Harmonic Mean Calculator

Created by Georgi Georgiev
Last updated: Oct 28, 2026

1. What is a harmonic mean
2. Harmonic mean formula
3. Calculating a harmonic mean
4. Weighted harmonic mean formula
5. How to calculate the harmonic mean of two numbers
6. Harmonic mean of three numbers
7. Practical applications

    What is a harmonic mean

The harmonic mean (archaic: subcontrary mean) is a specialized average of a set of numbers. It is one of the three Pythagorean means and is more complex to solve for than the more familiar arithmetic mean as it is in fact the reciprocal of the arithmetic mean of reciprocals. The harmonic mean is largely used in situations dealing with quantitative data such as finding the average of rates or ratios and has applications in physics, finance, chemistry, biology, geometry, and more.

    Harmonic mean formula

The formula for calculating the harmonic mean of a set of non-zero positive numbers is: where n is number of items and X₁…X₂ are the numbers from 1 to n.

Its essense of the equation is that we divide the number of items in the set by the sum of their reciprocals. This is the formula used in this harmonic mean calculator.

    Calculating a harmonic mean

Our harmonic mean calculator requires just a set of numbers. You can separate them with commas or blank spaces, then it will do the calculation for you. Note that the harmonic mean of a set of numbers is undefined if one of them is zero.

To find the harmonic mean of a set of numbers by hand, use the formula provided above. Begin by finding a common denominator. If the numbers are 3, 12, 20, 24 all of them are divisible by 120. Next up, find the sum of the reciprocals:

1/3 + 1/12 + 1/20 + 1/24 = 40/120 + 10/120 + 6/120 + 5/120 = 61/120.

As there are 4 items in the set, the final calculation is:

4 / (61/120) = 4 x 120/61 = 7.87

    Weighted harmonic mean formula

The weighted harmonic mean is the reciprocal of the weighted average of the reciprocals of a weights dataset corresponding to a set of numbers. It is calculated using the formula:

    How to calculate the harmonic mean of two numbers

In specific cases where the set consists of two numbers only, you can calculate the harmonic mean using this formula: . Even in this simple case it might be useful to turn to a calculator if the numbers are big or fractions.

In this case its relation to the arithmetic mean (A = (X₁ + X₂)/2) and the geometric mean (G = √X₁ x X₂) is simply:

H = G²/A

Since in a set of real, non-negative numbers the arithmetic mean is always greater than the geometric mean, we can conclude that for n = 2 the harmonic mean will always be lesser in value or equal to the geometric mean (H ≤ G). Another property of the Pythagorean means is that the geometric mean of two numbers is equal to the geometric mean of their arithmetic and harmonic averages (G = √A x H).

    Harmonic mean of three numbers

Three real, non-negative numbers A, G and H are respectively the average, geometric and harmonic means of three real, non-negative numbers if and only if this inequality is true:

    Practical applications

Example usage in Finance

In a comparative analysis of groups of companies there are financial calculations involving averaging multiples where price is in the numerator. In such cases it is advisable to turn to the weighted harmonic mean. For example, if you want to calculate the price-earnings ratio with a weighted arithmetic average, the results would be biased upwards since it gives greater weight to higher data points. However, the weighted harmonic mean assigns equal weight to all data points ^[1], therefore providing the truest average.

Let's say you need to calculate the P/E stock index of 3 companies. The first one has market capitalization of $10 billion and earnings of $1 billion; the second has market capitalization of $27 billion and earnings of $3 billion; finally, the third company has market capitalization of $100 million and earnings of $2 million. You calculate the P/E ratio of the first, second, and third company to be respectively 10, 9, and 50. The index you have to calculate is made up of the three stocks: 60% invested in the first company, 25% in the second, and 15% invested in the third. Its P/E ratio, found using the weighted harmonic mean, is:

P/E = (0.6 + 0.25 + 0.15)/ 0.6/10 + 0.25/9 + 0.15/50 = 1/ 0.09 = 11.1

Now let's calculate it using the weighted arithmetic mean:

P/E = 0.6 x 10 + 0.25 x 9 + 0.15 x 50 = 15.75

Evidently, the weighted arithmetic average overestimates the index, thus giving you incorrect information.

Example usage in Physics

You may want to measure the average speed of a vehicle travelling a round trip. In this case using the harmonic average will result in the correct estimate. Let's say the distance from A to B is 120 km. It is known that the vehicle has travelled from point A to point B at a speed of 80 km/h, whereas the trip back from B to A was at a speed of 60 km/h. The equation for finding the average speed is:

Average speed = total distance/ total time

Average speed = 2 x 120/(120/80 + 120/60) = 240/3.5 = 68.6 km/h

The above is exactly the harmonic mean which you can verify using our harmonic mean solver.

In applied chemistry and applied physics, the harmonic average is also useful in finding the exact densities and mass of particles such as in chemical analyses and nuclear physics tests.

Applications in Geometry

There are a few theorems regarding the usage of the harmonic mean in geometry:

• In each and any triangle, the harmonic mean of the three altitudes is equal to 1/3 of the radius (r) of the incircle;
• In a right-angled triangle with altitude (h) to the right angle from the hypotenuse, and legs a and b, the harmonic mean of a² and b² is twice as big as h^{2 [2] [3]};
• If m and n are the sides of two inscribed squares in a right-angled triangle, then the harmonic mean of the squared hypotenuse c² and m² is twice as big as n²;
• If a point D lies on the minor arc BC of the circumcircle of an equilateral triangle ABC, it is at a distance m from B, and at a distance n from C.  The intersection of AD and BC is at a distance o from D. Therefore, the harmonic mean of m and n is twice as big as o ^[4];
• The semi-latus rectum in an ellipse K is equal to the harmonic average of the minimum and maximum distances of K from a focus;
• In a trapezoid ABCD where AB and CD are parallel sides, the diagonals AC and BD intersect at point E. If point F lies on AD, and point G lies on BC, so that FEG is parallel to AB and CD, then FG is equal to the harmonic mean of AB and CD.

You can use this calculator to solve any of the above problems.

In the popular crossed ladders problem seen below, where two ladders are parallel to each other, one rises at a height A, the other at a height B. The two ladders cross at a height H from the ground. In this case, the harmonic mean of A and B is twice as big as H.

Application in other sciences

In computer sciences the harmonic mean is used to calculate the aggregate score for the evaluation of performance of machines, algorithms, and systems. In biology, more precisely in population genetics, the harmonic average is a means of calculating the fluctuation effects on generation size.