Harmonic Mean Calculator
Enter a list of positive numbers to calculate their harmonic mean — the correct type of average for rates like speed, where the same base quantity (like distance) is involved for each rate.
How the Harmonic Mean formula works
The harmonic mean formula:
Harmonic mean = n / (1/x₁ + 1/x₂ + ... + 1/xₙ)
Step-by-step calculation
- Take the reciprocal (1/x) of each value.
- Sum all the reciprocals.
- Divide the count of values by that sum.
Worked example
Driving the same distance at 60 mph and then 40 mph: Harmonic mean = 2 / (1/60 + 1/40) = 2 / (0.0167 + 0.025) = 2/0.0417 ≈ 48 mph — the true average speed for the whole trip, not the arithmetic average of 50 mph.
Frequently asked questions
Why is harmonic mean the correct average for speed over equal distances?
Because time spent, not distance, is what actually varies at different speeds — harmonic mean correctly weights each rate by the time it takes to cover the equal distance, while a simple arithmetic average would overstate the true average speed.
When would I use harmonic mean versus geometric mean?
Harmonic mean suits rates involving a shared numerator or denominator quantity (like distance in a speed calculation), while geometric mean suits multiplicative processes like sequential growth rates — they answer different kinds of averaging questions.