Z-Score vs Percentile

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 kth percentile is the value at or below which about $k$ percent of the observations lie. For example, if your test score is at the 85th percentile, roughly 85 percent of test-takers scored at or below you. Quartiles are percentiles too: the first quartile is the 25th percentile and the median is the 50th. Percentiles let you compare a single value to a whole distribution without knowing its shape.