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In statistics, the so-called 68–95–99.7 rule is a shorthand used to remember the percentage of values that lie within a band around the mean in a normal distribution with a width of one, two and three standard deviations, respectively; more accurately, 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 x 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, i.e. that it is empirically useful to treat 99.7% probability as "near certainty".〔this usage of "three-sigma rule" entered common usage in the 2000s, e.g. cited in ''Schaum's Outline of Business Statistics'', McGraw Hill Professional, 2003, p. 359, and in Erik W. Grafarend, ''Linear and Nonlinear Models: Fixed Effects, Random Effects, and Mixed Models'', Walter de Gruyter, 2006, p. 553.〕 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. 〔See: *D. J. Wheeler and D. S. Chambers, ''Understanding Statistical Process Control'', SPC Press, 1992. *Veronica Czitrom, Patrick D. Spagon ,''Statistical Case Studies for Industrial Process Improvement'', SIAM, 1997, ( p. 342 ). *F. Pukelsheim, "The three sigma rule", American Statistician 48 (1994), 88-91.〕 ==Cumulative distribution function== These numerical values "68%, 95%, 99.7%" come from the cumulative distribution function of the normal distribution. The prediction interval for any standard score corresponds numerically to (1−(1−Φ''µ'',''σ''2(standard score))·2). For example, Φ(2) ≈ 0.9772, or Pr(''x'' ≤ ''μ'' + 2''σ'') ≈ 0.9772, corresponding to a prediction interval of (1 − (1 − 0.97725)·2) = 0.9545 = 95.45%. Note that this is not a symmetrical interval – this is merely the probability that an observation is less than ''μ'' + 2''σ''. To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding): : This is related to confidence interval as used in statistics: ''x̅'' ± 2''σ'' is approximately a 95% confidence interval when ''x̅'' is the average of a sample. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「68–95–99.7 rule」の詳細全文を読む スポンサード リンク
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