In this series we have been playing with a number of images and models that all have some connection with weighted averages. Weighted averages have plenty of everyday uses, and the conceptions underlying weighted average turn out to be very rich and have all kinds of interesting tentacles into other things.
In this post we will be looking at what happens if we restrict ourselves to two points, and make sure that the weights add up to one. This is not a very restrictive step, as we already saw that weights of 6 and 4 work out the same as weights of 60% and 40%. We get the same weighted average in either case. What restricting the total of the weights to 100% allows us to do is to focus on just one of the weights; after all, if we know one, the other one will be what’s left over from 100%.
In the picture we show the same two numbers we’ve played with before, but this time we added a scale for one of the weights. The scale goes from 0 to 1 (or 0% to 100%.) The green lines in the figure indicate that a weight of 0 corresponds to 42, and a weight of 1 corresponds to 62. Weights in between 0 and 1 will correspond to numbers between 42 and 62, as shown below:
The green lines corresponding to weights of 30%, 50% and 80% have been sketched in. These correspond to the numbers 48, 52 and 58, respectively. The green lines give you a way to go back and forth between a weight and the numbers in the “middle” between 42 and 62 according to that weighted average. It really does work to draw the green lines straight, from the common point below the scale.
If the weight w varies from 0 to 1, then the number on the number line will vary from 42 to 62. The two are related by a formula, which I will present without derivation below. What is important here is the idea that every value of w corresponds to a number on the line between 42 and 62 and vice versa. (Numbers outside of the range of 42 to 62 happen to correspond to values of w either less than zero or more than 1.)
The formula: weighted average = 42 (1-w) + 62 w
(If you tried to come up with a formula yourself, you may have found different variations of the one shown here. Each variation highlights some idea at the expense of another. You may well have found this one: weighted average = 42 + w(62-42), which has the advantage of having all the ws in one place, but at the cost of losing some of the symmetry between the 42 and the 62.)