Heiman G. Basic Statistics for the Behavioral Sciences

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38 CHAPTER 3 / Frequency Distributions and Percentiles

Not only can we easily see the frequency of each score, but we can also determine

the combined frequency of several scores by adding together their individual s. For ex-

ample, the score of 13 has an of 4, and the score of 14 has an of 6, so their combined

frequency is 10.

Notice that, although there are 8 scores in the score column, is not 8. There are

18 scores in the original sample, so is 18. You can see this by adding together the fre-

quencies in the column: The 1 person scoring 17 plus the 4 people scoring 15 and so

on adds up to the 18 people in the sample. In a frequency distribution, the sum of the

frequencies always equals .

REMEMBER The sum of all frequencies in a sample equals N.

That’s how to create a simple frequency distribution. Such a distribution is also

called a regular frequency distribution or a plain old frequency distribution.

A QUICK REVIEW

■

A frequency distribution shows the number of times

participants obtained each score.

MORE EXAMPLES

The scores 15, 16, 13, 16, 15, 17, 16, 15, 17, and 15,

contain one 13, no 14s, four 15s, and so on, producing

the following frequency table:

Scores f

17 2

16 3

15 4

14 0

13 1

For Practice

1. What is the difference between and ?

2. Create a frequency table for these scores: 7, 9, 6, 6,

9, 7, 7, 6, and 6.

3. What is the here?

4. What is the frequency of 6 and 7 together?

Answers

1. f is the number of times a score occurs; N is the total

number of scores in the data.

2. Scores f

4. f 5 3 1 4 5 7

N 5 9

Graphing a Simple Frequency Distribution

When researchers talk of a frequency distribution, they often imply a graph. Essen-

tially, it shows the relationship between each score and the frequency with which it oc-

curs. We ask, “For a given score, what is its corresponding frequency?”, so we place

the scores on the axis and frequency on the axis.

REMEMBER A graph of a frequency distribution shows the scores on the

axis and their frequency on the axis.

Recall that a variable will involve one of four types of measurement scales—nominal,

ordinal, interval, or ratio. The type of scale involved determines whether we graph a

frequency distribution as a bar graph, a histogram, or a polygon.

Bar Graphs Recall that in nominal data each score identifies a category, and in or-

dinal data each score indicates rank order. A frequency distribution of nominal or ordi-

nal scores is graphed by creating a bar graph. In a bar graph, a vertical bar is centered

over each score on the axis, and adjacent bars do not touch.

Figure 3.1 shows two bar graphs of simple frequency distributions. Say that the up-

per graph is from a survey in which we counted the number of participants in each cat-

egory of the nominal variable of political party affiliation. The axis is labeled using

the “scores” of political party, and because this is a nominal variable, they can be

arranged in any order. In the frequency table, we see that six people were Republicans,

so we draw a bar at a height (frequency) of 6 and so on.

Say that the lower graph is from a survey in which we counted the number of partic-

ipants having different military ranks (an ordinal variable). Here the axis is labeled

from left to right, which corresponds to low to high. Again, the height of each bar is the

score’s frequency.

REMEMBER Create a bar graph to show the frequency distribution of nomi-

nal or ordinal scores.

Simple Frequency Distributions 39

Rep. Dem. Soc. Comm.

Political affiliation

Military rank

Sgt.

Gen.Col.Lt.

FIGURE 3.1

Simple frequency bar

graphs for nominal and

ordinal data

The height of each bar

indicates the frequency of

the corresponding score on

the x axis.

Nominal Variable of

Political Affiliations

Party f

Communist 1

Socialist 3

Democrat 8

Republican 6

Ordinal Variable of

Military Rank

Party f

General 3

Colonel 8

Lieutenant 4

Sergeant 5

40 CHAPTER 3 / Frequency Distributions and Percentiles

The reason we create bar graphs with nominal and ordinal scales is because re-

searchers assume that both are discrete scales: You can be in one group or the next, but

not in-between. The space between the bars in a bar graph indicates this. Later we will

see bar graphs in other contexts and this same rule always applies:

Create a bar graph whenever the variable is discrete.

On other hand, recall that interval and ratio scales are assumed to be continuous:They

allow fractional amounts that continue between the whole numbers. To communicate

this, these scales are graphed using continuous figures.

Histograms Create a histogram when plotting a frequency distribution containing a

small number of different interval or ratio scores. A histogram is similar to a bar graph

except that in a histogram adjacent bars touch. For example, say that we measured the

number of parking tickets some people received, obtaining the data in Figure 3.2. Again,

the height of each bar indicates the corresponding score’s frequency. Although you can-

not have a fraction of a ticket, this ratio variable is theoretically continuous (e.g., you

can talk about an average of 3.14 tickets per person). By having no gap between the bars

in our graph, we communicate that there are no gaps when measuring this variable.

Polygons Usually, we don’t create a histogram when we have a large number of dif-

ferent interval or ratio scores, such as if our participants had from 1 to 50 parking tick-

ets. The 50 bars would need to be very skinny, so the graph would be difficult to read.

We have no rule for what number of scores is too large, but when a histogram is unwork-

able, we create a frequency polygon. Construct a frequency polygon by placing a data

point over each score on the axis at a height corresponding to the appropriate fre-

quency. Then connect the data points using straight lines. To illustrate this, Figure 3.3

shows the previous parking ticket data plotted as a frequency polygon. Because each line

continues between two adjacent data points, we communicate that our measurements

continue between the two scores on the axis and therefore that this is a continuous

variable. Later we will create graphs in other contexts that also involve connecting data

points with straight lines. This same rule always applies:

Connect adjacent data points with straight lines whenever the variable

is continuous.

Number of parking tickets

234567

FIGURE 3.2

Histogram showing the

simple frequency of park-

ing tickets in a sample

Score f

Notice that a polygon includes on the axis the next score above the highest score

in the data and the next score below the lowest score (in Figure 3.3, scores of 0 and

8 are included). These added scores have a frequency of 0, so the polygon touches the

axis. In this way, we create a complete geometric figure—a polygon—with the axis

as its base.

REMEMBER Create a histogram or polygon to plot the frequency distribu-

tion for an interval or ratio variable.

Often in statistics you must a read a polygon to determine a score’s frequency, so be

sure you can do this: Locate the score on the axis and then move upward until you

reach the line forming the polygon. Then, moving horizontally, locate the frequency of

the score. For example, as shown by the dashed line in Figure 3.3, the score of 4 has an

equal to 4.

REMEMBER The height of the polygon above any score corresponds to that

score’s frequency.

Table 3.2 reviews the rules for constructing bar graphs, histograms, and polygons.

Simple Frequency Distributions 41

Number of parking tickets

2345678

FIGURE 3.3

Simple frequency

polygon showing the

frequency of parking

tickets in a sample

Score f

Graph When Used? How Produced?

Bar graph With nominal or Adjacent bars

ordinal scores do not touch

Histogram With small range of Adjacent bars

interval/ratio scores do touch

Polygon With large range of Straight lines;

interval/ratio scores add points above and

below actual scores

TABLE 3.2

When to create a bar

graph, histogram, or

polygon

Consider the scale of

measurement of scores on

the axis. X

42 CHAPTER 3 / Frequency Distributions and Percentiles

TYPES OF SIMPLE FREQUENCY DISTRIBUTIONS

Research often produces scores that form frequency polygons having one of several

common shapes, and so we have names to identify them. Each shape comes from an

idealized distribution of a population. By far the most important frequency distribution

is the normal distribution. (This is the big one, folks.)

The Normal Distribution

Figure 3.4 shows the polygon of the ideal normal distribution. (Let’s say these are test

scores from a population of college students.) Although specific mathematical proper-

ties define this polygon, in general it is a bell-shaped curve. But don’t call it a bell

curve (that’s so pedestrian!). Call it a normal curve or a normal distribution or say

that the scores are normally distributed.

Because it represents an ideal population, a normal curve is different from the

choppy polygon we saw previously. First, the curve is smooth because a population

produces so many different scores that the individual data points are too close to-

gether for straight lines to connect them. Instead, the data points themselves form a

smooth line. Second, because the curve reflects an infinite number of scores, we can-

not label the axis with specific frequencies. Simply remember that the higher the

curve is above a score, the higher is the score’s frequency. Finally, regardless of how

high or low an score might be, theoretically it might sometimes occur. Therefore,

as we read to the left or to the right on the axis, the frequencies approach—but

never reach—a frequency of zero, so the curve approaches but never actually touches

the axis. X

A QUICK REVIEW

■

Create a bar graph with nominal or ordinal scores,

a histogram with a few interval/ratio scores, and a

polygon with many different interval/ratio scores.

MORE EXAMPLES

After a survey, to graph (1) the frequency of males

versus females (a nominal variable), create a bar

graph; (2) the number of people who are first born,

second born, etc. (an ordinal variable), create a bar

graph; (3) the frequency of participants falling into

each of five salary ranges (a few ratio scores), create a

histogram; (4) the frequency for each individual

salary reported (many ratio scores), create a polygon.

For Practice

1. A has a separate, discrete bar above each

score, a contains bars that touch, and a

has dots connected with straight lines.

2. To show the number of freshmen, sophomores, and

juniors who are members of a fraternity, plot a .

3. To show the frequency of people who are above

average weight by either 0, 5, 10, or 15 pounds,

plot a .

4. To show the number of people preferring chocolate

or vanilla ice cream in a sample, plot a .

5. To show the number of people who are above av-

erage weight by each amount between 0 and 100

pounds, plot a .

Answers

1. bar graph; histogram; polygon

2. bar graph

3. histogram

4. bar graph

5. polygon

As you can see from Figure 3.4, the normal distribution has the following character-

istics. The score with the highest frequency is the middle score between the highest and

lowest scores. (Here it is the score of 30.) The normal curve is symmetrical, meaning

that the left half below the middle score is a mirror image of the right half above the

middle score. As we proceed away from the middle score either toward the higher or

lower scores, the frequencies at first decrease slightly. Farther from the middle score,

however, the frequencies decrease more drastically, with the highest and lowest scores

having relatively low frequency.

In statistics the scores that are relatively far above and below the middle score of the dis-

tribution are called the “extreme” scores. Then, the far left and right portions of a normal

curve containing the low-frequency, extreme scores are called the tails of the distribution.

In Figure 3.4, the tails are roughly below the score of 15 and above the score of 45.

The reason the normal curve is important is because it is a very common distribution

in psychology and other behavioral sciences: For most of the variables that we study, the

scores naturally form a curve similar to this, with most of the scores around the middle

score, and with progressively fewer higher or lower scores. Because of this, the normal

curve is also very common in our upcoming statistical procedures. Therefore, before you

proceed, be sure that you can read the normal curve. Can you see in Figure 3.4 that the

most frequent scores are between 25 and 35? Do you see that a score of 15 has a rela-

tively low frequency and a score of 45 has the same low frequency? Do you see that

there are relatively few scores in the tail above 50 or in the tail below 10? Above all,

you must be able to see this in your sleep.

On a normal distribution, the farther a score is from the central score of the

distribution, the less frequently the score occurs.

A distribution may not match the previous curve exactly, but it can still meet the mathe-

matical definition of a normal distribution. Consider the three curves in Figure 3.5. Curve

is generally what we think of as the ideal normal distribution. Curve is skinny relative to

the ideal because only a few scores around the middle score have a relatively high fre-

quency. On the other hand, Curve is fat relative to the ideal because more scores farther

below and above the middle have a high frequency. Because these curves generally have

that bell shape, however, for statistical purposes their differences are not critical.

Other Common Frequency Polygons

Not all data form a normal distribution and then the distribution is called nonnormal.

One common type of nonnormal distribution is a “skewed distribution.” A skewed dis-

tribution is similar to a normal distribution except that it has only one pronounced tail.

Types of Simple Frequency Distributions 43

FIGURE 3.4

The ideal normal curve

Scores farther above and be-

low the middle scores occur

with progressively lower

frequencies.

Test scores

Tail

10 15 20 25 30 35 40 45 50 55

. . .. . .

44 CHAPTER 3 / Frequency Distributions and Percentiles

As shown in Figure 3.6, a distribution may be either negatively skewed or positively

skewed.

A negatively skewed distribution contains extreme low scores that have a low fre-

quency but does not contain low-frequency, extreme high scores. The polygon on the

left in Figure 3.6 shows a negatively skewed distribution. This pattern might be found,

for example, by measuring the running speed of professional football players. Most

would tend to run at higher speeds, but a relatively few linemen lumber in at the slower

speeds. (To remember negatively skewed, remember that the pronounced tail is over

the lower scores, sloping toward zero, toward where the negative scores would be.)

On the other hand, a positively skewed distribution contains extreme high scores

that have low frequency but does not contain low-frequency, extreme low scores. The

right-hand polygon in Figure 3.6 shows a positively skewed distribution. This pattern

might be found, for example, if we measured participants’ “reaction time” for recogniz-

ing words. Usually, scores will tend to be rather low, but every once in a while a person

will “fall asleep at the switch,” requiring a large amount of time and thus producing a

high score. (To remember positively skewed, remember that the tail slopes away from

zero, toward where the higher, positive scores are located.)

REMEMBER Whether a skewed distribution is negative or positive corre-

sponds to whether the distinct tail slopes toward or away from zero.

Another type of nonnormal distribution is a bimodal distribution, shown in the left-

hand side of Figure 3.7. A bimodal distribution is a symmetrical distribution contain-

ing two distinct humps, each reflecting relatively high-frequency scores. At the center

FIGURE 3.5

Variations of bell-shaped curves

Curve B

Low . . . Middle . . . High

Low . . . Middle . . . HighLow . . . Middle . . . High

Curve A Curve C

scores scores scores

FIGURE 3.6

Idealized skewed

distributions

The direction in which the

distinctive tail slopes

indicates whether the skew

is positive or negative.

Low . . . Middle . . . High

Negative skew

scores

Low . . . Middle . . . High

Positive skew

scores

of each hump is one score that occurs more frequently than the surrounding scores, and

technically the center scores have the same frequency. Such a distribution would occur

with test scores, for example, if most students scored at 60 or 80, with fewer students

failing or scoring in the 70s or 90s.

Finally, a third type of distribution is a rectangular distribution, as shown in the

right-hand side of Figure 3.7. A rectangular distribution is a symmetrical distribution

shaped like . . . are you ready? . . . like a rectangle! There are no discernible tails be-

cause the frequencies of all scores are the same.

Labeling Frequency Distributions

You need to know the names of the previous distributions because descriptive statistics

describe the characteristics of data, and one very important characteristic is the shape

of the distribution that the data form. Thus, although I might have data containing many

different scores, if, for example, I tell you they form a normal distribution, you can

mentally envision the distribution and quickly and easily understand what the scores

are like: Few scores are very low or very high, with the most common, frequent scores

in the middle. Therefore, the first step when examining any data is to identify the shape

of the simple frequency distribution that they form.

REMEMBER The shape of the frequency distribution that scores form is an

important characteristic of the data.

Recognize, however, that data in the real world will never form the perfect shapes

that we’ve discussed. Instead, the scores will form a bumpy, rough approximation to

the ideal distribution. For example, data never form a perfect normal curve and, at best,

only come close to that shape. However, rather than drawing a different, approximately

normal curve every time, we simplify the task by envisioning the ideal normal curve as

our one “model” of any distribution that generally has this shape. Likewise, we envi-

sion the ideal shape when discussing the other common curves that we’ve seen.

Thus, we apply the names of our ideal distributions to actual data as a way of sum-

marizing and communicating their general shape. For example, Figure 3.8 contains

some frequency distributions that might be produced in research, and the correspon-

ding labels we might use. (Notice that we even apply these names to histograms or bar

graphs.) We assume that the sample represents a population that more closely fits the

corresponding ideal polygon: If we measure the population, the additional scores and

their corresponding frequencies should “fill in” the sample curve, smoothing it out to

be closer to the ideal curve.

Types of Simple Frequency Distributions 45

FIGURE 3.7

Idealized bimodal and

rectangular distributions

Low . . . Middle . . . High

Bimodal

scores

Low . . . Middle . . . High

Rectangular

scores

46 CHAPTER 3 / Frequency Distributions and Percentiles

A QUICK REVIEW

■

The most common frequency distributions are nor-

mal distributions, negatively or positively skewed

distributions, bimodal distributions, and rectangu-

lar distributions.

MORE EXAMPLES

The variable of intelligence (IQ) usually forms a

normal distribution: The most common scores are

in the middle, with higher or lower IQs occurring

progressively less often. If IQ was positively

skewed, there would be only one distinct tail, lo-

cated at the higher scores. If IQ was negatively

skewed, there would be only a distinct tail at the

lower scores. If IQ formed a bimodal distribution,

there would be two distinct parts of the curve con-

taining the highest-frequency scores. If IQ formed a

rectangular distribution, each score would have the

same frequency.

For Practice

1. Arrange the scores below from most frequent to

least frequent.

AB C D

FIGURE 3.8

Simple frequency distri-

butions of sample data

with appropriate labels

Low . . . Middle . . . High

Normal

scores

Low . . . Middle . . . High

Positively skewed

scores

Low . . . Middle . . . High

Bimodal

scores

Low . . . Middle . . . High

Negatively skewed

scores

continued

Relative Frequency and the Normal Curve 47

2. What label should be given to each of the following?

f f

Scores

(a)

Scores

(b)

Scores

(c)

Scores

(d)

Answers

1. C, B, A, D

2. a. positively skewed; b. bimodal; c. normal;

d. negatively skewed

RELATIVE FREQUENCY AND THE NORMAL CURVE

We will return to simple frequency distributions—especially the normal curve—throughout

the remainder of this course. However, counting the frequency of scores is not the only thing

we do. Another important procedure is to describe scores using relative frequency.

Relative frequency is the proportion of that is made up by a score’s simple frequency.

Recall that a proportion indicates a fraction of the total, so relative frequency indicates

the fraction of the entire sample that is made up by the times that a score occurs. Thus,

whereas simple frequency is the number of times a score occurs, relative frequency is the

proportion of time the score occurs. The symbol for relative frequency is

We’ll first calculate relative frequency using a formula so that you understand its

math, although later we’ll compute it using a different approach. Here is your first sta-

tistical formula.

rel. f.

The formula for computing a score’s relative frequency is

rel. f 5