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The harmonic mean is the correct way to average rates — like speed over equal distances — where a simple average would give a misleading answer.
This calculator finds the harmonic mean of any set of positive numbers.
How it works
Enter a list of positive numbers. The calculator adds up the reciprocal (1 divided by each value) of every number, then divides the count of numbers by that sum.
- Enter numbers (positive only, comma or space separated).
- Click Calculate to see your results.
Examples
Speeds of 10, 20, 30
The harmonic mean of 10, 20, and 30 is about 16.36 — for example, the correct average speed for a trip covering equal distances at each of these three speeds.
Who should use it
- Finding the true average speed for a trip covering equal distances at different speeds.
- Coursework involving the harmonic mean and when to use it.
Industry applications
- Physics and engineering (rate calculations)
- Finance (averaging ratios like P/E across a portfolio)
Advantages
- Simple, direct calculation for any count of positive numbers.
- Shows the full reciprocal-sum calculation in the step-by-step solution.
Limitations
- Only defined for positive numbers — can't be used with zero or negative values.
Common mistakes to avoid
- Using a simple average for rates over equal distances or equal amounts, which overstates the true average rate.
- Entering a zero, which makes the calculation undefined (division by zero).
Best practices
- Reach for the harmonic mean specifically for "rate per fixed unit" situations, like speed over equal distances or fuel efficiency over equal distances.
Tips
- Of the three Pythagorean means, harmonic mean ≤ geometric mean ≤ arithmetic mean always holds — the harmonic mean gives the most weight to smaller values in the set.