Geometric Mean Calculator Mean comparison
Geometric mean (GM) is correct when values multiply: growth factors, fold changes, index relatives. Harmonic mean (HM) is correct when rates share an equal denominator: equal-distance speeds, equal-budget prices per unit. For 30 mph and 60 mph over equal distance: HM = 40 mph (correct travel time), GM = 42.43 mph (wrong), AM = 45 mph (wrong by 33 minutes per 200 km).
Use geometric mean when each value multiplies the overall result: growth factors, ratios, normalized benchmarks. Use harmonic mean when each rate applies over the same distance, budget, or denominator — equal-distance speed averages, dollar-cost averaging, price-per-equal-unit comparisons. If neither fits, the decision guide covers median, mode, and arithmetic mean too.
Each scenario shows which mean gives the numerically correct answer and why the other is wrong.
HM is correct: 100/30 + 100/60 = 5 hrs = 200 km / 40 mph ✓. GM gives 200/42.43 = 4.72 hrs — 17 min too short.
Check the input type and formula side by side before calculating.
| Method | Correct for values that multiply: growth factors, ratios, fold changes | Correct for rates over equal denominators: speeds, prices-per-unit |
|---|---|---|
| Formula | GM = (x₁×x₂×⋯×xₙ)^(1/n) | HM = n / (1/x₁ + 1/x₂ + ⋯ + 1/xₙ) |
| What it preserves | The product: GM^n equals the product of all values. | The rate-over-equal-unit structure. Total distance / HM = actual travel time. |
| Best use | Growth factors, ratios, fold changes, index relatives, normalized scores. | Rates over equal distances or budgets: speeds, price-per-unit, fuel efficiency. |
| Ordering | HM ≤ GM ≤ AM for all positive values. | HM is always the lowest of the three Pythagorean means. |
| Input rule | All positive values. | All positive, non-zero values (zero makes 1/x undefined). |
Both means are valid operations on the numbers 30 and 60. Only one of them predicts how long the journey actually takes.
Situation: A delivery van drives a 120 km leg out at 30 km/h and the same 120 km back at 60 km/h.
| Step-by-step: how to use this calculator correctly | Geometric mean | Harmonic mean |
|---|---|---|
| 1. What the method assumes | Treats the two speeds as factors to be multiplied. | Treats distance as the fixed quantity: 120 km on each leg. |
| 2. Combine them | Product = 30 × 60 = 1,800 | Reciprocal sum = 1/30 + 1/60 = 0.05 |
| 3. Average | √1,800 = 42.43 km/h | 2 / 0.05 = 40.00 km/h |
| 4. Predict the round trip | 240 / 42.43 = 5.66 hours | 240 / 40.00 = 6.00 hours |
| 5. Check the stopwatch | 20 minutes short of the real time. | Matches exactly: 4 hours out + 2 hours back = 6 hours. |
Geometric mean is not bad arithmetic here — it is the wrong model. It preserves the product of the two speeds, but no physical quantity in this journey is conserved by that product. Harmonic mean preserves total time over fixed distance, which is precisely what the trip conserves.
Whether the denominator is fixed decides the formula. These three cases are where that question gets slippery.
Drive 30 km/h for one hour then 60 km/h for one hour and the correct average is 45 km/h, because you covered 90 km in 2 hours. The fixed quantity chooses the mean: fixed distance needs HM, fixed time needs arithmetic mean, and compounding factors need geometric mean.
A zero rate makes one reciprocal undefined, so HM has no value at all. A stopped vehicle contributes infinite time per kilometre — mathematically correct and practically useless. Drop zero-rate observations from the calculation and report them separately rather than substituting a small number.
P/E ratios, price relatives, and biological fold changes are already scale-relative, so multiplying them is meaningful while averaging their reciprocals is not. Reach for harmonic mean only when the reciprocal itself carries a physical meaning, such as hours per kilometre or dollars per unit.
The denominator structure of the data determines the correct mean.
If each rate applies over the same distance, budget, or quantity → use HM.
If each value multiplies the overall result (growth, fold change, ratio) → use GM.
If each rate is observed for the same duration → use AM, not HM.
Neither GM nor HM handles zero. Review whether zero is a data error before calculating.
100 km at 30 mph takes 3.33 hrs; 100 km at 60 mph takes 1.67 hrs. Total = 200 km in 5 hrs → average = 40 mph = HM(30,60) ✓. AM = 45 mph implies 4.44 hrs — 33 minutes too short. GM = 42.43 mph implies 4.72 hrs — 17 minutes too short.
Gene expression fold changes 1.2×, 1.5×, 2.0× multiply across conditions. GM = (1.2×1.5×2.0)^(1/3) = 3.6^(1/3) = 1.53×. This means a gene expressed 1.53× on average across the three conditions — the only value that, applied three times, gives 3.6× total ✓.
You invest $600 a month for two months, buying at $30 and then $60 per share. That buys 20 shares, then 10 — 30 shares for $1,200, an average cost of $40.00 per share, which is exactly HM(30, 60). The arithmetic mean of $45 overstates what you paid, because the cheaper month bought more shares and therefore carries more weight.
A price index moves by factors of 1.20, 0.90, and 1.15 over three years, a cumulative product of 1.242. GM = 1.242^(1/3) = 1.07491 → 7.49% per year, which reproduces the index level exactly. Harmonic mean of the same three factors returns 1.0661, a number that corresponds to no real index path at all.
A correct calculation can still mislead when the method or input type is wrong.
30 mph and 60 mph → GM = 42.43 mph but actual average is 40 mph. For 200 km: GM predicts 4.72 hrs, actual = 5 hrs. Error = 17 minutes. Use HM for equal-distance speeds.
Growth factors 1.06, 1.07, 1.08: HM = 1.0699. GM = 1.0700. GM^3 = 1.225 = actual product ✓. HM^3 = 1.224 — close but reproduces the wrong endpoint for compounding.
1/0 is undefined. If a speed is zero (vehicle stopped), that leg of the trip has infinite time — HM is undefined. Segment the trip differently or exclude the stopped interval with a documented reason.
Yes, for positive values: HM ≤ GM ≤ AM, with equality only when all values are identical. For {30, 60}: HM = 40, GM = 42.43, AM = 45. For {50, 50}: HM = GM = AM = 50.
Use HM when distances are equal (each leg covers the same distance). Use AM when time is equal (each leg lasts the same duration). Example: 30 mph for 1 hr and 60 mph for 1 hr → AM = 45 mph (correct: 45 km/hr over 2 hrs = 90 km ✓). Same distances → HM = 40 mph.
No, in standard use. GM requires all positive values (product must be positive for a real nth root). HM requires all positive, non-zero values (reciprocals must exist). For datasets with negatives, use AM or a signed-ratio approach specific to the domain.
For positive values that are not all identical, yes: HM < GM < AM always holds. For 30 and 60 the three come out at 40.00, 42.43, and 45.00. They are equal only when every value in the set is the same.
Write the rate as a fraction and ask which part stays constant across observations. Speed is kilometres per hour — if every leg covers the same kilometres, the numerator is fixed and you need harmonic mean. If every leg lasts the same hours, the denominator is fixed and you need arithmetic mean.
Only when the percentages are rates sharing a denominator, such as defects per 1,000 units at a constant batch size. Percentage returns compound instead of sharing a denominator, so they need geometric mean after conversion to growth factors.