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Math • Harmonic Mean Calculator

Harmonic Mean Calculator

Calculate the harmonic mean of a list of positive numbers — the appropriate average for rates like speed.

Arithmetic Mean (AM)
15.000
Geometric Mean (GM)
11.314
Harmonic Mean (HM)
8.533
Root Mean Square (RMS)
18.439
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Calculation Guide & Reference

Harmonic Mean Mathematics

Harmonic mean calculator computes the harmonic mean of a list of positive numbers, the appropriate average for rates like speed.

Standardized Mathematical Formula
HM = n / (1/x₁ + 1/x₂ + ... + 1/xₙ)

The harmonic mean is the number of values divided by the sum of their reciprocals — it's the correct average for rates, like average speed over equal distances.

Variables:
x₁...xₙ:The list of positive values
n:Count of values
How It Works (Step-by-Step)
  • 1Enter a list of positive numbers, separated by commas.
  • 2View the harmonic mean, alongside the arithmetic, geometric, and RMS means for comparison.
Real-World Numerical Example
Harmonic Mean of 4, 8, 16, 32

Find the harmonic mean of the numbers 4, 8, 16, and 32.

•Sum of reciprocals: 1/4 + 1/8 + 1/16 + 1/32 = 0.46875.
•Harmonic mean: 4 / 0.46875 = 8.533.
Result: The harmonic mean is 8.533 — the lowest of the three Pythagorean means for this data set.
Calculation Best Practices & Tips
Examine the discriminant (b² - 4ac) before solving quadratics to instantly know if roots are real, distinct, repeated, or complex.
In statistical analysis, always check both the mean and median to identify if data distribution is skewed by outliers.
When calculating percentage changes, ensure you divide by the original starting baseline value: ((New - Old) / Old) * 100.

Frequently Asked Questions (FAQ)

Use harmonic mean for rates measured over a fixed quantity of something else, like speed over a fixed distance or price-to-earnings ratios — a regular (arithmetic) average overstates the true rate in these cases.

For any set of positive numbers, harmonic mean ≤ geometric mean ≤ arithmetic mean — this ordering always holds, and all three are equal only when every value in the set is identical.

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