The Normal Distribution

The normal distribution — the bell curve — is the most important shape in statistics: symmetric around the mean, thin in the tails, and fully described by just two numbers, mean μ and standard deviation σ.

The 68–95–99.7 rule

Those three numbers are the fast mental version of the z table: the table simply answers the same question for any boundary, not just whole standard deviations.

Why it appears everywhere

The Central Limit Theorem: when many small independent effects add up — measurement errors, biological variation, averages of samples — their sum tends toward the normal shape regardless of each effect’s own distribution. That is why heights, lab errors and sample means look bell-shaped, and why so much of statistics is built on normal assumptions.

The standard normal

Setting μ = 0 and σ = 1 gives the standard normal distribution — the reference version every other normal converts into via z-scores. One table serves every normal problem because every normal problem can be standardized first.

When it’s the wrong model

Incomes, house prices and waiting times are typically skewed — using normal logic on them misleads. Check the shape before trusting the bell.