Visualizing the Geometric and Harmonic Means
Long after I learned how to calculate the geometric and harmonic means, I was still unclear about what exactly they were doing for me. I ended up clarifying this to myself by generating a series of visualizations which I’ll show you in this post.
First, a quick review of the algorithms themselves.
The arithmetic mean is what people are usually talking about when they say “average”. By far the most widely used, it’s simple to calculate: you sum the parts, then divide by how many there were.
The geometric mean has the same procedure but different operations. You multiply the parts, then take the root corresponding to how many there were. The geometric mean is often used when finding the mean of data which are measured in different units.
The harmonic mean is the arithmetic mean with two extra steps. First, find the multiplicative inverse of each number (for x, that’s 1÷x, or x⁻¹). Then sum and divide those inverses like you would for the arithmetic mean. Then, take the inverse again. The harmonic mean is used to calculate F₁ score.
Looking at the equations, I always thought the harmonic mean seemed less elegant than the other two, with all those exponents jammed in there. But as we’ll see, it’s exactly that exponentiation that will end up unifying our understanding of all three means.
A Gripping Plot
This plot shows the result of taking the arithmetic mean of two numbers. Each axis represents one of the numbers, and the shade of gray at each point represents the mean of the two.
The order of the two numbers doesn’t matter, so the axes are interchangeable.
All the plots we’ll look at today have the interesting property that their shape doesn’t change when you multiply all the numbers on their axes by the same factor¹. That means you can zoom in or out on the plot above and, as long as you keep the point (0, 0) in the lower left corner, it won’t look any different. (You’d also have to adjust the color scale, but it’s arbitrary anyway.) So from now on, instead of thinking about specific numbers, we can just think about what every plot with the origin in the lower left corner looks like:
To make it easier to see what’s going on, I’ll paint means that are close together the same shade:
Each stripe contains many pairs of numbers which have means within the same range. You can now see that the mean doesn’t change as long as you go along a straight, 45° line. That is, when you take some quantity away from one of the numbers, the arithmetic mean remains the same as long as you increase the other number by the same amount.
Now I’ll plot the geometric mean the same way.
We can immediately see the difference.
It’s a little darker. The geometric mean is always lower than or equal to the arithmetic mean.
It’s curved. An equal trade off between the two numbers no longer preserves their mean.
Because of the direction it curves in, pairs that are further apart from each other have lower means than those closer together. To clarify that, let’s look at the arithmetic and geometric means side by side:
Along the line traced in orange above, the two numbers we’re finding the mean of are the same. Whether arithmetic or geometric, the mean of two identical numbers is just that same number. So along the diagonal these images match perfectly, while on either side of it they obviously do not.
The point circled in teal represents the mean of two numbers with a large difference between them. When using the arithmetic mean (left), this point has the same value as the point 45° away from it on the orange line. But with the geometric mean (right), the values further away from the orange line bend outwards, producing a lower mean for the same point.
Now the harmonic mean:
It looks similar to the geometric. Let’s plot them side by side:
Again, they’re the same on the diagonal. But off to the side of the diagonal the harmonic mean is more curved than the geometric, and gives even lower results for pairs that are further apart from each other.
So the arithmetic mean has no curve, the geometric mean has some, and the harmonic mean has even more. The difference between them is in large part a matter of degree — of how curved they are. The stronger the curve, the lower the results of that mean when provided with dissimilar numbers. We can understand this better if we consider all the other degrees of curvature a mean can have.
Generalized Means Primer
All three Pythagorean means belong to a wider class of functions called generalized means. (These are closely related to the idea of Lᵖ spaces, so if you’re already comfortable with Lᵖ spaces it might be easier to think of these images as segments of circles centered on the origin in spaces of varying p.)
The equation for the generalized mean looks like the harmonic mean, but with the exponents replaced by a variable, conventionally p:
Or, algorithmically:
- Take everything to the pᵗʰ power
- Find the arithmetic mean as you normally would
- Take the result to the p⁻¹ᵗʰ power (find the pᵗʰ root)
When p is one, the first and last steps have no effect, so you’ve found the arithmetic mean.
When p is negative one, this equation simplifies to the equation for the harmonic mean.
It turns out the geometric mean is the limit of the generalized mean as p approaches zero. I won’t go through the proof for this because it would take too long and because I haven’t read it, but understand that the geometric mean lives in the same neighborhood as zero.
(There is no generalized mean at zero due to division by zero.)
OK, now we can visualize the generalized mean for any value of p.
All Means Possible
First, let’s retrace the steps we’ve already taken. We’ll start with the arithmetic mean and go down, pass the geometric mean at zero, and end with the harmonic mean at negative one.
Let’s keep going down.
As p approaches negative infinity, the contours become straight lines:
At this point the generalized mean becomes the function that returns the minimum of the two numbers it’s given. Notice how the result depends entirely on whichever number is lower.
OK, let’s go back to the arithmetic mean and travel in the opposite direction.
It gets lighter. The value is always equal to or higher than the arithmetic mean.
It curves the other way. Numbers that are further apart now have higher means than those closer together.
When p is two, the generalized mean is called the quadratic mean.
We use the quadratic mean often in statistics. The quadratic mean of some deviations is a standard deviation. The quadratic mean of some errors is a root-mean-square error, a name which explicitly describes the three steps listed earlier in the algorithm for the generalized mean.
Notably, at this point the stripes are perfectly round, but that’s just a side effect of your screen existing at a scale where the world deceptively appears to be a Euclidean space. 🤯
Let’s keep going up.
As p approaches positive infinity, the contours again become straight lines, and the mean becomes the maximum.
Finally, let’s see the whole thing.
You can think of generalized means as leaning in one direction or the other, either rewarding or punishing difference between the numbers they’re given.
In some situations you might need to use a mean with properties unique to that specific value of p. But for many applications, the important part is just whether p is at, below, or above one, which determines whether it curves and which direction it curves in.
I hope these animations above have given you a better intuition about what’s happening when you use a mean. If you have any corrections or insight to provide please leave a comment.
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I generated the images in this notebook (matplotlib, imageio).
Thanks to Sam Pierce Lolla for help editing.
[1] Because generalized means are homogeneous functions of degree one.
