Loss Models: From Data to Decisions, 4th Edition

3.2 Percentiles

One other value of interest that may be derived from the distribution function is the percentile function.⁴ It is the inverse of the distribution function, but because this quantity is not always well defined, an arbitrary definition must be created.

Definition 3.6 The 100pth percentile of a random variable is any value π_p such that F(π_p —) ≤ p ≤ F(π_p). The 50th percentile, π_0.5 is called the median.

This quantity is sometimes referred to as a quantile. Generally the former uses values between 0 and 1 while the latter uses values between 0 and 100. Thus the 70th percentile and the 0.7 quantile are the same. However, regardless of the term used, we will always use decimal values when using subscripts. So it will be π_0.5 and never π₅₀. If the distribution function has a value of p for one and only one x value, then the percentile is uniquely defined. In addition, if the distribution function jumps from a value below p to a value above p, then the percentile is at the location of the jump. The only time the percentile is not uniquely defined is when the distribution function is constant at a value of p over a range of values of the random variable. In that case, any value in that range (including both endpoints) can be used as the percentile.

EXAMPLE 3.6

Determine the 50th and 80th percentiles for Models 1 and 3.

For Model 1, the pth percentile can be obtained from ...