1. Find the range, that is, the difference between the largest and smallest observations. The range of weights in Table 1.1 is 245 133 = 112.
2. Find the class interval required to span the range by dividing the range by the desired number of classes (ordinarily 10). In the present example,
Class erval range
desired number of classes
intee = =112=
10 11 2. 3. Round off to the nearest convenient interval (such as 1, 2, 3, . . .
10, particularly 5 or 10 or multiples of 5 or 10). In the present example, the nearest convenient interval is 10.
4. Determine where the lowest class should begin. (Ordinarily, this number should be a multiple of the class interval.) In the present example, the smallest score is 133, and therefore the lowest class should begin at 130, since 130 is a multiple of 10 (the class interval).
5. Determine where the lowest class should end by adding the class interval to the lower boundary and then subtracting one unit of measurement. In the present example, add 10 to 130 and then subtract 1, the unit of measurement, to obtain 139—the number at which the lowest class should end.
6. Working upward, list as many equivalent classes as are required to include the largest observation. In the present example, list 130–139, 140–149, . . . , 240–249, so that the last class includes 245, the largest score.
7. Indicate with a tally the class in which each observation falls.
For example, the first score in Table 1.1, 160, produces a tally next to 160–169; the next score, 193, produces a tally next to 190–199;
and so on.
8. Replace the tally count for each class with a number—the frequency (f )—and show the total of all frequencies. (Tally marks are not usually shown in the final frequency distribution.)
9. Supply headings for both columns and a title for the table.
M i g h t E x c l u d e f r o m S u m m a r i e s
You might choose to segregate (but not to suppress!) an outlier from any summary of the data. For example, you might relegate it to a footnote instead of using exces- sively wide class intervals in order to include it in a frequency distribution. Or you might use various numerical summaries, such as the median and interquartile range, to be discussed in Chapters 3 and 4, that ignore extreme scores, including outliers.
M i g h t E n h a n c e U n d e r s t a n d i n g
Insofar as a valid outlier can be viewed as the product of special circumstances, it might help you to understand the data. For example, you might understand better why crime rates differ among communities by studying the special circumstances that produce a community with an extremely low (or high) crime rate, or why learning rates differ among third graders by studying a third grader who learns very rapidly (or very slowly).
Progress Check *2.4 Identify any outliers in each of the following sets of data collected from nine college students.
SUMMER INCOME AGE FAMILY SIZE GPA
$6,450 20 2 2.30
$4,820 19 4 4.00
$5,650 61 3 3.56
$1,720 32 6 2.89
$600 19 18 2.15
$0 22 2 3.01
$3,482 23 6 3.09
$25,700 27 3 3.50
$8,548 21 4 3.20
Answers on page 421.
2 . 4 R E L AT I V E F R E Q U E N C Y D I S T R I B U T I O N S
An important variation of the frequency distribution is the relative frequency distribution.
Relative frequency distributions show the frequency of each class as a part or fraction of the total frequency for the entire distribution.
This type of distribution allows us to focus on the relative concentration of observa- tions among different classes within the same distribution. In the case of the weight data in Table 2.2, it permits us to see that the 160s account for about one-fourth (12/53 = 23, or 23%) of all observations. This type of distribution is especially helpful when you must compare two or more distributions based on different total numbers of observations. For instance, as in Review Question 2.17, you might want to compare the distribution of ages for 500 residents of a small town with that for the approximately 300 million residents of the United States. The conversion to relative frequencies allows a direct comparison of the shapes of these two distributions without having to adjust for the radically different total numbers of observations.
C o n s t r u c t i n g R e l a t i v e F r e q u e n c y D i s t r i b u t i o n s
To convert a frequency distribution into a relative frequency distribution, divide the frequency for each class by the total frequency for the entire distribution. Table 2.5 illustrates a relative frequency distribution based on the weight distribution of Table 2.2.
Relative Frequency Distribution A frequency distribution showing the frequency of each class as a fraction of the total frequency for the entire distribution.
2 . 4 R E L AT I V E F R E Q U E N C Y D I S T R I B U T I O N S 2 9
The conversion to proportions is straightforward. For instance, to obtain the proportion of .06 for the class 130–139, divide the frequency of 3 for that class by the total fre- quency of 53. Repeat this process until a proportion has been calculated for each class.
P e r c e n t a g e s o r P r o p o r t i o n s ?
Some people prefer to deal with percentages rather than proportions because percentages usually lack decimal points. A proportion always varies between 0 and 1, whereas a percentage always varies between 0 percent and 100 percent. To convert the relative frequencies in Table 2.5 from proportions to percentages, multiply each proportion by 100; that is, move the decimal point two places to the right. For example, multiply .06 (the proportion for the class 130–139) by 100 to obtain 6 percent.
Progress Check *2.5 GRE scores for a group of graduate school applicants are distrib- uted as follows:
GRE f
725–749 1
700–724 3
675–699 14
650–674 30
625–649 34
600–624 42
575–599 30
550–574 27
525–549 13
500–524 4
475–499 2
Total 200
Convert to a relative frequency distribution. When calculating proportions, round numbers to two digits to the right of the decimal point, using the rounding procedure specified in Section A.7 of Appendix A.
Answers on page 421.
Table 2.5
RELATIVE FREQUENCY DISTRIBUTION
WEIGHT f RELATIVE f
240–249 1 .02
230–239 0 .00
220–229 3 .06
210–219 0 .00
200–209 2 .04
190–199 4 .08
180–189 3 .06
170–179 7 .13
160–169 12 .23
150–159 17 .32
140–149 1 .02
130–139 3 .06
Total 53 1.02*
* The sum does not equal 1.00 because of rounding-off errors.
2 . 5 C U M U L AT I V E F R E Q U E N C Y D I S T R I B U T I O N S
Cumulative frequency distributions show the total number of observations in each class and in all lower-ranked classes.
This type of distribution can be used effectively with sets of scores, such as test scores for intellectual or academic aptitude, when relative standing within the distribu- tion assumes primary importance. Under these circumstances, cumulative frequencies are usually converted, in turn, to cumulative percentages. Cumulative percentages are often referred to as percentile ranks.
C o n s t r u c t i n g C u m u l a t i v e F r e q u e n c y D i s t r i b u t i o n s
To convert a frequency distribution into a cumulative frequency distribution, add to the frequency of each class the sum of the frequencies of all classes ranked below it. This gives the cumulative frequency for that class. Begin with the lowest-ranked class in the frequency distribution and work upward, finding the cumulative frequen- cies in ascending order. In Table 2.6, the cumulative frequency for the class 130–139 is 3, since there are no classes ranked lower. The cumulative frequency for the class 140–149 is 4, since 1 is the frequency for that class and 3 is the frequency of all lower-ranked classes. The cumulative frequency for the class 150–159 is 21, since 17 is the frequency for that class and 4 is the sum of the frequencies of all lower-ranked classes.
Table 2.6
CUMULATIVE FREQUENCY DISTRIBUTION
WEIGHT f CUMULATIVE f
CUMULATIVE PERCENT
240–249 1 53 100
230–239 0 52 98
220–229 3 52 98
210–219 0 49 92
200–209 2 49 92
190–199 4 47 89
180–189 3 43 81
170–179 7 40 75
160–169 12 33 62
150–159 17 21 40
140–149 1 4 8
130–139 3 3 6
Total 53
C u m u l a t i v e P e r c e n t a g e s
As has been suggested, if relative standing within a distribution is particularly important, then cumulative frequencies are converted to cumulative percentages.
A glance at Table 2.6 reveals that 75 percent of all weights are the same as or lighter than the weights between 170 and 179 lbs. To obtain this cumulative percentage (75%), the cumulative frequency of 40 for the class 170–179 should be divided by the total frequency of 53 for the entire distribution.
Cumulative Frequency Distribution
A frequency distribution showing the total number of observations in each class and all lower-ranked classes.
2 . 6 F R E Q U E N C Y D I S T R I B U T I O N S F O R Q U A L I TAT I V E ( N O M I N A L ) D ATA 3 1
Progress Check *2.6
(a) Convert the distribution of GRE scores shown in Question 2.5 to a cumulative frequency distribution.
(b) Convert the distribution of GRE scores obtained in Question 2.6(a) to a cumulative percent frequency distribution.
Answers on page 421.
P e r c e n t i l e R a n k s
When used to describe the relative position of any score within its parent distribu- tion, cumulative percentages are referred to as percentile ranks. The percentile rank of a score indicates the percentage of scores in the entire distribution with similar or smaller values than that score. Thus a weight has a percentile rank of 80 if equal or lighter weights constitute 80 percent of the entire distribution.
A p p r o x i m a t e P e r c e n t i l e R a n k s ( f r o m G r o u p e d D a t a )
The assignment of exact percentile ranks requires that cumulative percentages be obtained from frequency distributions for ungrouped data. If we have access only to a frequency distribution for grouped data, as in Table 2.6, cumulative percentages can be used to assign approximate percentile ranks. In Table 2.6, for example, any weight in the class 170–179 could be assigned an approximate percentile rank of 75, since 75 is the cumulative percent for this class.
Progress Check *2.7 Referring to Table 2.6, find the approximate percentile rank of any weight in the class 200–209.
Answers on page 422.