Formula
z = (x − μ) ÷ σPercentile = Φ(z), the standard normal CDF How it works
A z-score tells you how many standard deviations a value is from the mean. A z-score of 0 is exactly average, +1.5 is one and a half standard deviations above, and −2 is two standard deviations below. Because z-scores strip away the original units, they let you compare results on different scales, such as an SAT score and an ACT score.
If the data follow a normal distribution, a z-score maps directly to a percentile using the standard normal table. A z of 1.5 corresponds to about the 93rd percentile, meaning the value beats roughly 93% of the population.
Values with |z| greater than about 2 are unusual (only around 5% of normal data are that far out), and |z| above 3 is rare enough that such values are often flagged as possible outliers.
Frequently asked questions
Can a z-score be negative?
Yes. A negative z-score just means the value is below the mean. z = −1 is one standard deviation below average, around the 16th percentile.
Should I use the sample or population standard deviation?
Use the population σ if you know it (for example, published test norms). If you only have a sample, the sample standard deviation s is the usual stand-in.