What it means
Most people know the arithmetic average, which adds the values and divides by how many there are. That method works well for quantities that add up, but it can mislead when the numbers are rates, such as speeds, prices per share or multiples.
The harmonic average gives the right answer in these cases because it gives more weight to the smaller values. A classic example is a fixed monthly investment.
If you invest the same dollar amount each month, you buy more shares when the price is low and fewer when it is high. Your average cost per share ends up lower than the simple average of the prices, and it equals the harmonic average.
Analysts also use it to average valuation multiples across a group of companies. If you average price to earnings ratios using the arithmetic method, a single very high ratio can distort the result.
The harmonic average treats the underlying earnings yield correctly and gives a more reliable figure. The calculation is simple.
Take the reciprocal of each value, add them up, divide by the number of values, and then take the reciprocal of the result. The harmonic average is always less than or equal to the arithmetic average, and the two are equal only when all values are the same.
There is one caution. The harmonic average cannot be used with zero values, and negative numbers cause problems, so data must be checked first.
When in doubt about which average to choose, ask whether the numbers represent a rate where the denominator is fixed or varies. Spreadsheets make the calculation easy.
Most packages have a built-in function for the harmonic mean, and the result should match the manual method. Checking one example by hand is a good habit, because it confirms that the right column of data has been selected.
In practice
Real-world examples.
Example
A retail investor invests $500 a month in a fund over six months while the unit price moves between $8 and $12. She wants to know her average cost per unit. She divides the total spent by the total units bought, which gives the harmonic average of the prices. The result is lower than the simple average of the prices because she bought more units when they were cheap.
Example
An equity analyst compares the valuations of four competitors with price to earnings ratios of 10, 12, 15 and 60. The arithmetic average of 24.25 is pulled up by the single high figure. The harmonic average is exactly 15, since 4 / (1/10 + 1/12 + 1/15 + 1/60) = 4 / 0.2667, which gives a better picture of the typical valuation.
Example
A logistics manager travels the same route in both directions, driving the outward leg at 60 kilometres an hour and the return at 40. The average speed for the whole trip is the harmonic average, 2 / (1/60 + 1/40) = 48 kilometres an hour. Using 50 would overstate how quickly the trips can be completed.
Formula
Calculation
Harmonic average = n / (1/x1 + 1/x2 + ... + 1/xn)
Suppose an investor buys $1,000 of a fund in January at $10 per unit and $1,000 in February at $20 per unit. In January she buys 1,000 / 10 = 100 units, and in February she buys 1,000 / 20 = 50 units.
Her total spending is $2,000 for 150 units, so the average cost is 2,000 / 150 = $13.33 per unit.
The harmonic average of the two prices is 2 / (1/10 + 1/20) = 2 / 0.15 = $13.33, which matches. The simple arithmetic average of the prices, (10 + 20) / 2 = $15, would overstate the cost.Case study
Seen in the real world.
Pinecrest Advisory is an illustrative, fictional firm that valued a private company using the price to earnings ratios of five listed rivals. The ratios were 8, 10, 12, 15 and 40.
The first analyst took a simple average and obtained 17, which suggested a high valuation. A senior colleague recalculated using the harmonic average, 5 / (1/8 + 1/10 + 1/12 + 1/15 + 1/40) = 5 / 0.4 = 12.5, which was closer to the group's typical figure.
The client avoided overpaying in this fictional story, and the firm adopted the harmonic method for all multiple-based valuations. It also noted the method in its valuation report so that the reader could follow the steps.
Watch out
Common mistakes.
- Using the arithmetic average for ratios and rates, which can overstate the result.
- Including zero or negative values, which makes the harmonic average undefined or misleading.
- Applying it to quantities that simply add up, such as total sales, where the arithmetic average is correct.
Questions
People also ask.
What is the harmonic average?
It is the number of values divided by the sum of their reciprocals.
Why is it lower than the arithmetic average?
Because it gives more weight to the smaller values, so large outliers have less effect.
When should I use it?
When averaging rates or ratios where a fixed amount is spent or travelled, such as price multiples or average cost per unit.
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