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 × 103 m
Geometric Mean= √(0.275 × 10-9 × 8.8 × 103)
= √(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 a1 to an is:

n√(a1 × a2 × ... × an)

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.