11.6 Percentiles

Percentiles split the data into 100 parts. Let us consider the pdf of Figure 11.15. The area to the left of point x _a is a; we denote this area as a percentile p _a. Typical percentile values are 0.05 or 5%, 0.10 or 10%, 0.95 or 95%, and 0.99 or 99% and these are denoted as p _0.05, p _0.10, , p _0.95, or p _0.99, respectively.

Figure 11.15: Sketch to illustrate the concept of percentile

For continuous functions, the jth percentile p _j is defined as

Example 11.15

Compute p _0.90, p _0.95, and p _0.99 for the cdf of the exponential distribution of Figure 11.16 if ? = 1/ 2.

Figure 11.16: Cdf for Example 11.15

Solution:

From (11.90),

or

Taking the natural log of both sides of the above expression, we get

or

Then,

For a pdf of the discrete type, we can get approximate values of percentiles, if we first arrange the sample in ascending order of magnitude; then, we compute the jth percentile p _j, j = 1, 2, 99 from the relation

where n = number of samples.

If i in (11.91) turns out to be an integer, the jth percentile is the average of the observations in positions i and i +