Harmonic Mean — Definition and Uses
The harmonic mean is an average best suited for rates, ratios, and multiples (for example, price-to-earnings ratios). It is defined as the reciprocal of the arithmetic mean of the reciprocals of a set of numbers. Compared with other averages, the harmonic mean gives relatively greater weight to smaller values.
Formula
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Unweighted harmonic mean of n positive values x1, x2, …, xn:
H = n / (sum of 1/xi) = n / (1/x1 + 1/x2 + … + 1/xn) -
Weighted harmonic mean with weights w1, w2, …, wn:
H_w = (sum wi) / (sum (wi / xi))
Note: The reciprocal of a number n is 1/n.
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Examples
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Simple example: For values 1, 4, 4:
H = 3 / (1 + 1/4 + 1/4) = 3 / 1.5 = 2 -
Finance example (average P/E for an index):
Two stocks have P/E ratios of 25 and 250. Portfolio weights: 10% and 90%. - Weighted arithmetic mean (WAM): 0.1×25 + 0.9×250 = 227.5
- Weighted harmonic mean (WHM): (0.1+0.9) / (0.1/25 + 0.9/250) ≈ 131.6
The WHM often gives a less biased average for ratios because it treats each data point equally in reciprocal space.
When to Use the Harmonic Mean
- Averaging rates (e.g., speeds over equal distances).
- Averaging multiples or ratios (commonly used for price-to-earnings and similar financial multiples).
- Situations where smaller values should have more influence on the average.
Comparison with Other Means
- Arithmetic mean: sum(xi) / n — best for additive quantities.
- Geometric mean: (product xi)^(1/n) — used for compounded growth rates and percentages.
- Pythagorean property (for positive values): Harmonic mean ≤ Geometric mean ≤ Arithmetic mean.
- The harmonic mean is the reciprocal of the arithmetic mean of reciprocals.
Advantages and Limitations
Advantages
– Appropriate for rates, ratios, and normalized multiples.
– Equalizes influence across data points when averaging reciprocals.
– Weighted form allows emphasizing more important observations.
Limitations
– Undefined if any xi = 0 (division by zero).
– Sensitive to extreme values; a single very small value strongly pulls the mean downward.
– Interpretation can be tricky with negative values; reciprocal sums may cancel or produce undefined results.
– Requires computing reciprocals, which can be less intuitive than arithmetic averaging.
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Key Takeaways
- Use the harmonic mean for averaging ratios and rates where equal treatment of observations in reciprocal terms is desired.
- It is calculated as n divided by the sum of reciprocals; the weighted form divides the sum of weights by the weighted sum of reciprocals.
- Be cautious when data contains zeros, extreme outliers, or negative values — these can make the harmonic mean undefined or misleading.