Z-Scores

A rigorous guide to standardization, sign and scale interpretation, normal probabilities, and comparison limits.

Standardization measures distance from the mean in standard deviations. A bell curve is divided at the mean and at one and two standard deviations on both sides. The standardized horizontal labels show how raw distance from the mean is expressed in units of sigma.

Precise definition

For a value $x$ in a distribution with mean $\mu$ and positive standard deviation $\sigma$, its z-score is $z=(x-\mu)/\sigma$. It reports how many standard deviations the value lies above or below the mean. Standardizing a random variable gives mean 0 and standard deviation 1 when those moments exist.

Notation and mathematical language

For sample data, $z=(x-\bar x)/s$ uses sample summaries and describes position within that sample. A z-score is dimensionless. If the original distribution is normal, $Z$ has the standard normal distribution and table probabilities apply exactly under the model.

Conceptual picture

Subtracting centres the value; dividing by standard deviation sets a common spread unit. This permits relative comparison across scales, but equal z-scores mean equal standardized position, not equal raw performance or equal percentile unless distribution shapes align.

Fully worked example

Interpretation and application

Z-scores standardize exams, measurements, anomaly screens, and model residuals. Comparing standardized scores across groups assumes the chosen reference groups and spreads are substantively appropriate; it does not create causal or fairness equivalence.