Weighted Mean vs Mean

A weighted mean lets some values count for more than others, which is what you need when categories differ in importance or in sample size. The formula is $\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$, read as weighted x-bar, where $w_i$ is the weight attached to the value $x_i$. If tests count 70 percent of a grade and homework counts 30 percent, a student with a 90 test average and an 80 homework average earns $0.7(90) + 0.3(80) = 63 + 24 = 87$, where the weights already total 1 so the denominator is 1 and drops out. The ordinary mean is just the special case in which every weight is equal.

The mean is the balance point of a distribution, the value where the data would balance if placed on a seesaw. For example, the mean of 4, 8, and 9 is $(4 + 8 + 9) / 3 = 7$. The sample mean is written $\bar{x} = \frac{1}{n}\sum x_i$, where $\sum x_i$ (the sum of the values) is divided by $n$ (the number of values). Because every value contributes, the mean is sensitive to outliers and skew.