Harmonic Mean Calculator

What is Harmonic Mean?

The harmonic mean is a type of average calculated as the reciprocal of the arithmetic mean of reciprocals. For a dataset of $n$ positive numbers $x_1, x_2, ..., x_n$, the harmonic mean $H$ is defined as $H = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}$. Unlike the arithmetic mean, the harmonic mean gives greater weight to smaller values, making it particularly useful for averaging rates, ratios, and speeds where the reciprocal relationship is meaningful.

The harmonic mean is always less than or equal to the geometric mean, which is always less than or equal to the arithmetic mean for any set of positive numbers. This fundamental relationship, known as the AM-GM-HM inequality, states that $\text{HM} \leq \text{GM} \leq \text{AM}$, with equality only when all values are identical. For other statistical measures, check out our Geometric Mean Calculator or Statistics Calculator.

The AM-GM-HM Inequality

The three Pythagorean means are related by the inequality $\text{HM} \leq \text{GM} \leq \text{AM}$ for any set of positive numbers. This means the harmonic mean is always the smallest of the three, and the arithmetic mean is always the largest. The differences between these means provide insight into the spread and distribution of the data. When values are tightly clustered, all three means are close together; when values are widely dispersed, the harmonic mean is significantly smaller than the arithmetic mean.

When to Use Harmonic Mean

The harmonic mean is the appropriate choice for averaging rates, ratios, and speeds where the quantities being averaged involve reciprocal relationships. For example, when traveling equal distances at different speeds, the average speed is the harmonic mean of the individual speeds, not the arithmetic mean. This is because you spend more time traveling at the slower speed, so the slower speed has a greater influence on the overall average.

In finance, the harmonic mean is used to calculate average price multiples like P/E ratios for portfolios. In machine learning, the F1 score is the harmonic mean of precision and recall, ensuring both metrics must be reasonably high for a good score. For financial analysis, see our Investment Calculator or CAGR Calculator.

Average Speed Example

If you drive 100 km at 40 km/h and return 100 km at 60 km/h, your average speed is the harmonic mean: $H = \frac{2}{\frac{1}{40} + \frac{1}{60}} = \frac{2}{\frac{3+2}{120}} = \frac{2 \times 120}{5} = 48$ km/h. This is less than the arithmetic mean of 50 km/h because you spend more time traveling at the slower speed. The harmonic mean correctly accounts for the time spent at each speed, providing a more accurate average for this scenario.

Important Considerations

The harmonic mean requires all positive, non-zero values. Division by zero is undefined, so zeros cannot be included. Negative numbers would make the sum of reciprocals potentially zero or negative, making the result undefined or meaningless. Very small values have a disproportionate effect on the harmonic mean because their reciprocals are very large, so it is important to check for outliers before using the harmonic mean. The tool automatically computes all three Pythagorean means so you can compare them and verify the inequality.