How this calculation works
Use the same positive list with equal observation weights. Arithmetic aggregation adds values, geometric aggregation averages logarithms and harmonic aggregation averages reciprocals. A different denominator interpretation can call for a different mean.
Arithmetic = Σx/n; geometric = exp(mean ln x); harmonic = n/Σ(1/x)A worked example
For 1,4,16, the arithmetic mean is 7, geometric mean is 4 and harmonic mean is 16/7 ≈ 2.285714. All three agree for a constant list.
Good to know
Strictly positive data from 10⁻¹² through 10¹². No weights, negative values or zero convention are inferred. Percentage changes must be converted to valid positive growth factors before any growth-factor interpretation; rates need compatible exposure weights.
Reference: SciPy: geometric mean definition (opens in a new tab) ↗
One dataset can have several meaningful averages
For 1,4,16, the arithmetic mean is (1+4+16)/3 = 7. Their product is 64, whose cube root is 4, giving the geometric mean. Their reciprocals sum to 21/16, so the harmonic mean is 3÷(21/16) = 16/7.
For strictly positive equal-weight inputs, harmonic mean is at most geometric mean, which is at most arithmetic mean. A constant list makes them equal. This ordering is a useful consistency check, not a rule that the largest or smallest answer is the best description of your task.
The weighted average calculator accepts a separate arithmetic weight for each value; this three-mean comparison deliberately assigns equal observation weights.
Rates need their exposure weights
Traveling equal distances at 30 and 60 distance units per hour gives an overall rate of 40, the harmonic mean: total distance is divided by the sum of the two travel times. Spending equal times at those rates instead gives an arithmetic mean of 45. The same two rate numbers describe different journeys when the exposures differ.
The fuel mileage calculator uses ratio-of-total quantities for its own trip model. The weighted average calculator accepts separate weights. This page deliberately uses equal observation weights and does not infer that your speeds, prices or ratios have equal distances, times or denominators.
Multiplicative factors are not raw percentage changes
A 50% rise followed by a 50% fall corresponds to factors 1.5 and 0.5. Their geometric mean is √0.75, below one, even though the arithmetic mean of the two signed percentage changes is zero. Convert valid changes to positive factors before interpreting a geometric mean as a growth factor.
This implementation averages logarithms instead of multiplying up to a thousand large values. Its harmonic calculation uses scaled reciprocals. Zero and negative inputs are excluded from this comparison so that no unstated zero convention, reciprocal singularity or complex logarithm is introduced.
A few questions, answered
Why does this reject zero?
This page compares strictly positive means; the harmonic reciprocal of zero is undefined and no limiting convention is assumed.
Are percentage signs accepted?
No. Enter numeric values with a clear meaning. For growth factors, 10% growth corresponds to 1.1, not 10.
Can I enter different weights?
Use the weighted average calculator for an arithmetic weighted mean. This comparison gives each observation equal weight.