How It Works
Subtract the mean from the value.
Z = (X − μ) / σ | where μ = mean, σ = standard deviation
- Divide by the standard deviation.
- Use the error function approximation to find the CDF (percentile).
Enter Your Numbers
The specific value you want to convert to a z-score.
Population or sample mean.
Population or sample standard deviation.
Summary of Computed Statistics
Z-score and normal-distribution figures recomputed from the value, mean, and standard deviation you entered.
| Statistic | Value |
|---|---|
| Value (X) | 75 |
| Mean (μ) | 70 |
| Standard deviation (σ) | 10 |
| Z-score | 0.5 |
| Percentile rank | 69.15% |
| P(X ≤ value) | 69.15% |
| P(X > value) | 30.85% |
Worked Example
Score of 75, mean=70, std dev=10.
X
75
Mean (μ)
70
Std Dev (σ)
10
Z-Score
(75−70)/10 = 0.5
Percentile
~69.1%
A score of 75 is 0.5 standard deviations above average — better than about 69% of the group.
Z-Scores: Turning a Raw Value Into a Standard Score
Standardizing one value against its group
This tool converts a raw value into a z-score — the number of standard deviations it sits from the mean — and then estimates the percentile and tail probabilities on a normal distribution. It is the standard way to compare a single result against the group it came from.
Z-scores show up in test scoring, quality control, lab reference ranges, and any setting where you want to know whether a value is typical or unusual relative to its distribution.
The formula and the steps
The formula is z = (X − μ) / σ, where X is your value, μ is the mean, and σ is the standard deviation. Two steps: subtract the mean to measure distance from center, then divide by the standard deviation to express that distance in standard-deviation units.
Dividing by σ is what makes the score comparable across different scales. A 20-point gap means very different things on a test with σ = 5 than on one with σ = 40, and the z-score captures that difference automatically.
From z-score to percentile
A z-score becomes a percentile by finding the area under the normal curve to its left. This calculator uses an error-function approximation of that cumulative area, so a z of 0 lands at the 50th percentile, a z of +1 near the 84th, and a z of −1 near the 16th.
The probability outputs split the curve at your value: P(X ≤ value) is the area to the left, and P(X > value) is the area to the right. Together they always sum to 100%.
What the sign and magnitude tell you
A positive z-score means above average, negative means below, and the magnitude tells you how far. By the empirical rule, most values on a bell curve fall between z = −2 and z = +2, so a score beyond that range is genuinely unusual.
The percentile translates the z-score into plain language: a 75th-percentile result beat about three-quarters of the group. Whether that is good or bad depends entirely on what you are measuring.
A worked example
Suppose a student scores 75 on a test where the mean is 70 and the standard deviation is 10. The z-score is (75 − 70) / 10 = 0.5, meaning the score is half a standard deviation above average. On a normal curve that maps to about the 69th percentile, so the student did better than roughly 69% of test-takers.
When the normal assumption breaks down
The percentile and probability outputs assume a normal distribution; for heavily skewed data they are approximations only. The error-function method is also an approximation, so the last decimal may differ slightly from a printed z-table.
A standard deviation of zero means every value is identical, so no value can be unusual — the calculator returns a z-score of 0 and 50% in that case. Make sure your mean and standard deviation come from the same dataset as the value you are scoring.
Assumptions & Best Uses
- Normal distribution.
- Standard deviation is positive.
Limitations
- Percentile estimate uses approximation; exact values require a z-table.
Frequently Asked Questions
What is a z-score?
A z-score measures how many standard deviations a value sits from the mean. Z=0 means exactly average; Z=1 means one standard deviation above (roughly the top 16%); Z=−1 means one below (roughly the bottom 16%).
How do I calculate a z-score by hand?
Subtract the mean from your value, then divide by the standard deviation: z = (X − μ) / σ. For X=75, μ=70, σ=10, that is (75 − 70) / 10 = 0.5.
What does a negative z-score mean?
A negative z-score means the value is below the mean. The size still tells you the distance: z = −2 is two standard deviations below average, which on a normal curve is around the 2nd percentile.
How does a z-score become a percentile?
The percentile is the area under the normal curve to the left of the z-score. This calculator computes that area with an error-function approximation; a z-score of 0.5 corresponds to about the 69th percentile.
Is a higher z-score always better?
It depends on context. For test scores, higher is usually better. For something like error rates or cholesterol, a lower or negative z-score may be preferable. The z-score only reports position relative to the mean, not whether that position is good.
What if my data is not normally distributed?
The z-score itself is still a valid measure of standardized distance, but the percentile and probability outputs assume a normal (bell-shaped) distribution. For skewed data, treat those percentages as rough guides rather than exact figures.
Sources & References
Figures on this page are checked against primary, authoritative sources. Links open in a new tab.
- Normal distribution and standard (z) scores — definition and cumulative probabilitiesNIST/SEMATECH e-Handbook of Statistical Methods · verified 2026-06-06
Note
This calculator is an educational tool. For graded coursework, exams, or professional work, double-check the method and rounding against your own requirements.
Built and maintained by Calculator Matters, an independent calculator project. Method checked against published formulas and primary sources · Last reviewed 3 June 2026 · How we calculate · Found an error? corrections@calculatormatters.com