Geometric Mean Calculator
Geometric mean of three positive values.
What it does
Averages differ by purpose. Arithmetic adds then divides, while geometric multiplies then roots. Growth factors, ratios, and normalized scores compound, so their center is geometric. Using arithmetic on growth overstates the typical year because spikes dominate sums.
The tool above handles three values. With 4, 9, and 16, the product is 576, the geometric mean is 8.32, and the arithmetic mean is 9.67. With 2, 8, and 32 the geometric mean is exactly 8 against an arithmetic 14, showing how geometric damps the large tail. Equal inputs such as 10, 10, 10 give 10 both ways.
Positivity is required because roots and logs of zero or negatives break the definition. That restriction is a feature: it forces the user to confirm that ratios and growth factors are the right model before averaging. For sums such as total cost, switch back to arithmetic.
This page fixes the count at three for a focused job. Larger sets need the nth root form with the same idea.
How to use
- Enter the first value above zero.
- Enter the second value above zero.
- Enter the third value above zero.
- Read the geometric mean first.
- Read the product for verification.
- Compare against the arithmetic mean.
Formula
- a
- First positive value.
- b
- Second positive value.
- c
- Third positive value.
Three positive values only. At any zero or negative input there is no geometric mean. Equal inputs make both means match exactly.
Worked example
Example: geometric mean of 4, 9, 16
Defaults: 4, 9, 16. All values must sit above zero.
- Inputs: 4, 9, 16.
- Product: 4 times 9 times 16 is 576.
- Geometric mean: cube root of 576 is 8.32.
- Arithmetic: (4 plus 9 plus 16) / 3 is 9.67.
- Geometric sits below arithmetic, as always for spread values. Tool shows the same.
Reference table
| Values | Geometric | Arithmetic |
|---|---|---|
| 4, 9, 16 | 8.32 | 9.67 |
| 2, 8, 32 | 8 | 14 |
| 10, 10, 10 | 10 | 10 |
| 1, 4, 9 | 3.3 | 4.67 |
| 3, 3, 12 | 4.76 | 6 |
Limits
- Positive values only. Zero and negatives excluded.
- Three values fixed. Larger sets need nth root.
- Ratios and growth only. Sums need arithmetic.
- Outliers still matter, just less than in arithmetic.
Common mistakes
- Averaging growth with arithmetic.
- Including zeros or negatives.
- Mixing percents with multipliers without converting.
- Reading product as the mean.
- Comparing means across different units.
Try it yourself
Quick checks. Answers sit tucked under each question.
Values 2, 8, 32?
Product is 512. Cube root is 8. Arithmetic is 14.
Values 10, 10, 10?
Product is 1000. Cube root is 10. Arithmetic is 10.
Questions
How is geometric mean figured?
Cube root of the product. Here cube root of 576 is 8.32.
When do I use geometric over arithmetic?
Use geometric for ratios, rates, and growth; arithmetic for sums. Mixing them misstates growth.
Can inputs be zero or negative?
No. Logs of zero or negatives break the mean. All inputs must exceed zero.
What if all values match?
Equal values make both means equal. Here 10, 10, 10 gives 10 and 10.
Does this extend past 3 values?
Yes. For n values take the nth root. This page fixes n at 3 for a focused tool.
Why is geometric lower here?
It compresses spikes. One large value lifts arithmetic far more than geometric.