Median, Quartiles and Percentiles of Continuous Random Variables

Find the median, quartiles, and percentiles of a continuous random variable by solving F(x) = p, either from a given CDF or by first building the CDF from a PDF.

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Tutorial

The Median of a Continuous Random Variable

For a continuous random variable $X$ with cumulative distribution function $F$ , the median is the value $m$ at which exactly half the probability lies to the left:

F(m) = P(X \leq m) = \dfrac{1}{2}.

To find the median, we set the CDF equal to $\dfrac{1}{2}$ and solve for $m$ .

For example, suppose $X$ has CDF

F(x) = \begin{cases} 0 & x < 0 \\ x & 0 \leq x \leq 1 \\ 1 & x > 1. \end{cases}

Setting $F(m) = \dfrac{1}{2}$ gives $m = \dfrac{1}{2}.$