Section 1.2 Percentiles
Before turning our attention to some basic graphical tools we use to visualize qualitative and quantitative variables, it is helpful to also briefly go over percentiles since they will be used in some of these tools. Many of us have probably encountered percentiles before in the context of standardized exam testing. A test score in and of itself is usually difficult to interpret. For example, if you learned that your score on a measure of shyness was 35 out of a possible 50, you would have little idea how shy you are compared to other people. More relevant is the percentage of people with lower shyness scores than yours. This percentage is called a percentile. If 65% of the scores were below yours, then your score would be the 65th percentile.
Subsection 1.2.1 Three Alternative 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. This is the way we defined it above and we will call this “Definition 1.” 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.” Though these two definitions appear very similar, they can sometimes 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 is a weighted average of the percentiles computed according to the first two definitions. The details of computing percentiles under this third definition are a bit complicated, but fortunately, statistical software can easily do the calculations for us. Since it is unlikely you will need to compute percentiles by hand, we leave the details of these computations to the appendix appearing at the end of this chapter. Despite its complexity, the third definition handles rounding more gracefully than the other two and has the advantage that it allows the median to be defined conveniently as the 50th percentile. Unless otherwise specified, when we refer to “percentile,” we will be referring to this third definition of percentiles.
