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03 // Bidirectional STEM Calculators // MATHHM = 3 / (1/a + 1/b + 1/c)

Harmonic Mean Calculator (3-Rate Average Solver)

The Harmonic Mean is the reciprocal of the arithmetic mean of reciprocals, providing the exact average for rates over equal distances (mph) and valuation multiples (P/E): HM = 3 / (1/a + 1/b + 1/c).

HM = 3 / (1/a + 1/b + 1/c)
Pick which value you want to calculate (or type into any field):Solving for: HM — Harmonic Mean
Calculated

True rate-weighted mean of a, b, and c

0.1100,000.00
unit

First positive rate or multiple

0.1100,000.00
unit

Second positive rate or multiple

0.1100,000.00
unit

Third positive rate or multiple

0.1100,000.00
unit
Calculated Result · Harmonic Mean (HM)
48

Step-by-Step Calculation

01Canonical Law: HM = 3 / (1/a + 1/b + 1/c)
02Given Inputs (Base Units): a = 30, b = 60, c = 80
03Solve for HM (Harmonic Mean): HM = 48

The Harmonic Mean is the reciprocal of the arithmetic mean of reciprocals, providing the exact average for rates over equal distances (mph) and valuation multiples (P/E): HM = 3 / (1/a + 1/b + 1/c).

Real-World Reference Values

Pythagorean Means InequalityHM ≤ GM ≤ AM

For (30, 60, 80): HM = 48.0 < GM = 52.41 < AM = 56.67

Equal-Distance Lap SpeedsAverage Speed = HM of Lap Speeds

Driving 1 mile at 30, 60, and 80 mph averages 48.0 mph

Portfolio P/E MultipleIndex P/E = HM of Constituent P/Es

Prevents high-P/E outliers from distorting index valuation

20-Row Comparison Table

Harmonic Mean (HM) · First Rate / Value (a) (a)

Harmonic Mean Calculator (3-Rate Average Solver) — First Rate / Value (a) → Harmonic Mean
#First Rate / Value (a) (a)Harmonic Mean (HM)Δ %
0130 Current48 0.00%
02725.3572 98.2148 +104.61%
031,450.61 100.4822 +109.34%
042,175.87 101.2615 +110.96%
052,901.13 101.6558 +111.78%
063,626.39 101.8938 +112.28%
074,351.64 102.0531 +112.61%
085,076.90 102.1672 +112.85%
095,802.16 102.2529 +113.03%
106,527.41 102.3197 +113.17%
117,252.67 102.3732 +113.28%
127,977.93 102.417 +113.37%
138,703.19 102.4535 +113.44%
149,428.44 102.4845 +113.51%
1510,153.70 102.511 +113.56%
1610,878.96 102.534 +113.61%
1711,604.21 102.5541 +113.65%
1812,329.47 102.5719 +113.69%
1913,054.73 102.5877 +113.72%
2013,779.99 102.6019 +113.75%
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Frequently Asked Questions

Why is the Harmonic Mean used for average speed over equal distances?+

Because you spend more time traveling the slow segment than the fast segment, weighting by time equals taking the harmonic mean of speeds.

How do I find the third lap speed c needed to achieve an overall average speed of 50 mph?+

Click SOLVE FOR on c, set HM = 50, and enter your first two lap speeds a and b.

How does this relate to parallel resistors in electronics?+

Three resistors in parallel have equivalent resistance R_eq = HM / 3.