What geometric average shows

How do you calculate a geometric average? I reccomend you to simply check wikipedias page about it, geometric mean.

But what is the use or simply just what's the deal about having a geometric average? In economy it is very useful when looking at the progress of the stock market for example.

I'll give you a quick example to understand the use:
Let's say you have a stock index which goes like this:
Y1: 100
Y2: 90
Y3: 105
Y4: 110

which is a developement of:
Y2: -10%
Y3: 16,66%
Y4: 4,76%

Now, if you invest 100 at year 1, your stocks will at the end of Y3 be worth 105.

If you simply take the average of those numbers the regular way: (-0,1+0,1666+0,0476)/3=0,038 ==> 3,8%

If you take the geometric average you get:
(0,9*1,167*1,048)^(1/3)=1,0325 - 1= 3,25%

Which one would you rely on to give you a rate on return on your investment?

100 * 1,038³= 111,8
100 * 1,0325³ = 110

As you can see, the geometric average gives you the correct return on your investment!