Chebyshev Theorem

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Chebyshev theorem is collection of many theorems like Bertrand’s postulate, Chebyshev inequality,

Chebyshev sum inequality, Chebyshev equioscillation theorem and prime number theorem. It was given by

Pafnuty Chebyshev, a Russian mathematician. This theorem help to find what percent of the values will fall

between the interval x1 and x2 for a given data set, where mean is given and standard deviation is known.

We need to find the range Mean-k*SD, Mean+K*SD) where SD is standard deviation. Chebyshev theorem

says that 1 – (1/k-squared) of the measure will fall within the above calculated range. 



Example 1:- The 5 values given are given as 2, 4, 6, 8, and 10. And standard deviation is given by

2. Find the Range in which 95 % value lies.



Solution 1:-. Given 5 input values are as follows:- 2, 4, 6, 8, and 10.

 So mean = (2+4+6+8+10)/5 = 30/5= 6

 Given Standard deviation (SD) = 2

 95 % of the values lie in between = (Mean- SD, Mean + SD)

 Therefore, 95 % of the values lie in between – (6-2, 6+2)

 Hence 95 % of the values lie in between in (4, 8).


 
Example 2:- The 5 values given are given as 1, 2, 3, 4, and 5. And standard deviation is given by 1.

Find the Range in which 95 % value lies.



Solution 2:- Given 5 input values are as follows:- 1, 2, 3, 4, and 5.

 So mean = (1+2+3+4+5)/5 = 15/5= 3

 Given Standard deviation (SD) = 1

 95 % of the values lie in between = (Mean- SD, Mean + SD)

 Therefore, 95 % of the values lie in between – (3-1, 3+1)

 Hence 95 % of the values lie in between in (2, 4).


 

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