Z-score

A z-score converts a raw value into a standardized measure of how far it sits from the mean, expressed in units of standard deviation. A z-score of 2 means the value is two standard deviations above the mean; a z-score of -1 means it is one standard deviation below.

It is calculated as z = (x - mean) / sd. Because it strips away the original units and rescales by spread, z-scores make it possible to compare values from different distributions on a common footing, a test score and a sales figure can both be expressed as z-scores and compared directly. In a normal distribution, a z-score also translates directly into a percentile or a probability via the standard normal table.

Z-scores are used to flag outliers, commonly anything beyond plus or minus 2 or 3, to standardize features before certain statistical or machine-learning models, and to compute significance in tests that assume normally distributed data. The main pitfall is applying z-scores to a distribution that is heavily skewed or has extreme outliers: the mean and standard deviation used in the formula are themselves distorted by those same outliers, so the resulting z-scores can understate how unusual a value really is.