Subject 7. Chebyshev's Inequality

The Russian mathematician, Pafnuty Chebyshev, developed a useful theorem of inequality dealing with standard deviation as a measure of dispersion.

Let c be any number greater than 1. For any sample or population of data, the proportion of observations that lie FEWER than c standard deviations from the mean is at least:

  • This applies to both populations and samples, and to discrete and continuous data, regardless of the shape of the distribution.
  • It gives a conservative estimate of the proportion of observations in an interval around the mean.
  • It highlights the importance of σ and places lower and upper limits.
  • Empirical rule: 68% within 1σ, 95% within 2σ, and 99% within 3σ.

Chebyshev's Inequality theorem is useful in that if we know the standard deviation, we can use it to measure the minimum amount of dispersion.


What approximate percent of a distribution will lie within ± two standard deviations of the mean?
Chebyshev's Inequality: 1-(1/c2) = 1 - (1/22) = 0.75 or 75%

Several other values are also useful to know.

  • A minimum of 36% of observations lie within 1.25 σ of the mean.
  • A minimum of 56% of observations lie within 1.5 σ of the mean.
  • A minimum of 75% of observations lie within 2 σ of the mean.
  • A minimum of 84% of observations lie within 2.5 σ of the mean.
  • A minimum of 89% of observations lie within 3 σ of the mean.
  • A minimum of 94% of observations lie within 4 σ of the mean.

User Contributed Comments 9

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sarath: Important concept.
itconcepts: why does the empirical rule differ from the "other useful values to know"
achu: empirical rule shows the dispersion of a normal distribution (within 1 sd =68%, 2sds 95%). As you can see, the normal distribution dispersion is 'tighter' than that shown by Cheby's inequality, but this is fine. At least 75% must be within 2sd's, but in a normal dist it goes up to 95%.
Cesarjr20: What would be a good book to read about this topic?
timmak: I would say any entry level university textbook on stats and probability should cover this topic in more detail.
Nuta: You can find good practical examples for understanding if you refer to 6 sigma rule.
Sophorior: Empirical rule: 68% within 1 sd;, 95% within 2 sd;, and 99% within 3 sd. this is valid for a normal distribution, not for any distribution (so it shoudn't be mentioned here...)
tschorsch: is this not a limit of maximum dispersion?
i.e. regardless of the distribution, no more than 1/k² of the values can be more than k std deviations from the mean
Ryan1234: That is correct.