Harmonic Mean — Definition, Formula & Examples

Harmonic Mean

A kind of average. To find the harmonic mean of a set of n numbers, add the reciprocals of the numbers in the set, divide the sum by n, then take the reciprocal of the result. The harmonic mean of {a₁, a₂, a₃, a₄, . . ., a_n} is given below.

Harmonic Mean formula: n divided by (1/a₁ + 1/a₂ + ... + 1/aₙ). Example: numbers 4 and 9 gives 2/(1/4+1/9) = 72/13 ≈ 5.54

See also

Mean

Key Formula

H = \frac{n}{\dfrac{1}{a_1} + \dfrac{1}{a_2} + \dfrac{1}{a_3} + \cdots + \dfrac{1}{a_n}}

Where:

$H$ = The harmonic mean of the set
$n$ = The number of values in the set
$a_1, a_2, \ldots, a_n$ = The individual positive values in the set

Worked Example

Problem: Find the harmonic mean of 2, 3, and 6.

Step 1: Count the number of values in the set.

n = 3

Step 2: Find the reciprocal of each value.

\frac{1}{2},\quad \frac{1}{3},\quad \frac{1}{6}

Step 3: Add the reciprocals together.

\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1

Step 4: Divide the number of values by that sum to get the harmonic mean.

H = \frac{3}{1} = 3

Answer: The harmonic mean of 2, 3, and 6 is 3.

Another Example

This example applies the harmonic mean to a real-world speed problem, showing why the arithmetic mean (50 km/h) would overestimate the true average speed when equal distances are traveled at different rates.

Problem: A car drives 60 km at 40 km/h and then 60 km at 60 km/h. Find the average speed for the whole trip using the harmonic mean.

Step 1: Identify the two speeds and note that the distances are equal, so the harmonic mean gives the correct average speed.

a_1 = 40, \quad a_2 = 60, \quad n = 2

Step 2: Find the reciprocal of each speed.

\frac{1}{40} = 0.025, \quad \frac{1}{60} \approx 0.01667

Step 3: Add the reciprocals.

\frac{1}{40} + \frac{1}{60} = \frac{3}{120} + \frac{2}{120} = \frac{5}{120} = \frac{1}{24}

Step 4: Divide the number of values by the sum of reciprocals.

H = \frac{2}{\frac{1}{24}} = 2 \times 24 = 48

Answer: The average speed for the whole trip is 48 km/h.

Frequently Asked Questions

What is the difference between harmonic mean and arithmetic mean?

The arithmetic mean adds all the values and divides by how many there are. The harmonic mean adds all the reciprocals, divides by the count, then takes the reciprocal of the result. For any set of unequal positive numbers, the harmonic mean is always smaller than the arithmetic mean. The harmonic mean is preferred when averaging rates or ratios, while the arithmetic mean suits ordinary quantities like heights or test scores.

When should you use the harmonic mean instead of the arithmetic mean?

Use the harmonic mean when you are averaging rates, speeds, prices per unit, or any quantity defined as a ratio. A classic case is finding average speed when equal distances are traveled at different speeds. Using the arithmetic mean in these situations gives an incorrect, inflated result.

Can the harmonic mean be used with negative numbers or zero?

No. The harmonic mean requires all values to be positive. If any value is zero, its reciprocal is undefined, so the harmonic mean does not exist. Negative values can cause the reciprocals to cancel in misleading ways, making the result meaningless as an average.

Harmonic Mean vs. Arithmetic Mean

	Harmonic Mean	Arithmetic Mean
Definition	n divided by the sum of reciprocals	Sum of values divided by n
Formula	H = n / (1/a₁ + 1/a₂ + … + 1/aₙ)	A = (a₁ + a₂ + … + aₙ) / n
Best used for	Averaging rates, speeds, ratios	Averaging ordinary quantities (scores, heights)
Relative size	Always ≤ arithmetic mean (for positive values)	Always ≥ harmonic mean (for positive values)
Effect of outliers	Strongly influenced by small values	Strongly influenced by large values

Why It Matters

You will encounter the harmonic mean in physics when computing average speed over equal distances, in finance when averaging price-to-earnings ratios, and in electronics when combining parallel resistances. It also appears in the AM–GM–HM inequality, a key result in algebra competitions: for positive numbers, the arithmetic mean is always greater than or equal to the geometric mean, which is always greater than or equal to the harmonic mean. Understanding when to use the harmonic mean prevents systematic errors in problems involving rates and ratios.

Common Mistakes

Mistake: Using the arithmetic mean to average speeds over equal distances.

Correction: When equal distances are traveled at different speeds, the correct average is the harmonic mean, not the arithmetic mean. The arithmetic mean overestimates the true average speed because more time is spent at the slower speed.

Mistake: Forgetting to take the final reciprocal after summing the reciprocals.

Correction: The formula has two reciprocal steps: first you take the reciprocal of each value, then after summing and dividing by n, you take the reciprocal of the entire result. Skipping the final step gives you the average of the reciprocals, not the harmonic mean.

Related Terms

Mean — General concept; harmonic mean is one type
Average — Umbrella term for all types of averages
Multiplicative Inverse of a Number — Reciprocals are central to the harmonic mean formula
Sum — Summing reciprocals is a key step
Set — The harmonic mean is computed over a set of values
Geometric Mean — Another type of mean; lies between harmonic and arithmetic
Arithmetic Mean — Most common mean; always ≥ harmonic mean