The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. It’s particularly useful for averaging rates and speeds.

When to Use Harmonic Mean

  • Average speeds: Traveling at different speeds for equal distances
  • Average rates: Different rates applied to equal bases
  • Ratios: When dealing with reciprocal quantities
  • Harmonic sequences: Data with harmonic relationships

Key principle: Use harmonic mean when the denominator (not numerator) should be averaged.

Harmonic Mean for Ungrouped Data

Formula:

HM = n / (1/x₁ + 1/x₂ + ... + 1/xₙ)
HM = n / Σ(1/xᵢ)

Example - Average Speed: You drive 100 miles at 60 mph, then 100 miles at 40 mph.

Incorrect (arithmetic mean): (60 + 40) / 2 = 50 mph Correct (harmonic mean): HM = 2 / (1/60 + 1/40) HM = 2 / (0.0167 + 0.025) HM = 2 / 0.0417 HM ≈ 48 mph

(Arithmetic mean overestimates because you spent more time at 40 mph)

Harmonic Mean for Grouped Data

Formula:

HM = Σfᵢ / Σ(fᵢ/xᵢ)

where:
fᵢ = frequency
xᵢ = class midpoint

Mean Hierarchy for Positive Values

For positive numbers:

Harmonic Mean ≤ Geometric Mean ≤ Arithmetic Mean

Example with values 2, 8:

  • Harmonic Mean = 3.2
  • Geometric Mean = 4
  • Arithmetic Mean = 5

When Not to Use

  • ❌ With zero values (undefined reciprocal)
  • ❌ With negative values (inconsistent meaning)
  • ❌ When arithmetic mean is more conceptually appropriate
  • ❌ For general “average” situations (use arithmetic mean)