If the variate of a distribution takes the values 1 2 , 2 2 , 3 2 , . . . n 2 with frequencies n , n - 1 , n - 2 , . . . 3,2 , 1 respectively, then the mean value of the distribution is

Step-by-step solution

x i : 1 2     2 2   3 2   4 2 . . . n - 1 2 n 2 f i : n   n - 1 n - 2 n - 3 . . .     2 1 ∴ ∑ i = 1 n x i f i = 1 2 ⋅ n + 2 2 n - 1 + 3 2 n - 2 + . . . + n - 1 2 x + n 2 ⋅ 1 = ∑ r = 1 n n + 1 r 2 - r 3 = n + 1 ∑ r = 1 n r 2 - ∑ r = 1 n r 3 = n n + 1 2 2 n + 1 6 - n 2 n + 1 2 4 n n + 1 2 12 2 2 n + 1 - 3 n = n n + 1 2 n + 2 12 Also, Σ f i = 1 + 2 + . . . . + n = n n + 1 2 Mean = Σ f i x i Σ f i = n n + 1 2 n + 2 12 × 2 n