Harmonic Mean Calculator

Frequently Asked Questions

What if one of my values is very small?

The harmonic mean gives heavy weight to small values because their reciprocals are large. A single value close to zero will pull the harmonic mean close to zero. This is by design for rate calculations, but check whether a small value is a genuine data point or an error.

When is the harmonic mean the same as the arithmetic mean?

Only when all values are identical. For example, if all speeds are 50 km/h, both means equal 50 km/h. Any variation in values makes the harmonic mean strictly less than the arithmetic mean.

Can I use the harmonic mean for growth rates?

No. Growth rates that compound over time call for the geometric mean. The harmonic mean applies when you are averaging rates over equal amounts of the quantity in the numerator (distance, money, items).

What is the harmonic mean of 2 and 2?

H = 2 ÷ (1/2 + 1/2) = 2 ÷ 1 = 2. When all values are equal, H = G = A = the common value.

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How This Calculator Works

The harmonic mean H of n positive numbers is the reciprocal of the arithmetic mean of their reciprocals:

H = n ÷ (1/x₁ + 1/x₂ + … + 1/x_n)

Enter the values separated by commas or spaces. All values must be strictly positive - a zero or negative number would make its reciprocal undefined or negative, breaking the formula. The harmonic mean is always ≤ the geometric mean ≤ the arithmetic mean, with equality only when all values are identical.

When to Use the Harmonic Mean

The harmonic mean is the correct average whenever a rate applies over a fixed quantity in the numerator - particularly distance. Classic examples:

Average speed: if you drive the same distance at 30 mph and then 60 mph, your average speed is the harmonic mean: H = 2 ÷ (1/30 + 1/60) = 2 ÷ (3/60) = 40 mph - not the arithmetic mean of 45 mph.

Price-to-earnings ratios: when you invest equal dollars in stocks with different P/E ratios, the harmonic mean gives the effective average P/E of the portfolio.

F1 score: the harmonic mean of precision and recall in machine learning classification tasks.

Why the Arithmetic Mean Gives Wrong Answers for Rates

Suppose you drive 100 km at 50 km/h and another 100 km at 100 km/h. Total distance = 200 km. Total time = 100/50 + 100/100 = 2 + 1 = 3 hours. True average speed = 200/3 ≈ 66.7 km/h.

The arithmetic mean gives (50 + 100)/2 = 75 km/h - an overestimate. The harmonic mean gives 2 ÷ (1/50 + 1/100) = 2 ÷ (0.02 + 0.01) = 2/0.03 ≈ 66.7 km/h - exactly right. The error arises because you spend more time at the lower speed, so it dominates the average more than its arithmetic contribution suggests.

Relationship to Other Means

For any set of positive numbers, the three classical means satisfy: H ≤ G ≤ A, where H is the harmonic mean, G the geometric mean, and A the arithmetic mean. Equality holds if and only if all values are the same. The harmonic mean gives the most weight to small values - a single very small number pulls H far below A and G.

When the values represent rates that apply to equal amounts (equal distances, equal invested dollars), the harmonic mean is correct. When they represent additive quantities or represent rates over equal time, the arithmetic mean is appropriate.

The F1 Score: Harmonic Mean in Machine Learning

In classification tasks, precision = TP/(TP+FP) measures how often a positive prediction is correct, and recall = TP/(TP+FN) measures how many true positives were found. When both matter equally, the F1 score combines them as their harmonic mean: F1 = 2 × precision × recall / (precision + recall).

The harmonic mean is chosen because F1 should be low if either precision or recall is low - a model that catches everything (recall = 1) but flags everything as positive (precision = 0.01) is useless. The harmonic mean penalizes extreme imbalance more harshly than the arithmetic mean would.

Harmonic Mean Calculator

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