Geometric Mean Calculator
Find the geometric mean of a set of positive numbers, or the average growth rate of a series of percentage changes. See every step and a clear comparison with the ordinary average.
Geometric Mean Calculator
Choose what you want to calculate, enter your numbers and get your result instantly.
Step-by-Step Solution
What Is the Geometric Mean?
The geometric mean is a kind of average that multiplies the numbers together instead of adding them. For n numbers, you multiply them all and then take the n-th root of the product.
It is the right average to use for things that grow by multiplication, such as investment returns, population growth and ratios. For the ordinary average that adds the numbers, see our Average Calculator.
How to Calculate the Geometric Mean
You can find the geometric mean by hand in three steps.
1. Multiply all the numbers
Find the product of all the values in your data set.
2. Count the numbers
Find n, the number of values.
3. Take the n-th root
Take the n-th root of the product. For two numbers this is the square root, for three numbers the cube root, and so on. The result is the geometric mean.
When the product is very large, our calculator uses logarithms behind the scenes, which gives the same answer without overflow.
Geometric Mean Calculator Examples
Example 1: two numbers
The product is 16, and the square root of 16 is 4. The geometric mean is 4. The ordinary average is 5.
Example 2: four numbers
The product is 64, and the fourth root of 64 is about 2.8284. The geometric mean is 2.8284. The ordinary average is 3.75.
Example 3: average growth rate
An investment gains 10% in the first year, loses 5% in the second year and gains 20% in the third year.
The growth factors are 1.10, 0.95 and 1.20, and their product is 1.254. The cube root of 1.254 is about 1.0784, so the average growth is about 7.84% per year. The total growth over the three years is 25.4%, so 1,000 becomes 1,254. The ordinary average of the three rates is 8.33%, which is higher than the true average growth.
Geometric Mean vs Arithmetic Mean
For positive numbers, the geometric mean is never larger than the arithmetic mean (the ordinary average). They are equal only when all the numbers are the same. The more spread out the numbers are, the bigger the gap between the two.
Use the arithmetic mean for values that add up, such as test scores or heights. Use the geometric mean for values that multiply, such as growth rates, price ratios and index numbers. If some values matter more than others, the Weighted Mean Calculator may be a better choice.
Average Growth Rate (Compound Growth)
When a quantity changes by a percentage each period, you cannot simply average the percentages. A gain of 50% followed by a loss of 50% does not bring you back to the start, because the second change applies to a bigger number.
The correct way is to convert each percentage change to a growth factor (1 plus the rate as a decimal), find the geometric mean of the factors, and subtract 1. Our calculator does this when you choose the growth rate option. Each rate must be greater than −100%, because a loss of 100% or more would take the value to zero or below.
Why Must the Numbers Be Positive?
The geometric mean works with products and roots. A zero in the data makes the product zero, so the geometric mean is zero and tells you very little. A negative number can make the product negative, and an even root of a negative number is not a real number. For these reasons, our calculator asks for values greater than zero.
More Geometric Mean Examples
These extra worked examples show a clean three-number case, why a 50% gain followed by a 50% loss is not an average of zero, how to find an average yearly growth rate, and why ratios need the geometric mean. Remember that for growth rates you convert each percentage to a factor, find the geometric mean of the factors, and subtract 1.
Example 4: Three numbers
Find the geometric mean of 4, 6 and 9.
Answer: the geometric mean is 6.
The product is 4 × 6 × 9 = 216. There are 3 numbers, so take the cube root: the cube root of 216 is 6, because 6 × 6 × 6 = 216. The ordinary average is (4 + 6 + 9) ÷ 3 ≈ 6.3333, which is a little higher, as expected for positive numbers that are not all equal.
Example 5: A 50% gain followed by a 50% loss
A value rises by 50% in the first period and then falls by 50% in the second period. What is the average growth rate?
Answer: the average growth rate is about −13.40% per period, not 0%.
The growth factors are 1.5 and 0.5, and their product is 0.75. The square root of 0.75 is about 0.8660, so the average growth rate is 0.8660 − 1 ≈ −0.1340, or −13.40% per period. Check with 100: after a 50% gain it is 150, and after a 50% loss it is 75, a total change of −25%. The simple average of +50% and −50% is 0%, which wrongly suggests no change at all.
Example 6: An average yearly growth rate
A company grows by 5%, 10%, 15% and 20% in four years in a row. What is the average yearly growth rate?
Answer: the average growth rate is about 12.36% per year.
The growth factors are 1.05, 1.10, 1.15 and 1.20. The product is 1.05 × 1.10 × 1.15 × 1.20 = 1.5939. With 4 values, take the fourth root: 1.5939 to the power of 1/4 is about 1.1236. Subtract 1 to get a growth rate of about 12.36% per year. The total growth over the four years is about 59.39%, so 1,000 becomes about 1,593.9. The simple average of the four rates is 12.5%, which is slightly too high.
Example 7: Averaging ratios
In one shop, item A costs 2 times as much as item B. In another shop, item A costs 0.5 times as much as item B. What is the fair average ratio?
Answer: the fair average ratio is 1.
The product of the two ratios is 2 × 0.5 = 1, and the square root of 1 is 1. So on average the two items cost the same. The ordinary average of 2 and 0.5 is 1.25, which would wrongly suggest that A is usually more expensive. Because 2 and 0.5 are exact opposites in terms of ratio, the geometric mean treats them fairly.
Common mistakes when finding a geometric mean
A common mistake is averaging percentage changes directly instead of using growth factors. Another is forgetting to subtract 1 at the end, which leaves you with a factor instead of a growth rate. Also use the right root: the cube root for 3 numbers, the fourth root for 4 numbers, and so on. The geometric mean needs numbers greater than zero. For values that simply add up, use our Average Calculator instead.
Created by Sajid Ali, SmartStatCalc · Last updated: 7 November 2026
Found a mistake or have a suggestion? Contact us.
How to Use Our Geometric Mean Calculator
- Choose whether you want the geometric mean of numbers or an average growth rate.
- Type or paste your values into the box.
- Click the Calculate Geometric Mean button.
- Read the result and compare it with the ordinary average.
- Use the step-by-step solution to see how the answer was found.
Click Reset at any time to clear your data and start again. To see more about your numbers, you can also try the Descriptive Statistics Calculator.
Why Use SmartStatCalc?
SmartStatCalc is built to make everyday statistics quick and easy to understand. Instead of working out large products and roots by hand, you can enter your numbers and see a clear result right away.
Who Can Use a Geometric Mean Calculator?
Students meet the geometric mean in statistics and math classes. Investors and analysts use it to find average returns and growth rates, and researchers use it for ratios and data that spreads across very different sizes.
Frequently Asked Questions
What is the geometric mean in simple words?
It is the n-th root of the product of n numbers. It is an average that works with multiplication instead of addition.
What is the geometric mean of 2 and 8?
The product is 16 and its square root is 4, so the geometric mean is 4.
Is the geometric mean always smaller than the arithmetic mean?
For positive numbers, it is smaller or equal. It is equal only when all the numbers are the same.
When should I use the geometric mean?
Use it for growth rates, returns, ratios and other values that multiply from one period to the next.
Can I use zero or negative numbers?
No. Zero makes the geometric mean zero, and negative numbers can give results that are not real numbers. The calculator asks for numbers greater than zero.
How do I find the average growth rate?
Convert each percentage to a growth factor, find the geometric mean of the factors, then subtract 1. Our calculator does this for you in the growth rate option.
Why is the average growth rate lower than the simple average of the rates?
Because growth compounds. Each percentage applies to a value that has already changed, so the simple average of the rates overstates the true average growth.
Is this Geometric Mean Calculator free?
Yes. SmartStatCalc's Geometric Mean Calculator is free to use online.
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Cite this calculator
If you use this calculator in your work, you can cite it like this:
SmartStatCalc. (2026). Geometric Mean Calculator. https://smartstatcalc.com/geometric-mean-calculator/