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) and so on.
Example: What's 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
In fact the area is the same!
Example: What's 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
In fact the volume is the same:
Example: What's 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's 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 × 103 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!"
Another cool one:
Example: What's the Geometric Mean of a Cell and the Earth?
- A skin cell is about 3 × 10-8 m across
- The Earth's diameter is 1.3 × 107 m
A child is about 0.6 m tall! So we could say, in a rough kind of way,
"A child is half-way between a cell and the Earth"
Another!
Example: What's the geometric mean of an Electron and our Galaxy?
A fun one to try yourself!
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 a1 to an is:
n√(a1 × a2 × ... × an)
Growth Rates
The Geometric Mean is great for calculating average growth rates.
Example: Investment Growth
Your investment grows by 10% the first year and 60% the second year. What's the average growth rate?
- First year multiplier: 1.10
- Second year multiplier: 1.60
- Multiply them: 1.10 × 1.60 = 1.76
- Take the square root: √1.76 ≈ 1.326...
The average growth is about 32.6% per year.
(Note: The Arithmetic Mean would give 35%, which is actually wrong in this case!)
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.