Harmonic Mean — Formula, Examples, and When to Use It

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of a set of positive values. For n values: HM = n / (1/x₁ + 1/x₂ + … + 1/xₙ). It is one of the three Pythagorean means and is the appropriate average for rates and ratios where the denominator of the rate stays constant across observations.

The general formula is HM = n / Σ(1/xᵢ): count the number of values, then divide by the sum of their reciprocals. For two numbers a and b, this simplifies to HM = 2ab / (a + b). For grouped data with frequencies: HM = Σf / Σ(f/X).

For any two positive numbers a and b, HM = 2ab / (a + b). This is the same as saying the harmonic mean is twice their product divided by their sum. Example: HM(30, 60) = 2 × 30 × 60 / (30 + 60) = 3600 / 90 = 40.

For three positive numbers a, b, c: HM = 3abc / (ab + bc + ca). This is derived by applying the general formula with n = 3. Alternatively, calculate 3 / (1/a + 1/b + 1/c) directly.

The arithmetic mean sums all values and divides by the count — correct for additive quantities like test scores. The harmonic mean inverts each value, averages the reciprocals, then inverts back — correct for rates where the denominator is fixed. For the same dataset, HM ≤ AM, with the gap widening as values become more spread out or asymmetric.

The geometric mean is the nth root of the product of n values — used for multiplicative or compounding data like investment returns. The harmonic mean involves reciprocals and is used for rates with a fixed denominator. Both are always less than the arithmetic mean for unequal positive values, but HM ≤ GM ≤ AM in strict order.

The three Pythagorean means are the arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM). They satisfy HM ≤ GM ≤ AM for any set of positive numbers that are not all equal. Each one corresponds to a different kind of average: additive (AM), multiplicative (GM), and rate-based (HM).

The name comes from the connection to harmonic series in music. The frequencies of overtones in a musical note — the fundamental plus harmonics — are in arithmetic progression, so their reciprocals (wavelengths) form a harmonic progression. The mean that naturally describes this reciprocal relationship became the harmonic mean.

Start with your values, say 10, 20, 40. Take their reciprocals: 1/10, 1/20, 1/40. Find the arithmetic mean of those reciprocals: (1/10 + 1/20 + 1/40) / 3 = 0.175 / 3 = 0.0583. The harmonic mean is the reciprocal of that result: 1 / 0.0583 = 17.14. That three-step process is the defining property.

It is a three-word mnemonic for calculating the harmonic mean: (1) Invert each value by taking its reciprocal. (2) Average those reciprocals using the arithmetic mean. (3) Invert the result. The final number is the harmonic mean. This sequence correctly handles rate data because inverting speeds gives times, averaging times is correct, and inverting back gives the average speed.

HM = N / Σ(1/X), where N is the number of values and X represents each individual value. Sum all the reciprocals of the data values, then divide N by that sum. All values must be positive.

For a discrete frequency distribution: HM = Σf / Σ(f/X). For a continuous frequency distribution: HM = Σf / Σ(f/X̄), where X̄ is the class midpoint = (lower + upper limit) / 2. In both cases, divide the total frequency by the sum of frequency-over-value terms.

HM = Σf / Σ(f/X). Create a column of f/X values by dividing each frequency by its corresponding value. Sum all f/X values to get Σ(f/X). Sum all frequencies to get Σf. Then HM = Σf ÷ Σ(f/X).

HM = Σf / Σ(f/X̄). The only change from the discrete formula is that you use the class midpoint X̄ = (lower + upper limit) ÷ 2 in place of a discrete value. This applies to data presented as class intervals such as 10–20, 20–30, etc.

WHM = Σwᵢ / Σ(wᵢ/xᵢ), where wᵢ is the weight assigned to value xᵢ. Sum all the weights for the numerator. For each value, multiply its weight by its reciprocal to get wᵢ/xᵢ, then sum those for the denominator.

Step 1: Confirm all values are positive. Step 2: Calculate the reciprocal 1/xᵢ for each value. Step 3: Add all reciprocals: Σ(1/xᵢ). Step 4: Count the values: n. Step 5: HM = n ÷ Σ(1/xᵢ). Keep full precision on reciprocals throughout; round only the final answer.

Use =HARMEAN(range). For example, if speeds are in A1:A5, enter =HARMEAN(A1:A5). The function accepts up to 255 arguments. It returns a #NUM! error if any value is zero or negative, so clean your data first. HARMEAN is in every version of Excel from 97 onwards.

The function is HARMEAN, spelled in full with no abbreviation. Syntax: =HARMEAN(number1, [number2], …). You can pass individual cells, comma-separated values, or a cell range. There is no equivalent shortcut or abbreviation; HARMEAN is the only built-in option.

Three options: (1) statistics.harmonic_mean(data) from Python's standard library. (2) scipy.stats.hmean(data) from SciPy. (3) Manual: len(data) / sum(1/x for x in data). With NumPy: len(data) / np.sum(1/np.array(data)). All produce identical results for positive data.

Using the psych package: harmonic.mean(data). Base R without any package: length(data) / sum(1/data). For grouped data: sum(f) / sum(f/x) where f and x are frequency and value vectors.

Yes, for any set of positive values that are not all identical: HM < AM. When all values are equal, HM = GM = AM. This is part of the Pythagorean means inequality: HM ≤ GM ≤ AM. The gap widens as values become more spread out.

For two positive numbers a and b, the product of their arithmetic mean and harmonic mean equals the square of their geometric mean: AM × HM = GM². Proof: AM = (a+b)/2, HM = 2ab/(a+b), so AM × HM = [(a+b)/2] × [2ab/(a+b)] = ab = (√ab)² = GM². This is an exact identity, not an approximation.

When all values equal some constant c, the harmonic mean also equals c. This is consistent with all means: HM = n / (n/c) = c. The HM ≤ GM ≤ AM inequality becomes an equality (all three equal c) when and only when all values are identical.

Because smaller values have larger reciprocals. The reciprocal of 1 is 1, but the reciprocal of 0.1 is 10. When you sum reciprocals, the small values dominate the total. That large sum in the denominator of n/Σ(1/xᵢ) produces a small HM, pulled strongly toward the small values in your dataset.

Not in its standard definition, which requires all values to be positive. If a dataset contains negative values, some reciprocals are negative, and the formula can produce misleading or meaningless results for rate applications. The harmonic mean is defined for strictly positive values.

Effectively, no. For positive values, HM = n / Σ(1/xᵢ) is always positive. If a zero were included, 1/0 is undefined and the formula cannot be evaluated. The result cannot be zero from positive inputs.

No. A zero value makes 1/x undefined — you cannot take the reciprocal of zero. The harmonic mean is undefined for any dataset that contains a zero. This is not a rounding or precision issue; it is a fundamental mathematical constraint. If your data contains zeros, the HM cannot be computed.

Use the harmonic mean when you are averaging a rate — something expressed as "X per Y" — and the denominator Y is the same for all observations. The clearest cases: average speed over equal distances, average P/E with equal dollar investment, the F1 score, and parallel resistors. If the denominator varies across observations, use the weighted harmonic mean instead.

Arithmetic mean: when values are additive quantities (heights, scores, prices) or when equal time is spent at each rate. Harmonic mean: when values are rates with a fixed denominator, equal distance is covered at each speed, or equal amounts of money are invested at each P/E ratio. The easiest test: does adding more time at a slow rate change the total time more than expected? If yes, use the harmonic mean.

Geometric mean: compounding or multiplicative processes — investment returns over multiple years, population growth rates, ratios that multiply together. Harmonic mean: rate data where the denominator is held fixed — speed over equal distances, efficiency ratios, the F1 score. Neither is used for additive quantities; that's the arithmetic mean's domain.

Because travel time for a given distance = distance ÷ speed, and the total time for equal distances is proportional to the sum of reciprocals of the speeds. The harmonic mean accounts for the fact that you spend more time at slower speeds — it weights each speed by the time spent at it, not equally. The arithmetic mean ignores this and gives an answer that is always too high when speeds differ.

The F1 score measures a classifier's performance using both precision and recall. Using the arithmetic mean (0.9 + 0.1) / 2 = 0.5 would suggest a decent model even when recall is disastrously low. The harmonic mean gives 2 × 0.9 × 0.1 / (0.9 + 0.1) = 0.18 — correctly revealing the problem. The harmonic mean penalizes imbalance between the two metrics far more aggressively than the arithmetic mean.

When you invest equal dollar amounts in multiple stocks, you buy more shares of cheaper (low P/E) stocks and fewer shares of expensive (high P/E) ones. The harmonic mean reflects this correctly: it weights each P/E by the inverse of its value, matching how dollars actually buy shares. The arithmetic mean overstates the typical valuation because it treats each stock as contributing equally regardless of how many shares your money bought.

For parallel resistors, the reciprocal rule 1/Req = Σ(1/Rᵢ) rearranges to Req = n / Σ(1/Rᵢ) — which is exactly the harmonic mean formula divided by n. So Req = HM(R₁, …, Rₙ) / n. For two resistors: Req = HM/2 = R₁R₂/(R₁+R₂). The same mathematics applies to series capacitors and lenses because they all obey reciprocal addition rules.

The effective population size Nₑ — the size of an idealized population with the same genetic drift as the real one — is the harmonic mean of census population sizes across generations. This correctly accounts for bottlenecks: a single generation with very few individuals restricts genetic diversity for all future generations, and the harmonic mean heavily weights that small value.

Main advantages: (1) Gives the mathematically correct average for rates with a fixed denominator. (2) Uses every value in the dataset. (3) Resistant to inflated high values. (4) Uniquely defined — no ambiguity. (5) Algebraically tractable, obeying AM × HM = GM². (6) Extends naturally to weighted and grouped data.

Main disadvantages: (1) Cannot be computed with any zero values. (2) Sensitive to small values in ways that can distort results if those small values are measurement errors. (3) Less intuitive than the arithmetic mean. (4) Requires confirming that your situation actually has a fixed denominator — misapplied, it gives wrong answers. (5) Cannot be calculated if any value is unknown.

Beyond the zero-value and negative-value restrictions, the harmonic mean's main limitation is scope: it is correct only for rates with a fixed denominator. Applied to any other kind of data it produces systematically incorrect results. It also does not handle missing values — you need every observation to compute it.

A harmonic progression is a sequence of numbers whose reciprocals form an arithmetic progression. If a, a+d, a+2d, … is an AP, then 1/a, 1/(a+d), 1/(a+2d), … is an HP. Any term in an HP is the harmonic mean of its two neighbors. The harmonic progression and the harmonic mean are related through reciprocals but serve different mathematical purposes.

The harmonic series is 1 + 1/2 + 1/3 + 1/4 + 1/5 + … — the sum of reciprocals of all positive integers. Despite the terms approaching zero, the sum grows without bound (the series diverges). This was first proved by Nicole Oresme in the 14th century. The series shares its name with the harmonic mean and harmonic progressions through the common theme of reciprocals.