# 68–95–99.7 Rule

The 68–95–99.7 Rule is an empirical rule that applies to normal distributions.

• Approximately 68% of the observations ($x$ values) fall between $\mu - \sigma$ and $\mu+\sigma$
• Approximately 95% of the $x$ values fall between $\mu - 2\sigma$ and $\mu+2\sigma$
• Approximately 99.7% of the $x$ values fall between $\mu -3\sigma$ and $\mu+3\sigma$

## References

### 2016

\begin{align} \Pr(\mu-\;\,\sigma \le x \le \mu+\;\,\sigma) &\approx 0.6827 \\ \Pr(\mu-2\sigma \le x \le \mu+2\sigma) &\approx 0.9545 \\ \Pr(\mu-3\sigma \le x \le \mu+3\sigma) &\approx 0.9973 \end{align}
In the empirical sciences the so-called three-sigma rule of thumb expresses a conventional heuristic that "nearly all" values are taken to lie within three standard deviations of the mean, i.e. that it is empirically useful to treat 99.7% probability as "near certainty".

The usefulness of this heuristic of course depends significantly on the question under consideration, and there are other conventions, e.g. in the social sciences a result may be considered "significant" if its confidence level is of the order of a two-sigma effect (95%), while in particle physics, there is a convention of a five-sigma effect (99.99994% confidence) being required to qualify as a "discovery".

The "three sigma rule of thumb" is related to a result also known as the three-sigma rule, which states that even for non-normally distributed variables, at least 98% of cases should fall within properly-calculated three-sigma intervals.

• How many heads you would expect to see; these are “successes” in this binomial experiment.
• The standard deviation.
• The upper and lower limits for the number of heads you would get 68% of the time, 95% of the time and 99.7% of the time