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State what is being summarized
For equal-period returns +20% and −20%, the arithmetic mean is zero. The geometric mean is approximately −2.020410% because total growth factor is 0.96 over two periods. Both calculations use the same observations but target different summaries.
Compare the operations
| Feature | Arithmetic mean | Geometric mean |
|---|---|---|
| Operation | Add rates, divide by two | Multiply factors, take square root, subtract one |
| Example result | 0% | -2.02041% |
| Endpoint reconstruction | Does not generally reproduce it | Reproduces 0.96 under a constant rate |
Keep history separate from a forecast
An arithmetic sample mean can be used as an input to a one-period statistical estimate under additional assumptions. A historical geometric mean describes observed linked growth. Neither label by itself validates a forecast model or proves future returns will equal that value.
Retain frequency and domain
If observations are monthly, both means are initially monthly descriptions. Annual geometric conversion needs the covered horizon. A complete loss has a zero linked factor and no finite log return, so logarithmic shortcuts need their domain checked.