SL 4.9—Normal distribution

The normal model turns a continuous measurement into areas.

A normal variable is described by mean μ and standard deviation σ; probabilities are areas between cut-offs, not point probabilities.

Example

If $X\sim N(70,5^2)$, approximately 68% of values lie from 65 to 75, 95% from 60 to 80 and 99.7% from 55 to 85. For other probabilities or an inverse-normal cut-off, enter the mean and standard deviation directly in technology.

Sketch and shade the required area, use normal or inverse-normal technology with the stated $\mu$ and $\sigma$, and check that the output lies on the expected side of the mean.

The 68–95–99.7 rule is approximate. In this SL 4.9 objective, inverse-normal calculations use the given mean and standard deviation without first transforming to $z$; standardisation belongs to SL 4.12.