Harmonic Mean Calculator

Definition and formula

For positive values x₁, x₂, …, xₙ, the harmonic mean H is:

H = n / (1/x₁ + 1/x₂ + ... + 1/xₙ)

The formula divides the number of values by the sum of their reciprocals. It agrees with the definition published in the NIST Dataplot reference and the equal-weight form documented by SciPy.

This calculator accepts only strictly positive values. That domain keeps every reciprocal defined and matches the positive-rate use cases and the comparison of means described below.

Worked example: equal distances

Suppose a vehicle travels the same distance at 60 km/h and 30 km/h. The arithmetic mean of the two speeds is 45 km/h, but it is not the average speed for the whole trip because the slower leg takes more time.

H = 2 / (1/60 + 1/30)
  = 2 / 0.05
  = 40 km/h

The average speed over the two equal-distance legs is 40 km/h. If the vehicle instead spent the same amount of time at each speed, the arithmetic mean of 45 km/h would be appropriate.

Harmonic, geometric and arithmetic means

For the same set of strictly positive values, the harmonic mean is no greater than the geometric mean, and the geometric mean is no greater than the arithmetic mean. Equality occurs when all values are the same.

The calculator displays the harmonic and arithmetic means. Use the related geometric mean calculator if you also need the middle value.

When the harmonic mean is appropriate

• Speeds over equal distances: each speed applies to the same distance.
• Prices with equal spending: each purchase uses the same amount of money, so the average cost per unit is harmonic.
• Comparable rates with a common numerator: the reciprocal formulation reflects the shared exposure.
• Precision and recall: the F1 score is a specialized harmonic mean of precision and recall, as documented by scikit-learn. Classification libraries define their own behavior when either quantity is zero; this general calculator instead requires positive inputs.

Do not choose the harmonic mean merely because the numbers are called rates. If observations have different distances, spending amounts or exposures, an unweighted harmonic mean can be misleading. Use an appropriately weighted calculation instead.

Input rules and numerical behavior

• The data set can contain at most 500 values and 20,000 characters.
• ASCII digits and the native digits used by the supported locales are accepted.
• The active locale’s decimal mark is accepted. In a comma-decimal locale, write values such as 1,5; 2,5; 5 or put each value on a new line.
• Grouping separators and scientific notation are not accepted because they can be ambiguous in a list.
• Every token must be a finite number greater than zero. One invalid token stops the calculation and removes any stale result.
• The computation scales both reciprocal and arithmetic sums before calculating the final means. This reduces avoidable floating-point overflow and underflow for very large or very small positive values.
• Results use the active locale’s digits and separators. Extremely small or large values may use localized scientific notation.

Privacy and funnel storage

The calculation runs in this browser. Values and results are not bound to our servers and are not sent to the server. On the single-page calculator, the values stay in page memory.

Funnel mode stores the data set only in this tab’s session storage for up to 30 minutes so it can move between three pages. The fixed expiry is not extended when you continue. At most three current harmonic-mean jobs are retained, and only a random 32-character job identifier appears in the URL. Start over removes the current local record after confirmation. If tab storage is blocked, expired or damaged, the tool shows a recovery state rather than inventing a result.