Harmonic Mean Calculator
Calculate the harmonic mean of a list of positive numbers.
How to use this tool
- Enter numbers (comma-separated) in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your harmonic mean and the full breakdown beneath it.
Harmonic mean = n / (1/x₁ + 1/x₂ + … + 1/xₙ). Ideal for averaging rates and speeds.
Formula
HM = n / Σ(1/xᵢ)
Where n is the count of values and the sum runs over all non-zero entries. Equivalently, HM = n / (1/x₁ + 1/x₂ + … + 1/xₙ).
How it works
This calculator computes the harmonic mean — the reciprocal of the arithmetic mean of the reciprocals — which is the appropriate average when the quantity of interest is a rate and the denominators (not numerators) are held equal across observations.
Common applications include averaging speeds over equal distances, computing average price-to-earnings ratios in finance, and combining resistors in parallel. Zero and negative values are excluded because their reciprocals are undefined or lead to meaningless results.
Worked example
Worked example
- Input: 1, 2, 4 (n = 3).
- Reciprocals: 1/1 = 1, 1/2 = 0.5, 1/4 = 0.25.
- Sum of reciprocals: 1 + 0.5 + 0.25 = 1.75.
- HM = n / sum = 3 / 1.75 ≈ 1.7143.
Harmonic mean = 1.7143
Key terms
- Harmonic mean (HM)
- The reciprocal of the arithmetic mean of reciprocals; always less than or equal to the geometric mean, which is less than or equal to the arithmetic mean (HM ≤ GM ≤ AM).
- Rate averaging
- When averaging rates such as speeds over equal distances, the harmonic mean gives the correct overall rate, whereas the arithmetic mean overstates it.
- Reciprocal
- For a number x, its reciprocal is 1/x. The harmonic mean is computed by averaging these reciprocals and then taking the reciprocal of the result.
- HM–GM–AM inequality
- For any set of positive numbers, harmonic mean ≤ geometric mean ≤ arithmetic mean, with equality only when all values are equal.
- Parallel resistance
- When resistors R₁, R₂, … are connected in parallel, the equivalent resistance equals the harmonic mean of the individual resistances divided by n, illustrating a natural physical application of the harmonic mean.
Frequently asked questions
- When is the harmonic mean appropriate?
- Use the harmonic mean when averaging rates (e.g., speeds, fuel efficiency). If you drive 60 mph for one hour and 40 mph for one hour, the harmonic mean gives the true average speed: 2 / (1/60 + 1/40) = 48 mph.