Let the variance of the data set 1, 2, 3, 4, x be an integer, where x 1, 2, 3, , 50 . The sum of all possible values of x is:

Step-by-step solution

The given data set is 1, 2, 3, 4, x . The number of observations is 5 . Mean, = 1 + 2 + 3 + 4 + x 5 = 10 + x 5 Variance, ^2 = x_i^2 5 - ^2 ^2 = 1^2 + 2^2 + 3^2 + 4^2 + x^2 5 - ( 10 + x 5 )^2 ^2 = 30 + x^2 5 - 100 + 20x + x^2 25 ^2 = 150 + 5x^2 - 100 - 20x - x^2 25 = 4x^2 - 20x + 50 25 For the variance to be an integer, 4x^2 - 20x + 50 must be a multiple of 25 . 4x(x - 5) + 50 is a multiple of 25 . Since 50 is a multiple of 25 , 4x(x - 5) must be a multiple of 25 . Since 4 and 25 are coprime, x(x - 5) must be divisi