How are values converted to the standard normal distribution?

To convert a value from any normal distribution with a mean ($\mu$) and a standard deviation ($\sigma$) to the standard normal distribution—which has a mean of zero and a standard deviation of one—you must rescale the variable using the following formula:

$$z=\frac{x-\mu }{\sigma }$$

In this equation, $x$ represents the original data value, $\mu$ is the population mean, and $\sigma$ is the population standard deviation. The resulting value, often called a z-score, indicates the relative position of the data point by representing the number of standard deviations it is located from the mean.123

Once converted, these z-scores allow for the use of standard statistical tables to compute probabilities for any normal distribution. In descriptive statistics, this same calculation is frequently used to identify outliers, where values with a z-score less than $-3$ or greater than $+3$ are typically considered unusual.132