Z-Score Calculator
Standardize any value and instantly see how unusual it is. Enter a raw score with its mean and standard deviation to get the z-score, or enter a z-score to find its probability and percentile — complete with a bell-curve visual, one- and two-tailed p-values, and every step shown.
Textbook-accurate normal-distribution math · Reviewed July 2026 · Formula and probabilities follow the NIST standard normal distribution
Choose What to Calculate
The single value you want to standardize — a test score, measurement, or observation.
Use the population mean and standard deviation if you have them; the sample mean (x̄) and sample SD (s) work too.
Your Result
Standard Normal Curve
Your z-score marked on the bell curve
- Area below (percentile)
- Your z-score
What Is a Z-Score?
A z-score — also called a standard score — tells you how many standard deviations a value sits from the mean of its distribution. It converts a raw number into a position on a universal scale, which is what makes it so powerful: once you’ve standardized two values, you can compare them directly even if they came from completely different measurement systems. A z-score of 0 means the value is exactly average. A positive z-score means it’s above average; a negative z-score means it’s below. The size of the number tells you how far: a z-score of 1.5 is one and a half standard deviations above the mean, while −2.0 is two standard deviations below it.
Because a z-score is dimensionless — the units cancel out in the calculation — you can compare an SAT score against a GPA, a runner’s time against a swimmer’s, or a patient’s blood pressure against their cholesterol. That’s why z-scores turn up everywhere from classrooms and hospitals to finance and quality control.
The Z-Score Formula
The z-score formula is refreshingly simple:
z = (x − μ) / σ
where x is your raw value, μ (mu) is the mean of the distribution, and σ (sigma) is the standard deviation. In words: subtract the mean from your value to see how far above or below average it is, then divide by the standard deviation to express that distance in standard-deviation units. If you’re working from a sample rather than a full population, you simply substitute the sample mean (x̄) and sample standard deviation (s) — the arithmetic is identical.
A worked example
Suppose a student scores 84 on a test where the class mean is 70 and the standard deviation is 10. Their z-score is (84 − 70) / 10 = 1.4, meaning they scored 1.4 standard deviations above the class average. Using the standard normal distribution, that z-score corresponds to roughly the 92nd percentile — the student outperformed about 92% of the class. The calculator above produces exactly this, along with the bell-curve visual and the one- and two-tailed probabilities.
How to Interpret a Z-Score
Reading a z-score is a two-part process: the sign gives the direction, and the magnitude gives the rarity.
| Z-score | Meaning | Approx. percentile |
|---|---|---|
| z = 0 | Exactly at the mean | 50th |
| z = +1 | 1 SD above average | 84th |
| z = +2 | 2 SD above average — unusual | 97.7th |
| z = −1 | 1 SD below average | 16th |
| z = −2 | 2 SD below average — unusual | 2.3rd |
| |z| > 3 | Extremely rare — likely outlier | Beyond 99.7% |
Can a z-score be negative?
Yes — and it’s completely normal. A negative z-score simply means the value falls below the mean. A z-score of −1.5 is one and a half standard deviations below average, nothing more sinister than that. About half of all values in any distribution have negative z-scores, so seeing one is expected, not a problem.
Is a z-score good or bad?
It depends entirely on what you’re measuring. For test scores or athletic performance, a high positive z-score is good — you’re well above average. For things you want to minimize, like error rates, cholesterol, or defect counts, a negative z-score is the better outcome. And in quality control, where you want values close to a target, a z-score near zero is ideal. Always ask whether more of the measured quantity is desirable before deciding whether a z-score is good news.
Z-Scores and the Empirical Rule (68-95-99.7)
When data follows a normal distribution, z-scores map onto a memorable set of percentages known as the empirical rule, or the 68-95-99.7 rule:
- About 68% of values fall within one standard deviation of the mean (z between −1 and +1).
- About 95% fall within two standard deviations (z between −2 and +2).
- About 99.7% fall within three standard deviations (z between −3 and +3).
This rule is why a z-score beyond ±3 is so striking — fewer than 3 in 1,000 values land that far out in a normal distribution. It’s also the basis for using |z| > 3 as a common outlier threshold: anything that extreme is rare enough to warrant a second look.
Z-Score to Percentile and P-Value
Once you have a z-score, the standard normal distribution lets you convert it into a probability. The left-tail probability, P(X ≤ x), is the area under the curve to the left of your z-score — multiply it by 100 and you get the percentile. The right-tail probability, P(X ≥ x), is simply one minus that. The two-tailed p-value is the combined area in both tails beyond ±z, which is the version used in most significance tests. The calculator reports all of these at once so you don’t need a z-table.
The 1.96 rule and statistical significance
One z-score shows up constantly in research: ±1.96. Exactly 95% of a normal distribution lies between −1.96 and +1.96, leaving 5% split across the two tails. That’s why a result with |z| > 1.96 is declared “statistically significant at the 5% level” in a two-tailed test, and why ±1.96 marks the boundaries of a 95% confidence interval.
Z-Score vs. T-Score: When to Use Each
A close relative of the z-score is the t-score. The rule of thumb: use a z-score when you know the population standard deviation and your sample is large (typically more than 30 observations). Use a t-score when the population standard deviation is unknown and you’re estimating it from a small sample — the t-distribution has slightly heavier tails to account for that extra uncertainty. As sample size grows, the t-distribution converges on the normal, and the two scores become nearly identical.
Where Z-Scores Are Used
- Education & testing: standardizing and comparing scores across different exams and scales.
- Healthcare: growth charts and bone-density (T-score) results are z-score based, showing how a measurement compares to a reference population.
- Finance: the Altman Z-score gauges bankruptcy risk, and standardized returns help compare investments.
- Quality control: flagging products whose measurements drift too far from the target.
- Research & hypothesis testing: the z-test uses z-scores to judge whether a result is due to chance.
Because the z-score is built directly on the mean and standard deviation, it’s worth being confident in those two inputs first. You can compute them for any dataset with our Standard Deviation Calculator or get a full breakdown from the Descriptive Statistics Calculator.
Frequently Asked Questions
A z-score tells you how many standard deviations a value is from the average. A z-score of 0 means the value equals the mean, a positive z-score means it’s above the mean, and a negative z-score means it’s below. For example, a z-score of +2 means the value is two standard deviations above average. Because z-scores are on a universal, unitless scale, they let you compare values that came from completely different measurement systems.
Use the formula z = (x − μ) / σ, where x is your value, μ is the mean, and σ is the standard deviation. Subtract the mean from your value, then divide by the standard deviation. For instance, if a score is 85, the mean is 75, and the standard deviation is 10, the z-score is (85 − 75) / 10 = 1.0, meaning the score is one standard deviation above the mean. If you only have a sample, use the sample mean and sample standard deviation.
Yes. A negative z-score is completely normal and just means the value falls below the mean. A z-score of −1.5, for example, is one and a half standard deviations below average. Roughly half of all values in any distribution have negative z-scores, so they are entirely expected. The negative sign indicates direction (below the mean), not that anything is wrong with the value.
There’s no universally “good” z-score — it depends on what you’re measuring. For test scores or performance metrics, a higher positive z-score is better because it means you’re above average. For quantities you want to minimize, like error rates or cholesterol, a negative z-score is preferable. In quality control, where values should hit a target, a z-score near zero is ideal. Always consider whether more or less of the measured quantity is desirable.
Convert a z-score to a percentile using the standard normal distribution: the percentile is the area under the curve to the left of the z-score, multiplied by 100. A z-score of 0 is the 50th percentile, +1 is about the 84th, and −1 is about the 16th. This calculator does the conversion automatically, reporting the percentile along with the left-tail, right-tail, and two-tailed probabilities, so you don’t need to look up a z-table.
Use a z-score when you know the population standard deviation and your sample is large, generally more than 30 observations. Use a t-score when the population standard deviation is unknown and you’re estimating it from a smaller sample, because the t-distribution has heavier tails that account for the added uncertainty. As the sample size increases, the t-distribution approaches the normal distribution, and the two scores become almost identical.
Methodology & formulas used
This calculator uses the standard normal distribution, reviewed July 2026. All computation runs locally in your browser; your data is never uploaded.
- Z-score: z = (x − μ) / σ.
- Left-tail probability / percentile: Φ(z), the standard normal cumulative distribution function (CDF). Percentile = Φ(z) × 100.
- Right-tail probability: 1 − Φ(z).
- Two-tailed p-value: 2 × (1 − Φ(|z|)).
- CDF computation: the error function is evaluated with the Abramowitz & Stegun 7.1.26 rational approximation (accurate to about 1×10⁻⁷), and Φ(z) = ½[1 + erf(z / √2)].
Probabilities and percentiles assume the underlying data is normally distributed. For data that isn’t approximately normal, treat the percentile and p-values as approximations. Results are rounded for display but computed at full precision.
References
- NIST/SEMATECH e-Handbook of Statistical Methods. Standard Normal Distribution. itl.nist.gov
- Abramowitz, M. & Stegun, I.A. (1964). Handbook of Mathematical Functions, formula 7.1.26 (error function approximation). Error function overview
- Wikipedia. Standard score (z-score) — definition and applications. en.wikipedia.org