Harmonic Mean Calculator: Average Speed & Rates

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Formula at a glance

Harmonic Mean Calculator: Harmonic mean — HM = n / (1/x₁ + 1/x₂ + … + 1/xₙ). Example: Average speed at 60 and 40 km/h → Harmonic mean: 48 — How the Harmonic Mean Calculator works: Harmonic mean — HM = n / (1/x₁ + 1/x₂ + … + 1/xₙ)

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📘 About the Harmonic Mean Calculator

The harmonic mean is the number of values divided by the sum of their reciprocals. It is the right average for rates when the same amount of work is done at each rate: driving 60 km at 60 km/h and 60 km at 40 km/h gives an average speed of 48 km/h, not 50. It is also used for P/E ratios, F1 scores and resistors in parallel.

Enter the values (and weights if the amounts differ). The calculator shows the harmonic mean, the reciprocals behind it, and the arithmetic and geometric means for comparison.

🧮 Formula

Harmonic mean

HM = n / (1/x₁ + 1/x₂ + … + 1/xₙ)

Weighted

HM = Σw / Σ(w/x)

Two numbers

HM = 2ab/(a + b)

✏️ Worked examples

Average speed at 60 and 40 km/h

You enter
Numbers	60, 40

You get
Harmonic mean	48
Arithmetic mean	50
Geometric mean	48.98979486
Example meaning	Driving equal distances at 60, 40 gives an average speed of 48, not 50

HM = 48.

Three rates

You enter
Numbers	10, 20, 40

You get
Harmonic mean	17.14285714
Arithmetic mean	23.33333333
Geometric mean	20
Example meaning	Driving equal distances at 10, 20, 40 gives an average speed of 17.1429, not 23.3333

HM = 17.14285714.

🧭 How to use this calculator

What you enter

NumbersSeparate numbers with commas, spaces or new lines.
Weights (optional)

What you get

Harmonic mean
Arithmetic mean
Geometric mean
Example meaning

❓ Frequently asked questions

Why is the average speed not the simple average?

You spend longer at the slower speed. Over equal distances, the harmonic mean accounts for that.

When should I use the harmonic mean?

For averaging rates (per hour, per unit, per rupee) over equal amounts of the thing in the denominator.

Is the harmonic mean always smallest?

For positive numbers, yes: harmonic ≤ geometric ≤ arithmetic mean, with equality only when all values are equal.

1/x̄ Harmonic Mean Calculator (Average Speed & Rates)