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Difference between revisions of "Three-sigma rule"

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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]].
 
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" />,
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Let $X$ be a normally $N(a,\sigma^2)$ distributed random variable. For any $k>0$,
  
<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>
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$$\operatorname P\{|X-a|<k\sigma\}=2\Phi(k)-1,$$
  
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
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where $\Phi(\cdot)$ is the distribution function of the standard normal law; whence, in particular, for $k=3$ 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>
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$$\operatorname P\{a-3\sigma<X<a+3\sigma\}=0.99730.$$
  
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.
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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====
 
====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>
 
<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>

Latest revision as of 00:44, 24 December 2018

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)
How to Cite This Entry:
Three-sigma rule. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Three-sigma_rule&oldid=17366
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article