Question

For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4 respectively. The variance of the combined data set is ______ 

Options

• `5/2`

• `11/2`

• 6

• `13/2`

Answer 1

For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4 respectively. The variance of the combined data set is `underline(11/2)`.

Explanation:

Here, `sigma_1^2 = 4, sigma_2^2 = 5, overlineX_1 = 2, overlineX_2 = 4` and n₁ = n₂ = 5

∴ `overlineX = (n_1overlineX_1 + n_2overlineX_2)/(n_1 + n_2) = 3`

`d_1 = overlineX_1 - overlineX = 2 - 3 = -1`,

`d_2 = overlineX_2 - overlineX = 4 - 3 = 1`

Let `sigma^2` be the combined variance. Then,

`sigma^2 = (n_1(sigma_1^2 + d_1^2) + n_2(sigma_2^2 + d_2^2))/(n_1 + n_2)`

= `((4 + 1) + (5 + 1))/2 = 11/2`