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Z-Score Calculator
Find how many population standard deviations a value is above or below the mean.
About the Z-Score Calculator
A z-score (standard score or z value) expresses the distance of one value from a population mean in standard deviation units. It is dimensionless. Use a mean and standard deviation describing the same population and measurement as the raw score.
How to use it
Enter a raw score, population mean and positive population standard deviation. Negative scores and means are allowed. Calculate to see the signed standard score and substitution into the formula.
Formula / Explanation
z = (X − μ) / σ. A positive result is above the mean, a negative result is below it, and zero is exactly at the mean. The tool uses the supplied statistics; it does not estimate them from a dataset.
Example
With X = 85, μ = 75 and σ = 10, z = (85 − 75) / 10 = 1: one standard deviation above the mean. With X = 60 and the same population statistics, z = −1.5. For X = −2.5, μ = −1 and σ = 0.5, z = −3.
Does a z-score give a percentile or probability?
Not by itself. Converting a score to a standard normal percentile requires a normal-distribution assumption or another justified model. This tool does not calculate probabilities, p-values or statistical significance.
Is a score beyond ±2 always unusual?
No universal label applies to every distribution. A distance of two standard deviations describes scale, but how common it is depends on the distribution. Do not use a fixed cutoff as a diagnosis or automatic outlier rule.
Why must the standard deviation be positive?
Standard deviation measures spread and cannot be negative. With zero spread the formula divides by zero, so a z-score is undefined.
Can I use a sample standard deviation?
This form asks for population statistics. A sample-based standardized value can be computed arithmetically, but uncertainty, sample estimates and statistical tests need appropriate methods; this is not a t-test calculator.
How are values displayed?
Results show up to 10 significant digits; very small or large values use scientific notation. Calculations use JavaScript floating-point arithmetic.
Where can I read about standardization?
See the NIST standardization reference. Use the Scientific Calculator for other arithmetic.
Basic Calculator
Perform basic math operations like addition, subtraction, multiplication, and division. Includes memory functions and square root support.
How to Use
Enter a raw score, population mean and positive population standard deviation. Negative scores and means are allowed. Calculate to see the signed standard score and substitution into the formula.