Empirical Rule for Normal Distributions - Probability & Statistics Step by Step

For an approximately normal distribution, about 68%, 95%, and 99.7% of values fall within 1, 2, and 3 standard deviations of the mean.

empirical-rule-normal Use μ ± kσ for k = 1, 2, and 3.

Example

Find the one-, two-, and three-sigma intervals and percentages. These fixed percentages hold only when the data follow a normal distribution; the book assumes normality rather than deriving it, and the deeper result that makes the normal model appear is the Central Limit Theorem, which it does not prove.

highlighted = computed this step

Step 1 — Mean and standard deviation

List the mean and standard deviation.

\mu= \hl{100} \quad \sigma= \hl{15}

Step 2 — Within one standard deviation

Compute the interval and percent for one sigma.

\begin{array}{cccc}\text{k} & \text{lo} & \text{hi} & \text{pct} \\ 1 & \hlmath{85} & \hlmath{115} & \hlmath{68\%} \\ 2 & \square & \square & \square \\ 3 & \square & \square & \square\end{array}

Step 3 — Within two standard deviations

Compute the interval and percent for two sigma.

\begin{array}{cccc}\text{k} & \text{lo} & \text{hi} & \text{pct} \\ 1 & 85 & 115 & 68\% \\ 2 & \hlmath{70} & \hlmath{130} & \hlmath{95\%} \\ 3 & \square & \square & \square\end{array}

Step 4 — Within three standard deviations

Compute the interval and percent for three sigma.

\begin{array}{cccc}\text{k} & \text{lo} & \text{hi} & \text{pct} \\ 1 & 85 & 115 & 68\% \\ 2 & 70 & 130 & 95\% \\ 3 & \hlmath{55} & \hlmath{145} & \hlmath{99.7\%}\end{array}

Step 5 — Empirical rule

State the empirical rule intervals.

\begin{array}{cccc}\text{k} & \text{lo} & \text{hi} & \text{pct} \\ 1 & 85 & 115 & 68\% \\ 2 & 70 & 130 & 95\% \\ 3 & 55 & 145 & 99.7\%\end{array}