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Quantiles are points taken at regular intervals from the cumulative distribution function (CDF) of a random variable. Dividing ordered data into q essentially equal-sized data subsets is the motivation for q-quantiles; the quantiles are the data values marking the boundaries between consecutive subsets. Put another way, the kth q-quantile for a random variable is the value x such that the probability that the random variable will be less than x is at most k / q and the probability that the random variable will be more than x is at most (qk) / q. There are q − 1 of the q-quantiles, one for each integer k satisfying 0 < k < q.


Specialized quantiles

Some q-quantiles have special names:[citation needed]

  • The 2-quantile is called the median
  • The 3-quantiles are called tertiles or terciles → T
  • The 4-quantiles are called quartiles → Q
  • The 5-quantiles are called quintiles → QU
  • The 6-quantiles are called sextiles → S
  • The 10-quantiles are called deciles → D
  • The 12-quantiles are called duo-deciles → Dd
  • The 20-quantiles are called vigintiles → V
  • The 100-quantiles are called percentiles → P
  • The 1000-quantiles are called permilles → Pr

More generally, one can consider the quantile function for any distribution. This is defined for real variables between zero and one and is mathematically the inverse of the cumulative distribution function..

Quantiles of a population

For a population of discrete values or for a continuous population density the kth q-quantile is the data value where the cumulative distribution function crosses k / q. That x is a kth q-quantile for a variable X if

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