Chapter 15.1, Problem 18WE

a.

To determine

Find the mean of $m+n$ scores?

Answer to Problem 18WE

  $M=\frac{\left(m\times M_1\right)+\left(n\times M_2\right)}{m+n}$

Explanation of Solution

Given information:

The mean of $m$ scores is $M_1$ . The mean of $n$ scores is $M_2$ .

Calculation:

The mean is arithmetic average of the numbers. It is computed by dividing the sum of the observations by the numbers of observations in the distribution. So the eman of $m$ scores is $M_1$ , the sum of these scores is given by $m\times M_1$ . Similarly, hence the mean of $n$ scores is $M_2$ , the sum of these scores is given by $n\times M_2$ . Hence the sum of the $m+n$ scores is $\left(m\times M_1\right)+\left(n\times M_2\right)$

And hence the mean of these $m+n$ scores is given by

  $M=\frac{\left(m\times M_1\right)+\left(n\times M_2\right)}{m+n}$

b.

To determine

Determine the combined mean equal to $\frac{M_1+M_2}{2}$ .

Answer to Problem 18WE

Both the data sets contain equal number of observations.

Explanation of Solution

Given information:

The mean of $m$ scores is $M_1$ . The mean of $n$ scores is $M_2$ .

Calculation:

The combined mean of these $m+n$ scores is

  $M=\frac{\left(m\times M_1\right)+\left(n\times M_2\right)}{m+n}$

First data set is equal to the number of scores in the second data set, that is, both data sets contain, say $m$ scores. Then the combined mean is given by replacing $n$ by $m$ in the above formula for combined mean:

  $\begin{array}{l}M=\frac{\left(m\times M_1\right)+\left(m\times M_2\right)}{m+m}\\ =\frac{m\left(M_1+M_2\right)}{m+m}\\ =\frac{m\left(M_1+M_2\right)}{2m}\\ =\frac{M_1+M_2}{m}\end{array}$

Hence, the combined mean of two data sets is equal to the mean of the means of the data sets.

Hence, if both the data sets contain equal number of observations.