Normal distribution

The normal distribution is a symmetric, bell-shaped curve where values cluster around a central mean and taper off equally in both directions. It is fully described by two numbers: the mean, which sets its center, and the standard deviation, which sets its width.

Its defining property is the empirical rule: about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. Converting any value to a z-score, how many standard deviations it sits from the mean, makes it possible to read off its position in a standard normal distribution regardless of the original units.

The normal distribution matters because so much of classical statistics, including the t-test, confidence intervals, and many forms of hypothesis testing, assumes data or sample means are approximately normal. That assumption is often justified by the central limit theorem, which shows that the average of many independent samples tends toward a normal shape even when the underlying data is not. The common pitfall is assuming raw data is normally distributed without checking: many real-world measures, like income or time between failures, are skewed or heavy-tailed, and applying normal-distribution methods to them without adjustment can produce misleading results.