SL 4.9—Normal distribution

A normal distribution $N(\mu,\sigma^2)$ is continuous, bell-shaped and symmetric about mean $\mu$; mean, median and mode coincide, and $\sigma$ controls spread. Total area under the curve is 1 and an exact point has probability 0.

Approximately 68%, 95% and 99.7% of values lie within $1\sigma$, $2\sigma$ and $3\sigma$ of $\mu$. Use technology for lower-tail, upper-tail and interval probabilities, and inverse normal to find a boundary from a given cumulative probability.

Example

If $X\sim N(300,40^2)$, then about 68% of observations lie from $260$ to $340$. An inverse-normal calculation for cumulative probability $0.95$ returns the value $x$ satisfying $P(X\le x)=0.95$.

At this SL objective, standardized $z$ transformation and continuity correction are not required. Check that a normal model is contextually reasonable, and enter standard deviation $\sigma$, not variance $\sigma^2$, when technology requests it.