Calculate the z-score for any value given a mean and standard deviation. Returns the standard score and the corresponding percentile rank using the exact normal distribution CDF.
The z-score standardizes any value by measuring its distance from the mean in standard deviation units. This makes values from different distributions directly comparable on the same scale.
The conversion from z to percentile uses the cumulative distribution function (CDF) of the standard normal. There is no closed-form solution, so calculators use polynomial approximations or numerical integration.
Worked example 1: Score of 85, class mean 75, SD 10. z = (85 - 75) / 10 = 1.00. Percentile = 84.13%. This score is 1 SD above the mean, placing it in the top 16% of the class.
Worked example 2: Score of 62, mean 75, SD 10. z = (62 - 75) / 10 = -1.30. Percentile = 9.68%. This score is 1.3 SDs below the mean, placing it in the bottom 10%.
A z-score assumes you know the population mean and standard deviation (or that your sample is large enough to estimate them reliably). A t-score is used when the population standard deviation is unknown and sample size is small (typically n < 30). As sample size increases, the t-distribution converges to the normal distribution and z and t give nearly identical results.
| Situation | Use |
|---|---|
| Population mean and SD known | Z-score |
| Large sample (n > 30), estimating population | Z-score (t converges to z) |
| Small sample (n ≤ 30), unknown population SD | T-score |
| Standardized tests (SAT, GRE, IQ) | Z-score (population norms known) |
| Lab values, bone density scans | Z-score (age/sex matched norms) |
| Hypothesis testing with small samples | T-score |
The Correlation Coefficient Calculator also uses the standard normal for significance testing when sample size is large. For small samples, the t-distribution is the correct reference.
Each row shows the cumulative probability (area to the left) for a given z-score. A percentile value of 84.13% at z = +1.00 means 84.13% of values in a normal distribution fall below that point.
| Z-Score | Percentile | Interpretation |
|---|---|---|
| -3.00 | 0.13% | Extremely rare, bottom 0.13% |
| -2.58 | 0.49% | Bottom 1% threshold |
| -2.00 | 2.28% | Below 98% of distribution |
| -1.65 | 5.00% | Bottom 5% threshold |
| -1.00 | 15.87% | Below 84% of distribution |
| 0.00 | 50.00% | Exactly at the mean |
| +1.00 | 84.13% | Above 84% of distribution |
| +1.65 | 95.00% | Top 5% threshold |
| +2.00 | 97.72% | Above 98% of distribution |
| +2.58 | 99.51% | Top 1% threshold |
| +3.00 | 99.87% | Extremely rare, top 0.13% |
The Binomial Distribution Calculator uses the normal approximation for large n, which relies on this same table when np and n(1-p) are both at least 5.
Researches and verifies the formulas, methodology, and source data behind each calculator on CalculatorFlux. All tools are built and checked against the cited references before publication.