The normal distribution

Heights, measurement errors, test scores, the sum of many small random effects — they all pile up into the same shape: the normal distribution, the bell curve. It's the most important distribution in all of statistics.

The 68–95–99.7 rule

A normal distribution is defined by just two numbers — its mean μ (center) and standard deviation σ (spread) — and it always obeys the same rule:

• about 68% of values fall within 1σ of the mean
• about 95% within 2σ
• about 99.7% within 3σ

This holds for any normal distribution, whatever μ and σ are. So a value more than 2σ from the mean is genuinely unusual — it happens only about 5% of the time.

The z-score: a universal ruler

To compare values from different normal distributions, convert each to a z-score — how many standard deviations it sits from its mean:

$z = \frac{x - \mu}{\sigma}$

A z-score of +1.5 means "1.5 standard deviations above average," whether we're talking about heights or test scores. It puts everything on one common scale, which is what makes percentiles and comparisons possible.

Why the bell is everywhere

The normal curve isn't a coincidence. Whenever an outcome is the sum of many small, independent influences — dozens of genes plus diet shaping height, many tiny errors in a measurement — those influences average out into a bell. That deep fact (the Central Limit Theorem) is the next lesson.

Why this matters

The normal distribution and the z-score are the shared language of quality control, standardized testing, finance, and every "how unusual is this?" question. Master the 68–95–99.7 rule and you can eyeball probabilities for a huge slice of the real world without a single heavy calculation.