Harmonic Mean
Reciprocal average — rates, speeds, ratios
🔒 No server · Method · · Disclaimer · Harmonic Mean
Harmonic Mean turns the numbers behind reciprocal average into a result, step by step. Drop Data (comma separated) into Harmonic Mean — the figure refreshes as you type. Harmonic Mean puts the headline number first. Harmonic Mean suits budgeting and planning checks. Harmonic Mean is privacy-first: Math inputs stay offline. Harmonic Mean sits in the Math section next to free online harmonic mean calculator and harmonic mean calculator average rates speeds. Run Harmonic Mean twice to find the driving input. Try Harmonic Mean now and keep it handy.
Harmonic mean of a data set
The harmonic mean is the average for rates and ratios, such as average speed or the price-to-earnings multiple of equal-sized holdings, because it weights small values more heavily than the arithmetic mean does. The page sums the reciprocals and divides by the count.
The harmonic mean averages reciprocals, which is why it is the correct average for rates: driving a fixed distance at two speeds does not average the speeds the way addition would suggest.
The formula
H = n ÷ Σ(1/xᵢ) — the count divided by the sum of the reciprocals.
Reciprocals: 1/2 = 0.5000, 1/4 = 0.2500, 1/8 = 0.1250. Sum = 0.8750. H = 3 ÷ 0.8750 = 3.4286.
Worked example
Data 2, 4, 8
Reciprocals 0.5, 0.25, 0.125
Sum 0.875
H = 3 ÷ 0.875 = 3.4286
Arithmetic mean for comparison: (2+4+8)/3 = 4.67
Why it is the correct average for rates
Drive 10 km at 40 km/h and 10 km at 60 km/h. The trip takes 15 + 10 = 25 minutes for 20 km, so the average speed is 48 km/h — the harmonic mean, not 50. The slower leg occupies more time, and averaging speeds arithmetically ignores that.
The same logic applies to fuel economy, price-to-earnings ratios across a portfolio, and averaging resistor values in parallel.
When to use it — and when not to
- Use it for rates with a fixed numerator: km per litre, km/h over equal distances, parallel resistances.
- Use the arithmetic mean for values that add naturally, such as heights or prices paid.
- Use the geometric mean for growth rates and ratios that multiply, such as compound returns.
- Zero breaks it. Any zero in the data makes the reciprocal infinite, so the harmonic mean is undefined.
Why the result is always the lowest of the three
For positive data, harmonic ≤ geometric ≤ arithmetic, with equality only when every value is identical. The harmonic mean is pulled down by the smallest values because their reciprocals are the largest, which is exactly the weighting a rate average needs.
Frequently Asked Questions
Why is the harmonic mean 3.43 and not 4.67?
Because it averages reciprocals. The reciprocals 0.5, 0.25 and 0.125 sum to 0.875, and 3 ÷ 0.875 = 3.4286 — lower than the arithmetic average because small values weigh more.
When should I use the harmonic mean?
When averaging rates: speed over equal distances, fuel consumption, price-to-earnings ratios, or parallel resistances. It is the average that accounts for the time or quantity spent at each rate.
Can I use it with a zero in the data?
No. The reciprocal of zero is undefined, so the harmonic mean does not exist. Remove zero values or use a different average.
Why is 40 and 60 km/h an average of 48, not 50?
Because equal distances are driven at unequal times. The slower leg takes longer, so it dominates the trip. Total distance over total time gives 48 km/h, which is the harmonic mean.
How does it compare with the geometric mean?
For positive values, harmonic ≤ geometric ≤ arithmetic. Use geometric for growth rates that compound, harmonic for rates with a fixed numerator, arithmetic for plain quantities.
Distance = Rate × Time and other Math tools
Distance = Rate × Time · Percentage Calculator · LCM & GCD Calculator · Average Calculator · Complex Number Calculator · Binomial Probability · Math Guides