Three Sigma Rule in the empirical sciences express 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 rule states that even for non-normally distributed variables, at least 88.8% of cases should fall within properly-calculated three-sigma intervals. It follows from Chebyshev’s Inequality. For unimodal distributions, the probability of being within the interval is at least 95%. There may be certain assumptions for a distribution that force this probability to be at least 98%. These numerical values “68%, 95%, 99.7%” come from the cumulative distribution function of the normal distribution. The “68–95–99.7 rule” is often used to quickly get a rough probability estimate of something, given its standard deviation, if the population is assumed to be normal. It is also as a simple test for outliers if the population is assumed normal, and as a normality test if the population is potentially not normal. To use as a test for outliers or a normality test, one computes the size of deviations in terms of standard deviations and compares this to expected frequency.
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