A Lesser Known Average - The Geometric Mean

Hi there. In this statistics education post, I talk about the geometric mean which is not as popular as the usual average known as the arithmetic mean.

Math symbols and text are rendered images from the use of QuickLaTeX.com.

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Topics

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• A Review Of The Arithmetic Mean
• The Geometric Mean
• Geometric Mean Used In Finance
• Notes

 

A Review Of The Arithmetic Mean

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Typically when people mention an average it is usually referred to the arithmetic average or arithmetic mean. This average is computed by adding up all the observed values and then dividing that total by the number of observations.

In math notation, it can be written as:

In more layman terms it is more like:

Keep that in mind that the average represents partial information of the population. For example, the sample average of height of Canadians does NOT represent the average height of a random person in North America.

Example

Julia's test scores from three tests are 92, 88, and 81. Her average of these test scores are (92 + 88 + 81) divided by 3. This comes to 261 divided by 3 which is equal to 87.

 

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The Geometric Mean

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While the usual arithmetic mean adds all the numbers and divides by the number of numbers, the geometric means involves multiplying the numbers and then taking the n-th root. (The second root where n = 2 is the square root.)

In more simpler terms, the geometric average is expressed as:

 

Example

The geometric average of five numbers 4, 8, 3, 5 and 1 is the fifth root of 480. It is approximately 3.438.

 

Geometric Mean Used In Finance

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You might ask yourself, what's the point of the geometric mean. Isn't the usual average of adding up the numbers/observations and dividing by the number of observations enough?

According to this Investopedia page, the geometric mean is used for portfolio returns. In their example near the end of the page, it is stated that the geometric average is closer to reality than the usual arithmetic mean.

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Note

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There may be some manipulative people out there who misreport statistical findings in order to mislead people.

You may get the rare case where someone says average where you think they refer to the usual arithmetic average when they actually refer to the geometric average!

 

Thank you for reading.