Geometric Mean Calculator

Geometric Mean Calculator

Use this geometric mean calculator to determine the geometric mean of a set of numbers or percentages. It can handle any set of numbers, from small datasets to large arrays, including integers and decimal values. Our calculator can also take the negative values. By Converting these negative values into positive growth rates it finds the geometric mean of return on investment.

What Is A Geometric Mean?

The geometrical mean is the measure that represents the central tendency of positive real numbers by taking the nth root of the product of “n” numbers. 

Geometric Mean Formula:

Where:

• n is the total number of values
• (a₁, a₂, a₃,... aₙ) indicates the numbers from first to nth term

How To Calculate Geometric Mean?

• Write down all the numbers and multiply them together
• Take the nth root of the product

Geometric Mean Example:

Find the geometric mean of 12, 23, 34.

Solution:

Step 1: Multiply the values together

= 12 × 23 × 34

Step 2: Take the nth root of the product

Geometric Mean = 3 (9384)

Geometric Mean = 21.0926

The geometric mean of the small data set can be calculated easily as we have done in the above example. But what about the large arrays of data? Don't overwhelm yourself in performing the long calculation, just access the Geomen calculator and get the geometric mean of your data by making a few clicks. 

Geometric Means for Negative Numbers:

According to the above definition and formula, the geometric mean can be calculated only for the positive numbers. But you can find out the geometric mean of negative numbers in certain contexts when they represent ratios or rates. Negative values are used to indicate a decline or loss in financial calculations and growth analysis. 

Example:

Calculate the geometric mean of −20, 30, and -15.

Solution:

Consider the given values as percentages -20%, 30%, and -15%

Now convert these percentages into growth factors:

-20% decline = 1 x 1 - 20 100

-20% decline = -0.8

30% growth = 1 x 1 + 30 100

30% growth = 1.3

-15% decline = 1 x 1 - 15 100

-15% decline = -0.85

Multiply these growth factors:

= -0.8 x 1.3 x -0.85

= 0.884

Take the cube root:

= 3 0.884

≈ 0.9597

Convert it into the percentage:

Geometric Mean ≈ (0.9597 -1) x 100 ≈ -0.0402 x 100 ≈ -4.02

If you are dealing with negative numbers, you can perform such calculations instantly using an online geometric mean calculator with steps.

Geometric Mean With Logarithm:

• Take the logarithm of numbers
• Sum the logarithm and divide by the total values
• Now, take the antilog of the result to find out the geometric mean

Geometric Mean With Zero:

If a zero is present in the data set then the geometric mean is not meaningful. However, in some cases, adjustments can be made to accommodate the zero value. A zero can be turned into 100% or 1. Sometimes zeros are used to represent “no response” and can be removed from the data while finding the geometric means. Our geometric average calculator can not automatically adjust the zero.

When To Use Geometric Mean?

The geometric mean is used when it's necessary to find:

• The average rate of return or growth rate
• Ideal when dealing with the multiplication of values or exponential growth

References:

From the source of Wikipedia: Geometric mean and Calculation, Relationship with logarithms.

From the source of Investopedia: Breaking Down the Geometric Mean in Investing. 

From the source of LibreTexts: Geometric Mean.