Z-Score vs Standard Deviation

A z-score standardizes a value by measuring its distance from the mean in standard deviations. You compute it as $z = \frac{x - \mu}{\sigma}$, where $x$ is the value, $\mu$ (mu) is the mean, and $\sigma$ (sigma) is the standard deviation. For example, a score of 85 in a distribution with mean 70 and standard deviation 5 has $z = (85 - 70)/5 = 3$, so it sits 3 standard deviations above the mean. A negative z-score means the value is below the mean.

The standard deviation summarizes spread as the typical gap between a value and the mean. The sample standard deviation is $s = \sqrt{\frac{1}{n-1}\sum (x_i - \bar{x})^2}$, where $x_i$ are the values, $\bar{x}$ (x-bar) is the mean, and $n$ is the number of values. For example, a small $s$ means the data cluster tightly around the mean, while a large $s$ means they spread out widely. It is the square root of the variance, which returns the measure to the original units.