# Geometric Mean – Formula, Calculations & Examples

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The geometric mean is a critical mathematical concept, mainly used in fields like statistics and geometry, offering an alternative perspective on averages compared to the more commonly known arithmetic mean. Unlike the arithmetic mean, the geometric mean multiplies all values together and then takes the nth root. This approach is particularly useful when dealing with quantities that increase or decrease exponentially, such as growth rates and ratios. Understanding the geometric mean is imperative for interpreting data accurately and serves as a foundational tool for more advanced statistical models.

## Geometric Mean- In a Nutshell

• The geometric mean, calculated by multiplying the values, represents the average rate of return for a set of numbers.
• The geometric mean is the measure for portfolios and other data sets when there is a serial correlation.
• Bond yields, stock prices, and market risk premiums strongly correlate with one another.
• By factoring in compounding yearly, the geometric average gives a more accurate calculation of the real return for statistics that fluctuate widely from year to year.

## Definition: Geometric mean

When you take the square root of the product of a set of numbers, you get the GM, which is the average value or mean that characterizes the central tendency of the numbers in the collection. The process is simple:

• Multiply all the values by n and extract the nth root of the resulting integers.
• If we take any two integers, say 8 and 1, their geometric mean will be:

Therefore, the nth root of the product of n integers may be considered the geometric mean. Take into consideration that this is not the same as the arithmetic mean.

To calculate the arithmetic mean, we sum up all the numbers in the data and divide them by the total. However, in geometric mean, you multiply the provided data values by the radical index and then take root to get the complete outcome of data values.

Take the square root if you have two data points, the cube root if you have three, the fourth root if you have four, and so on, as the number of data points increases.

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## Geometric mean: Formula

If there are n observations in a data collection, the GM is calculated as the nth root of the product of the numbers. Let’s say we’re trying to get the geometric mean of a set of observations, and those observations are denoted by the letters , …,. The formula to determine the geometric mean is provided below:

or

Alternative representations include:

or

whereby n= the total number of terms that are multiplied.

Taking logarithms on both sides:

Therefore, the geometric mean =

## Geometric mean: Calculation

The geometric mean is found by performing two basic operations:

• Add up all the numbers to see what they multiply.
• Discover the nth root of the sum (n is the total number of values).

It is important to keep in mind the following before trying to calculate this central tendency indicator:

• Only positive numbers will do if you are looking for the GM.
• The GM will be 0 if any numbers in the data set are zero.

In Excel, “=GEOMEAN” is the keyboard shortcut for determining the geometric mean. To compute the GM of a set of numbers, insert the expression into a cell and then specify the cells or numbers that include those values.

## Geometric mean: Example with varying values

Two machines are compared in terms of performance across three distinct tasks and evaluation criteria. Calculating the geometric and arithmetic means of each machine’s process rating scores yields an estimate of its average efficiency.

Example

 Procedure 1 Procedure 2 Procedure 3 Machine X 7 80 2100 Machine Y 3 94 2350

### Geometric mean of machine X

Step 1:

To obtain their product, multiply all of the values together. In this case, the sum is:

Step 2:

Determine the nth root of the product, where n is the number of possible values.

### Geometric mean of machine Y

Step 1:

Multiply the values given to get the product. In this example, the sum is:

Step 2:

Determine the nth root of the product, where n is the number of possible values.

## Geometric mean: Example with percentage

You are curious about the average voter turnout in the last five U.S. elections and have obtained the following information.

Example

 Year 2000 2004 2008 2012 2016 Voter turnout (%) 50.3 55.7 57.1 54.9 60.1

Step 1:

Multiply the values together and get the product:

Step 2:

Find the nth root of the product:

Across the last five U.S. elections, the average voter participation has been 54.64 percent.

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## Geometric mean vs. Arithmetic mean

When illustrating percentage change over time or compound interest, the GM outperforms the arithmetic mean. You may now use mathematical and geometric comparisons from the examples above to find the most efficient machines between X and Y.

 Arithmetic Mean Geometric Mean Machine X 729 105.55 Machine Y 815.67 87.18

Even if Machine X has a better efficiency according to the arithmetic means, Machine Y has a higher efficiency according to the geometric means. Because the arithmetic mean is inclined toward larger numbers than the vast majority of your data, the geometric mean is a better approximation.

## When to use the geometric mean

The geometric mean provides more reliable results for data or percentages that are favorably skewed than the more popular arithmetic mean. There is a concentration of lower scores and a more dispersed right tail in a positively skewed distribution.

Some examples of skewed statistics include the distribution of income. In a positively skewed dataset, most values are often rather low, but a few extremely high values or outliers may skew the arithmetic mean to the right.

The GM is often less than the arithmetic mean, so it is a more accurate representation of smaller numbers. Geometric mean assumes that variables cannot take on negative values and always have a genuine zero. It works best for comparing ratios. Any negative percentage shifts must be couched in a positive light.

## FAQs

#### What does the geometric mean calculate?

The GM is used when calculating an average since it multiplies all numbers to determine their root.

#### Why are there discrepancies between the geometric and arithmetic mean?

The arithmetic mean, commonly known as “the mean,” is the most used measure of central tendency. The geometric mean uses multiplication and finding the root of numbers to conclude, while the arithmetic means averages them.

#### Is it possible to use negative numbers in a geometric mean calculation?

No, it’s impossible to compute the geometric mean for a set that contains negative values.

#### What do central tendency measurements indicate?

The data collection’s average or median may be determined using central tendency measures. The mean, median, and mode are the three most prevalent indicators of central tendency.

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