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

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 $1 + r$, where $r$ is the rate of return in each year.  In contrast, the arithmetic mean is appropriate for quantities that are additive in nature; for example, your average annual salary from the past 3 years is the 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, $G$.  The average factor of growth from the 3 years is

$G = [(1 + r_1)(1 + r_2) ... (1 + r_n)]^{1/n}$,

where $r_i$ is the rate of return in year $i$, $i = 1, 2, 3, ..., n$.  The average annual rate of return is $G - 1$.  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

$G = [(1 + 0.05) \times (1 + 0.10) \times (1 + 0.15)]^{1/3} = 1.099242$.

Thus, our annual rate of return is $G - 1 = 1.099242 - 1 = 0.099242 = 9.9242\%$.

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.

1. The first set, $S_1$, consists of your data $x_1, x_2, ..., x_n$, and this set has a sample size of $n$.
2. The second, $S_2$,  set also has a sample size of $n$, but all $n$ values are the same – let’s call this common value $y$.
• What number must $y$ be such that the sums in $S_1$ and $S_2$ are equal?  This value of $y$ is the arithmetic mean of the first set.
• What number must $y$ be such that the products in $S_1$ and $S_2$ are equal?  This value of $y$ is the geometric mean of the first set.

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