Arithmetic, Geometric, Harmonic Means.

Question

I understand the procedure to calculate the Arithmetic, Geometric & Harmonic means very well.

But I have a few questions regarding them.

1. What I want to know is why these 3 are used?

2. What do they tell us? What is their significance practically?

3. What is the intuition behind them? Why do they work the way they do?

I googled and read about it but I don't understand anything from that.

Can anyone explain me in easiest words?

Thanks a lot.

Answer 1

Any mean is an attempt at answering the question "We have a collection of numbers. If we wanted to swap all these numbers out with equally many copies of a single number, what number ought that to be if the end result should be the same?"

Arithmetic mean is the one to go for if the way you combine different numbers is adding them up. For instance, I have 10 dollars, you have 20 dollars. Since the relevant quantity is the total sum of our respective fortunes, we have on average $\frac{10 + 20}2 = 15$ dollars each.

Geometric mean is the one to go for if it is the product of the numbers which ought to be unchanged. I can't come up with any good concrete, real-life examples, but if you have a rectangle with sides $4$ and $9$, say, and the thing you really care about is the area, then the most fair way to average those two sides is the geometric mean: your rectangle is as good as a square with sides $\sqrt{4\cdot 9} = 6$. (Also, there are many contexts where to me, $10$ feels like a more natural mid-point between $1$ and $100$ than $50.5$ does. But that may just be personal.)

Harmonic mean is the one to use if you're interested in, for instance, how much time a task takes. If I finish a task in 2 minutes, and you can finish it in 3, then on average, we finish one task every $\dfrac{2}{\frac12 + \frac13} = \frac{12}{5}$ minutes.

There are many other means out there (median, mode, quadratic mean, max, min, and so on), but they all answer that same question in different contexts. If you're uncertain which one to use, forget all about formulas and what the different means are called, and just answer that question yourself.