For two data sets each of size 5 the variance are given to ve 4 and 5 and tge corresponding means are given to be 2 and 4 respectively the variance of the combined data set is ?

hello,

Solution of the given problem is:

Let, the elements of set 1 be: x1, x2, x3, x4, x5 and for set 2 be: y1, y2, y3, y4, y5

Thus, variance of set 1:

The variance of x1 be v(x1), x2 be v(x2), y1 be v(y1), y2 be v(y2) and so on.

Thus,

For set 1:

Mean = (x1 + x2 + x3 + x4 + x5) / 5

=> 2 = (x1 + x2 + x3 + x4 + x5) / 5

=> x1 + x2 + x3 + x4 + x5 = 10

Variance = sigma(x _i ² )/number of elements – (mean) ²

=> 4 = sigma(x _i ² )/5 - 4

=> sigma(x _i ² ) = 40

For set 2:

Mean = (y1 + y2 + y3 + y4 + y5) / 5

=> 4 = (y1 + y2 + y3 + y4 + y5) / 5

=> y1 + y2 + y3 + y4 + y5 = 20

Variance = sigma(y _i ² )/number of elements – (mean) ²

=> 5 = sigma(y _i ² )/5 - 16

=> sigma(x _i ² ) = 105

Now, if we combine the total number of elements from both the sets

Total elements = 10

Thus,

mean = (x1 + x2 + x3 + x4 + x5 + y1 + y2 + y3 + y4 + y5)/10

= (10 + 20)/10

= 3.

Now, variance = sum(square of elements)/number of elements – (mean) ²

= (40 + 105)/10 – 9 = 55/10 = 11/2.