Geometric mean

Imagine you want to buy a new car, say a Ford Mustang. You have been eyeing it for a long time, and every year you note down how much the price has gone up compared with the year before. In the first year the car cost £40,000. In the second year the price went up by 25%, the next year by 5%, in the fourth year by 10%, and now, in the fifth year, by 20%. We can work out what the car costs today by turning the percentages into multipliers, so “a 25% increase” is written as “the original price times 1.25”. The current price of the Ford Mustang is therefore

$$40000\cdot 1.25\cdot 1.05\cdot 1.1\cdot 1.2=69300$$

After five years the Mustang costs £69,300. Now we could ask: what was the average yearly price increase? In the first step the price was multiplied by 1.25, in the second by 1.05 and so on. What was the average multiplier? If we used the ordinary arithmetic mean, we would get nonsense. The arithmetic mean of the values 1.25, 1.05, 1.1 and 1.2 would be

$$\frac{1.25+1.05+1.1+1.2}{4}=\frac{4.6}{4}=1.15$$

Let's check by working out the price of the car if it went up by a factor of 1.15 four years in a row:

$$40000\cdot 1.15\cdot 1.15\cdot 1.15\cdot 1.15=69960.25$$

We got a different number! So 1.15 (that is, 15%) is not the right answer: it is not the correct average yearly price increase. The arithmetic mean cannot answer this question. We have to use the geometric mean.

How to work out the geometric mean

By how many per cent did the car's price rise over those years? We find out by multiplying all the multipliers together:

$$1.25\cdot 1.05\cdot 1.1\cdot 1.2=1.7325$$

So the price of the Ford Mustang went up by a factor of 1.7325 over the four increases. Now what we are really asking is: which number, when multiplied by itself four times, gives 1.7325? That is the fourth root of 1.7325. Let's try it:

$$\sqrt[4]{1.25\cdot 1.05\cdot 1.1\cdot 1.2}=\sqrt[4]{1.7325}\approx 1.147277$$

Now let's multiply the price of the car by this number four times:

$$40000\cdot \sqrt[4]{1.7325}\cdot \sqrt[4]{1.7325}\cdot \sqrt[4]{1.7325}\cdot \sqrt[4]{1.7325}=69300$$

Now it works: the number $\sqrt[4]{1.7325}$ is the correct average yearly price increase.

Formula for the geometric mean

If we have n numbers x₁, x₂, …, x_n, we work out their geometric mean G as

$$G=\sqrt[n]{x_1\cdot x_2\cdot \dots \cdot x_n}$$

Equivalently, we can write it like this:

$$G=\sqrt[n]{\prod _{i=1}^n{x}_i}$$

For example, we would work out the geometric mean of the numbers 1, 2, 3, 4, 5 as

$$\sqrt[5]{1\cdot 2\cdot 3\cdot 4\cdot 5}=\sqrt[5]{120}\approx 2.6$$