Harmonic Mean Calculator
Find the harmonic mean of three numbers.
Results update as you type.
How it works
The harmonic mean is a type of average that is especially suited to rates, speeds, and ratios, where the ordinary arithmetic mean would give a misleading answer. It is defined as the number of values divided by the sum of their reciprocals, which makes it always less than or equal to the arithmetic mean of the same numbers. This harmonic mean calculator takes three positive or negative non-zero numbers and returns their harmonic mean. The classic use is averaging speeds over equal distances: if you drive one leg at 30 and another at 60, your average speed for the round trip is the harmonic mean, not the simple average, because you spend more time at the slower speed. The harmonic mean also appears in finance for averaging price-to-earnings ratios, in physics for parallel resistances, and in statistics as part of the F-score. For the numbers 1, 2, and 4, the harmonic mean is 3 divided by the sum of 1, one-half, and one-quarter, which comes to about 1.714. Because dividing by zero is undefined, none of the values may be zero. Analysts, scientists, and students use a harmonic mean calculator whenever they need the right kind of average for rate-based data.
harmonic mean = 3 ÷ (1⁄n₁ + 1⁄n₂ + 1⁄n₃)
The harmonic mean is the number of values divided by the sum of their reciprocals. It is the reciprocal of the arithmetic mean of the reciprocals, which is why it gives more weight to smaller values and less to larger ones. This makes it the correct average for quantities defined as rates, such as speed or price per unit, because averaging the reciprocals correctly accounts for how those rates combine over a fixed distance or amount.
Source:Standard statistics
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Worked example
How to use this calculator
- 1
Enter your values
Enter number 1, number 2 and number 3.
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Harmonic mean updates instantly as you type — no button to press.
Frequently asked questions
What is the harmonic mean used for?
The harmonic mean is the correct average for rates and ratios measured over a fixed quantity, such as average speed across equal distances, average price-to-earnings ratios, or combined rates of work. It gives more weight to smaller values, which reflects how rates actually combine. Using an arithmetic mean in these situations overstates the true average, so the harmonic mean provides the mathematically appropriate result in rate-based problems.
How does it differ from the arithmetic mean?
The arithmetic mean adds the values and divides by how many there are, treating each equally. The harmonic mean instead averages the reciprocals and inverts the result, which gives smaller numbers more influence. For any set of positive numbers that are not all identical, the harmonic mean is always smaller than the arithmetic mean. The two agree only when every value in the set is exactly the same.
Why can't any value be zero?
The formula relies on taking the reciprocal of each number, and the reciprocal of zero is undefined because you cannot divide one by zero. If any input were zero, its reciprocal would be infinite and the harmonic mean would collapse. For this reason the calculator returns no result when any of the three values is zero. All inputs must be non-zero for the harmonic mean to be defined.
Can I use it for average speed?
Yes, and this is its most famous application. When you travel equal distances at different speeds, your overall average speed is the harmonic mean of those speeds, not the arithmetic mean. This is because you spend more time covering the distance at the slower speed, so it should count more heavily. Enter the individual speeds and the calculator returns the true average speed for equal-distance legs.
Is the harmonic mean always smaller than the average?
For a set of positive numbers, the harmonic mean is always less than or equal to the geometric mean, which in turn is less than or equal to the arithmetic mean. Equality occurs only when all the numbers are identical. This ordering, known as the mean inequality, means the harmonic mean sits at the low end and is pulled strongly toward the smallest values in the set.