Harmonic Mean Calculator

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

Inputs

Result

Loading calculator…
—

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.

Related calculators

Weighted Average Calculator
WA = Σ(value × weight) / Σ(weight). Generalizes the arithmetic mean.
Arithmetic Sequence Calculator
aₙ = a₁ + (n−1)d. Find any term of an arithmetic sequence.
Geometric Sequence Calculator
aₙ = a₁ × r^(n−1). Find any term of a geometric sequence.
Sum of Arithmetic Series
Sₙ = n × (a₁ + aₙ) / 2 = n/2 × (2a₁ + (n−1)d).
Sum of Geometric Series
Sₙ = a₁(1 − rⁿ) / (1 − r) for r ≠ 1. Infinite: S∞ = a₁/(1−r) if |r|<1.
Fibonacci Number Calculator
F(n) = F(n−1) + F(n−2). 0, 1, 1, 2, 3, 5, 8, 13, 21, ...

More tools you might like

Hand-picked tools that pair well with this one — same audience, same intent.

Arithmetic Sequence Calculator
Calc

aₙ = a₁ + (n−1)d. Find any term of an arithmetic sequence.

Geometric Sequence Calculator
Calc

aₙ = a₁ × r^(n−1). Find any term of a geometric sequence.

Sum of Arithmetic Series
Calc

Sₙ = n × (a₁ + aₙ) / 2 = n/2 × (2a₁ + (n−1)d).

Sum of Geometric Series
Calc

Sₙ = a₁(1 − rⁿ) / (1 − r) for r ≠ 1. Infinite: S∞ = a₁/(1−r) if |r|<1.

Fibonacci Number Calculator
Calc

F(n) = F(n−1) + F(n−2). 0, 1, 1, 2, 3, 5, 8, 13, 21, ...

Capital Gains Tax Calculator
Finance

Short-term vs long-term capital gains tax — federal rates only.