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

Given: The mean and standard deviation of a set of n₁ observations are and s₁, respectively while the mean and standard deviation of another set of n₂ observations are 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

Now mean of the combined series is given by

And the corresponding square of standard deviation is

Therefore, square of standard deviation becomes,

Now,

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

Also,

But

Substituting value from equation (i), we get

Substituting this value in equation (iii), we get

Similarly, we have

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

Also,

But

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

So the combined standard deviation

Hence proved