Answer :

Given: The mean and standard deviation of a set of n_{1} observations are and s_{1}, respectively while the mean and standard deviation of another set of n_{2} observations are and s_{2}, respectively

To show: the standard deviation of the combined set of (n_{1} + n_{2}) observations is given by

As per given criteria,

For first set

Let x_{i} where i=1, 2, 3,4 , …, n_{1}

For second set

And y_{j} where j=1, 2, 3, 4, …, n_{2}

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

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