Geometric Mean: Definition, Formula & Finding

What is the Geometric Mean?

The geometric mean is a measure of central tendency that averages a set of products. Its formula takes the n^th root of the product of n numbers.

Like the arithmetic mean, the geometric mean finds the center of a dataset. While the arithmetic mean finds the center by summing the values and dividing by the number of observations, the geometric mean finds the center by multiplying and then taking a root of the product.

Based on the calculation methods, the arithmetic mean is the better statistic when adding data is appropriate, while the geometric mean is better when you need to multiply the data.

Use the geometric mean when your subject area requires you to multiply your values or uses exponents. For example, use the geometric mean for interest rates, rates of return, and data that follow the lognormal distribution. All these applications involve multiplication (i.e., products) rather than addition. For example, interest rate calculations include multiplying the principal by the rate. Additionally, the multiplication of random variables often produces lognormal distributions.

In this post, learn when to use the geometric mean and how to interpret it. I’ll also show you how to find the geometric mean using its formula. We’ll work through a real-life example of when it’s appropriate to use the geometric mean.

Related posts: Measures of Central Tendency and Arithmetic Mean

How to Find the Geometric Mean

The geometric mean formula is the following:

Where:

• X = the values of your variable.
• n = the number of values in your dataset.

To describe this process using words, the geometric mean formula takes the nth root of the product of n values.

To find the geometric mean, multiply all your values together and then take a root of it. The root depends on the number of values in your dataset. If you have two values, take the square root. With three values take the cube root. With four values, take the 4^th root, and so on.

All of your data must have positive values. For any given dataset, the geometric mean is almost always less than the arithmetic mean. The exception occurs when your dataset contains identical numbers (e.g., all 5s). In that case, the geometric mean equals the arithmetic mean.

Let’s work through several examples with step-by-step instructions.

How to Find Geometric Mean with Two Numbers

Imagine you have two numbers of 5 and 20 and need to find the geometric mean. Let’s use the geometric mean formula with these numbers.

• Multiply the two numbers: 5 *20 = 100.
• Because there are two values, take the square root of 100, which equals 10.

Therefore, we find the geometric mean of these two numbers is the following:

In geometry, imagine a rectangle that has two sides of 5 and 20. The area equals 100. The geometric mean tells you the size of the square (which must have equal sides) that produces the same area as the rectangle.

For this example, a square with equal sizes of 10 produces the same area as the 5 X 20 rectangle.

5 X 20 = 10 X 10 = 100

How to Find Geometric Mean with Three Numbers

Suppose you have three numbers of 5, 12, and 22 and need to find the geometric mean. We’ll enter these values into the geometric mean formula.

• Multiply the three numbers: 5 *12 * 22 = 1320.
• Because there are three values, take the cube root of 1320, which equals 10.97.

Therefore, we find the geometric mean of these three numbers is the following:

In geometry, imagine you have a rectangular box with three sides of 5 X 12 X 22. The volume of this box equals 1320. The geometric mean tells you the size of the cube (which must have equal sides) that produces the same volume as the rectangular box.

5 X 12 X 22 = 10.97 X 10.97 X 10.97 = 1320

Interpreting the Geometric Mean

First, let’s look at the more familiar arithmetic mean. This statistic calculates a number that sums to the total value of your dataset given its sample size.

Suppose your dataset has these five values: 8, 10, 12, 14, 16. They sum to 60.

The arithmetic mean of these values is 60 / 5 = 12. Therefore, five 12s sum to 60:

8 + 10 + 12 + 14 + 16 = 12 + 12 + 12 + 12 + 12 = 60.

The geometric mean formula calculates a number that produces the same product as your sample. Using the above dataset, the product is the multiplication of all five values: 8 X 10 X 12 X 14 X 16 = 215,040.

The geometric mean is the 5^th root of this product:

We find the geometric mean of these values is 11.655. Therefore:

8 X 10 X 12 X 14 X 16 = 11.655 X 11.655 X 11.655 X 11.655 X 11.655 = 215,040

Alternatively, 8 X 10 X 12 X 14 X 16 = 11.655⁵.

Finding the geometric mean is appropriate when you’re multiplying a set of varying numbers and need to find a constant number that produces the same product. I’ll show you a real-life example in the next section!

When to Use It

As you can saw above, the geometric mean formula involves multiplying values. Hence, when your subject area involves multiplication, consider using the geometric mean. This need often occurs when you’re working with interest rates and growth rates because they involve multiplication.

When working with interest rates, you multiply the principal by the interest rate, and you need to factor in compounding over time. Suppose you have a principle of $1,000 and a 10% annual interest rate. In the first year, you’ll earn $1,000 X 10% = $100 in interest and have a total of $1,100. For the second year, you apply the interest rate to the principle of $1000 plus the $100 of previous interest. That’s compounding, and the geometric mean accounts for it.

Use the geometric mean when working with the interest/growth rates rather than the actual dollar or population amounts. Depending on the type of growth rate in your study, the geometric mean formula finds the compounded annual growth rate, the average annualized rate, or the average growth rate.

Interest Rate Example

Let’s work through an interest rate example. However, throughout this example, you could replace the words “interest rate” with “growth rate” to see how you’d use the geometric mean with population growth rates.

Suppose you have nine years of interest rates: 10%, 13%, 9%, 14%, 11%, 13%, 15%, 9%, 11%. Your starting balance is $10,000 and the ending balance is $26,955.74. You want to calculate the average annualized interest rate. In other words, while the actual interest rate fluctuated over the nine years, what is the average rate that would produce the same total growth?

To solve this type of question, add 1 to all your interest rates (e.g., 10% becomes 1.10, etc.) Then find the geometric mean by incorporating these values into the formula. For our example, we need to calculate the geometric mean for this dataset: 1.10, 1.13, 1.09, 1.14, 1.11, 1.15, 1.09, 1.11.

Here’s the geometric mean formula for these data:

Hence, we find the geometric mean is 1.1165. This value indicates that the average annualized rate of return is 11.65%. If you start with a balance of $10,000 and have a constant annual rate of return of 11.65% for nine years, you’ll finish with a balance of $26,955.74. That’s the same final balance as with the actual, varying interest rates.

In other words, the geometric mean finds the constant interest rate you multiply each year to achieve the same growth as the variable interest rates over those years.