Scores, counts, dollars, temperatures → use AM. AM×n = sum.
Mean comparison
Geometric Mean vs Arithmetic Mean: Which Average Is Correct?
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.
Short answer
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.
Same four numbers, AM vs GM: when does the gap matter?
For additive data the gap is negligible. For multiplicative data, AM overstates the true compounding result and produces the wrong ending value.
Example input 72, 80, 88, 96
Gap = 0.57 (0.7%). Both are defensible for additive scores. AM is the standard choice.
AM = 84 84.00
GM = 83.43 83.43
Side-by-side comparison
Use this table when you need a fast, practical distinction before choosing a calculator.
| Feature | Option A | Option B |
|---|---|---|
| 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. |
AM or GM decision: three questions
Answer these in order. Stop at the first match.
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.
Examples that make the choice clear
Class scores: AM is correct (additive, same unit)
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.
Investment returns: GM is correct (multiplicative, compounding)
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 ✓.
Common mistakes
These are the errors most likely to make the correct formula return a misleading answer.
Using AM to average investment returns
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.
Using GM with zero or negative values
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.
Mixing raw percentages and factors before taking GM
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.
Frequently asked questions
Is geometric mean always lower than arithmetic mean?
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.
Which mean should I use for percentages?
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.
Why does arithmetic mean overstate investment returns?
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.