A Harmonic Mean Calculator

The harmonic mean is the average for rates. Instead of adding the values and dividing, you flip each one, average the flipped numbers, then flip the answer back.

First valueA positive number. For speeds use the slowest leg first so you can see its pull.Second valueThe second positive value, in the same units as the first.Third valueThe third positive value. Use the reciprocal form for rates.

Harmonic mean

49.0909

Arithmetic mean: 60
Geometric mean: 54.5136
Sum of the reciprocals: 0.0611
Harmonic over arithmetic: 0.8182

Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.

The formula this calculator uses

H = n / (1/x1 + 1/x2 + ... + 1/xn)

H: The harmonic mean of the values
n: How many values there are
x1 ... xn: The positive values being averaged
1/xi: The reciprocal of each value

How to check the result by hand

1
Check every value is positive
The harmonic mean divides by each value, so a zero breaks it outright and a negative value gives an answer that no longer describes the data. Speeds, prices and rates are all positive by nature.
2
Flip each value into a reciprocal
One over 30 is 0.033333, one over 60 is 0.016667 and one over 90 is 0.011111. Notice how the slowest speed contributes the largest reciprocal, which is what pulls the mean downwards.
3
Add the reciprocals
The three reciprocals add to 0.061111. This sum is the total time per unit of distance for the three legs, and it is the quantity that actually accumulates.
4
Divide the count by the sum
3 divided by 0.061111 is 49.0909. Compare it with the arithmetic mean of 60: the difference of 10.909 is exactly the penalty for treating a rate as if it were an ordinary number.

For worked examples, common mistakes and the limits of this formula, read the full How To Calculate A Harmonic Mean page.