Geometric Mean Calculator Mean comparison
Arithmetic mean (AM = sum/n) is correct for additive values. Geometric mean (GM = nth root of product) is correct when values multiply or compound. For scores 72, 80, 88, 96: AM = 84 (correct). For annual return factors 1.06, 0.92, 1.15: GM = 1.038 = 3.8%/yr (correct) — AM of 1.043 overstates by 0.5 pp and compounds to the wrong ending value.
Use arithmetic mean when values add together (scores, counts, temperatures). Use geometric mean when values multiply, compound, or represent ratios — growth factors, index relatives, CAGR inputs. AM always ≥ GM for positive values; a large gap signals you may have the wrong formula, and standard deviation will tell you how wide that spread really is.
For additive data the gap is negligible. For multiplicative data, AM overstates the true compounding result and produces the wrong ending value.
Gap = 0.57 (0.7%). Both are defensible for additive scores. AM is the standard choice.
Check the input type and formula side by side before calculating.
| Method | Correct for values that add: scores, counts, temperatures | Correct for values that multiply: growth factors, ratios, indexes |
|---|---|---|
| Formula | AM = (x₁+x₂+⋯+xₙ) / n | GM = (x₁×x₂×⋯×xₙ)^(1/n) |
| What it preserves | The sum. AM × n = total of all values. | The product. GM^n = product of all values. |
| Outlier behavior | Sensitive: {4,9,16,100} → AM = 32.25 (above 75% of values). | Less sensitive: same data → GM = 15.46 (near the central cluster). |
| Investment returns | Wrong for compounding. Returns 50%, −50%: AM = 0% but actual = −25% loss. | Correct for compounding. GM of 1.50×0.50 = √0.75 − 1 = −13.4%/yr. |
| Input requirement | Accepts any real number including negative and zero. | Requires all positive values. Zero or negative values make GM undefined in standard use. |
Both averages below are computed correctly. Only one of them answers the question an investor is actually asking: what single annual rate would have produced this balance?
Situation: A $10,000 position returns +90%, −60%, +40%, −20%, and +30% across five years.
| Step-by-step: how to use this calculator correctly | Arithmetic mean | Geometric mean |
|---|---|---|
| 1. Prepare the values | Use the percentages as given: 90, −60, 40, −20, 30. | Convert each to a growth factor: 1.90, 0.40, 1.40, 0.80, 1.30. |
| 2. Combine them | Sum = 90 − 60 + 40 − 20 + 30 = 80 | Product = 1.90 × 0.40 × 1.40 × 0.80 × 1.30 = 1.10656 |
| 3. Average | 80 / 5 = 16.00% per year | 1.10656^(1/5) = 1.02046 → 2.05% per year |
| 4. Project the $10,000 forward | 10,000 × 1.16^5 = $21,003.42 | 10,000 × 1.02046^5 = $11,065.60 |
| 5. Check against the account | Overstates the real balance by $9,937.82. | Matches it exactly: 10,000 × 1.90 × 0.40 × 1.40 × 0.80 × 1.30 = $11,065.60. |
The arithmetic mean nearly doubles the true ending value. It is not a rounding difference — it happens because a −60% year removes more capital than a +90% year adds back, and summing percentages hides that asymmetry entirely.
The add-versus-multiply test covers most cases. These three are the situations where it needs a second thought.
If every value is identical, AM and GM are equal. Returns of 5%, 5%, 5% give 5% either way. The gap only opens when values differ, and it widens with the spread — which makes the size of the gap a rough read on volatility, not just a curiosity.
A −100% return becomes a factor of 0, and one zero collapses the product no matter what the other years did. Geometric mean then reports −100% per year, which is correct: once capital is gone, later gains have nothing to compound. Arithmetic mean would average −100% with +50% and report −25%, which describes nothing real.
GM describes a path you already have. If you are forecasting a single future year from a distribution of possible outcomes, the arithmetic mean is the unbiased estimate for that one draw. Use geometric mean to report what happened, arithmetic mean to model what one more year might do.
Answer these in order. Stop at the first match.
Scores, counts, dollars, temperatures → use AM. AM×n = sum.
Growth factors, ratios, index relatives → use GM. GM^n = product.
Large gap signals multiplicative structure or an outlier. Investigate before choosing AM.
Only AM handles negatives. Convert returns to factors before using GM.
Scores 72, 80, 88, 96: AM = 84. Verify: 84×4 = 336 = total ✓. GM = 83.43. The 0.7% gap is trivial and AM is the standard because scores add to a class total.
Annual returns 6%, −35%, 10%. Convert to factors: 1.06, 0.65, 1.10. GM = (1.06×0.65×1.10)^(1/3) = 0.7579^0.333 = 0.9144 → CAGR = −8.56%/yr. AM of factors = 0.937 → AM-implied return = −6.3%/yr. Only GM reproduces the actual ending value: $1,000×0.9144³ = $765 = $1,000×1.06×0.65×1.10 ✓.
A town grows 8%, then 3%, then shrinks 2%. Factors 1.08, 1.03, 0.98 give a product of 1.090152, so GM = 1.090152^(1/3) = 1.02919 → 2.92% per year. Arithmetic mean of the factors is 1.03 → 3.00% per year. The 0.08 point gap is small here because the values are close, but it compounds: over 30 years the AM figure overstates the population by about 2.5%.
A fund beats its index by ratios of 1.12, 0.95, and 1.08 over three years. Product = 1.14912, so GM = 1.04742 → 4.74% average annual outperformance. Arithmetic mean gives 1.05 → 5.00%. Because these are ratios to a moving benchmark, only the geometric figure reproduces the cumulative 14.9% outperformance actually recorded.
A correct calculation can still mislead when the method or input type is wrong.
Returns 50%, −50%: AM = 0% (implies no change). Actual: $1,000×1.5×0.5 = $750. Loss = 25%. Use GM of factors: √(1.5×0.5)−1 = −13.4%/yr.
A single zero in the dataset makes the entire product = 0, so GM = 0 regardless of other values. Convert returns to factors (e.g., −100% → 0.0) and check whether zero means the investment was wiped out.
GM of 6%, 7%, 8% (as percentages) = (6×7×8)^(1/3) = 6.99 — meaningless. Convert to factors first: 1.06×1.07×1.08 → GM factor = 1.07 → 7%/yr.
Yes, for any set of positive values that are not all identical. AM-GM inequality: AM ≥ GM, with equality only when all values are equal. For {4, 9, 16, 25}: AM = 13.50, GM = 10.95. For {9, 9, 9, 9}: AM = GM = 9.
Depends on whether the percentages add or multiply. Portfolio weights (15%, 25%, 60%) add to 100% → AM-type logic. Annual return percentages compound → convert to factors (1.15, 1.25, 1.60) and use GM. GM of raw percentages is meaningless.
Because loss and gain are not symmetric on a compounding base. A 50% loss requires a 100% gain just to break even. AM treats +50% and −50% as net 0%. GM accounts for the changing base: GM = √(1.5×0.5)−1 = −13.4%/yr, which correctly reproduces the ending value.
Yes. For investment returns the gap is approximately half the variance: AM − GM ≈ σ²/2. A portfolio with 20% annual volatility gives up roughly 2 percentage points a year to this effect, often called volatility drag. A widening gap between the two averages is a direct signal that dispersion is increasing.
Not directly. An odd count of negatives makes the product negative, and there is no real odd root of a negative number in standard use. For investment losses, convert to growth factors first — a −40% return becomes 0.60, which is positive. Raw negative data such as temperature changes is not a candidate for geometric mean at all.
=AVERAGE() is the arithmetic mean. For geometric mean use =GEOMEAN(), which requires strictly positive inputs, so feed it growth factors rather than percentage returns. =GEOMEAN(1.06, 0.92, 1.15) returns 1.0382; subtract 1 to read it as a 3.82% annual rate.