Combined Mean and Weighted Mean

Combined mean. When two (or more) groups of data are merged into one, the mean of the combined group is not simply the average of the two individual means — it must be weighted by how many observations each group actually contributes. For two groups of sizes $n_1$ and $n_2$ with means $\bar{x}_1$ and $\bar{x}_2$:

This extends the same way to three or more groups: $\bar{x} = \dfrac{\Sigma(n_i\bar{x}_i)}{\Sigma n_i}$. This formula is exactly what §1's fourth requisite of a good average — "capable of further algebraic treatment" — means in practical, computable terms.

Weighted mean. A simple arithmetic mean treats every observation as equally important. A weighted mean is used when some observations genuinely matter more than others — for example, a student's marks in different subjects carrying different credit hours, or a firm's average cost of a raw material bought in different quantities at different prices. If each value $x$ carries a weight $w$:

$\bar{x}_w = \dfrac{\Sigma wx}{\Sigma w}$ …