Geometric Mean Quick Framework
Use this when data compounds (growth %, ratios, indices). Avoid when additive effects dominate.
1. Definition Core
- GM = (∏ xᵢ)^{1/n} for n positive numbers.
- Log form (stable): GM = exp( (1/n) Σ ln xᵢ ).
- Requires xᵢ > 0 (zero → GM = 0 only if any term = 0; negatives invalid for real result).
2. When It Beats Arithmetic Mean
- Sequential percentage returns (investment, growth).
- Rates that multiply (population, chemical kinetics scaling).
- Normalization of ratios / indices.
- Skewed distributions—reduces influence of large outliers.
3. Decision Flow
- All inputs positive? If no → stop (or clean dataset).
- Is the process multiplicative / compounding? If yes → GM.
- Need central tendency for uniform additive errors? Use arithmetic mean instead.
4. Comparisons
- Always: Harmonic ≤ Geometric ≤ Arithmetic (for positive unequal numbers).
- Equality only when all numbers identical.
5. Stability Tips
- Use log form for large n or wide magnitude spread.
- Filter zeros: treat as data error unless representing true extinction (then GM=0 and interpretation changes).
- Convert percentages: +12% → 1.12, -5% → 0.95 before multiplication.
6. Quick Patterns
- Two numbers: GM = √(ab).
- Scaled set: Multiply every xᵢ by k → GM scales by k.
- If numbers form geometric progression a, ar, ar²,... → GM = a·r^{(n-1)/2}.
7. Sanity Checks
- GM ≤ arithmetic mean; if larger, inputs mishandled.
- Monotonic: adding a value greater than current GM increases GM.
- Dimensional consistency: all inputs must share same unit or be ratios.
8. Mini Example
Values: 2, 8, 4, 16 → product 1024; 4th root = 5.6569. AM = 7.5 (higher as expected). HM ≈ 4.266 (lower).
9. Pitfalls
- Including negatives (invalid) or unconverted percentages.
- Using GM for sums (e.g., average test scores not multiplicative).
- Rounding intermediate log sums too early.
10. Action Tip
Convert everything to growth factors first; then apply log-sum method for maximum numerical stability.