The Normal Distribution
The normal distribution, also called the Gaussian, is the famous symmetric bell curve. It is defined by two parameters: the mean that sets its center and the standard deviation that sets its width.
The empirical rule
For a normal distribution, the spread follows a tidy pattern often called the 68 95 99 rule.
- About 68 percent of values fall within one standard deviation of the mean.
- About 95 percent fall within two standard deviations.
- About 99.7 percent fall within three standard deviations.
This lets you judge how unusual a value is just from its distance to the mean.
Standardizing with z scores
A z score rescales any normal value by subtracting the mean and dividing by the standard deviation. The result is a standard normal with mean zero and standard deviation one, making different normals directly comparable.
Why it dominates
The normal arises naturally when many small independent effects add together, a fact guaranteed by the central limit theorem. It is also mathematically convenient, so models from linear regression to many noise assumptions lean on it.
Key idea
The normal distribution is a symmetric bell defined by mean and standard deviation, with about 68, 95, and 99.7 percent of data within one, two, and three deviations.