# Normal distribution

The normal distribution is the symmetric bell curve, set by its mean mu and standard deviation sigma. About 68 percent of values lie within 1 sigma of the mean, 95 percent within 2 sigma and 99.7 percent within 3 sigma.

Statistics (university) · Part 04: Continuous distributions and the normal curve · Card 4 of 8 · Last updated 2026-10-19

## Analogy

Heights in a large crowd: most people cluster near the average, and very tall or very short people are rare.

## Example

**The 68-95-99.7 rule**

Heights: mean 170, SD 8
68%: 162 to 178 cm
95%: 154 to 186 cm
99.7%: 146 to 194 cm

## In practice

Measurement errors, heights and exam scores often follow a bell shape, and many statistical tests rely on it.

## Practice

### Flashcards

- **State the 68-95-99.7 rule.** About 68%, 95% and 99.7% of values lie within 1, 2 and 3 SDs of the mean.

## Questions people ask

### Why does the bell curve appear so often?

When many small independent effects add together, the total tends to be bell-shaped, a result called the central limit theorem. Height depends on many genes and many environmental factors, so it looks normal. Not everything is normal, though: incomes and waiting times are skewed.

### What does it mean to write X ~ N(170, 64)?

It says X follows a normal distribution with mean 170 and variance 64. Many texts write the variance, as here, so the standard deviation is 8. Others write N(mean, SD), so always check which convention the source uses before you take a square root or square a number.

## Related concepts

- [Standard normal and z-scores](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-5)
- [Normal probabilities and percentiles](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-6)
- [Normal approximation to the binomial](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-7)
- [Sums of normal variables](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-8)

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