Harmonic Mean Calculator
The harmonic mean is the reciprocal of the arithmetic mean of reciprocals, used for averaging rates and ratios. It is essential in physics (average speed), economics (average cost per unit), and statistics when the reciprocal of a value has a natural interpretation. This calculator computes the harmonic mean, geometric mean, and arithmetic mean for comparison. The harmonic mean is always the smallest of the three means (for positive numbers), reflecting its emphasis on smaller values due to the reciprocal calculation.
Harmonic mean formula
HM = n / (1/x1 + 1/x2 + ... + 1/xn)
All values must be non-zero
Example: Average speed
- Drive 100 km at 60 km/h, then 100 km at 40 km/h.
- Arithmetic mean of speeds: (60 + 40) / 2 = 50 km/h (incorrect).
- Harmonic mean of speeds: 2 / (1/60 + 1/40) = 48 km/h (correct).
- Total distance: 200 km. Total time: (100/60) + (100/40) hours = 4.17 hours. Average: 200 / 4.17 = 48 km/h.
The three means relationship
For positive numbers, the harmonic mean is always less than or equal to the geometric mean, which is always less than or equal to the arithmetic mean:
HM ≤ GM ≤ AM
The three are equal only when all values are identical. This relationship (AM-GM-HM inequality) is fundamental in mathematics and shows why different means are appropriate for different types of data.
Harmonic mean: frequently asked questions
What is the harmonic mean?
The harmonic mean is n divided by the sum of reciprocals: HM = n / (1/x1 + 1/x2 + ... + 1/xn). It is used for rates and ratios where the denominator matters, such as average speeds or average costs per unit.
When should I use the harmonic mean?
Use the harmonic mean when averaging rates (e.g., kilometers per hour), costs per unit, or any ratio where the denominator is important. For example, if you drive 60 km/h for the first 100 km and 40 km/h for the next 100 km, the average speed is the harmonic mean of 60 and 40, not the arithmetic mean.
Why is the harmonic mean lower than the geometric and arithmetic means?
The harmonic mean emphasizes smaller values more because it uses reciprocals. The ordering is always: HM ≤ GM ≤ AM. This makes the harmonic mean appropriate for quantities measured in reciprocal units (rates, frequencies).
Can I use harmonic mean for negative numbers?
The harmonic mean is technically defined for negative numbers, but it is rarely used. Negative rates are unusual in most practical applications. All positive or all negative values work; mixed signs can produce unexpected results.
What is the relationship between harmonic mean and average cost per unit?
If you buy items at different prices per unit and want to know your average cost per unit, use the harmonic mean of the unit prices weighted by quantity. This accounts for the fact that you bought more units at lower prices.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: NIST Handbook.
- American Statistical Association: ASA.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.