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Free online harmonic mean calculator. Easily calculate the harmonic mean of positive numbers for average speed, exchange rates, and financial data.
Separate numbers by commas, spaces, newlines, or semicolons. All values must be greater than 0.
Please enter a set of numbers greater than 0 to calculate the harmonic mean.
Overview
Understand what the tool solves, how it works, and the boundaries of its data.
The harmonic mean averages positive values by averaging their reciprocals, then taking the reciprocal of that average. It is useful when a quantity is expressed as a rate and the underlying exposure is equal across observations, such as equal travel distances at different speeds. It is not automatically the right average for every list of rates.
IUPAC defines the harmonic mean as the number of observations divided by the sum of their reciprocals. For positive inputs x1 through xn, the calculation is H = n / (1/x1 + 1/x2 + … + 1/xn). The calculator also reports the arithmetic mean, geometric mean, number of accepted values, and reciprocal sum.
Type numbers separated by commas, spaces, line breaks, or semicolons. The calculator accepts only parsed values greater than zero for its result. If a parsed value is zero or negative, it does not produce a mean. The displayed values use locale number formatting and are rounded to at most six decimal places, so the screen is a presentation of the calculation rather than an unlimited-precision export.
For speeds of 20 and 30 miles per hour over equal distances, the harmonic mean is 2 ÷ (1/20 + 1/30) = 24 miles per hour. Equal distance is the condition that makes this rate average appropriate. If the travel distances differ, calculate total distance divided by total travel time instead of averaging the two speeds with this simple formula.
For 20 and 30: reciprocal sum = 1/20 + 1/30 = 1/12; harmonic mean = 2 ÷ (1/12) = 24.
Enter two or more positive values, using any of the supported separators.
Review the harmonic, arithmetic, and geometric means, the count, and the sum of reciprocals shown in the result area.
Check that the observations share the same measurement units and that the underlying quantities make a harmonic average meaningful.
When each leg covers the same distance, enter the speeds to average them across the combined trip; unequal legs need a total-distance and total-time calculation.
For rates tied to equal amounts of exposure, the reciprocal structure may match the way the combined quantity is accumulated. Confirm the denominator being held equal first.
Use the companion means to compare how different definitions respond to the same positive dataset, while interpreting each statistic according to the data design.
No. The tool requires every parsed value to be greater than zero; a zero or negative value prevents a result.
No. It matches the familiar average-speed case when the distances are equal. For unequal distances, use total distance divided by total time.
For a positive dataset, small values contribute large reciprocals and pull the harmonic mean downward. The tool shows both results so you can see how the averaging rule changes the answer.
The implementation calculates it from the same positive values used for the other means. A missing result indicates no valid set was available, such as when an input is empty or includes a nonpositive number.
A correctly computed mean can still be inappropriate for the question. The harmonic mean can be misused when reciprocals do not represent the underlying measurement relationship. Confirm equal denominators or the relevant rate model, keep units consistent, and avoid comparing datasets with different definitions as if the result alone made them comparable.
Guide
Follow the workflow and verify inputs and outputs with practical examples.
Enter positive numeric values separated by commas, spaces, line breaks, or semicolons. The result reports the harmonic mean, arithmetic mean, geometric mean, count, and reciprocal sum. One zero or negative parsed value prevents calculation. Values are displayed to at most six decimal places.
Use cases
See how the tool fits into real work and everyday tasks.
Enter speeds for equal-length legs to obtain the combined average speed; use total distance over total time if the legs differ.
Use the result fields to see how harmonic, geometric, and arithmetic means differ on the same positive observations.
Q&A
Find concise answers to common questions and confusing cases.
The reciprocal of zero is undefined, and the tool requires all inputs to be positive.
Only when the data and the equal underlying quantity justify averaging reciprocals; equal-distance speeds are a common example.
Each value contributes its reciprocal to the denominator, so a small positive value contributes a relatively large amount.
Notes
Review scope, result limitations, and important precautions before use.
The calculator verifies only that values can be parsed and are positive. It cannot decide whether the values represent comparable observations or whether a harmonic mean answers your question. Confirm units, the underlying denominator, and the study or measurement design before interpreting the result.
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