Geometric Mean Calculator
Find the geometric mean of a set of positive numbers.
Comma-separated positive numbers
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How it works
The geometric mean is a special kind of average that is appropriate for numbers that multiply together, such as growth rates, ratios, and index values, and this calculator computes it from a list of positive numbers. It works by multiplying all the values and taking the nth root of the product, which for the numbers 1, 3, and 9 gives the cube root of 27, or 3. Unlike the ordinary arithmetic mean, which sums values, the geometric mean respects compounding, making it the right choice for averaging percentage changes over time, investment returns across years, or population growth rates. For example, if an investment grows 10 percent one year and 30 percent the next, the geometric mean of the growth factors, not the arithmetic mean of the percentages, gives the true average annual growth. The geometric mean is always less than or equal to the arithmetic mean of the same numbers, and it is only defined for positive values because a zero or negative number would make the product zero or complex. It is widely used in finance, biology, and any field dealing with rates of change or normalised ratios. Enter your positive values to compute the geometric mean.
GM = (x₁ · x₂ · … · xₙ)^(1/n)
The geometric mean multiplies all the values together and takes the nth root of the product, where n is how many values there are. Because it is based on products rather than sums, it is the correct average for quantities that combine multiplicatively, such as growth rates and ratios. All the values must be positive, since the product of a negative number cannot be handled by a real root in general.
Source:Descriptive statistics
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Worked example
How to use this calculator
- 1
Enter your values
Enter values (positive).
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
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Frequently asked questions
What is the geometric mean?
The geometric mean is the nth root of the product of n numbers. It is a type of average suited to values that combine by multiplication rather than addition, such as growth rates and ratios. For example, the geometric mean of 4 and 9 is the square root of 36, which is 6. It requires all values to be positive.
When should I use the geometric mean instead of the arithmetic mean?
Use the geometric mean when averaging quantities that multiply or compound, such as annual investment returns, growth rates, or ratios and indices. The arithmetic mean is right for quantities that add, like test scores or heights. Using the arithmetic mean on compounding rates overstates the true average, which is why finance uses the geometric mean for returns.
Why must the numbers be positive?
The geometric mean involves taking a root of a product. If any value were zero, the whole product would be zero, and if a value were negative, the product could be negative and its root would not be a real number in general. To keep the result meaningful and real, the geometric mean is defined only for positive numbers.
How does it compare to the arithmetic mean?
For any set of positive numbers, the geometric mean is always less than or equal to the arithmetic mean, and they are equal only when all the values are identical. This is known as the AM-GM inequality. The gap between them widens as the values become more spread out, reflecting the geometric mean's sensitivity to proportional differences.
How is it used for average growth?
To find the average growth rate over several periods, convert each rate to a growth factor, such as 1.1 for 10 percent growth, take the geometric mean of the factors, and subtract one. This gives the constant rate that would produce the same overall result, which the arithmetic mean of the percentages cannot do because growth compounds.