Three-sigma rule

A rule of thumb, according to which, in certain problems in probability theory and mathematical statistics, an event is considered to be practically impossible if it lies in the region of values of the normal distribution of a random variable at a distance from its mathematical expectation of more than three times the standard deviation.

Let $X$ be a normally $N(a,\sigma^2)$ distributed random variable. For any $k>0$,

$$\operatorname P\{|X-a|<k\sigma\}=2\Phi(k)-1,$$

where $\Phi(\cdot)$ is the distribution function of the standard normal law; whence, in particular, for $k=3$ it follows that

$$\operatorname P\{a-3\sigma<X<a+3\sigma\}=0.99730.$$

The latter equation means that the values of $X$ can differ from its expectation $a$ by a quantity exceeding $3\sigma$ on the average in not more than 3 times in a thousand trials. This circumstance is sometimes used by an experimenter in certain problems of probability theory and mathematical statistics, by assuming that the event $\{|X-a|>3\sigma\}$ is practically impossible and, consequently, the event $\{|X-a|<3\sigma\}$ is practically certain. In this case one says that the experimenter has applied the "three-sigma" rule.

References

[1]	N.V. Smirnov, I.V. Dunin-Barkovskii, "Mathematische Statistik in der Technik" , Deutsch. Verlag Wissenschaft. (1969) (Translated from Russian)