedures for cre-

ating research publications established by the American Psychological Association (APA).

We will also follow “APA format” when we discuss how to report various statistics. (Note:

Computer programs, such as SPSS, sometimes do not operate according to these rules.)

You won’t always see frequency tables and graphs in published reports, because they are

very expensive to print. Instead, often researchers simply state that “the scores were nor-

mally distributed” or whatever, and you are expected to mentally envision the distribution.

One way in which researchers shrink the size of published tables and graphs is to

create a grouped distribution.

A WORD ABOUT GROUPED FREQUENCY DISTRIBUTIONS

In the previous examples, we examined each score individually, creating ungrouped

distributions. When we have too many scores to produce a manageable ungrouped dis-

tribution, we create a grouped distribution. In a grouped distribution, scores are com-

bined to form small groups, and then we report the total , , , or percentile of all

scores in each group.

For example, look at the grouped distribution shown in Table 3.6. In the score col-

umn, “0-4” contains the scores 0, 1, 2, 3, 4, and “5–9” contains scores 5 through 9, and

so on. The for each group is the sum of the frequencies for the scores in that group.

Thus, the scores between 0 and 4 have a total of 7, while, for the highest scores be-

tween 40 and 44, the total is 2. Likewise, the combined relative frequency of scores

between 0 and 4 is .28, while for scores between 40 and 44 it is .08. Each cumulative

cfrel. ff

A QUICK REVIEW

■

Cumulative frequency ( ) indicates the number of

participants in a sample that scored at or below a

particular score.

■

Percentile indicates the percent of a sample that

scored at or below a particular score.

■

The proportion of the normal curve to the left of a

score is the proportion of participants scoring below

that score, which translates into the score’s percentile.

MORE EXAMPLES

If the score of 80 has a of 35, it means that 35 par-

ticipants had a score of 80 or below 80. If 80 is at the

90th percentile, it means that 90% of the sample had a

score of 80 or below 80 and, on a normal distribution,

80 is located over toward the right-hand tail, with 90%

of the curve located to the left of it.

For Practice

On an exam, 10 students scored 19, 15 students scored

20, no students scored 21, and everyone else scored 21.

1. The for the score of 20 is .

2. The for the score of 21 is .

3. If 60 students took the exam, the percentile for the

score of 20 is .

4. What does this percentile indicate?

5. In a normal curve showing these grades, how

much of the curve is to the right of (above) 20 and

how much is to the left of (below) 20?

Answers

1. 15

2. 15

3. Then percentile.

4. 25% of the class scored 20 or below 20.

5. 75% of the curve is above 20, and 25% of the curve is

below 20.

1.252110025 25th115>6025 .25.

frequency is the number of scores that are at 44 or below the highest score in the group.

Thus, 7 scores are at 4 or below while 25 scores are at 44 or below. (Because 44 is the

highest score, we know that is 25.) Finally, each percentile indicates the percent of

scores that are below the highest score in the group, so the score of 4 is at the 28th per-

centile, and the score of 44 is at the 100th percentile.

Appendix A.1 shows examples of graphs of grouped distributions (Figure A.1) with

the details of how to create grouped distributions.

Chapter Summary 55

Score f rel. f cf Percentile

40–44 2 .08 25 100

35–39 2 .08 23 92

30–34 0 .00 21 84

25–29 3 .12 21 84

20–24 2 .08 18 72

15–19 4 .16 16 64

10–14 1 .04 12 48

5–9 4 .16 11 44

0–4 7 .28 7 28

TABLE 3.6

Grouped Distribution

Showing f, rel.f, cf, and

Percentile

The left-hand column identi-

fies the lowest and highest

score in each group.

All of the procedures in this chapter indicate how often certain scores occur, but

each provides a slightly different perspective that allows you to interpret the data in

a slightly different way. Which particular procedure you should use is determined

by which provides the most useful information. However, you may not automati-

cally know which is the best technique for a given situation. So, understand that re-

searchers often explore their data, and you should too. Try different techniques and

then choose the approach that allows you to make the most sense out of your data.

Using the SPSS Appendix: Appendix B shows you how to use the SPSS computer

program to produce the distributions that we’ve discussed. As described in Appendix

B.1 you’ll first need to label your variables and input your raw scores. Then, as in

Appendix B.2, you can create frequency tables and plot bar graphs, histograms and

polygons. You can also compute percentiles.

CHAPTER SUMMARY

1. The number of scores in the data is symbolized by .

2. A simple frequency distribution shows the frequency of each score. The symbol

for simple frequency is .

3. When graphing a simple frequency distribution, if the variable involves a nominal

or an ordinal scale, create a bar graph. If the variable involves a few different

interval or ratio scores, create a histogram. With many different interval or ratio

scores, create a polygon.

4. In a normal distribution forming a normal curve, extreme high and low scores are

relatively infrequent, scores closer to the middle score are more frequent, and the

middle score occurs most frequently. The low-frequency, extreme low and

extreme high scores are in the tails of a normal distribution.

PUTTING IT

ALL TOGETHER

56 CHAPTER 3 / Frequency Distributions and Percentiles

5. A negatively skewed distribution contains low-frequency, extreme low scores,

but not low-frequency, extreme high scores. A positively skewed distribution

contains low-frequency, extreme high scores, but not low-frequency, extreme

low scores.

6. A bimodal distribution is symmetrical, with two areas showing relatively high-

frequency scores. In a rectangular distribution, all scores have the same

frequency.

7. The relative frequency of a score, symbolized by , is the proportion of time

that the score occurred. A relative frequency distribution is graphed in the same

way as a simple frequency distribution except that the axis is labeled in

increments between 0 and 1.0.

8. The proportion of the total area under the normal curve occupied by particular

scores equals the combined relative frequency of those scores.

9. The cumulative frequency of a score, symbolized by , is the frequency of all

scores at or below the score.

10. Percentile is the percent of all scores at or below a given score. On the normal

curve the percentile of a score is the percent of the area under the curve to the left

of the score.

11. In an ungrouped distribution, the , , , or percentile of each individual score

is reported.

12. In a grouped distribution, different scores are grouped together, and the , , ,

or percentile for each group is reported.

cfrel. ff

rel. f

KEY TERMS

bar graph 39

bimodal distribution 44

cumulative frequency 52

distribution 36

frequency 36

frequency polygon 40

grouped distribution 54

histogram 40

negatively skewed distribution 44

normal curve 42

normal distribution 42

cfrel. fNf

percentile 52

positively skewed distribution 44

proportion of the total area under the

curve 50

rectangular distribution 45

relative frequency 47

relative frequency distribution 48

simple frequency 37

simple frequency distribution 37

tail 43

ungrouped distribution 54

REVIEW QUESTIONS

(Answers for odd-numbered problems are in Appendix D.)

1. What do each of the following symbols mean? (a) ; (b) ; (c) ; (d) .

2. (a) What is the difference between a bar graph and a histogram? (b) With what

kind of data is each used?

cfrel. ffN

3. (a) What is the difference between a histogram and a polygon? (b) With what kind

of data is each used?

4. (a) What is the difference between a score’s simple frequency and its relative

frequency? (b) What is the difference between a score’s cumulative frequency

and its percentile?

5. (a) What is the advantage of computing relative frequency instead of simple

frequency? (b) What is the advantage of computing percentile instead of

cumulative frequency?

6. (a) What is the difference between a skewed distribution and a normal

distribution? (b) What is the difference between a bimodal distribution and

a normal distribution? (c) What does a rectangular distribution indicate about

the frequencies of the scores?

7. What is the difference between a positively skewed distribution and a negatively

skewed distribution?

8. (a) Why must the for the highest score in a sample equal ? (b) Why must the

sum of all s in a sample equal ?

9. What is the difference between graphing a relationship as we did in Chapter 2 and

graphing a frequency distribution?

10. What is the difference between how we use the proportion of the total area under

the normal curve to determine relative frequency and how we use it to determine

percentile.

11. What does it mean when a score is in a tail of a normal distribution?

12. (a) How is percentile defined in a small sample? (b) How is percentile defined for

a large sample or population when calculated using the normal curve?

APPLICATION QUESTIONS

13. In reading psychological research, you encounter the following statements.

Interpret each one. (a) “The IQ scores were approximately normally distributed.”

(b) “A bimodal distribution of physical agility scores was observed.” (c) “The

distribution of the patients’ memory scores was severely negatively skewed.”

14. From the data 1, 4, 5, 3, 2, 5, 7, 3, 4, and 5, Poindexter created the following

frequency table. What five things did he do wrong?

Ncf

Application Questions 57

Score f cf

110

211

323

425

538

719

N 5 6

15. The distribution of scores on your statistics test is positively skewed. What does

this indicate about the difficulty of the test?

16. (a) On a normal distribution of exam scores, Poindexter scored at the 10th

percentile, so he claims that he outperformed 90% of his class. Why is he correct

or incorrect? (b) Because Foofy’s score is in a tail of the distribution, she claims

she had one of the highest scores on the exam. Why is she correct or incorrect?