# Normal Distribution

**Type:** Concept  
**Domain:** Probability & Statistics  
**Codex URL:** /mathematics/probability-statistics/normal-distribution/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
The symmetric bell curve, arising from the Central Limit Theorem. Also called Gaussian distribution.

## Formula
f(x) = (1/σ√(2π))·e^(−(x−μ)²/2σ²)

## 68-95-99.7 rule
68% within 1σ, 95% within 2σ, 99.7% within 3σ of the mean.

## Central Limit Theorem
Sum/average of many independent random variables approaches normal distribution, regardless of their individual original distributions.

## Historical development
de Moivre (1733) — binomial approximation. Laplace — extended. Gauss (1809) — independently, astronomical measurement errors (hence "Gaussian distribution").

## Not everything is normal
Income, city populations, word frequencies: typically heavy-tailed (log-normal, power-law), not normal. Checking assumptions matters.

## Historical misapplication note
Normal-distribution reasoning was misapplied in early 20th-century eugenics — a scientifically and ethically discredited application, now a cautionary historical example.

## Sources
### Tier 1
- Feller, W. (1968/1971). *An Introduction to Probability Theory*. Vols 1-2.
- Billingsley, P. (1995). *Probability and Measure*. 3rd ed.
### Tier 2
- Stigler, S.M. (1986). *The History of Statistics*. Harvard University Press.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*