Percentile

Percentiles are statistical parameters which describe the distribution of a (real) value in a population (or a sample). Roughly speaking, the k-th percentile separates the smallest p percent of values from the largest (100-p) percent.

Special percentiles are the median (50th percentile), the quartiles (25th and 75th percentile), and the deciles (the k-th decile is the (10k)-th percentile). Percentiles are special cases of quantiles: The k-th percentile is the same as the (k/100)-quantile.

Definition

The value x is k-th percentile if

$P(\omega \leq x)\geq {k \over 100}{\textrm {\ \ and\ \ }}P(\omega \geq x)\leq 1-{k \over 100}$

Special cases

For a continuous distribution (like the normal distribution) the k-th percentile x is uniquely determined by

$P(\omega \leq x)={k \over 100}{\textrm {\ \ and\ \ }}P(\omega \geq x)=1-{k \over 100}$

In the general case (e.g., for discrete distributions, or for finite samples) it may happen that the separating value has positive probability:

$P(\omega =x)>0\Rightarrow P(\omega \leq x)>{k \over 100}{\textrm {\ \ and\ \ }}P(\omega \geq x)>1-{k \over 100}$

or that there are two distinct values for which equality holds $x_{1}<x_{2}$ such that

$P(\omega \leq x_{1})={k \over 100}{\textrm {\ \ and\ \ }}P(\omega \geq x_{2})=1-{k \over 100}$

Then every value in the (closed) intervall between the smallest and the largest such value

$\left[\min \left\{x{\Bigl \vert }P(\omega \leq x)={k \over 100}\right\},\max \left\{x{\Bigl \vert }P(\omega \geq x)=1-{k \over 100}\right\}\right]$

is a k-th percentiles.

Example

The following examples illustrates this:

• Take a sample of 101 values, ordered according to their size:

$x_{1}\leq x_{2}\leq \dots \leq x_{100}\leq x_{101}$.

Then the unique k-th percentile is $x_{k+1}$.

• If there are only 100 values

$x_{1}\leq x_{2}\leq \dots \leq x_{99}\leq x_{100}$.

Then any value between $x_{k}$ and $x_{k+1}$ is a k-th percentile.

Educational institutions (i.e. universities, schools...) frequently report admission test scores in terms of percentiles. For instance, assume that a candidate obtained 85 on her verbal test. The question is how did this student compared to all others students? If she is told that her score correspond to the 80th percentile, we know that approximately 80% of the other candidates scored lower than him and that approximately 20% of the students had higher score than her.