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

What approximate percent of a distribution will lie within ± two standard deviations of the mean?

Chebyshev's Inequality: 1-(1/c

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.

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. |