68–95–99.7 rule

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68–95–99.7 rule

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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 withina band around the mean in a normal distribution with a width of two, four and six standard deviations, respectively; more precisely, 68.27%, 95.45% and 99.73% 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 Χ is an observation from a normally distributed random variable, μ is the mean of the distribution, and σ is its standard deviation: 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, and thus it is empirically useful to treat 99.7% probability as near certainty.The usefulness of this heuristic especially depends on the question under consideration. 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 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% by the Vysochanskij–Petunin inequality. There may be certain assumptions for a distribution that force this probability to be at least 98%.