Question

The mean and standard deviation of a set of n₁ observations are \(\overline x_1\) and s₁, respectively while the mean and standard deviation of another set of n₂ observations are \(\overline x_2\) and s₂, respectively. Show that the standard deviation of the combined set of (n₁ + n₂) observations is given by

Answer 1

Given: The mean and standard deviation of a set of n₁ observations are \(\overline x_1\) and s₁, respectively while the mean and standard deviation of another set of n₂ observations are \(\overline x_2\) and s₂, respectively

To show: the standard deviation of the combined set of (n₁ + n₂) observations is given by

As per given criteria,

For first set

Let x_i where i=1, 2, 3,4 , …, n₁

For second set

And y_j where j=1, 2, 3, 4, …, n₂

And the means are

But the algebraic sum of the deviation of values of first series from their mean is zero.

Substituting value from equation (i), we get

Substituting this value in equation (iii), we get

But the algebraic sum of the deviation of values of second series from their mean is zero.

Substituting value from equation (i), we get

Substituting this value in equation (v), we get

Substituting equation (iv) and (vi) in equation (ii), we get