The Normal Distribution (HL)

The normal distribution is the most important continuous distribution in statistics. Its graph is the familiar symmetric, bell-shaped curve, and it models many natural quantities such as heights, measurement errors, and exam scores. A normal distribution is completely specified by its mean $\mu$, which locates the center, and its standard deviation $\sigma$, which controls the width. We write $X \sim N(\mu, \sigma^2)$.

Properties

The curve is symmetric about $\mu$, where the mean, median, and mode coincide. The total area under the curve equals $1$, and probabilities correspond to areas, so $P(X < \mu) = 0.5$. A memorable feature is the empirical (68–95–99.7) rule: about $68\%$ of values lie within one standard deviation of the mean, about $95\%$ within two, and about $99.7\%$ within three