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. 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 more often 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 subsets 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}.
Quantiles of a population 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. For a population, of discrete values or for a continuous population density, the k-th q-quantile is the data value where the cumulative distribution function crosses k/q. That is, x is a k-th q-quantile for a variable X if
Pr[X < x] ≤ k/q or, equivalently, Pr[X ≥ x] ≥ 1 − k/q and
Pr[X ≤ x] ≥ k/q where Pr is the probability function. For a finite population of N equally probable values indexed 1, …, N from lowest to highest, the k-th q-quantile of this population can equivalently be computed via the value of Ip = N k/q. If Ip is not an integer, then round up to the next integer to get the appropriate index; the corresponding data value is the k-th q-quantile. On the other hand, if Ip is an integer then any number from the data value at that index to the data value of the next index can be taken as the quantile, and it is conventional (though arbitrary) to take the average of those two values (see Estimating quantiles from a sample). If, instead of using integers k and q, the "p-quantile" is based on a real number p with 0 < p < 1 then p replaces k/q in the above formulas. This broader terminology is used when quantiles are used to parameterize continuous probability distributions. Moreover, some software programs (including Microsoft Excel) regard the minimum and maximum as the 0th and 100th percentile, respectively. However, this broader terminology is an extension beyond traditional statistics definitions.
Examples The following two examples use the Nearest Rank definition of quantile with rounding. For an explanation of this definition, see percentiles.
Even-sized population Consider an ordered population of 10 data values [3, 6, 7, 8, 8, 10, 13, 15, 16, 20]. What are the 4-quantiles (the "quartiles") of this dataset?
So the first, second and third 4-quantiles (the "quartiles") of the dataset [3, 6, 7, 8, 8, 10, 13, 15, 16, 20] are [7, 9, 15]. If also required, the zeroth quartile is 3 and the fourth quartile is 20.
Odd-sized population Consider an ordered population of 11 data values [3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20]. What are the 4-quantiles (the "quartiles") of this dataset?
So the first, second and third 4-quantiles (the "quartiles") of the dataset [3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20] are [7, 9, 15]. If also required, the zeroth quartile is 3 and the fourth quartile is 20.
Relationship to the mean For any population probability distribution on finitely many values, and generally for any probability distribution with a mean and variance, it is the case that
μ − σ ⋅ 1 − p p ≤ Q ( p ) ≤ μ + σ ⋅ p 1 − p , {\displaystyle \mu -\sigma \cdot {\sqrt {\frac {1-p}{p}}}\leq Q(p)\leq \mu +\sigma \cdot {\sqrt {\frac {p}{1-p}}}\,,}
where Q(p) is the value of the p-quantile for 0 < p < 1 (or equivalently is the k-th q-quantile for p = k/q), where μ is the distribution's arithmetic mean, and where σ is the distribution's standard deviation. In particular, the median (p = k/q = 1/2) is never more than one standard deviation from the mean. The above formula can be used to bound the value μ + zσ in terms of quantiles. When z ≥ 0, the value that is z standard deviations above the mean has a lower bound
μ + z σ ≥ Q ( z 2 1 + z 2 ) , f o r z ≥ 0. {\displaystyle \mu +z\sigma \geq Q\left({\frac {z^{2}}{1+z^{2}}}\right)\,,\mathrm {~for~} z\geq 0.}
For example, the value that is z = 1 standard deviation above the mean is always greater than or equal to Q(p = 0.5), the median, and the value that is z = 2 standard deviations above the mean is always greater than or equal to Q(p = 0.8), the fourth quintile. When z ≤ 0, there is instead an upper bound
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