# 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!"

Another cool one:

### Example: What is 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 × 10
^{7}m

**Geometric Mean**

**= √(3 × 10**

^{-8}× 1.3 × 10^{7})**= √(3.9 × 10**

^{-1})**= √0.39**

**≈ 0.6 m**

A child is about **0.6 m** tall! So we could say:

"A child is half-way between a cell and the Earth"

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.