Standard Normal Distribution vs Normal Distribution

The standard normal distribution is a normal curve with mean $\mu = 0$ (mu) and standard deviation $\sigma = 1$ (sigma). Any normal value converts to it through the z-score $z = \frac{x - \mu}{\sigma}$, which rescales the data onto this common curve. For example, a value 2 standard deviations above its mean maps to $z = 2$ on the standard normal. A z-table or calculator then gives the area, and therefore the probability, to the left of that z-score.

The normal distribution is a bell-shaped curve centered at its mean $\mu$ (mu) and spread out by its standard deviation $\sigma$ (sigma). Many natural measurements, such as heights or measurement errors, are approximately normal. For example, adult heights cluster near an average, with fewer people far above or below it. The empirical rule says about 68, 95, and 99.7 percent of the data fall within 1, 2, and 3 standard deviations of the mean.