Geometric Mean
The Geometric Mean is a special type of average where we multiply the numbers together and then take a square root (for two numbers), cube root (for three numbers) etc.
Example: What is the Geometric Mean of 2 and 18?
- First we multiply them: 2 × 18 = 36
- Then (as there are two numbers) take the square root: √36 = 6
In one line:
Geometric Mean of 2 and 18 = √(2 × 18) = 6
It is like the area is the same!
Example: What is the Geometric Mean of 10, 51.2 and 8?
- First we multiply them: 10 × 51.2 × 8 = 4096
- Then (as there are three numbers) take the cube root: ^{3}√4096 = 16
In one line:
Geometric Mean = ^{3}√(10 × 51.2 × 8) = 16
It is like the volume is the same:
Example: What is the Geometric Mean of 1, 3, 9, 27 and 81?
- First we multiply them: 1 × 3 × 9 × 27 × 81 = 59049
- Then (as there are 5 numbers) take the 5th root: ^{5}√59049 = 9
In one line:
Geometric Mean = ^{5}√(1 × 3 × 9 × 27 × 81) = 9
I can't show you a nice picture of this, but it is still true that:
1 × 3 × 9 × 27 × 81 = 9 × 9 × 9 × 9 × 9
Example: What is the Geometric Mean of a Molecule and a Mountain
Using scientific notation:
- A molecule of water (for example) is 0.275 × 10^{-9} m
- Mount Everest (for example) is 8.8 × 10^{3} m
Geometric Mean | = √(0.275 × 10^{-9} × 8.8 × 10^{3}) | |
= √(2.42 × 10^{-6}) | ||
≈ 0.0016 m |
Which is 1.6 millimeters, or about the thickness of a coin.
We could say, in a rough kind of way,
"a millimeter is half-way between a molecule and a mountain!"
So the geometric mean gives us a way of finding a value in between widely different values.
Definition
For n numbers: multiply them all together and then take the nth root (written ^{n}√ )
More formally, the geometric mean of n numbers a_{1} to a_{n} is:
^{n}√(a_{1} × a_{2} × ... × a_{n})
Useful
The Geometric Mean is useful when we want to compare things with very different properties.
Example: you want to buy a new camera.
- One camera has a zoom of 200 and gets an 8 in reviews,
- The other has a zoom of 250 and gets a 6 in reviews.
Comparing using the usual arithmetic mean gives (200+8)/2 = 104 vs (250+6)/2 = 128. The zoom is such a big number that the user rating gets lost.
But the geometric means of the two cameras are:
- √(200 × 8) = 40
- √(250 × 6) = 38.7...
So, even though the zoom is 50 bigger, the lower user rating of 6 is still important.
It works well as, for example, a 25% change in either number changes the result by about 12%.