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

52 CHAPTER 2 | Frequency Distributions
percentages that appear in the table. Using the table in Example 2.7, for example, you
should be able to answer the following questions:
1. What is the 95th percentile? (Answer: X = 4.5.)
2. What is the percentile rank for X = 3.5? (Answer: 70%.)
However, there are many values that do not appear directly in the table, and it is impossible
to determine these values precisely. Referring to the table in Example 2.7 again,
1. What is the 50th percentile?
2. What is the percentile rank for X = 4?
Because these values are not specifically reported in the table, you cannot answer the
questions. However, it is possible to estimate these intermediate values by using a procedure
known as interpolation.
Before we apply the process of interpolation to percentiles and percentile ranks, we will
use a simple, commonsense example to introduce this method. Suppose that your friend
offers you $60 to work for 8 hours on Saturday helping with spring cleaning in the house
and yard. On Saturday morning, however, you realize that you have an appointment in the
afternoon and will have to quit working after only 4 hours. What is a fair amount for your
friend to pay you for 6 hours of work? Because this is a working-for-pay example, your
automatic response probably is to calculate the hourly rate of pay —how many dollars per
hour you are getting. However, the process of interpolation offers an alternative method for
finding the answer. We begin by noting that the original total amount of time was 8 hours.
You worked 4 hours, which corresponds to 1 2 of the total time. Therefore, a fair payment
would be 1 2 of the original total amount: 1 2 of $60 is $30.
The process of interpolation is pictured in Figure 2.13. In the figure, the line on the left
shows the time for your agreed work, from 0 up to 8 hours, and the line on the right shows
the agreed pay, from 0 to $60. We also have marked different fractions along the way.
Using the figure, try answering the following questions about time and pay.
1. How long should you work to earn $45?
2. How much should you be paid after working 2 hours?
If you got answers of 6 hours and $15, you have mastered the process of
interpolation.
Notice that interpolation provides a method for finding intermediate values—that is,
values that are located between two specified numbers. This is exactly the problem we
faced with percentiles and percentile ranks. Some values are given in the table, but others
are not. Also notice that interpolation only estimates the intermediate values. The basic
assumption underlying interpolation is that there is a constant rate of change from one end
of the interval to the other. In the working-for-pay example, we assume a constant rate of
pay for the entire job. Because interpolation is based on this assumption, the values we
calculate are only estimates. The general process of interpolation can be summarized as
follows:
1. A single interval is measured on two separate scales (for example, time and dollars).
The endpoints of the interval are known for each scale.
2. You are given an intermediate value on one of the scales. The problem is to find the
corresponding intermediate value on the other scale.
3. The interpolation process requires four steps:
a. Find the width of the interval on both scales.