Harmonic Mean Calculator

H = n / Σ(1/xᵢ). Used for averaging rates and ratios.

Inputs

Result

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General calculation reads

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How Not to Be Wrong
Jordan Ellenberg - math thinking
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The Joy of X
Steven Strogatz - accessible math
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Scientific calculator (Casio fx-991EX)
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How to use this calculator

  • Enter positive values, comma-separated.
  • Read all three means.

About this calculator

Harmonic mean: H = n / Σ(1/xᵢ). Always ≤ geometric ≤ arithmetic for positive values (AM-GM-HM inequality). Used for averaging rates: avg speed of round trip (60 mph there + 40 mph back is harmonic mean = 48 mph, NOT arithmetic 50). Also in p/r ratios, density of subscribers, parallel resistance (1/R = 1/R₁ + 1/R₂ + ...).

Frequently asked

When to use harmonic?+
Averaging rates of constant numerator (e.g. distance) where denominator varies (time). Round trip speed.
AM ≥ GM ≥ HM?+
Always for positive values. Equality when all values equal.
Round-trip speed?+
60 mph going + 40 mph back: avg = 2 × 60 × 40 / (60+40) = 48 mph. NOT 50 mph (would be wrong).
Parallel resistance?+
1/R_total = Σ(1/Rᵢ). H of resistances × 1/n = parallel total. Same math.
F1 score (ML)?+
F1 = 2PR/(P+R) is harmonic mean of precision and recall. Penalizes extreme imbalance.

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