Percentiles
Prerequisites
none
Learning Objectives
 Define percentiles
 Use three formulas for computing percentiles
Two Simple Definitions of Percentile
There is no universally accepted definition of
a percentile. Using the 65th percentile as an example, the 65th
percentile can be defined as the lowest score that is greater
than 65% of the scores. The 65th percentile can also be defined
as the smallest score that is greater than or equal to 65% of
the scores. This we will call "Definition 2." Unfortunately,
these two definitions can lead to dramatically different results,
especially when there is relatively little data. Moreover,
neither of these definitions is explicit about how to handle
rounding. For instance, what rank is required to be higher
than 65% of the scores when the total number of scores is 50?
This is tricky because 65% of 50 is 32.5. How do we find the
lowest number that is higher than 32.5% of the scores? A third
way to compute percentiles (presented below), is a weighted
average of the percentiles computed according to the first two
definitions. This third definition handles rounding more gracefully
than the other two and has the advantage that it allows the median
(discussed in Chapter 3) to be defined conveniently as the 50th
percentile.
A Third Definition
Unless otherwise specified, when we refer to "percentile,"
we will be referring to this third definition of percentiles.
Let's begin with an example. Consider the 25th percentile for
the 8 numbers in Table 1. Notice the numbers are given ranks ranging
from 1 for the lowest number to 8 for the highest number.
The first step is to compute the rank (R) of the
25th percentile. This is done using the following formula:
R = P/100 x (N + 1)
where P is the desired
percentile (25 in this case) and N is the number of numbers (8
in this case). Therefore,
R = 25/100 x (8 + 1) = 9/4 = 2.25.
If R were an integer, the Pth percentile would
be the number with rank R. When R is not an integer, we compute
the Pth percentile by interpolation as follows:
 Define IR as the integer portion of R (the number to the
left of the decimal point). For this example, IR = 2.
 Define FR as the fractional portion or R. For this example,
FR = 0.25.
 Find the scores with Rank IR and
with Rank IR + 1. For this example,
this means the score with Rank 2 and the score with Rank 3.
The scores are 5 and 7.
 Interpolate by multiplying the difference between the scores
by FR and add the result to the lower
score. For these data, this is (0.25)(7  5) + 5 = 5.5.
Therefore, the 25th percentile is 5.5. If we had
used the first definition (the smallest score greater than 25%
of the scores) the 25th percentile would have been 7. If we had
used the second definition ( the smallest score greater than or
equal to 25% of the scores) the 25th percentile would have been
5.
