Harmonic Mean Calculator
Calculate the harmonic mean—the reciprocal of the arithmetic mean of reciprocals. The harmonic mean is the mathematically correct method for averaging rates, speeds over equal distances, and financial multiples.
List of positive values or speeds (e.g. 40, 60 for round-trip speed)
Correct central tendency for rates: n / Σ(1/x)
How to Use This Calculator
- Enter positive rates or speeds separated by commas or spaces.
- View the harmonic mean calculation.
- Compare across the three classical Pythagorean means (A ≥ G ≥ H).
Formula & Calculation Methodology
The harmonic mean calculates the reciprocal of the mean of reciprocals. For two values, H = 2ab / (a + b).
Worked Example
Round trip driving at 40 mph out and 60 mph return
Averaging parallel electrical resistance: 10Ω, 20Ω, 30Ω
Frequently Asked Questions
Why is the average speed of 40 mph and 60 mph equal to 48 mph instead of 50 mph?↓
Because you spend more time driving at the slower speed than at the faster speed. Total distance divided by total time yields 48 mph, which matches the harmonic mean exactly.
What is the Pythagorean Means relationship?↓
For any positive numbers, the Arithmetic Mean (AM) is greater than or equal to the Geometric Mean (GM), which is greater than or equal to the Harmonic Mean (HM): AM ≥ GM ≥ HM.
Why can’t zero be used in the harmonic mean?↓
Because the reciprocal of zero (1 / 0) is undefined, creating an infinite sum in the denominator.
What is a limitation of the harmonic mean?↓
It gives disproportionate weight to very small numbers and cannot handle negative numbers or zeros.
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Source Citation & Disclaimer
Informational tool only; not licensed tax, legal, or financial advice.