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Harmonic Mean Calculator

· Last updated：2026-04-25

Calculate the harmonic mean of positive values, with optional arithmetic and geometric mean comparison

Details

What this tool helps you do

Calculate the harmonic mean for a dataset of positive numbers.

The harmonic mean is:

H = n / sum(1 / x)

It is especially useful for averaging rates, ratios, speeds, unit prices, and other values where reciprocal relationships matter.

Execution

Run this tool

Fill in the form, run the tool, and review the result in one place.

Samples

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Written by the Elysia Tools team
Math502
Speed53

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category
Math & Numbers
Input types
textarea, number, checkbox
Output type
json
Sample coverage
4
API ready
Yes

Overview

The Harmonic Mean Calculator is a specialized statistical tool designed to find the exact harmonic average of a dataset containing positive numbers. By calculating the reciprocal of the arithmetic mean of reciprocals, it provides accurate averages for rates, ratios, speeds, and unit prices where traditional arithmetic means fall short. You can also compare the result against arithmetic and geometric means to gain deeper insights into your data distribution.

When to use

  • When calculating the average speed of a journey with multiple segments covering the same distance at different speeds.
  • When determining the average cost per unit of shares or items purchased with fixed investment amounts over time.
  • When analyzing financial ratios, such as Price-Earnings (P/E) ratios, across a portfolio of assets.

How it works

  1. 1Enter your dataset of positive numbers into the text area, separating values with commas, spaces, or newlines.
  2. 2Specify the desired number of decimal places for the output precision (between 0 and 10).
  3. 3Toggle the mean comparison option to simultaneously calculate and compare the harmonic, arithmetic, and geometric means.
  4. 4The tool computes the formula H = n / sum(1 / x) and returns the results in a structured JSON format.

Use cases

  • Financial analysts calculating the average P/E ratio of an investment portfolio to avoid upward bias.
  • Engineers and physicists determining average speeds or rates of flow across equal distances or volumes.
  • Data scientists comparing Pythagorean means to understand the skewness and distribution of a dataset.

Examples

1. Calculating Average Travel Speed

Logistics Planner

Background

A delivery truck travels from point A to point B at 60 mph, and returns along the exact same route at 40 mph.

Problem

Using a standard arithmetic mean (50 mph) is incorrect because the truck spends more time traveling at the slower speed.

How to use

Input the speeds '60, 40' into the dataset field and set decimal places to 2.

{
  "dataset": "60, 40",
  "decimalPlaces": 2,
  "includeMeanComparison": false
}

Outcome

The tool calculates the harmonic mean as 48.00 mph, providing the true average speed for the entire round trip.

2. Averaging Portfolio P/E Ratios

Financial Analyst

Background

An analyst is evaluating a small portfolio of three stocks with Price-to-Earnings (P/E) ratios of 15, 20, and 35.

Problem

Calculating the arithmetic mean overstates the portfolio's valuation because it gives disproportionate weight to high P/E stocks.

How to use

Enter '15, 20, 35' into the dataset, enable 'Include Mean Comparison', and set decimal places to 4.

FAQ

What is the difference between harmonic and arithmetic mean?

The arithmetic mean adds all values and divides by the count, which is best for standard quantities. The harmonic mean divides the count by the sum of the reciprocals of the values, making it accurate for rates, ratios, and speeds.

Can I use negative numbers or zero in the dataset?

No, the harmonic mean is only defined for strictly positive numbers. Zeros will cause a division by zero error, and negative numbers can yield invalid or misleading results.

Why is the harmonic mean always lower than the arithmetic mean?

According to the inequality of arithmetic and geometric means, for any set of positive numbers with at least one variation, the harmonic mean is always the smallest, followed by the geometric, then the arithmetic mean.

How do I format my input data?

You can paste your numbers separated by commas, spaces, or line breaks. The tool will automatically parse the numeric values from your input.

What does the 'Include Mean Comparison' feature do?

When enabled, the tool calculates the arithmetic and geometric means alongside the harmonic mean, allowing you to easily compare the three standard Pythagorean means for your dataset.

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{
  "dataset": "15, 20, 35",
  "decimalPlaces": 4,
  "includeMeanComparison": true
}

Outcome

The tool outputs a harmonic mean of 20.5882, alongside the arithmetic mean (23.3333), showing the more accurate, lower valuation metric.