The mean and variance of 20 observations are 10 and 5 respectively. It was later found that two observations, 8 and 12 , were incorrect. If these two incorrect observations are replaced by two correct observations a and b , the new mean remains 10 but the new variance becomes 6.2 . The value of |a - b| is equal to:

Step-by-step solution

Let the original sum of observations be x_i and the original sum of squares be x_i^2 . Given n = 20 , mean x = 10 , and variance ^2 = 5 . x_i = 20 10 = 200 x_i^2 = n( ^2 + x ^2) = 20(5 + 10^2) = 20 105 = 2100 When 8 and 12 are replaced by a and b , the new mean is 10 . New sum = 200 - 8 - 12 + a + b = 20 10 = 200 a + b = 20 The new variance is 6.2 . New sum of squares = 20(6.2 + 10^2) = 20 106.2 = 2124 Also, new sum of squares = 2100 - 8^2 - 12^2 + a^2 + b^2 2100 - 64 - 144 + a^2 + b^2 = 2124 1892 + a^2 + b^2 = 212