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:
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?
Several other values are also useful to know.
|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.|