8.8 | Probability, Mean, and Median

A median of a quantity $$x$$ distributed through a population is a value $$T$$ such that half the population has values of $$x$$ less than (or equal to) $$T$$, and half the population has values of $$x$$ greater than (or equal to) $$T$$. Thus, a median satisfies
$$\int_{-\infty}^T p(x) \;dx = 0.5$$
where $$p$$ is the density function. In other words, half the area under the graph of $$p(x)$$ lies to the left of $$x=T$$.
If a quantity has density function $$p(x)$$, then the mean value of the quantity is given by
$$\int_{-\infty}^\infty p(x) \; dx.$$
A normal distribution has a density function of the form
$$p(x) = \frac{1}{\sigma \sqrt{2 \pi}}e^{-(x- \mu)^2/(2 \sigma^2)}$$
where $$\mu$$ is the mean of the distribution and $$\sigma$$ is the standard deviation, with $$\sigma >0$$.