The means of two samples of size 40 and 50 were found to be 54 and 63 respectively. Their standard deviations were 6 and 9 respectively. The variance of the combined sample of size 90 is

Step-by-step solution

n 1 = 40 , n 2 = 50 x ¯ 1 = 54 , x ¯ 2 = 63 σ 1 = 6 , σ 2 = 9 x ¯ = n 1 x ¯ 1 + n 2 x ¯ 2 n 1 + n 2 = 40 × 54 + 50 × 63 90 = 5 9 d 1 = x ¯ - x ¯ 1 = 5 d 2 = x ¯ 2 - x ¯ = 4 Combined S . D = n 1 d 1 2 + n 2 d 2 2 + n 1 σ 1 2 + n 2 σ 2 2 n 1 + n 2 = 40 × 25 + 50 × 16 + 40 × 36 + 50 × 81 90 = 9 Variance = 81