How this calculator works
A z-score (or standard score) says how many standard deviations a value is above or below the mean of its data set. A positive z-score means the value is above average, a negative one means below average, and the size of the number shows how unusual the value is.
This calculator converts a z-score into a percentile using the standard normal cumulative distribution function, and also reports the two-tail p-value, the probability of seeing a value at least that far from the mean in either direction. These are the building blocks behind confidence intervals and statistical significance testing.
How do you calculate a z-score?
Z-score: z = (x − μ) ÷ σPercentile: Φ(z) × 100, where Φ is the standard normal cumulative distribution functionTwo-tail p-value: 2 × (1 − Φ(|z|))Φ(z) is computed here with the Abramowitz and Stegun polynomial approximation of the error function (erf), accurate to about 1.5 × 10⁻⁷ — far beyond what's needed for percentile and p-value reporting.
Worked example: a test score of 68
- A test has a mean score of 65 and a standard deviation of 3. A student scores 68.
- Z-score: (68 − 65) ÷ 3 = 1.
- A z-score of 1 corresponds to about the 84.13th percentile of a standard normal distribution — the student scored better than roughly 84% of test-takers, assuming scores are normally distributed.
- As a sanity check on the method: a z-score of 1.96 works out to the 97.5th percentile, and a z-score of 0 works out to exactly the 50th percentile, matching standard statistical tables.
Frequently asked questions
What does a z-score of 0 mean?
A z-score of 0 means the value is exactly equal to the mean. It sits at the 50th percentile of a normal distribution — half the data is expected to fall below it and half above.
Why is 1.96 a commonly cited z-score?
A z-score of ±1.96 marks the boundary of the middle 95% of a standard normal distribution, which is why it's the standard cutoff for a 95% confidence interval and for a two-tailed hypothesis test at the 0.05 significance level.
What is the difference between percentile and p-value here?
Percentile answers "what fraction of the distribution is below this value?" A two-tail p-value answers a different question: "what is the probability of a value at least this extreme, in either direction?" That's why the p-value uses the absolute value of z.
Does this calculator require the data to be normally distributed?
Yes. The percentile and p-value results assume the underlying data follows (or approximates) a normal distribution. The z-score itself is just a standardized distance from the mean and can be computed for any data set, but converting it to a percentile only makes sense under that normality assumption.
What approximation is used for the normal distribution's percentile?
This calculator uses the Abramowitz and Stegun (1964) rational approximation of the error function, a widely published numerical method with a maximum error of about 0.000015%, which is why results match standard z-tables to within 0.01 percentage points.