View source for 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|normal distribution]] of a random variable at a distance from its [[Mathematical expectation|mathematical expectation]] of more than three times the [[Standard deviation|standard deviation]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092750/t0927501.png" /> be a normally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092750/t0927502.png" /> distributed random variable. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092750/t0927503.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092750/t0927504.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092750/t0927505.png" /> is the distribution function of the standard normal law; whence, in particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092750/t0927506.png" /> it follows that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092750/t0927507.png" /></td> </tr></table>

The latter equation means that the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092750/t0927508.png" /> can differ from its expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092750/t0927509.png" /> by a quantity exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092750/t09275010.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092750/t09275011.png" /> is practically impossible and, consequently, the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092750/t09275012.png" /> is practically certain. In this case one says that the experimenter has applied the  "three-sigma"  rule.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.V. Smirnov,   I.V. Dunin-Barkovskii,   "Mathematische Statistik in der Technik" , Deutsch. Verlag Wissenschaft.  (1969)  (Translated from Russian)</TD></TR></table>