Z-Score Calculator

Z-score: standardized value

The z-score normalizes a value by expressing how many standard deviations it lies above or below the mean: z = (x − μ) / σ. Example: in a test with mean 70 and standard deviation 10, a grade of 80 gives z = 1 — one standard deviation above average. It lets you compare quantities measured on different scales (grades, salaries, biometrics). For the standard normal distribution: z = 0 is the mean (50th percentile), z = 1 covers about 84.1%, z = 2 about 97.7%, z = 3 about 99.87% — the empirical 68–95–99.7 rule.

Applications

Outlier detection (|z| > 3 is the usual rule of thumb), feature standardization for machine learning (StandardScaler), statistical process control (Six Sigma), psychometric tests (IQ standardized to μ = 100, σ = 15), the ENEM TRI scale (mean 500, σ ≈ 100), and the z-test in hypothesis testing.

FAQ

Can z-score be negative? Yes — a negative z means the value is below the mean. z = −1.5 means 1.5 standard deviations below average.

Z-score vs percentile — what's the relationship? Under a normal distribution, each z maps to a fixed percentile (z = 1 → ~84th percentile). The mapping only holds if the data is approximately normal.

When does z fail as an outlier criterion? In small or heavy-tailed samples, μ and σ are themselves distorted by the outlier. Robust alternatives include the modified z-score (using the median and MAD) and the IQR rule.