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by , PhD

This lecture introduces and discusses the notion of quantile of the probability distribution of a random variable. We start by giving a formal definition of quantile and we then discuss its meaning.

Table of Contents


In order to understand the definition of quantile, you need to remember how the distribution function [eq1] of a random variable X is defined:[eq2]

Quantiles are defined as follows.

Definition Let X be a random variable having distribution function [eq3]. Let [eq4]. The p-quantile of X, denoted by [eq5] is[eq6]

When the distribution function is continuous and strictly increasing on R, then the smallest x that satisfies[eq7]is the unique x that satisfies[eq8]

Furthermore, the distribution function has an inverse function $F_{X}^{-1}$ and we can write[eq9]

Example If a random variable X has a standardized Cauchy distribution, then it distribution function is[eq10]which is a continuous and strictly increasing function. The p-quantile of X is[eq11]

However, in many cases the distribution function is not continuous or not invertible (or both) and we need to apply the definition above in order to derive the quantiles of the random variable. The following example illustrates one such case.

Example Let X be a discrete random variable with support[eq12]and probability mass function[eq13]The distribution function of X is[eq14]which is clearly not invertible. Now, suppose we want to compute the p-quantile for $p=0.2$. There is no x such that[eq15]However, the smallest x such that [eq16]is $x=0$ because [eq17] for $x<0$ and [eq18] for $x=0$. Thus, we have[eq19]


When there is an x such that [eq20], the quantile [eq21] can be interpreted as a cut-off point: with probability p, the realization of the random variable will be less than or equal to [eq22]; with probability $1-p$, it will be greater than [eq23].

Quantile function

When [eq24] is regarded as a function of p, that is, [eq25], it is called quantile function.

The quantile function is often denoted by[eq26]

As we have already shown above, when the distribution function is continuous and strictly increasing on R, then the quantile function coincides with the inverse of the distribution function.

More details

More details about quantiles can be found in the following subsections.

Special quantiles

Some quantiles have special names:

Other definitions

Although the definition of quantile given above is the one that is usually employed in probability theory and mathematical statistics, there are also other slightly different definitions that can be given. For a review, see


Note that in the definition of quantile, we have imposed [eq35]. This is because for $p=0$, we have that [eq36]whatever the distribution of X. Instead, for $p=1$, we have that [eq37] is not well-defined in general: for example, if X has a normal distribution, then[eq38]is not well-defined because the set[eq39]is empty.

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