The harmonic mean

Last time we looked at the geometric mean in contrast to the “normal” arithmetic mean. There is even a third way to determine the “middle” of two numbers, the harmonic mean. The harmonic mean of two numbers a and b is 2/(1/a + 1/b). In other words it is the inverse of the arithmetic mean of the inverse numbers 1/a and 1/b. It might seem like this should be the same as the arithmetic mean but we will look at an example to see that it isn’t.

As one easily sees the arithmetic mean of one and a million is about half a million. Note that the mean of two and a million or three and a million or fifty and a million will still be very close to half a million (at least in a geometric sense). On the other hand, doubling the million instead of the one will also very nearly double the mean. We have seen this property already. For the harmonic mean we look at the inverse numbers, one and 1/1.000.000 (~0). Their mean is about ½, the inverse and thus the harmonic mean is two, compared to the arithmetic mean of half a million. Furthermore, adding a few more millions to the second number barely does something to the mean because the inverse was already basically zero. Doubling the one however will mean that its inverse is now ½, resulting in a harmonic mean of four, double what we had before.

In this example it becomes clear that for the arithmetic mean the bigger number is more important and, if it is much bigger, can determine the mean almost on its own. For the harmonic mean however, the small number is all that (really) matters. In the geometric mean both numbers are equally important, as one easily confirms, quadrupling one of them will double the geometric mean. Particularly in the case of one and a million the geometric mean is a thousand, between the harmonic and arithmetic one. This is no coincidence. As the discussion above suggests, the harmonic mean will always be smaller than the geometric mean which is again smaller than the arithmetic mean. In fact, they can only be the same when the two numbers are identical.

Finally, I want to point out that we compared the means in a geometric sense where, by construction, the geometric mean was right in the middle while the arithmetic mean was much bigger and the harmonic mean smaller. Similarly, in the usual sense of distance (the difference between the numbers) the arithmetic mean is the exact middle and the other two are way smaller. As you might imagine, there is also a “harmonic” way to measure distance where the geometric and arithmetic mean appear too big. Similarly to calculating the mean this is the difference of the inverse numbers (1/a – 1/b).