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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.

[edit] Definition

The value x is k-th percentile (for a given k = 1,2,...,99) if

 P(\omega\le x) \ge {k\over100}    \textrm{\ \ and \ \ }
            P(\omega\ge x) \ge 1-{k\over100} \, \textrm{.}

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

[edit] Special cases

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

 P(\omega\le x) = {k\over100}    \textrm{\ \ and \ \ }
            P(\omega\ge x) = 1-{k\over100}

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\le x) > {k\over100}    \textrm{\ \ or \ \ }
            P(\omega\ge x) > 1-{k\over100}

or that there is a gap in the range of the variable such that, for two distinct x1 < x2, equality holds:

 P(\omega\le x_1) = {k\over100}    \textrm{\ \ and \ \ }
            P(\omega\ge x_2) = 1-{k\over100}

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

 \left [ \min \left\{ x \Bigl\vert P(\omega\le x) = {k\over100} \right\},
                 \max \left\{ x \Bigl\vert P(\omega\ge x) = 1-{k\over100} \right\} \right]

is a k-th percentile.

[edit] Examples

The following examples illustrate this:

 x_1 \le x_2 \le \dots \le x_{100} \le x_{101} .
Then the unique k-th percentile is xk + 1.
 x_1 \le x_2 \le \dots \le x_{99} \le x_{100} .
Then any value between xk and xk + 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.

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