Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.

Chebyshev's Theorem

The Russian mathematician P.L. Chebyshev (1821-1894) proved a theorem which is valid for any distribution of data:

      For any number k greater than 1, at least 1-1/k2 of the data will fall within k standard deviations of the mean.

This theorem produces a few useful rules:

  • no information can be obtained on the fraction of values falling within 1 standard deviation of the mean
  • at least 75% will fall within 2 standard deviations
  • at least 88.8 % will fall within 3 standard deviations
The following figure is a graphical presentation of these rules.