So you’ve got some numbers, and you want to produce one number that represents their typical value. If you’ve taken a little bit of math or statistics, you might reach for the mean—the arithmetic mean, to be precise. Add the numbers together and divide by the number of numbers you have. Easy enough.
But perhaps you’re a bit of a Chidi, and furthermore you made the mistake of learning about the geometric mean at some point. The geometric mean is another measure of central tendency, as statisticians say. The geometric mean is like the arithmetic mean, but you change addition to multiplication and division to taking roots. To find the geometric mean of two numbers, multiply them together and take the square root of their product. (This operation can only be performed on two numbers that are either both positive or both negative. But the world is negative enough. Let’s just think about positive numbers!) True to its name, this mean has a nice geometric interpretation: it is the side length of a square with the same area as a rectangle having your two numbers as side lengths. To find the geometric mean of a lot of numbers—let’s say n of them—multiply them all together and take the nth root.
Knowing about both the arithmetic and geometric means, you are wracked with internal turmoil: Which mean will best represent your numbers?
The arithmetic mean is nice. It seems very balanced and equitable. But the geometric mean has its merits as well. It is a useful tool when you’re working with processes that work multiplicatively instead of additively, like interest rates. It pulls larger numbers closer to smaller numbers more than the arithmetic mean does, whether you are taking the geometric mean of just two numbers or many numbers. The geometric mean might be a better representative than the arithmetic mean or even the median for a data set that has a lot of smaller values and a few large ones—say, income distributions.
Which will it be? Decisions are so hard!
Why not both?
The arithmetic-geometric mean lets you find a number between your two favorite positive numbers that is a compromise between the arithmetic and geometric means, letting your inner Chidi rest easy.
Finding the arithmetic-geometric mean is an iterative process. Each step produces two numbers: the arithmetic and geometric means of the previous two numbers. So starting with, say, 1 and 2, the first step produces the two numbers 3/2 and √2. At the next step, you find the arithmetic mean of 3/2 and √2, which is approximately 1.457, and the geometric mean of 3/2 and √2, which is approximately 1.456. At that point, the two values you're getting are already very close together, and subsequent iterations will produce two numbers that are arbitrarily close together. The limit of both the arithmetic and geometric means produced in this process is the same, so it is called the arithmetic-geometric mean. The arithmetic-geometric mean of 1 and 2 is 1.45679…; a bit disappointing in that it would be more fun if it started 1.456789, but a satisfying answer nonetheless.
The approximations of the arithmetic-geometric mean of two numbers get very close together very quickly, so the process has been used to find good approximations for irrational numbers, as in this paper about how to use it to approximate π.
What if you have more than two numbers? As far as I can tell, no one has ever defined the arithmetic-geometric mean for an arbitrary set of positive numbers, but that didn’t stop me. I’m not going to use the name arithmetic-geometric mean for the generalization to make sure nobody thinks it’s an “official” math term. Instead, I’ll call it the ditherer’s mean.
For the arithmetic-geometric mean of two numbers, we had an iterative process that gave us two numbers at every step. One way of thinking about it is that we replaced the smallest number with the geometric mean of the previous numbers and the largest number with the arithmetic mean of them. We’ll do the same thing for the ditherer’s mean.
To take the ditherer’s mean of n numbers, we want an iterative process that gives us n numbers at each step. So at each step, we replace the smallest number from the previous list of numbers with the geometric mean of the previous numbers and the largest number with the arithmetic mean of the numbers.
Let’s take a look at a set of 4 numbers to get a feel for how the process works. We’ll start with the numbers 1, 5, 20, and 26. The arithmetic mean of these numbers is 13, and the geometric mean is approximately 7.14. So we replace the largest and smallest numbers in our first list with 13 and 7.14. Now we have the numbers 5, 7.14, 13, and 20. We repeat the process. The arithmetic mean of those four numbers is about 11.285. The geometric mean is about 9.82. Now our list is 7.14, 9.82, 11.285, and 13. The arithmetic mean is 10.31 and the geometric mean is 10.07. Keep going: 9.82, 10.07, 10.31, 11.285. Then 10.07, 10.31, 10.35, 10.37. Progress! A few more iterations, and it’s clear the numbers are getting closer and closer together, landing around 10.3.
The arithmetic-geometric mean of two numbers has been a useful concept for mathematics. The iterative process that produces it converges very quickly, so it has been used to compute approximations quickly and accurately, as in this paper about computing π using the arithmetic-geometric mean. As far as I can tell, mathematicians have not yet found use for the ditherer’s mean, but I hope it will help some indecisive people take an average and move on with their lives.
There you have it: Now you can find an average of a set of positive numbers without having to choose between their arithmetic and geometric means. Isn’t it wonderful the way math always gives you a tidy answer with no room for uncertainty or ambiguity?