Arithmetic, Geometric and Harmonic Means: AM-GM-HM Guide

The arithmetic mean (AM) is the familiar average: the sum of all values divided by how many there are. The geometric mean (GM) is the n-th root of the product of n values, and the harmonic mean (HM) is the number of values divided by the sum of their reciprocals. Each "average" answers a different question.

It matters because using the wrong mean gives wrong answers: averaging speeds over equal distances with the AM is incorrect, and averaging growth rates with the AM overstates the result. Matching the mean to the situation is one of the most practical skills in statistics.

Key Concepts

Arithmetic mean (x̄): Σx/n — appropriate for additive quantities like marks, heights, temperatures. Geometric mean: (x₁·x₂·…·xₙ)^(1/n) — appropriate for multiplicative growth like investment returns and population growth; all values must be positive. Harmonic mean: n/(Σ(1/x)) — appropriate for rates over fixed quantities, like average speed over equal distances. Ordering: for positive unequal values, AM > GM > HM always; the three are equal only when all values are equal.

Formulas

AM: x̄ = (x₁ + x₂ + … + xₙ)/n. Weighted AM: x̄ = Σ(wᵢxᵢ)/Σwᵢ, used with frequency weights. GM: G = (Πxᵢ)^(1/n) = antilog(Σlog xᵢ / n). GM of two numbers: G = √(ab). HM: H = n/Σ(1/xᵢ). HM of two numbers: H = 2ab/(a+b). Relation for two numbers: G² = A·H, i.e. GM² = AM × HM. Units match the original data (e.g. km/h for speeds).

Worked Example

A car travels 120 km at 40 km/h and returns the same 120 km at 60 km/h. Wrong approach: AM of speeds = (40 + 60)/2 = 50 km/h. Correct approach (HM): total distance = 240 km; time going = 120/40 = 3 h; time returning = 120/60 = 2 h; total time = 5 h; average speed = 240/5 = 48 km/h. Using the formula: H = 2/(1/40 + 1/60) = 2/(0.025 + 0.0167) = 2/0.0417 ≈ 48 km/h. The AM overstates the answer because more time is spent at the slower speed. GM check: for 40 and 60: G = √(40×60) = √2400 ≈ 48.99, sitting between HM (48) and AM (50), as expected.

Common Misconceptions

1. "The AM is always the right average." It fails for rates over equal distances (use HM) and for compound growth (use GM). 2. "GM and HM can use zero or negative values." Both need positive values — the HM divides by x and the GM needs a real n-th root of the product. 3. "AM ≥ GM ≥ HM is only approximate." For positive data it is a strict inequality unless all values are identical.

Real-World Applications

AM is used for average marks, average temperature and average salary. GM is used for average annual investment growth (CAGR), population growth rates and index numbers. HM is used for average speed over equal distances, average price when buying fixed quantities, and parallel resistance calculations in electronics.

AM suits additive data, GM suits multiplicative growth, and HM suits rates over fixed amounts, with AM > GM > HM for unequal positive values. Choosing the mean that matches the situation — not defaulting to the arithmetic mean — is what makes the average meaningful.