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?

In one line:

Geometric Mean of 2 and 18 = √(2 × 18) = 6

It is like the area is the same!

geometric mean 2x18 = 6x6

Example: What is the Geometric Mean of 10, 51.2 and 8?

In one line:

Geometric Mean = 3√(10 × 51.2 × 8) = 16

It is like the volume is the same:

geometric mean 10x51.2x8 = 16x16x16

Example: What is the Geometric Mean of 1, 3, 9, 27 and 81?

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

length continuum

Using scientific notation:

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.

top view camera

Example: you want to buy a new camera.

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:

So, even though the zoom is 50 bigger, the lower user rating of 6 is still important.