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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 equalsized data subsets is the motivation for qquantiles; the quantiles are the data values marking the boundaries between consecutive subsets. Put another way, the k^{th} qquantile 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 (q − k) / q. There are q − 1 of the qquantiles, one for each integer k satisfying 0 < k < q.
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Specialized quantiles
Some qquantiles have special names:^{[citation needed]}
 The 2quantile is called the median
 The 3quantiles are called tertiles or terciles → T
 The 4quantiles are called quartiles → Q
 The 5quantiles are called quintiles → QU
 The 6quantiles are called sextiles → S
 The 10quantiles are called deciles → D
 The 12quantiles are called duodeciles → Dd
 The 20quantiles are called vigintiles → V
 The 100quantiles are called percentiles → P
 The 1000quantiles 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 qquantile is the data value where the cumulative distribution function crosses k / q. That x is a kth qquantile for a variable X if
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