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# Geometric Mean Calculator

The Geometric Mean Calculator will calculate:

1. The geometric mean of a list of numbers

Geometric Mean Calculator Parameters: You can enter between 2 and 9 inputs (n) to calculate the geometric mean of the numbers entered.

 🖹 Normal View🗖 Full Page View Calculator Precision (Decimal Places)0123456789101112131415 Number of Inputs (n) Input 1 (a1) Input 2 (a2)
Geometric Mean Formula and Calculations The Geometric Mean (GM(n)) is GM(n) = √a1 × a2 × … × anGM(n) = √a1 × a2 GM(n) = √GM(n) = √GM(n) = Input 1 (a1) Input 2 (a2)

Please note that the formula for each calculation along with detailed calculations is shown further below this page. As you enter the specific factors of each geometric mean calculation, the Geometric Mean Calculator will automatically calculate the results and update the formula elements with each element of the geometric mean calculation. You can then email or print this geometric mean calculation as required for later use.

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## Theoretical description

Geometric mean (GM) is a type of mean or average that shows the central tendency (otherwise known as the "typical value") of a set of values. Unlike arithmetic mean (which is obtained by adding all terms and then dividing the sum by the total number of terms), geometric mean is obtained by multiplying all terms with each other and then calculating the n-th root of the product. Mathematically, we can write

GM = a1 × a2 × … × an

For example,

GM (2, 4 and 8) is 4 because 2 × 4 × 8 = 64 = 4;

GM (4, 5 and 50) is 10 because 4 × 5 × 50 = 1000 = 10; etc.

The concept of geometric mean is very important in many fields, especially in finance. For example, return, otherwise known as growth, is one of the important parameters used in finance in order to determine the both the actual and future profitability of an investment. When the return or growth amount is compounded, the investor needs to use the geometric mean to calculate the final value of the investment. Let's explain this point through an example:

### Example

An investor invested \$400, 000 in stock market in January 2017. His annual return was 4% in 2017, 8% in 2018, 10% in 2019, -32% in 2020 (due to pandemic/banking crisis/economic global downturn etc.) and 14% in 2021. What is the mean annual return (profit or loss) of the investment?

### Solution

From finance, it is known that the return of an investment represents the difference (in percent) between the actual and the initial equity. Using the recurrent (one by one) method (which is correct but long, and that will serve us as a proof for the other method), and giving that for example 5% = 0.05 and so on, we obtain

E1 = E0 + 4% = E0 + 0.04 ∙ E0 = 1.04 ∙ E0 = 1.04 ∙ \$400,000 = \$416,000
E2 = E1 + 8% = E1 + 0.08 ∙ E1 = 1.08 ∙ E1 = 1.08 ∙ \$416,000 = \$449,280
E3 = E2 + 10% = E2 + 0.10 ∙ E2 = 1.10 ∙ E2 = 1.10 ∙ \$449,280 = \$494,208
E4 = E3-32% = E3-0.32 ∙ E3 = 0.68 ∙ E3 = 0.68 ∙ \$494,208 = \$336,061.44
E5 = E4 + 14% = E4 + 0.14 ∙ E4 = 1.14 ∙ E4 = 1.14 ∙ \$336,061.44 = \$383,110

Thus, the return of investment is

Return(in \$) = \$383,110-\$400,000 = -\$16,890

When expressed in percent, we obtain

Return(in %) = -\$16,890 / \$400,000 × 100% = -4.22%

As you see, this is a very long an inefficient way. Therefore, we use the geometric mean method to calculate the return of investment much easier. Thus, writing everything as a decimal and in terms of E0, we obtain

Return (in decimal) = E1 ∙ E2 ∙ E3 ∙ E4 ∙ E5 - E0
= (√(1.04 ∙ 1.08 ∙ 1.10 ∙ 0.68 ∙ 1.14) ∙ E0 - E0
= 0.9578 ∙ E0 - E0
= -0.0422 ∙ E0 × 100%
= -4.22%

As you see, the result is the same but using the geometric mean we don't need to calculate the return every year and other complicated operations.

If we used the arithmetic mean for this purpose, we would find a value that is too far from the truth. In this case, we would obtain

AM = 1.04 + 1.08 + 1.10 + 0.68 + 1.14 / 5 ∙ E0
= 1.008 ∙ E0

This provides a return of 0.008 · E0, which when written as a percentage gives +0.8%.

## Ratio and Proportion Math Tutorials associated with the Geometric Mean Calculator

The following Math tutorials are provided within the Ratio and Proportion section of our Free Math Tutorials. Each Ratio and Proportion tutorial includes detailed Ratio and Proportion formula and example of how to calculate and resolve specific Ratio and Proportion questions and problems. At the end of each Ratio and Proportion tutorial you will find Ratio and Proportion revision questions with a hidden answer that reveal when clicked. This allows you to learn about Ratio and Proportion and test your knowledge of Math by answering the revision questions on Ratio and Proportion.

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