Chebychev`s inequality

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Definition: Chebychev`s inequality

Chebychev`s inequality is a statistical theorem that states that in any probability distribution,atleast 1-(1/N2) of data from a sample must fall within N standard deviations from the mean ,where N is any positive real number greater than 1.The inequality can be expressed as

PR ([X-µ] ≥ kα) ≤ 1/k2

Where µ is the expected value and α is the standard deviation.

For example, if N=2, then 1-(1/22) = ¾ = 75%.So as per Chebychev`s inequality, at least 75% of the data values of any distribution must be within two standard deviations of the mean.
If N=3, then 1-(1/32) = 8/9 = 89%.This implies that 89% of the values must be within three standard deviations from the mean.

The use of this inequality is such that it gives a worst case scenario when the only known information about sample data is the mean and standard deviation. The formula helps determine the number of values that reside in and outside the standard deviation. This inequality has been proven mathematically also.


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