Chebyshev Theorem

Chebyshev Theorem Chebyshev theorem is collection of many theorems like Bertrands 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).