Z-score Calculator
Standard Score & Percentile Tool
Enter a value (x), the mean (μ), and the standard deviation (σ) to find the z-score and percentile.
Enter a z-score, the mean (μ), and the standard deviation (σ) to solve for the raw value x.
Understanding Z-Scores
A z-score standardizes a value by expressing it in terms of how many standard deviations it sits away from the mean — making it possible to compare apples and oranges on the same scale.
A score of 85 means very different things on a test with mean 75 and one with mean 95. Converting to a z-score puts every value on the same standardized scale, centered at 0.
If the data is approximately normally distributed, a z-score converts directly into a percentile — the share of the population that falls at or below that point — using the standard normal cumulative distribution function.
The Formulas
Z-Score
Reverse (Solve for x)
Percentile uses the standard normal CDF: Φ(z) = 0.5 × (1 + erf(z / √2))
Reading a Z-Score
A z-score of 0 sits exactly at the mean (the 50th percentile). A z-score of +1 is one standard deviation above the mean; for normally distributed data, that lands around the 84th percentile. A z-score of -2 is two standard deviations below the mean, around the 2nd percentile. Roughly 68% of normally distributed data falls within one standard deviation of the mean (z between -1 and +1), and about 95% falls within two.
A z-score of 0 is exactly the 50th percentile.
About 68% of normal data falls between z = -1 and z = 1.
About 95% of normal data falls between z = -2 and z = 2.
Key Takeaways
- A z-score expresses distance from the mean in standard-deviation units, making values from different distributions comparable.
- The Find X tab reverses the formula, letting you solve for the raw value behind a known z-score.
- Percentile assumes an approximately normal distribution. For skewed data, the percentile figure is only an approximation.
- Building a confidence interval? The Confidence Interval Calculator uses the same standard normal z-values.
Frequently Asked Questions
- Pick the Find Z-Score tab to compute a z-score and percentile from a raw value, mean, and standard deviation, or the Find X tab to reverse the calculation and solve for the raw value from a known z-score.
- Fill in the fields for that tab.
- Click "Calculate" to see the result.
A z-score tells you how many standard deviations a value is from the mean of its data set. A z-score of 0 means the value equals the mean; positive means above the mean; negative means below. It's a standardized way to compare values from different distributions on the same scale.
z = (x − μ) / σ subtracts the mean (μ) from your value (x) to find its raw distance from center, then divides by the standard deviation (σ) to express that distance in standard-deviation units instead of the original units. To reverse it and solve for x: x = μ + zσ.
Assuming your data follows a normal (bell-curve) distribution, the percentile is the percentage of the distribution that falls at or below a given z-score — equivalently, the area under the standard normal curve to the left of that z-score. This calculator computes it using a standard closed-form approximation to the normal cumulative distribution function (CDF) based on the error function (erf).
Standard deviation measures spread, and a spread of zero or less than zero has no meaningful interpretation — every value would be identical, or the concept simply doesn't apply. The calculator requires a standard deviation greater than 0 for both directions of the calculation.
The calculator uses a well-known rational approximation to the error function with a maximum error of about 0.00000015 (1.5 × 10⁻⁷) — more than precise enough for any practical statistics, testing, or quality-control use case.