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Concept
68–95–99.7 rule
Summary
In statistics, the 68–95–99.7 rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within
an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively.
In mathematical notation, these facts can be expressed as follows, where Pr() is the probability function, Χ is an observation from a normally distributed random variable, μ (mu) is the mean of the distribution, and σ (sigma) is its standard deviation:
The usefulness of this heuristic especially depends on the question under consideration.
In the empirical sciences, the so-called three-sigma rule of thumb (or 3σ rule) expresses a conventional heuristic that nearly all values are taken to lie within three standard deviations of the mean, and thus it is empirically useful to treat 99.7% probability as near certainty.
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.
A weaker three-sigma rule can be derived from Chebyshev's inequality, stating that even for non-normally distributed variables, at least 88.8% of cases should fall within properly calculated three-sigma intervals. For unimodal distributions, the probability of being within the interval is at least 95% by the Vysochanskij–Petunin inequality. There may be certain assumptions for a distribution that force this probability to be at least 98%.
We have that
doing the change of variable , we have
and this integral is independent of and . We only need to calculate each integral for the cases .
These numerical values "68%, 95%, 99.7%" come from the cumulative distribution function of the normal distribution.
The prediction interval for any standard score z corresponds numerically to (1 − (1 − _,^2(z)) · 2).
For example, (2) ≈ 0.9772, or Pr( ≤ + 2) ≈ 0.
Official source
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