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title: "Tchebysheffs Rule"
canonical: "https://modelassist.epixanalytics.com/space/EA/26574957/Tchebysheffs%20Rule"
format: markdown
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If a data set has mean and standard deviation s, we are used to saying that 68% of the data will lie between *(x* *- s)* and *(x* *+ s)*, 95% lie between *(x* *- 2s)* and *(x* *+ 2s)*, etc.. However, that is *only* true when the data follow a [Normal distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575246). The same applies for a probability distribution. So, when the data, or probability distribution, are *not* normally distributed, how can we interpret the standard deviation?
Tchebysheffs rule applies to *any* probability distribution or data set. It states:
For any number *k* greater than 1, at least (1-1/*k**2*) of the measurements will fall within *k* standard deviations of the mean.
Substituting k=1, Tchebysheffs rule says that at least 0% of the data or probability distribution lie within one standard deviation of the mean. Well, we already knew that! However, substitute k = 2 tells us that 75% of the data or distribution lie within 2 standard deviations of the mean. That is useful information because it applies for all distributions.
This is a fairly conservative rule, in that if we know the distribution type we could specify a much higher percentage (e.g. 95% for 2 standard deviations for a Normal distribution, compared with 75% with Tchebysheffs rule) but is certainly helpful in interpreting the standard deviation of a data set or probability distribution that is grossly non-Normally distributed.
This figure compares Tchebysheffs Rule with the results of a few distributions. You can see that for any k, knowing the distribution type allows you to specify a much higher fraction of the distribution to be contained in the range mean +/- k standard deviations. The bimodal distribution tested was:
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