Harmonic Mean Definition, Formula, and Examples

What Is the Harmonic Mean?

The harmonic mean is a numerical average calculated by dividing the number of observations, or entries in the series, by the reciprocal of each number in the series. Thus, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals.

For example, to calculate the harmonic mean of 1, 4, and 4, you would divide the number of observations by the reciprocal of each number, as follows: 

$\frac{3}{\left(\frac{1}{1}\ +\ \frac{1}{4}\ +\ \frac{1}{4}\right)}\ =\ \frac{3}{1.5}\ =\ 2$(11​ + 41​ + 41​)3​ = 1.53​ = 2

The harmonic mean has uses in finance and technical analysis of markets, among others.

Understanding the Harmonic Mean

The harmonic mean helps to find multiplicative or divisor relationships between fractions without worrying about common denominators. Harmonic means are often used in averaging things like rates (e.g., the average travel speed given a duration of several trips).

The weighted harmonic mean is used in finance to average multiples like the price-to-earnings (P/E) ratio because it gives equal weight to each data point. Using a weighted arithmetic mean to average these ratios would give greater weight to high data points than low data points because P/E ratios aren't price-normalized while the earnings are equalized.

The harmonic mean is the weighted harmonic mean, where the weights are equal to 1. The weighted harmonic mean of x₁, x₂, x₃ with the corresponding weights w₁, w₂, w₃ is given as:

$\displaystyle{\frac{\sum^n_{i=1}w_i}{\sum^n_{i=1}\frac{w_i}{x_i}}}$∑i=1n​xi​wi​​∑i=1n​wi​​

Harmonic Mean vs. Arithmetic Mean and Geometric Mean

Other ways to calculate averages include the simple arithmetic mean and the geometric mean. Taken together, these three types of mean (harmonic, arithmetic, and geometric) are known as the Pythagorean means. The distinctions between the three types of Pythagorean mean makes them suitable for different uses.

An arithmetic average is the sum of a series of numbers divided by the count of that series of numbers. If you were asked to find the class (arithmetic) average of test scores, you would simply add up all the test scores of the students, and then divide that sum by the number of students. For example, if five students took an exam and their scores were 60%, 70%, 80%, 90%, and 100%, the arithmetic class average would be 80%.

The geometric mean is the average of a set of products, the calculation of which is commonly used to determine the performance results of an investment or portfolio. It is technically defined as "the nth root product of n numbers." The geometric mean must be used when working with percentages, which are derived from values, while the standard arithmetic mean works with the values themselves.

Example of the Harmonic Mean

As an example, take two firms. One has a market capitalization of $100 billion and earnings of $4 billion (P/E of 25), and the other has a market capitalization of $1 billion and earnings of $4 million (P/E of 250). In an index made of the two stocks, with 10% invested in the first and 90% invested in the second, the P/E ratio of the index is: 

$\begin{aligned}&\text{Using the WAM:\ P/E}\ =\ 0.1 \times25+0.9\times250\ =\ 227.5\\\\&\text{Using the WHM:\ P/E}\ =\ \frac{0.1\ +\ 0.9}{\frac{0.1}{25}\ +\ \frac{0.9}{250}}\ \approx\ 131.6\\&\textbf{where:}\\&\text{WAM}=\text{weighted arithmetic mean}\\&\text{P/E}=\text{price-to-earnings ratio}\\&\text{WHM}=\text{weighted harmonic mean}\end{aligned}$​Using the WAM: P/E = 0.1×25+0.9×250 = 227.5Using the WHM: P/E = 250.1​ + 2500.9​0.1 + 0.9​ ≈ 131.6where:WAM=weighted arithmetic meanP/E=price-to-earnings ratioWHM=weighted harmonic mean​

As can be seen, the weighted arithmetic mean significantly overestimates the mean price-to-earnings ratio.

Advantages and Disadvantages of the Harmonic Mean

The harmonic mean is effective because it incorporates all the entries in the series, and remains impossible to compute if any item is disallowed. Using the harmonic mean also allows a more significant weighting to be given to smaller values in the series, and it can also be calculated for a series that includes negative values. In comparison with the arithmetic mean and geometric mean, the harmonic mean generates a straighter curve.

However, there are also a few downsides to using the harmonic mean. First and foremost, because it requires using the reciprocals of the numbers in the series, the calculation of harmonic mean can be complex and time-consuming. In addition, because of the impossibility of finding the reciprocal of zero, it is not feasible to calculate the harmonic mean if the series contains a zero value. Finally, any extreme values on the high or low end of the series have an intense impact on the results of the harmonic mean.

The Bottom Line

The harmonic mean is calculated by dividing the number of entries in a series by the reciprocal of each number in the series. The harmonic mean stands out from the other types of Pythagorean mean—the arithmetic mean and geometrical mean—by using reciprocals and giving greater weight to smaller values. The harmonic mean is best used for fractions such as rates, and in finance, it is useful for averaging data like price multiples and identifying patterns such as Fibonacci sequences.