What is a Z-score?
A Z-score (also called a standard score) tells you how many standard deviations a particular value is from the mean of its distribution. The formula is simple: z = (x − μ) / σ, where x is the observed value, μ is the population mean, and σ is the standard deviation. A positive Z-score means the value is above the mean; a negative one means it is below. A Z-score of zero means the value equals the mean exactly.
How to use this calculator
Enter the raw value (x), the mean (μ), and the standard deviation (σ) in the input fields. Click Calculate Z-score to see the result. The tool returns the Z-score, the corresponding cumulative probability, the percentile, and a plain-language interpretation telling you whether the value sits above or below the mean and what percentage of values fall beneath it.
Why Z-scores matter
Z-scores let you compare values from different distributions on the same scale. A student who scores 85 on a test with a mean of 70 and a standard deviation of 10 has a Z-score of 1.5—performing better than roughly 93 % of test-takers. The same logic applies in finance (measuring how far a stock return deviates from its historical average), quality control (detecting out-of-spec measurements), and medical research (flagging abnormal lab results).
Interpreting the result
Most data in a normal distribution falls within three standard deviations of the mean. About 68 % of values have |z| < 1, about 95 % have |z| < 2, and about 99.7 % have |z| < 3. Values beyond ±3 are rare and often considered outliers. The cumulative probability shown by this calculator tells you exactly what fraction of the distribution lies to the left of the given value.
Tips
Make sure your standard deviation is greater than zero—the Z-score is undefined when σ = 0. If you are working with sample data rather than a full population, use the sample standard deviation (s) in place of σ. This free calculator runs entirely in your browser with no sign-up, no download, and instant results on any device. Bookmark the page for quick access whenever you need Z-score calculations.