What is the logic behind the use of the harmonic mean? I'm a self-taught student in mathematics, and I only began self-study about half a year ago. I was thinking about the harmonic mean, and the logic behind its use.
Firstly, let's begin with the arithmetic mean which equals the sum of the set of numbers divided by the number of numbers.
For example: say you have a set $(4, 3, 2)$, and you want the mean. You simply divide:
$\frac{4+3+2}{3}$. You get 3 in this case. Thus, in a way, the logic behind is to generalize it. It's to take all of the numbers, and distribute them equally to these 3 "slots" if that make sense. In doing this, you "generalize" all the numbers to one single number if that makes sense.
The geometric mean is easy to understand too. Imagine you have a set of numbers which are constantly being multiplied.
Say you have a set whose first number is multiplied by 1.3, 1.2, and 1.5.
So, starting with 4:
$(4, 8, 12, 36)$.
We can "generalize" this as we did with the arithmetic mean, because:
$4*8*12*36=x^4$.
So, we can "generalize" those numbers to the variable x like we did with the arithmetic mean.
In the words of a Stackoverflow user by the name of Domagoj Pandzha (who I thank for allowing me to rationalize the geometric mean):
"We are, essentially looking for a value which, when multiplied by itself 3 times (the number of percentages per respective years), gives the same final value as if we simply repeatedly multiplied each year with the respective 1.3, 1.4 and 1.5."
So,
$\sqrt[4]{x^4}$ = $\sqrt[4]{4*8*12*36}$
Makes sense to me to use it, since we're trying to average out the products here.
But the harmonic mean I struggle with in my mind. What is the logic behind its use?
Thanks.
 A: I will begin by staring that I am new to this site and mathematics.
Geometric mean is similar to arithmetic mean. Arithmetic mean explains, if I have a set of n numbers and I add them together, what if I stead I was to add a single number to itself that many times, what would this number be. For example 3+7+2=12, but adding 4 to itself three times does the same. (3+7+2)/3 is the simple way to do this, since deciding by three is equivalent to finding what added to itself 3 times gives (3+7+2). Maybe you already know this but it helps with geometric mean. The idea is very similar. Let's say I have a finite set of n numbers. I multiply them together. What number, if take it and multiply it to itself n times, give the same value as the product of the set of n numbers? As an example, take (2x3x4). The answer is 24. If I multiply 8 to itself three times,I get 24 also.Therefore the geometric mean is 8. Notice the the nth root symbol just represents the number that, when repeatedly multiplied to itself n times, gives the number inside the square root. Therefore this symbol is what we need and the formula for the geometric mean is the nth root of (the product of all the numbers involved in the geometric mean). The point that is to be made in these two explanations of arithmetic and geometric mean is that often things with a name in mathematics as far as I know are given names because it is useful to give names to these things. The harmonic mean honestly can just be thought of as exactly what the formula says it is. It is useful in things like rates and such,and so was given a formula. You can think of it as the reciprocal the average of (1/a+1/b+1/c...)/n but in the end it is just a name given to a predetermined way of combining numbers. Just like how derivatives are named, integrals are named, the speed of light is named, things in combinatorics are named, things in probability are named. Sometimes the meaning is just the definition of the numbers within the formula, and there is only an interpretation numerically. There are uses for the harmonic mean like again rate and outliers, if you search them up. You can connect the formula to those. But really, a named thing in math often is just something that is usefully referred to. That's my thoughts take what you will :).
