## Mathematics and Applied Statistics Lesson of the Day – The Geometric Mean

June 30, 2014 Leave a comment

Suppose that you invested in a stock 3 years ago, and the annual rates of return for each of the 3 years were

- 5% in the 1st year
- 10% in the 2nd year
- 15% in the 3rd year

What is the average rate of return in those 3 years?

It’s tempting to use the arithmetic mean, since we are so used to using it when trying to estimate the “centre” of our data. However, the arithmetic mean is not appropriate in this case, because the annual rate of return implies a * multiplicative* growth of your investment by a

**factor**of , where is the rate of return in each year. In contrast, the arithmetic mean is appropriate for quantities that are

*in nature; for example, your average annual salary from the past 3 years is the*

**additive****sum**of last 3 annual salaries divided by 3.

If the arithmetic mean is not appropriate, then what can we use instead? Our saviour is the **geometric mean**, . The **average factor of growth** from the 3 years is

,

where is the rate of return in year , . The average annual rate of return is . **Note that the geometric mean is NOT applied to the annual rates of return, but the annual factors of growth.**

Returning to our example, our average factor of growth is

.

Thus, our annual rate of return is .

Here is a good way to think about the difference between the arithmetic mean and the geometric mean. Suppose that there are 2 sets of numbers.

- The first set, , consists of your data , and this set has a sample size of .
- The second, , set also has a sample size of , but all values are the same – let’s call this common value .

- What number must be such that the
**sums**in and are equal? This value of is theof the first set.**arithmetic mean** - What number must be such that the
**products**in and are equal? This value of is theof the first set.**geometric mean**

Note that the **geometric means is only applicable to positive numbers**.

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