Chebyshev's theorem

Chebyshev's theorem is a name given to several theorems proven by Russian mathematician Pafnuty Chebyshev

• Bertrand's postulate
• Chebyshev's inequality
• Chebyshev's sum inequality
• The statement that if the function $\scriptstyle \pi (x)\ln x/x$ has a limit at infinity, then the limit is 1 (where π is the prime-counting function). This result has been superseded by the prime number theorem.

The proportion (or fraction)of any set of data lying with K standard deviations of the mean is always at least 1-1/K², Where K is any positive number greater than 1.

This concept is helpful in understanding the or interpreting a value of a standard deviation. Chebyshev's theorem applies to to any data set and it's results are very approximate.

Applying Chebyshev's concept can be stated as follows in this example:

Heights of women have a mean of 163cm and stand deviation of 6cm. What can we conclude from chebyshev's theorem?

SOLUTION: Applying chebyshev's theorem with the following information we reach the following conclusions

    * At least 3/4 (75%) of ALL heights of women fall within 2 standard deviations of the mean (between 151 - 175 cm)
    * Alteast 8/9 (89%) of ALL heights of women are within 3 standard deviations of the mean (between 145 - 181cm)

Take note that Chebyshev's theorem also coincides with the EMPIRICAL RULE (68-95-99.7).