Statistics
How To Calculate A Geometric Mean
The geometric mean is the average you use when values multiply rather than add. Multiply all the numbers together and take the root that matches how many there are.
Quick Answer
Geometric mean = (x1 x x2 x ... x xn)^(1/n)
- x1 ... xn
- The values being averaged, all positive
- n
- How many values there are
- product
- The result of multiplying all the values
Multiply every value, then raise the product to the power of one over the count. Three growth factors of 1.08, 1.12 and 0.95 multiply to 1.14912, and the cube root is 1.047422, an average growth rate of 4.7422 percent.
What Is A Geometric Mean?
The geometric mean of n positive numbers is the nth root of their product. Three values are multiplied together and the cube root is taken; four values take a fourth root, and so on. It answers what single value, repeated, gives the same result.
It is the correct average for anything that compounds. Investment returns, population growth, inflation rates and interest all multiply period by period, so the average has to be found multiplicatively rather than additively.
The arithmetic mean overstates growth. Returns of 50 percent and minus 50 percent average to zero arithmetically, but the portfolio actually ends where it started times 0.75, a loss of 25 percent. The geometric mean of 1.5 and 0.5 is 0.866, which is the honest answer.
The geometric mean is never larger than the arithmetic mean, and the two are equal only when every value is identical. The gap widens as the values spread further apart, which is why volatile assets look better under simple averaging.
A useful special case is two numbers: the geometric mean is the square root of the product. This is the altitude rule in geometry, where the height of a right triangle to the hypotenuse is the geometric mean of the two segments it creates.
It requires positive numbers. A zero value forces the product to zero and the mean collapses, and a negative value makes the root undefined for an even count. Growth rates are therefore converted to factors first, such as 1.08 for 8 percent growth.
Converting growth rates to factors is the standard trick. Multiply the factors, take the root, and subtract one to get back to a percentage. The result is the constant rate that would have produced the same final value.
Logarithms are the computational shortcut. The log of a product is the sum of the logs, so averaging logs and exponentiating gives the geometric mean. This is how it was done before calculators and why it appears throughout statistics.
It is the natural centre for log-normal distributions. Body weights, incomes and latency measurements are usually log-normal, and the geometric mean describes them better than the arithmetic mean because it is not dragged by the tail.
Aspect ratios and scaling use it too. When an image is resized to keep the same area, the linear scale factor is the geometric mean of the width and height factors, and print sizes are often quoted this way.
Benchmarking suites report the geometric mean of per-test speedups. That stops a single enormous ratio from dominating the summary, which is exactly what happens with an arithmetic mean.
In practice, compute the product, then take the root with a power of one over n. Watch for overflow with many large values, and switch to the log method if the product grows beyond what the calculator holds.
Formula
GM = (x1 x x2 x ... x xn)^(1/n)
Formula page →Multiply all values and take the nth root of the product.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| GM | Geometric mean | same as input | The multiplicative average of the values. |
| x1 ... xn | Values | same as input | The positive numbers being averaged. |
| n | Count | unitless | How many values are being averaged. |
g = ((1 + r1) x (1 + r2) x ... x (1 + rn))^(1/n) - 1
Formula page →Convert each return to a factor, average multiplicatively, then subtract one.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| g | Average growth rate | decimal | The constant rate giving the same final value, as a decimal. |
| ri | Period returns | decimal | Each period return as a decimal, so 8 percent is 0.08. |
| n | Periods | unitless | How many periods are averaged. |
GM = exp( (ln x1 + ln x2 + ... + ln xn) / n )
Formula page →Averaging logarithms avoids overflow when the product is very large.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| GM | Geometric mean | same as input | The multiplicative average. |
| ln | Natural logarithm | unitless | The log base e of each value. |
| exp | Exponential | unitless | Raises e to the averaged logarithm. |
How To Calculate A Geometric Mean
- 1
Convert to growth factors
Add one to each rate. Returns of 8 percent, 12 percent and minus 5 percent become 1.08, 1.12 and 0.95, all positive so the root is defined.
- 2
Multiply the factors
1.08 times 1.12 is 1.2096, and 1.2096 times 0.95 is 1.14912. This is the total growth over the three periods.
- 3
Take the cube root
Raise 1.14912 to the power of one third. The cube root is 1.047422, the factor that repeated three times gives the same total.
- 4
Subtract one for the rate
1.047422 minus 1 is 0.047422, so the average growth rate is 4.7422 percent per period. The arithmetic mean would have been 5 percent.
Examples
Example 1: Three annual returns of 8 percent, 12 percent and minus 5 percent
- First value
- 1.08
- Second value
- 1.12
- Third value
- 0.95
| Step | Calculation | Result |
|---|---|---|
| Growth factors | 1.08, 1.12, 0.95 | 1.08, 1.12, 0.95 |
| Product of the factors | 1.08 x 1.12 x 0.95 | 1.14912 |
| Cube root | 1.14912 ^ (1/3) | 1.047422 |
| Average growth rate as a percentage | (1.047422 - 1) x 100 | 4.7422 |
Result: Three returns of 8 percent, 12 percent and minus 5 percent multiply to 1.14912, whose cube root is 1.047422. The average growth rate is therefore 4.7422 percent, not the 5 percent the arithmetic mean suggests.
Example 2: The geometric mean of 4, 9 and 16
- First value
- 4
- Second value
- 9
- Third value
- 16
| Step | Calculation | Result |
|---|---|---|
| Product | 4 x 9 x 16 | 576 |
| Cube root | 576 ^ (1/3) | 8.320335 |
| Arithmetic mean for contrast | (4 + 9 + 16) / 3 | 9.666667 |
| Difference | 9.666667 - 8.320335 | 1.346332 |
Result: The product of 4, 9 and 16 is 576, and its cube root is 8.320335. The arithmetic mean of the same three numbers is 9.666667, which is 1.346332 higher and shows how the two averages separate as values spread out.
Calculator
Geometric mean
1.0474
- Average growth rate
- 4.74%
- Arithmetic mean
- 1.05
- Product of the values
- 1.1491
- Geometric over arithmetic
- 0.9975
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the A Geometric Mean calculator page.
Common Mistakes
Using the arithmetic mean for growth rates
Returns of 50 percent and minus 50 percent average to zero, but the portfolio loses 25 percent. The geometric mean of 1.5 and 0.5 is 0.866, which is the true average factor.
Averaging percentages instead of factors
Multiply the factors, not the rates. Averaging 0.08, 0.12 and minus 0.05 directly gives a meaningless number because the signs do not compound correctly.
Taking the wrong root
Three values need a cube root, four need a fourth root. Using a square root for three values gives a value that is too large by the remaining root.
Feeding in zero or negative values
A zero forces the product to zero, and a negative value breaks the root for an even count. Convert to positive growth factors before averaging.
Expecting the geometric mean to exceed the arithmetic mean
It never does. If your geometric mean is larger, the arithmetic mean was computed from different values or an arithmetic slip occurred.
Mixing up the order of operations
The product is taken first and the root second. Taking the root of each value and then multiplying gives the same answer only for two values, and even then by coincidence of scale.
FAQ
What is the geometric mean formula?
Multiply all the values together and raise the product to the power of one over the count. For three values that is the cube root of the product.
When should I use the geometric mean instead of the arithmetic mean?
Whenever the quantities multiply, such as investment returns, growth rates or ratios. The geometric mean is the only average that gives the same result when repeated.
Can the geometric mean be negative?
No. It is only defined for positive numbers, because a negative product has no real even root. Convert rates to positive growth factors before averaging.
How do you find the geometric mean of two numbers?
Take the square root of their product. For 4 and 16 the product is 64 and the geometric mean is 8, which is the altitude rule from right triangle geometry.
Why is the geometric mean smaller than the arithmetic mean?
Multiplication rewards balance. A set with one very large and one very small value multiplies to less than the same total spread would add to, so the root lands below the simple average.
How do I average growth rates correctly?
Add one to each rate to get factors, multiply them, take the nth root, then subtract one. That constant rate reproduces the same final value as the original sequence.
References
- [1]Wikipedia, Geometric mean — https://en.wikipedia.org/wiki/Geometric_mean
- [2]Wikipedia, Arithmetic-geometric mean inequality — https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means
- [3]Wikipedia, Compound annual growth rate — https://en.wikipedia.org/wiki/Compound_annual_growth_rate