Math · Statistics
Z-score calculator.
How many standard deviations is a value from the mean, and what percentile is that? Enter the value, mean and standard deviation. Everything runs in your browser.
A score of 130 on a test with mean 100 and standard deviation 15 has a z-score of (130 − 100) / 15 = 2.00. On a normal distribution that is the 97.72nd percentile: 97.72% of values fall below it and 2.28% above.
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The formula
A z-score (standard score) is the number of standard deviations a value is above or below the mean: z = (x − μ) / σ. Positive means above the mean and negative means below. For the mean of a sample of n values the denominator is the standard error σ / √n.
The percentile comes from the standard normal cumulative distribution function Φ(z), the area under the bell curve to the left of z. It is exact only if the data are normally distributed; for skewed data the z-score still measures distance from the mean in standard deviations, but the percentile is only an approximation.
Worked example
x = 130, μ = 100, σ = 15: z = 30 / 15 = 2. Φ(2) = 0.97725, so the percentile is 97.72, with 2.28% of values above. The probability of a value at least 2 standard deviations from the mean in either direction is 4.55%. For x = 85 the z-score is −1 and the percentile is 15.87. Landmarks: Φ(1.960) = 0.975, so 95% of a normal distribution lies within 1.96 standard deviations of the mean, and 68.27% lies within one.
Limits
Computing a z-score assumes you know the population mean and standard deviation. If you only have sample estimates, t-based methods are used instead, for example for a sample mean, which matters for small samples. Percentiles assume a normal distribution.
Sources, opened 8 October 2026: Wikipedia, Standard score (definition z = (x − μ) / σ, positive above and negative below the mean, and the sample-based estimate z = (x − x̄) / s); NIST/SEMATECH e-Handbook, 1.3.6.7.1 Cumulative Distribution Function of the Standard Normal Distribution (cumulative probabilities including Φ(1.960) = 0.975).
Frequently asked questions
How do you calculate a z-score?
Subtract the mean from the value and divide by the standard deviation: z = (x - mean) / sd. For x = 130, mean 100 and sd 15 that is 30 / 15 = 2.
How do I convert a z-score to a percentile?
Look up the area under the standard normal curve to the left of z, or let this calculator do it. A z-score of 2 is the 97.72nd percentile and a z-score of -1 is the 15.87th.
What does a z-score of 1.96 mean?
In a normal distribution about 95% of values lie within 1.96 standard deviations of the mean, so 1.96 is the usual cut-off for a two-sided 95% interval. The cumulative probability at 1.96 is 0.975.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean. Its percentile is below 50.
What is a z-score for a sample mean?
Enter the sample size and the calculator divides the standard deviation by the square root of n. This gives the z-score of a sample mean when the population standard deviation is known.
Does this work for data that is not normal?
The z-score itself always measures distance from the mean in standard deviations. The percentile it reports is only accurate if the data are roughly normal.
Planning and learning aid
Educational calculator for normally distributed data with a known mean and standard deviation.