How to find arithmetic mean of elements of a set from arithmetic means of its distinct subsets?

Question

Consider three sets $S_1$, $S_2$, and $S_3$, containing integers. Knowing arithmetic means $a_1$, $a_2$, $a_3$ of all elements for each of these sets, find arithmetic mean of the values of all three sets together.

Answer 1

You must know the size of each set (or some other information) to calculate the arithmetic mean of all of them.

Let set $S_n$ have $k_n$ elements.

The sum of elements in $S_n$ hence is $a_nk_n$.

It follows that the sum of all elements of $t$ sets is

$$\sum_{n = 1}^t a_nk_n$$

The arithmetic mean of all sets hence is

$$\frac{\displaystyle\sum_{n = 1}^t a_nk_n}{\displaystyle\sum_{n= 1}^t k_n}$$