Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 2.4 | Percentiles, Percentile Ranks, and Interpolation 51
individuals (14 out of 20) had scores of X = 3 or lower. Cumulative percentages can be
computed by
c% 5 cf
N s100%d
X f Cf c%
5 1 20 100%
4 5 19 95%
3 8 14 70%
2 4 6 30%
1 2 2 10%
■
The cumulative percentages in a frequency distribution table give the percentage of
individuals with scores at or below each X value. However, you must remember that the X
values in the table are usually measurements of a continuous variable and, therefore, represent
intervals on the scale of measurement (see page 20). A score of X = 2, for example,
means that the measurement was somewhere between the real limits of 1.5 and 2.5. Thus,
when a table shows that a score of X = 2 has a cumulative percentage of 30%, you should
interpret this as meaning that 30% of the individuals have been accumulated by the time
you reach the top of the interval for X = 2. Notice that each cumulative percentage value is
associated with the upper real limit of its interval. This point is demonstrated in Figure 2.12,
which shows the same data that were used in Example 2.7. Figure 2.12 shows that two
people, or 10%, had scores of X = 1; that is, two people had scores between 0.5 and 1.5.
You cannot be sure that both individuals have been accumulated until you reach 1.5, the
upper real limit of the interval. Similarly, a cumulative percentage of 30% is reached at 2.5
on the scale, a percentage of 70% is reached at 3.5, and so on.
■ Interpolation
It is possible to determine some percentiles and percentile ranks directly from a frequency
distribution table, provided the percentiles are upper real limits and the ranks are
cf = 20
cf = 19
FIGURE 2.12
The relationship between
cumulative frequencies
(cf values) and upper real
limits. Notice that two
people have scores of
X = 1. These two individuals
are located between the
real limits of 0.5 and 1.5.
Although their exact locations
are not known, you can be
certain that both had scores
below the upper limit of 1.5.
cf = 0
cf = 2
X = 1
f = 2
cf = 6
X = 2
f = 4
cf = 14
X = 3
f = 8
X = 4
f = 5
X = 5
f = 1
0.5 1.5 2.5 3.5 4.5 5.5