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Quantile
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{{Short description|Statistical method of dividing data into equal-sized intervals for analysis}} [[File:Iqr with quantile.png|thumb|Probability density of a [[normal distribution]], with quantiles, {{math|''Q''<sub>1</sub>}}, {{math|''Q''<sub>2</sub>}}, and {{math|''Q''<sub>3</sub>}}, are shown. The area below the red curve is the same in the intervals {{math|(ββ,''Q''<sub>1</sub>)}}, {{math|(''Q''<sub>1</sub>,''Q''<sub>2</sub>)}}, {{math|(''Q''<sub>2</sub>,''Q''<sub>3</sub>)}}, and {{math|(''Q''<sub>3</sub>,+β)}}.]] In [[statistics]] and [[probability]], '''quantiles''' are cut points dividing the [[Range (statistics)|range]] of a [[probability distribution]] into continuous intervals with equal probabilities or dividing the [[Observation (statistics)|observations]] in a [[Sample (statistics)|sample]] in the same way. There is one fewer quantile than the number of groups created. Common quantiles have special names, such as ''[[quartile]]s'' (four groups), ''[[decile]]s'' (ten groups), and ''[[percentile]]s'' (100 groups). The groups created are termed halves, thirds, quarters, etc., though sometimes the terms for the quantile are used for the groups created, rather than for the cut points. '''{{mvar|q}}'''-'''quantiles''' are values that [[Partition of a set|partition]] a [[finite set]] of values into {{mvar|q}} [[subset]]s of (nearly) equal sizes. There are {{math|''q'' β 1}} partitions of the {{mvar|q}}-quantiles, one for each [[integer]] {{mvar|k}} satisfying {{math|0 < ''k'' < ''q''}}. In some cases the value of a quantile may not be uniquely determined, as can be the case for the [[median]] (2-quantile) of a uniform probability distribution on a set of even size. Quantiles can also be applied to [[Continuous function|continuous]] distributions, providing a way to generalize [[rank statistics]] to continuous variables (see [[percentile rank]]). When the [[cumulative distribution function]] of a [[random variable]] is known, the {{mvar|q}}-quantiles are the application of the'' [[quantile function]]'' (the [[inverse function]] of the [[cumulative distribution function]]) to the values {{math|{1/''q'', 2/''q'', β¦, (''q'' β 1)/''q''}}}.
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