68–95–99.7 Rule

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

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



[math]\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}[/math]
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