Median - Medians of Probability Distributions

Medians of Probability Distributions

For any probability distribution on the real line R with cumulative distribution function F, regardless of whether it is any kind of continuous probability distribution, in particular an absolutely continuous distribution (and therefore has a probability density function), or a discrete probability distribution, a median is by definition any real number m that satisfies the inequalities

or, equivalently, the inequalities

in which a Lebesgue–Stieltjes integral is used. For an absolutely continuous probability distribution with probability density function ƒ, the median satisfies

Any probability distribution on R has at least one median, but there may be more than one median. Where exactly one median exists, statisticians speak of "the median" correctly; even when the median is not unique, some statisticians speak of "the median" informally.