68-95-99.7 rule

Formulation

The 68-95-99.7 rule indicates the percentage of the values that lie within one, two, and three standard deviations interval from the mean in a normal distribution. It is written mathematically as:

$$\begin{array}{rl}p(\mu -1\sigma \le X\le \mu +1\sigma )& \approx 0.68\\ p(\mu -2\sigma \le X\le \mu +2\sigma )& \approx 0.95\\ p(\mu -3\sigma \le X\le \mu +3\sigma )& \approx 0.997\end{array}$$

where

$$X\sim \mathcal{N}(\mu ,\sigma )$$

Proof

$$\begin{array}{rl}p(\mu -n\sigma \le X\le \mu +n\sigma )& ={\int }_{\mu -n\sigma }^{\mu +n\sigma }\frac{1}{\sqrt{2\pi }\sigma }\exp \left(-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2\right)dx\\ & =\frac{1}{\sqrt{2\pi }}{\int }_{\mu -n\sigma }^{\mu +n\sigma }\frac{1}{\sigma }\exp \left(-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2\right)dx\\ & =\frac{1}{\sqrt{2\pi }}{\int }_{-n}^n\exp \left(-\frac{u^2}{2}\right)du\end{array}$$

Substituting the values yield the values.

References

• Wikipedia contributors. (2024, January 26). 68–95–99.7 rule. Wikipedia. https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule