The normal distribution is a symmetric, bell-shaped probability distribution defined by two parameters: the mean (μ), which sets the center, and the standard deviation (σ), which sets the width. It's symmetric: the mean, median, and mode are all equal. The tails extend infinitely in both directions but with rapidly decreasing probability density.
The normal distribution describes many real-world phenomena because of the Central Limit Theorem: the average of a large number of independent, identically distributed random variables (regardless of the original distribution) will be approximately normally distributed. This is why measurement errors, biological traits like height and weight, and many economic variables cluster around a bell shape.
The 68-95-99.7 empirical rule
For any normally distributed dataset: 68.27% of values fall within 1 standard deviation of the mean (between μ−σ and μ+σ); 95.45% fall within 2 standard deviations; and 99.73% fall within 3 standard deviations. These are exact figures for a perfect normal distribution. In practice, statisticians round to 68%, 95%, and 99.7%.
A practical example: adult human height in the US for men is approximately normally distributed with mean 69.1 inches and standard deviation 2.9 inches. The 68% rule predicts that 68% of men are between 66.2 and 72.0 inches tall. The 95% rule predicts that 95% are between 63.3 and 74.9 inches. In reality these proportions hold closely, validating the normal model.
Z-scores: measuring distance from the mean
A Z-score converts any value from a normal distribution into a standardized measure of how many standard deviations it is from the mean: Z = (x − μ) / σ. A Z-score of 0 means the value equals the mean. A Z-score of 2.0 means the value is 2 standard deviations above the mean, which the 95% rule tells us is in approximately the top 2.5% of the distribution.
Z-scores let you compare values from different normal distributions. A score of 85 on a test with mean 70 and standard deviation 10 (Z = 1.5) is more impressive than a score of 90 on a test with mean 80 and standard deviation 15 (Z = 0.67), even though 90 is a higher raw score. Standardized tests like the SAT use Z-score-based transformations to produce their scaled scores.