# Quantile

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## quantile

[′kwän‚tīl]*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Quantile

one of the numerical characteristics of random quantities used in mathematical statistics. If a distribution function of a random quantity *X* is continuous, then the quantile K_{p} of order *p* is defined as that number for which the probability of the inequality *X* < *K*<_{p} is equal to *p*. From the definition of quantile it follows that the probability of the inequality *K*_{p} < *X* < *K*′’_{p} is equal to *p′* − *p*. The quantile K_{½} is the median of the random quantity *X*. The quantiles *K*_{¼} and *K*_{¾} are called quartiles, and *K*_{0.1}, *K*_{0.2}, … *E*_{0.9} are called deciles. Knowledge of the quantiles for suitably selected values of *p* makes it possible to visualize the distribution function.

For example, for the normal distribution (see Figure 1)

the graph of the function Φ (*x*) may be plotted by means of the deciles *K*_{0.1} = −1.28, *K*_{0.2} = −0.84, *K*_{0.3} = –0.52, *K*_{0.4} = –0.25 *K*_{0.5} = 0.25, *K*_{0.7}= 0.52, *K*_{0.8} = 0.52, *K*_{0.8} = 0.84, and *K*_{0.9} =1.28. The quartiles of the normal distribution Φ(*x*) are K_{¼} = − 0.67 and *K* = _{0.67}.