Z-Score Calculator
Compute the Z-score (standard score) of a value given population mean and standard deviation. Includes corresponding percentile. Everything in your browser.
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.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.