The normal distribution is a symmetric bell-shaped density defined by its mean and standard deviation, with predictable probabilities given by the empirical rule.

What this dot point is asking

The normal distribution is the most important continuous distribution in statistics. Many natural and measured quantities - heights, exam scores, measurement errors - follow it closely, and it is the limiting shape of the binomial distribution for large $n$.

Properties of the bell curve

The notation $X\sim N(\mu,\sigma^2)$ means $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$ (so standard deviation $\sigma$).

The empirical rule (68-95-99.7)

For any normal distribution, fixed proportions of the data fall within whole numbers of standard deviations of the mean:

Because the curve is symmetric, each tail beyond $\pm 1\sigma$ holds about $16\%$, beyond $\pm 2\sigma$ about $2.5\%$, and beyond $\pm 3\sigma$ about $0.15\%$.

Common errors

Why it matters

The normal distribution underpins the rest of Topic 5 (z-scores and exact probabilities) and all of Topic 6 (sampling distributions and confidence intervals), where sample means are approximately normal. Recognising when data is normal and using the empirical rule for quick estimates is a core Stage 2 skill.