# Percentile

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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 (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 (for a given k = 1,2,...,99) if

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

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

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

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

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

or that there is a gap in the range of the variable such that, for two distinct ${\displaystyle x_{1}, equality holds:

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

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

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

is a k-th percentile.

## Examples

The following examples illustrate this:

• Take a sample of 101 values, ordered according to their size:
${\displaystyle x_{1}\leq x_{2}\leq \dots \leq x_{100}\leq x_{101}}$.
Then the unique k-th percentile is ${\displaystyle x_{k+1}}$.
• If there are only 100 values
${\displaystyle x_{1}\leq x_{2}\leq \dots \leq x_{99}\leq x_{100}}$.
Then any value between ${\displaystyle x_{k}}$ and ${\displaystyle x_{k+1}}$ 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.