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Anonymous ID: 9d78e3 Dec. 3, 2018, 9:45 a.m. No.4130121   🗄️.is 🔗kun

>>4130062

>68–95–99.7 rule

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 a band around the mean in a normal distribution with a width of two, four and six 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:

 

Pr ( μ − σ ≤ X ≤ μ + σ ) ≈ 0.6827 Pr ( μ − 2 σ ≤ X ≤ μ + 2 σ ) ≈ 0.9545 Pr ( μ − 3 σ ≤ X ≤ μ + 3 σ ) ≈ 0.9973 {\displaystyle {\begin{aligned}\Pr(\mu -\;\,\sigma \leq X\leq \mu +\;\,\sigma )&\approx 0.6827\\Pr(\mu -2\sigma \leq X\leq \mu +2\sigma )&\approx 0.9545\\Pr(\mu -3\sigma \leq X\leq \mu +3\sigma )&\approx 0.9973\end{aligned}}} {\displaystyle {\begin{aligned}\Pr(\mu -\;\,\sigma \leq X\leq \mu +\;\,\sigma )&\approx 0.6827\\Pr(\mu -2\sigma \leq X\leq \mu +2\sigma )&\approx 0.9545\\Pr(\mu -3\sigma \leq X\leq \mu +3\sigma )&\approx 0.9973\end{aligned}}}

 

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 depends significantly 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%. There may be certain assumptions for a distribution that force this probability to be at least 98%.