# Lesson WHAT IS Geometric mean

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## Geometric mean

#### Definition of Geometric Mean

If two positive real numbers, and , are given, then Geometric mean is
.

For example, Geometric mean of numbers 2 and 8 is equal to = .
Geometric mean of numbers 3 and 4 is equal to = = 3.464 (approximately).

Geometric mean is, usually, different from Arithmetic mean. See a separate lesson explaining What is the difference between an arithmetic and geometric mean.

The term “Geometric mean” is originated from Geometry. If you draw the perpendicular from the right angle of the right triangle to its hypotenuse, then the height of the triangle is exactly equal to Geometric mean of two parts of hypotenuse: , see Figure below.

When someone calculates Geometric mean, he or she usually does it for positive numbers.

Example 1. Calculate Geometric mean of numbers 5 and 7.

Example 2. Calculate Geometric mean of numbers 9 and 11.

Note. Geometric mean of two positive numbers is always lesser or equal to their Arithmetic mean.

Geometric mean of two positive numbers is equal to their Arithmetic mean if and only if both numbers are equal.

Geometric mean of three real positive numbers, , and , is

Geometric mean of the set of n real positive numbers, , , …, , is

Again, Geometric mean of n numbers is defined for positive numbers only (no negative numbers, no zero).

#### Geometric mean in Finance

Geometric mean is often used in finance to calculate an average return on investment, from returns during several given years. Below, the typical use of Geometric mean in finance is shown.

Suppose that you invested \$1000.00 in a mutual fund for four years. If your return rates each year were =10%, =14%, =16% and =-10%, what would your average return rate be during this period?

If you are calculating the average return rate simply as Arithmetic mean of annual rates, you would get an answer of 7.5%. But this is not correct calculation.

The correct calculation is as follows.

After the first   year you will have 1000.00*(1+0.1) dollars.
After the second year you will have 1000.00*(1+0.1)*(1+0.14) dollars.
After the third   year you will have 1000.00*(1+0.1)*(1+0.14)*(1+0.16) dollars.
After the fourth year you will have 1000.00*(1+0.1)*(1+0.14)*(1+0.16)*(1-0.1) dollars.

If we designate the average annual return rate as r, then your return after four years is
, and an equation to calculate the unknown value of r is

.

Hence, 1+r is simply Geometric mean of four numbers , , and :

.

This means that

.

So, the average annual rate is 6.5%.

From this example you can see that Geometric mean is an appropriate tool to calculate average growth rate for processes with variable (in time) growth rate.

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