Normal distribution
Also known as: Gaussian distribution, bell curve
The normal distribution is a symmetric, bell-shaped probability distribution defined by its mean and standard deviation. Many natural measurements follow it, and it underlies z-scores, the empirical rule, and much of inferential statistics.
The normal distribution is a continuous probability distribution with a symmetric, bell-shaped curve centered at its mean. Two parameters describe it completely: the mean μ sets the center, and the standard deviation σ sets the spread. The mean, median, and mode all coincide at the peak, and the curve's tails extend indefinitely in both directions without touching zero.
Its most practical feature 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. Any normal value can be standardized with a z-score, z = (x − μ) / σ, which counts how many standard deviations the value sits from the mean. If test scores are normal with mean 500 and standard deviation 100, a score of 650 has z = 1.5, and a z-table (or calculator) converts that to a percentile.
The normal distribution is the workhorse of inferential statistics because of the central limit theorem: the sampling distribution of a sample mean becomes approximately normal as the sample size grows, regardless of the population's shape. That result justifies z-tests, confidence intervals, and countless approximations, and it explains why so many real-world measurements — heights, measurement errors, manufacturing tolerances — look bell-shaped.
The normal distribution appears on a remarkable range of exams. AP Statistics tests z-scores, normal calculations, and sampling distributions; the FE Mechanical exam covers it among common probability distributions and in hypothesis testing; and the GRE quantitative section expects comfort with percentiles and standard deviation on a bell curve.
Key takeaways
- The normal distribution is symmetric and bell-shaped, fully described by its mean and standard deviation.
- The empirical rule: roughly 68%, 95%, and 99.7% of values fall within 1, 2, and 3 standard deviations of the mean.
- Z-scores standardize any normal value: z = (x − μ) / σ.
- The central limit theorem makes sample means approximately normal, which underpins much of inferential statistics.
- AP Statistics, the FE exams, and the GRE all test normal distribution calculations.