Harmonic Mean Calculator

Harmonic Mean Calculator

Calculate the harmonic mean for rates, ratios, speeds, prices per unit and other positive values where reciprocal averaging gives the better result.

Average rates with reciprocal weighting

Harmonic mean is not the usual average: It is calculated by averaging reciprocals, then taking the reciprocal of that average.

What this calculator does: A harmonic mean calculator computes the harmonic mean by dividing the number of values by the sum of the reciprocals of those values.

Use it for rates: Harmonic mean is useful for average speed over equal distances, price per unit, productivity rates and ratios.

Positive values only: Zero and negative values make the standard harmonic mean invalid in this calculator.

Commas, spaces or new lines

Enter positive numbers. Use the same unit for every value.

Switch to weighted mode if some values represent larger groups or distances.

How to use the harmonic mean calculator

  1. Enter the values: Add positive numbers separated by commas, spaces or new lines.
  2. Choose the mode: Use simple harmonic mean for equal importance, or weighted harmonic mean when values represent different amounts.
  3. Keep units consistent: Do not mix miles per hour with kilometers per hour, dollars per item with euros per item, or different time units.
  4. Check the comparison: The calculator also shows arithmetic and geometric means so you can see how the harmonic mean differs.
  5. Use it for rates: Harmonic mean is most helpful when averaging ratios, speeds, rates or unit prices.

Visual mean comparison

Harmonic mean treats small rates as important because each value is converted into a reciprocal before the final average is calculated.

usual average adds values directly

Arithmetic mean

Best for ordinary quantities where each value contributes directly to the total.

growth average multiplies factors

Geometric mean

Useful for growth rates and compounded changes across multiple periods.

rate average averages reciprocals

Harmonic mean

Best when the numbers are rates and the denominator units are comparable.

Harmonic mean formula

The harmonic mean formula is n / (1 / x1 + 1 / x2 + 1 / x3 + ...). Harmonic means work best for rates, speeds and ratios because they reduce the impact of large outlier values and give more weight to slower or smaller rates.

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

Weighted H = sum(w) / sum(w / x)

Use the weighted formula when each rate applies to a different weight, such as different distances, quantities or exposure amounts.

When to use harmonic mean

Swipe to view table
Situation Use harmonic mean? Why
Average speed over equal distances Yes Speed is a rate, and equal distances make reciprocal averaging appropriate.
Average price per unit for equal spending Often Unit price is a ratio, so lower prices can affect the true average strongly.
Ordinary test scores Usually no Scores are direct quantities, so arithmetic mean is normally clearer.
Investment growth factors Usually no Geometric mean is normally better for compounded growth.
Productivity rates with equal tasks Yes Rates per task can be averaged through reciprocal time per task.

Example harmonic mean calculation

Suppose you travel the same distance at 40 mph, 60 mph and 80 mph. Because the distances are equal, the harmonic mean gives the correct average speed.

Values

40, 60, 80

Formula

3 / sum(1 / x)

Result

55.38 mph

The arithmetic mean is 60 mph, but that overstates the equal-distance average because the slower section takes more time.

Common mistakes with harmonic mean

Using zero values

A reciprocal of zero is undefined, so remove invalid zero entries or choose a different average.

Mixing units

Convert all values to the same unit before calculating, such as all mph or all km/h.

Wrong weighting

Use weights only when they represent the correct exposure, distance, quantity or group size.

Which mean should you choose?

Search questions about averages often use similar wording, but the correct mean depends on what the numbers represent. Use this guide before choosing a formula.

Question type Best mean Use when
What is the usual average value? Arithmetic mean The data is made of direct quantities, such as scores, heights, counts or temperatures.
What is the average growth factor? Geometric mean The values multiply over time, such as investment returns, growth rates or index changes.
What is the average rate over equal units? Harmonic mean The values are rates or ratios, and each value applies to the same distance, task count, spend amount or exposure.
What if some rates matter more? Weighted harmonic mean Each rate has a matching weight, such as distance, quantity, portfolio weight or group size.

Harmonic mean examples by field

These examples help you map a real-world question to the value you should enter and the result you should expect from the calculator.

Travel and speed

Enter speeds when each speed covers the same distance. The result is the average speed for the full trip.

Unit price and buying

Enter prices per unit when the same amount of money is spent at each price. The result estimates the effective unit price.

Productivity rates

Enter rates such as tasks per hour when each worker, machine or process completes the same number of tasks.

Finance ratios

Use weighted harmonic mean for ratios such as price-to-earnings when each ratio has a portfolio or exposure weight.

Frequency and cycle rates

Enter frequencies when each observation covers the same number of cycles. The result reflects average cycles per time.

Benchmark comparisons

Use harmonic mean when comparing per-unit performance and you want smaller bottleneck rates to affect the result properly.

Dataset checklist before calculating

A clean harmonic mean result depends less on the calculator and more on whether the input values describe the same kind of rate. Check these points before trusting the output.

Same unit

Convert every value into the same unit, such as all mph, all km/h, all dollars per item or all tasks per hour.

Same denominator logic

Confirm that the rates share a comparable base, such as equal distances, equal task counts or a meaningful set of weights.

Positive values

Remove zero, blank or negative entries unless you are using a specialized statistical method outside this calculator.

Outlier review

Check very small values carefully because the harmonic mean is intentionally sensitive to low rates.

Weight meaning

In weighted mode, each weight should describe exposure, amount, distance, quantity or group size for the matching value.

Result label

Label the result with the original unit, such as mph or dollars per unit, so the average is easy to interpret later.

Frequently asked questions

What is the harmonic mean in statistics?

The harmonic mean is a statistics average calculated from reciprocal values. For a positive dataset, divide the number of values in the sample by the sum of 1 divided by each value. This math is especially useful when the data describes a rate, ratio, speed or frequency.

When should I use harmonic mean instead of arithmetic mean?

Use harmonic mean when the calculation averages rates over equal units, such as speed over equal distances or cost per unit under certain buying patterns. For ordinary totals in a dataset, arithmetic mean is usually more appropriate, so the best average depends on what each number represents.

Can a harmonic mean calculator use zero or negative numbers?

This calculator requires positive numbers. Zero cannot be used because its reciprocal is undefined, and negative values often break the practical interpretation of rates or ratios. If your data includes zero, the harmonic mean formula is usually not the right method for that sample.

What is a weighted harmonic mean?

A weighted harmonic mean gives some values more importance in the final result. It divides the total weight by the sum of each weight divided by its value, which is useful when a rate applies to different distances, amounts, exposure levels or group sizes.

Why is harmonic mean usually smaller in a comparison?

Because reciprocals make smaller values contribute more strongly to the calculation. In a positive data set, harmonic mean is usually less than or equal to geometric mean and arithmetic mean, so the comparison helps show how much smaller rates are shaping the result.

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Disclaimer: This harmonic mean calculator is informative and intended for general math, education and estimation. Check formulas, assumptions and units before using any result for technical, financial or academic work.

Last updated: May 12, 2026