Quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile than the number of groups created. Common quantiles have special names, such as quartiles (four groups), deciles (ten groups), and percentiles (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. q-quantiles are values that partition a finite set of values into q subsets of (nearly) equal sizes. There are q − 1 partitions of the q-quantiles, one for each integer k satisfying 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 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 q-quantiles are the application of the quantile function (the inverse function of the cumulative distribution function) to the values {1/q, 2/q, ..., (q − 1)/q}. Some q-quantiles have special names: The only 2-quantile is called the median The 3-quantiles are called tertiles or terciles → T The 4-quantiles are called quartiles → Q; the difference between upper and lower quartiles is also called the interquartile range, midspread or middle fifty → IQR = Q3 − Q1. The 5-quantiles are called quintiles or pentiles → QU The 6-quantiles are called sextiles → S The 7-quantiles are called septiles → SP The 8-quantiles are called octiles → O The 10-quantiles are called deciles → D The 12-quantiles are called duo-deciles or dodeciles → DD The 16-quantiles are called hexadeciles → H The 20-quantiles are called ventiles, vigintiles, or demi-deciles → V The 100-quantiles are called percentiles or centiles → P The 1000-quantiles have been called permilles or milliles, but these are rare and largely obsolete As in the computation of, for example, standard deviation, the estimation of a quantile depends upon whether one is operating with a statistical population or with a sample drawn from it.