Quantiles are values taken
at regular intervals from the inverse function of the cumulative
distribution function (CDF) of a random variable. Dividing ordered
data into q essentially equal-sized data subsets is the motivation
for q-quantiles; the quantiles are the data values marking the
boundaries between consecutive subsets. The quantiles can be used as
cutoff values for grouped data in approximately equal size groups.
Quantiles can also be applied to continuous data, providing a way to
generalise rank statistics to continuous variables.
A kth q-quantile for a
random variable is a 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 greater than x is at
most (q−k)/q = 1−(k/q). There are q−1 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 of a uniform probability distribution on a set of
even size.
