Tchebycheffs theorem
As we all know standard deviation is a most widely used measure of variation. It has certain mathematical properties that facilitate development of statistical theory. It carries the same units as the variable, whereas the variable is expressed in squared units.
Deeper insight into the role and meaning of standard deviation can be gained through examination and use of a theorem developed by the Russian mathematician tchebycheff. According to this theorem, it is given a group of N within K standard deviations of the mean. The symbol K represents number of standard deviations. The theorem can be quantified for any desired value of K each different value of K produces a new interval with different minimum proportions of observations encompassed.
To illustrate, let K = 1, 2 and 3 when K = 1, this interval created is the mean ± 1 standard deviation, i.e. µ ± s. For this value of K the minimum proportion of observations contained in the interval is
1 + (1/K)^2 + 1 -1/1 = 1-1 = 0
When K = 2, the interval is µ ± 2s and the minimum proportion observation in the interval is:
1 – (1/)2 = 1 –1/4= 3/4.
When K = 3, interval is µ ± 3s and the minimum proportion of items in the interval is:
1 – (1/3)2 = 1 - 1/9 = 8/9