Standardize values to compare positions across different scales.
Overview
A z-score measures how many standard deviations a value lies from the mean: z = (x − μ) / σ. Positive z is above mean; negative z below. Unitless z enables comparing scores from different tests.
Under normal distributions, z maps to percentiles: z=0 is 50th percentile, z≈1.96 often links to 95% two-tailed tails in large samples. Heavy-tailed or skewed data break strict percentile claims without transformation.
Use sample mean x̄ and sample standard deviation s when μ and σ are unknown, recognizing sample z estimates population position with uncertainty, especially for small n.
Z-scores underpin outlier screens, process control charts, and machine learning feature scaling. Always report whether σ is population or sample-based.
Steps
The observation to standardize.
μ or sample mean.
σ or sample s (non-zero).
Subtract mean and divide by SD.
|z|>2 often flagged as unusual in normal models.
Use normal table or calculator if appropriate.
Math
z = (x − μ) / σ
Centers x at zero and scales spread to one standard deviation unit.
Worked cases
(78−70)/4
z = 2.0.
(62−70)/4
z = −2.0.
Higher z ranks higher within each distribution
Math score slightly stronger relatively.
Why use this
Questions