# 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 *k* percent
of values from the largest (100-*k*) percent.

Special percentiles are the median (50th percentile),
the quartiles (25th and 75th percentile),
the quintiles (20th, 40th, 60th and 80th percentile),
and the deciles (the *k*-th decile is the (10*k*)-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 (for a given *k* = 1,2,...,99) if

In this definition, *P* is a probability distribution on the real numbers.
It may be obtained either

- from a (theoretical) probability measure (such as the normal or Poisson distribution), or
- from a finite population where it expresses the probability of a random element to have the property,

i.e., it is the relative frequency of elements with this property (number of elements with the property divided by the size of the population),or - from a sample of size
*N*where it also is the relative frequency which is used to estimate the corresponding percentile for the population from which it was taken.

## Special cases

For most standard continuous distributions (like the normal distribution) the
*k*-th percentile *x* is uniquely determined by

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

or that there is a gap in the range of the variable such that, for two distinct , equality holds:

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

is a *k*-th percentile.

## Examples

The following examples illustrate this:

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

- .

- Then the unique
*k*-th percentile is .

- If there are only 100 values

- .

- Then any value between and is a
*k*-th percentile.

**Example from the praxis:**

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 other 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 she
and that approximately 20% of the students had a higher score than she had.