Quick answer
A z-score is z = (x − μ) ÷ σ, the number of standard deviations a value lies from the mean. A score of 85 with a mean of 70 and standard deviation of 10 has z = 1.5, which is the 93.32nd percentile under a normal distribution.
Enter the value, the mean and the standard deviation. The calculator returns the z-score and the percentile.
Related tools: Confidence Interval Calculator: Confidence interval for a mean (t or z). Standard Deviation Calculator: Sample or population standard deviation with variance and mean. Mean Median Mode Calculator: Mean, median and mode of a data set.
How to use the Z-Score Calculator
- Enter value (x).
- Enter mean (μ).
- Enter standard deviation (σ) (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Value (x) | A number | — |
| Mean (μ) | A number | — |
| Standard deviation (σ) | A number | > 0 |
Formula and method
Formula
z = (x − μ) ÷ σ
IEEE-754 double precision; shown to at most 6 decimal places.
Number of standard deviations from the mean; percentile uses the standard normal CDF (Abramowitz–Stegun erf approximation, error < 1.5×10⁻⁷).
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Value (x) = 85; Mean (μ) = 70; Standard deviation (σ) = 10.
| Result | Value |
|---|---|
| Z-score | 1.5 |
| Percentile (normal) | 93.3193% |
Example 2
Inputs: Value (x) = 70; Mean (μ) = 70; Standard deviation (σ) = 10.
| Result | Value |
|---|---|
| Z-score | 0 |
| Percentile (normal) | 50% |
The first line is the headline result (z-score). Below it the z-Score Calculator lists percentile (normal), which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Reading a z-score
z = 0 is exactly average, z = 1 is one standard deviation above and z = −1 one below. For a normal distribution about 68% of values fall within ±1, 95% within ±2 and 99.7% within ±3.
The percentile assumes normality
The percentile comes from the standard normal distribution, so it is accurate only if the data are roughly bell-shaped. For skewed data the z-score is still valid as a distance, but its percentile is not. σ must be positive.
How the result changes with value
Holding every other input at the first worked example, this table changes only value. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Value | Z-score | Percentile (normal) |
|---|---|---|
| 42.5 | -2.75 | 0.298% |
| 85 | 1.5 | 93.3193% |
| 170 | 10 | 100% |
| 340 | 27 | 100% |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Standard deviation (σ): entering "0" shows "Standard deviation (σ) must be greater than 0." beside that field and no result.
Choosing between the Z-Score Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Confidence Interval Calculator](/statistics/confidence-interval-calculator/) | sample mean, standard deviation and sample size | Confidence interval |
| [Standard Deviation Calculator](/statistics/standard-deviation-calculator/) | data set and data type | Population standard deviation (σ) |
| [Mean Median Mode Calculator](/statistics/mean-median-mode-calculator/) | data set | Mean |
| [Sample Size Calculator](/statistics/sample-size-calculator/) | estimate a, confidence level and margin of error | Required sample size |
| [T-Score (Standardized) Calculator](/statistics/t-score-calculator/) | raw score, mean and standard deviation | T-score |
Background: descriptive statistics
Population statistics divide by n and sample statistics divide by n − 1 (Bessel's correction), because a sample tends to understate variability. Use the population version when your data is every value in the group, and the sample version when it is a subset.
Defaults differ between tools, and so do quartile conventions, which can give slightly different values for small data sets; this page states the method it uses.
Assumptions and limitations
- σ must be positive
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for standard deviation (σ) to confirm this.
Frequently asked questions
What is a good z-score?
It depends on the context; z = 1.5 is above average and z = −1.5 below.
How do I go from z to a value?
Use x = μ + zσ: multiply z by the standard deviation and add the mean.
What is the difference between a z-score and a t-score?
A t-score is used when σ is estimated from a small sample.
What does the Z-Score Calculator calculate?
Standardize a value using a mean and standard deviation. It returns Z-score and Percentile (normal).
What formula does the Z-Score Calculator use?
z = (x − μ) ÷ σ. Number of standard deviations from the mean; percentile uses the standard normal CDF (Abramowitz–Stegun erf approximation, error < 1.5×10⁻⁷).
What do I need to enter in the z-Score Calculator?
You enter Value (x), Mean (μ) and Standard deviation (σ).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. σ must be positive.
What is an example of a z-Score result?
With value (x) 85, mean (μ) 70 and standard deviation (σ) 10, the z-score is 1.5.
Related calculators
- Confidence Interval Calculator — Confidence interval for a mean (t or z).
- Standard Deviation Calculator — Sample or population standard deviation with variance and mean.
- Mean Median Mode Calculator — Mean, median and mode of a data set.
- Sample Size Calculator — Sample size for estimating a mean or a proportion.
- T-Score (Standardized) Calculator — Convert a raw score to a standardized T-score (mean 50, SD 10).
- Mean Median Mode Range Calculator — Mean, median, mode and range of a data set.
- Median Calculator — Median of a data set.
- Relative Standard Deviation Calculator — Relative standard deviation (RSD) as a percentage.
Sources
- Standard statistics definition (e.g. NIST/SEMATECH e-Handbook of Statistical Methods)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.