Introduction to the Normal Distribution

The normal distribution is the most important continuous distribution in statistics. Heights, weights, measurement errors, examination marks and many natural and economic quantities are found to be approximately normally distributed, and — through the Central Limit Theorem — averages of large samples tend to be normal whatever the original distribution. Its graph is the familiar symmetric bell-shaped curve.

Pictured on a horizontal $X$-axis, the curve rises to a single peak directly above the mean $\mu$ and falls away in mirror image on either side, so a vertical line through $x=\mu$ is its axis of symmetry; both tails approach the axis without ever touching it. Reading outward from the centre, the curve encloses the areas described by the empirical (68–95–99.7) rule: the band from $\mu-\sigma$ to $\mu+\sigma$ holds about $68\%$ of the total area, the wider band from $\mu-2\sigma$ to $\mu+2\sigma$ about $95\%$, and the band from $\mu-3\sigma$ to $\mu+3\sigma$ about $99.7\%$ — so almost all of the distribution lies within three standard deviations of the mean, with each successive band adding a thinner slice of area near the tails.

A continuous variable $X$ follows a normal distribution with mean $\mu$ and variance $\sigma^2$, written $X \sim N(\mu, \sigma^2)$, if its density is