Definition
A normal distribution (also called a Gaussian distribution or bell curve) is a probability distribution that is symmetric around the mean. In a normal distribution, most values cluster near the center, with fewer and fewer values appearing as you move further away in either direction.
The 68-95-99.7 Rule
The normal distribution follows a predictable pattern known as the empirical rule.
Adult heights in a population have a mean of 170 cm and a standard deviation of 7 cm.
68% of people are between 163 cm and 177 cm (within 1 SD)
95% of people are between 156 cm and 184 cm (within 2 SDs)
99.7% of people are between 149 cm and 191 cm (within 3 SDs)
Someone who is 195 cm tall is more than 3 standard deviations above the mean - very unusual.
Why It Matters
The normal distribution is the foundation of modern statistics. Most hypothesis tests, confidence intervals, and prediction methods assume normality. Thanks to the central limit theorem, sample means follow a normal distribution even when the raw data does not, which is why these methods work so broadly.
In quality control, the normal distribution helps set acceptable tolerances. In education, it helps interpret standardized test scores. Understanding this distribution helps you determine what is typical and what is unusual in almost any dataset.
The normal distribution is the most important probability distribution in statistics. Its bell shape and the 68-95-99.7 rule make it a powerful tool for understanding data.