The 68-95-99.7 rule

For a Gaussian there is a practical rule that lets you estimate in your head the probabilities of intervals centred on the mean, without resorting to tables.

Property — The $68$–$95$–$99{,}7$ rule

For a Gaussian variable:

• $P(\mu-\sigma \le X \le \mu+\sigma) \approx 68\%$
• $P(\mu-2\sigma \le X \le \mu+2\sigma) \approx 95\%$
• $P(\mu-3\sigma \le X \le \mu+3\sigma) \approx 99{,}7\%$

In words: about two thirds of the data fall within one standard deviation of the mean, $95\%$ within two, and practically all (the $99{,}7\%$) within three. It is the criterion by which, in the laboratory, one judges whether a measurement is “compatible” with an expected value.

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Topics: Distribuzioni probabilita
Concepts: Deviazione standard · Distribuzione normale
Skills: Calcolo probabilita · Stimare
People: Carl Friedrich Gauss