Weighted averages

Picture the balance beam again, but now some weights are heavier than others. A heavy weight pulls the balance point toward it hard. A light weight barely moves it. That's how one 40%-weighted final exam can outweigh three 20%-weighted quizzes. One more habit is worth knowing. The weights don't have to add up to any particular number. Doubling every single weight at once changes nothing about where the balance point ends up. A 2-pound weight and a 4-pound weight balance the beam in the same spot as a 20-pound weight and a 40-pound weight. Only how the weights compare to each other matters, not their actual size. That also means weights don't need to be written as percentages at all. A plain count of how many times something shows up works exactly the same way as a weight.

For values x_1, \ldots, x_n with weights w_1, \ldots, w_n, the weighted mean is \bar{x}_w = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i}. Three load-bearing facts follow directly from that formula:

• Setting every w_i = 1 collapses this exactly to the plain arithmetic mean from the last section — weighted mean is the general case, not a different formula.
• Only relative weight matters, not absolute size. Scaling every weight by the same constant c leaves \bar x_w unchanged: in one line, \dfrac{\sum (c\,w_i)x_i}{\sum (c\,w_i)}=\dfrac{c\sum w_i x_i}{c\sum w_i}=\dfrac{\sum w_i x_i}{\sum w_i}, since the c cancels. That's why weights can be given as raw counts, percentages, or credit hours interchangeably — rescaling them never changes the answer.
• Weights don't have to be percentages of 1 — any nonnegative numbers work, since dividing by \sum w_i normalizes them automatically. A frequency — how many times a value occurs — is itself a weight in this sense.