Scores, counts, dollars, temperatures → use AM. Verify: AM × n should equal the total.
Decision guide
Which Average Should I Use? AM, GM, HM, Median, or Mode
Arithmetic mean (AM) is correct for additive values: scores 72+80+88+96 → AM = 84. Geometric mean (GM) is correct for multiplicative values: growth factors 1.06×0.65×1.10 → GM = 0.9144 = −8.56%/yr. Harmonic mean (HM) is correct for equal-denominator rates: 30 mph and 60 mph over equal distance → HM = 40 mph. Median is correct for skewed data: incomes {$30k,$34k,$35k,$38k,$200k} → median = $35k (mean = $67.4k is 93% too high). Mode is correct for categories: survey {A,B,B,C,B,A} → mode = B.
Short answer
Use AM when values add (scores, counts). Use GM when values multiply (growth factors, ratios). Use HM when rates share an equal denominator (equal-distance speed, equal-budget price-per-unit). Use median when outliers skew the mean. Use mode when values are categorical or when frequency is the question.
Five data types, five correct averages — and the wrong answer each other would give
Each example shows the correct average and what the most common wrong choice would produce.
Example input 72, 80, 88, 96
AM is correct because scores add to a class total. AM×4 = 336 = sum ✓. GM is close but algebraically unjustified for additive data.
AM = 84 ✓ 84
GM = 83.43 (only 0.7% off — but no reason to prefer it) 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 |
|---|---|---|
| Arithmetic mean | AM = sum/n. Preserves the sum. | Scores, counts, temperatures. Wrong for returns (AM of 50%, −50% = 0% but actual = −25%). |
| Geometric mean | GM = (product)^(1/n). Preserves the product. | Growth factors, ratios, fold changes. GM of 1.06×0.65×1.10 = 0.9144 → −8.56%/yr ✓. |
| Harmonic mean | HM = n / Σ(1/xᵢ). Preserves equal-denominator rates. | Speeds over equal distance, equal-budget price-per-unit. HM(30,60) = 40 mph ✓. |
| Median | Middle value when sorted. Ignores magnitudes. | Skewed data: income, home prices, response times. {$30k–$200k}: median = $35k ✓. |
| Mode | Most frequent value. Can be multiple or none. | Categories, survey answers, repeated choices. {A,B,B,C,B,A}: mode = B ✓. |
Five-question decision flow: stop at the first match
Answer these questions in order about your dataset. The first 'yes' determines the correct average.
Growth factors, ratios, fold changes → use GM. Convert % returns to factors first (6% → 1.06).
Equal-distance speeds, equal-budget prices → use HM. If time is equal instead of distance, use AM.
Skewed income, home prices → use median. Compare AM with median: if AM > median by >20%, report both.
Survey answers, product choices → use mode. Report frequency distribution alongside mode.
Examples that make the choice clear
Returns: AM = 0% (wrong) vs GM = −13.4%/yr (correct)
Annual returns: +50%, −50%. AM of factors: (1.50+0.50)/2 = 1.00 → 0%/yr return (implies breakeven). Actual: $1,000×1.5×0.5 = $750. Loss = 25% over 2 years. GM: √(1.5×0.5) = √0.75 = 0.866 → −13.4%/yr. Only GM reproduces the actual ending value: $1,000×0.866² = $750 ✓.
Speed: AM = 45 mph (wrong) vs HM = 40 mph (correct)
Drive 100 km at 30 mph (takes 3.33 hrs), then 100 km at 60 mph (takes 1.67 hrs). Total: 200 km in 5 hrs → 40 mph = HM(30,60) ✓. AM = 45 mph implies 200/45 = 4.44 hrs → 33 min faster than reality. The error exists because the slower leg takes disproportionately more time.
Common mistakes
These are the errors most likely to make the correct formula return a misleading answer.
Defaulting to arithmetic mean for investment returns
Returns 50%, −50%: AM implies breakeven. Actual = 25% loss. Convert to factors and use GM: √(1.5×0.5)−1 = −13.4%/yr. AM overstates return by 13.4 percentage points.
Using GM when values include zero
A single zero makes the product = 0, so GM = 0 regardless of other values. If zero represents a complete wipeout (−100% return → factor = 0.0), GM correctly reports total loss. If zero is missing data, remove it first.
Ignoring skew: reporting only mean for income data
Incomes {$30k,$34k,$35k,$38k,$200k}: AM = $67.4k, median = $35k. AM overstates the typical salary by 93%. Always compare AM with median — if the gap exceeds ~20%, report both and explain the outlier.
Frequently asked questions
What is the safest average to use?
There is no universally safe average. AM is the most common but is wrong for compounding (overstates returns by ignoring base changes) and for skewed data (overstates typical value). The safe approach: ask 'do values add, multiply, rate, rank, or repeat?' before choosing. Each question maps to exactly one correct formula.
Which average for growth rates?
Use GM when you have period-by-period growth factors: convert returns to factors (6% → 1.06, −35% → 0.65), take GM, subtract 1. Use CAGR when you have beginning value, ending value, and years: CAGR = (EV/BV)^(1/n) − 1. Both are geometric-mean operations on different input formats.
Which average for speed?
Use HM when distances are equal (each leg covers the same distance). Use AM when times are equal (each leg lasts the same duration). 30 mph and 60 mph: equal distance → HM = 40 mph ✓; equal time → AM = 45 mph ✓. They are different questions and different correct answers.