Standard Normal Distribution

A normal distribution $X \sim N(\mu, \sigma^2)$ is fully described by just two parameters — its mean $\mu$ and standard deviation $\sigma$. In practice, though, different data sets rarely share the same mean and standard deviation: one set might have a mean of $50$ and a standard deviation of $3$, while another has a mean of $100$ and a standard deviation of $5$. Comparing values across such differently-scaled normal distributions directly isn't meaningful, since a given value means something different in each. What is needed is a common scale on which any normal variate can b …