Harmonic Mean Calculator

About Harmonic Mean Calculator

Harmonic mean is appropriate for averaging rates. It's always ≤ geometric mean ≤ arithmetic mean (for positive values).

How to use this calculator

1. Enter the positive numbers you want to average.
2. Make sure each value is greater than 0, since the harmonic mean is only defined for positive numbers.
3. Submit the values to get the harmonic mean instantly.
4. Use the displayed formula and example to understand how the result was found.

The formula explained

The harmonic mean is computed by dividing the number of values, \(n\), by the sum of the reciprocals of the values. This finds an average that works well for rates and ratios.

• H = the harmonic mean
• n = the number of values
• \(x_i\) = the i-th value in the data set

Step by step method

1. Count how many numbers you have, so you know \(n\).
2. Find the reciprocal of each number, then add those reciprocals.
3. Divide \(n\) by that reciprocal sum to get the harmonic mean.

Worked example

Problem. Find the harmonic mean of 4, 5, and 10.

1. There are 3 numbers, so \(n=3\).
2. Compute the reciprocals: \(\frac{1}{4}=0.25\), \(\frac{1}{5}=0.2\), and \(\frac{1}{10}=0.1\). Add them: \(0.25+0.2+0.1=0.55\).
3. Now divide: \(H=\frac{3}{0.55}=5.4545\ldots\).

Tips and common mistakes

• All inputs must be positive, because zero or negative values make the harmonic mean undefined in this tool.
• Do not average the numbers first and then take a reciprocal, because the harmonic mean uses the reciprocals of each value before dividing.

Frequently asked questions