Harmonic Mean Calculator – Speeds & Fuel

The harmonic mean is built for rates that alternate over equal stretches. If a car does one lap at 30 mpg and the next at 20 mpg over the same distance, the combined fuel economy is H = 2/(1/30 + 1/20) = 2/(0.08333) = 24 mpg — not the arithmetic 25, because the thirsty lap consumes more fuel per mile. The calculator prints each reciprocal (1/30 ≈ 0.0333, 1/20 = 0.05), sums them, divides n by the total, and shows the arithmetic mean beside it for contrast. Data validation blocks zeros and negatives since 1/x must exist, and the result always lands between the smallest and largest inputs, skewed toward the low end — one slow leg dominates blended averages. Multiperiod examples like 60, 40 and 30 mph over equal legs produce H = 3/(1/60+1/40+1/30) = 40 mph, well below the arithmetic 43.3.

1. 1

   Enter your rates

   Two or more positive values — speeds, mpg, prices per unit

2. 2

   Invert each one

   Every 1/x is computed and displayed

3. 3

   Average the reciprocals

   Sum them, divide by n, then flip the result

4. 4

   Compare with the arithmetic mean

   The gap explains why blending matters

• 1

  Equal distances or amounts → harmonic; equal times → arithmetic

• 2

  H is pinned near the minimum: one slow rate drags the blend

• 3

  Never average a rate over its own denominator arithmetically

• Averaging 30 and 20 mpg to 25; the true blend is 24

• Including a zero rate, which makes 1/x undefined

• Using the harmonic mean for amounts (distances, prices totals) instead of rates

• Forgetting the final reciprocal — leaving the answer as Σ(1/x)/n