Z-Score Calculator
Find the z-score of a value, the percentile and tail probabilities that go with it, and every step of the working — from a mean and standard deviation, from a raw data set, or backwards from a percentile.
Written by Suhaib Hassan. Formulas and worked examples are checked against the NIST/SEMATECH Engineering Statistics Handbook before publishing. Our methodology.
It standardises with z = (x − μ) ÷ σ, then evaluates the standard normal distribution numerically — a series near zero and a continued fraction in the tails.
Most z-score tools stop at one number. This one shows the tail areas and the working, and also handles raw data sets and inverse (percentile → value) lookups.
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Area under the curve to the left of z
Complementary upper-tail area
Extreme-value area in both tails
| # | Value | z-score |
|---|
Population vs sample z-scores
The z-score formula is the same either way; what changes is where the mean and standard deviation come from. Use σ when you know the whole population, and the sample statistics when you only have a sample.
Population z-score μ and σ known
Use when the mean and standard deviation describe the entire population — for example a published test norm.
Sample z-score x̄ and s from data
Use when the mean x̄ and standard deviation s are estimated from a sample. s divides by n − 1 (Bessel's correction).
The empirical rule: how much of a normal distribution lies within each band
Exact areas for a normal distribution — the familiar 68–95–99.7 figures are rounded versions of these. Your |z| falls in the highlighted band.
| Band around the mean | Area inside the band | Area in each tail | Typical use |
|---|---|---|---|
| μ ± 1σ (|z| ≤ 1) | 68.2689% | 15.8655% | The central bulk of the observations |
| μ ± 1.645σ | 90.0000% | 5.0000% | 90% confidence interval; one-sided 5% test |
| μ ± 1.96σ | 95.0000% | 2.5000% | 95% confidence interval; two-sided 5% test |
| μ ± 2σ | 95.4500% | 2.2750% | A common "unusual value" screen |
| μ ± 2.576σ | 99.0000% | 0.5000% | 99% confidence interval; two-sided 1% test |
| μ ± 3σ | 99.7300% | 0.1350% | Very rare values under normality |
| Beyond ±3σ | under 0.27% outside | under 0.135% | Extreme values — check the data |
Worked example: a standardised test score
A test has a mean of 500 and a standard deviation of 100, and results are roughly normally distributed. A candidate scores 650. How did they do relative to everyone else?
- 1
State what you know
Raw score x = 650, mean μ = 500, standard deviation σ = 100.
- 2
Subtract the mean
How far the score sits from the average.
x − μ = 650 − 500 = +150 - 3
Divide by the standard deviation
Express that distance in standard deviations.
z = 150 ÷ 100 = +1.50 - 4
Read off the percentile
The standard normal CDF at 1.50 is 0.9332, so the candidate scored higher than about 93.3% of test-takers.
Φ(1.50) = 0.9332 → 93.32nd percentile
Where z-scores are used
Standardising puts values from different scales on one common footing, which is why the idea appears across fields.
Machine learning
Feature standardisation rescales each variable to mean 0 and standard deviation 1, so large-valued features do not dominate distance-based or gradient-based models.
Also called "z-score normalisation"Growth & health screening
Paediatric growth references and bone-density reports express a measurement as a z-score against an age- and sex-matched reference group.
Reference-group comparisonQuality control
Process monitoring flags a measurement that lands several standard deviations from the process mean as a possible special cause worth investigating.
Outlier screeningWhat is a z-score?
A z-score (standard score) says how many standard deviations a value lies from the mean of its distribution. A z-score of 0 is exactly average, +1 is one standard deviation above, and −2 is two below. Because the result has no units, z-scores let you compare a height in centimetres with an exam mark or a lab value.
How to calculate a z-score
z = (x − μ) ÷ σ, where x is the raw score, μ the mean and σ the standard deviation. To go the other way, x = μ + zσ. The percentile comes from the standard normal cumulative distribution function Φ(z), assuming the data is approximately normal.
- Left tail = Φ(z), the share of values below x.
- Right tail = 1 − Φ(z), the share above x.
- Two-tailed = 2 × the smaller tail, the share at least as far from the mean in either direction.
By Suhaib Hassan · Reviewed 28 September 2026 · How CalculatePilot verifies formulas →
Interpreting your result
Between −1 and +1 is ordinary variation, and beyond ±2 is unusual: roughly 5% of a normal distribution lies that far out. Beyond ±3 is rare (about 0.27%). The percentile is only meaningful if the data is close to bell-shaped — for skewed data a high z-score does not mean the same tail probability.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods — 1.3.6.6.1 Normal distribution
- Abramowitz, M. & Stegun, I. A. (eds.), Handbook of Mathematical Functions, National Bureau of Standards, 1964 — chapter 26, normal probability function.
- ISO 3534-1:2006, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in probability.
Frequently asked questions
What does a negative z-score mean?
It means the value is below the mean. A z-score of −1.5 is one and a half standard deviations below average, which on a normal distribution is about the 6.68th percentile.
How do I convert a z-score to a percentile?
Evaluate the standard normal cumulative distribution function Φ(z) and multiply by 100. For example Φ(1.00) = 0.8413, so z = 1 is the 84.13th percentile. This calculator does that for you and also shows both tail areas.
What is the difference between a z-score and a t-score?
A z-score uses a known population standard deviation and the normal distribution. A t statistic uses a standard deviation estimated from a small sample and the t distribution, which has heavier tails. Separately, "T-score" in psychometrics means a standardised score with mean 50 and standard deviation 10, which is unrelated to the t distribution.
What are the critical z-values for 95% and 99% confidence?
For a two-sided interval, z = 1.960 for 95% confidence and z = 2.576 for 99% confidence. For 90% it is 1.645. Use the inverse mode with a percentile of 97.5 or 99.5 to reproduce them.
Should I use the sample or the population standard deviation?
Use the population standard deviation (divide by N) only when your data is the whole population. If it is a sample drawn from a larger group, divide by n − 1 to get the unbiased estimate of the variance, which is the usual choice.
Does the percentile work for any data?
No. The z-score itself is always valid, but the percentile and tail areas assume an approximately normal distribution. For strongly skewed or heavy-tailed data they can be badly off.
Related statistics calculators
Next steps once you have a z-score.