Weighted mean

A weighted mean lets some values count for more than others. It is $\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$, said as weighted x-bar, where $w_i$ is the weight attached to the value $x_i$. Set every weight to 1 and the formula collapses to the ordinary mean, so the plain mean is not a different kind of average. It is the special case in which the values genuinely deserve equal say.

A shop sells 120 mugs at a mean price of 8 dollars and 30 teapots at a mean of 28 dollars. The mean price across all 150 items is $\frac{120(8) + 30(28)}{150} = \frac{1800}{150} = 12$ dollars. The weights are the counts, and 12 is simply the total money divided by the total items.

"Average the two means: $(8 + 28)/2 = 18$." That is 6 dollars too high, and the 6 splits unevenly. Promoting the teapots from 20 percent of the items to 50 percent of the say contributes $(0.50 - 0.20)(28 - 12) = 4.80$ of it, while demoting the mugs from 80 percent to 50 percent contributes $(0.50 - 0.80)(8 - 12) = 1.20$. Four fifths of the error comes from over-weighting the small expensive group and only one fifth from under-weighting the large one.

With non-negative weights that are not all zero, the weighted mean always lands between the smallest and largest value being averaged, which is a quick sanity check: 12 sits between 8 and 28. What the formula does not do is combine spread. Averaging two group standard deviations the same way is wrong, and rebuilding a combined one needs the original values or their sum of squares.

The Fall 2026 course does not name the weighted mean, but topic 2.9 contains one. The expected value of a discrete random variable, $\mu_X = \sum x_i P(x_i)$, is a weighted mean whose weights are probabilities and already add to 1, which is why the denominator disappears there.