How to use it
In single value mode, enter the observation, the mean and the standard deviation of the distribution it came from. In whole data set mode, paste your raw numbers instead: the mean and standard deviation are computed from the data, a z-score is produced for every value, and anything beyond two standard deviations is flagged in the table.
The header strip shows the raw deviation (x − μ) alongside the inputs, so you can see the numerator of the calculation before it is scaled.
What a z-score actually is
A z-score converts a raw measurement into a count of standard deviations from the mean. That is the whole idea, and it is more powerful than it sounds, because it makes measurements from completely different scales directly comparable.
Subtracting the mean centres the distribution on zero. Dividing by the standard deviation rescales it so that one unit of z equals one standard deviation. The result is dimensionless — a z-score has no units, which is exactly why a height and a test mark can be compared once both are standardised.
Reading a z-score is direct. A z of 0 means the value is exactly average. A z of +1.5 means it sits one and a half standard deviations above the mean. A z of −0.4 means slightly below average. The sign gives direction, the magnitude gives distance.
A worked example
A student scores 82 on a test where the class mean is 70 with a standard deviation of 8.
- Deviation: 82 − 70 = 12 marks above the mean
- Standardise: 12 ÷ 8 = z = +1.50
- Area below: the normal CDF at 1.50 is 0.9332
- Percentile: the student outperformed about 93.3% of the class
Now suppose the same student scored 74 in a second subject where the mean was 65 and the standard deviation 5. Raw marks say 82 beats 74. Standardised, the second result is z = (74 − 65) ÷ 5 = +1.80, comfortably stronger. Relative to the cohort, the lower raw mark was the better performance. That is the comparison a z-score exists to make.
Converting z to a percentile
Once you have a z-score, the proportion of a normal distribution lying below it comes from the standard normal cumulative distribution function. This calculator evaluates it with the Abramowitz & Stegun 7.1.26 rational approximation, accurate to about 1.5 × 10−7 — tighter than any printed table.
| z | Area below | Percentile | Two-tailed p |
|---|---|---|---|
| −3.00 | 0.0013 | 0.1 | 0.0027 |
| −2.00 | 0.0228 | 2.3 | 0.0455 |
| −1.96 | 0.0250 | 2.5 | 0.0500 |
| −1.00 | 0.1587 | 15.9 | 0.3173 |
| 0.00 | 0.5000 | 50.0 | 1.0000 |
| +1.00 | 0.8413 | 84.1 | 0.3173 |
| +1.645 | 0.9500 | 95.0 | 0.1000 |
| +1.96 | 0.9750 | 97.5 | 0.0500 |
| +2.00 | 0.9772 | 97.7 | 0.0455 |
| +3.00 | 0.9987 | 99.9 | 0.0027 |
Interpretation bands
| |z| | Reading | Share of a normal distribution |
|---|---|---|
| 0 – 1 | Typical | 68.3% — entirely ordinary |
| 1 – 2 | Somewhat unusual | 27.1% — noticeable, not remarkable |
| 2 – 3 | Unusual | 4.3% — conventionally flagged |
| > 3 | Extreme | 0.27% — check for data error |
The |z| > 3 convention for outlier detection is common but imperfect, particularly on small samples. The problem is circular: an extreme value inflates the very standard deviation used to judge it, which can mask it. With fewer than about 30 observations, the IQR fence method is generally more reliable. On larger samples the z-score approach works well.
Where z-scores are used
- Standardised testing. Percentile ranks on aptitude and entrance exams are z-scores converted to a friendlier scale. IQ scores are z-scores rescaled to a mean of 100 and a standard deviation of 15, so an IQ of 130 is simply z = +2.
- Clinical growth charts. Paediatric height, weight and head circumference are reported as z-scores against age- and sex-specific reference populations, where they are called SD scores.
- Quality control. Process control charts set limits at ±3 standard deviations. A measurement beyond that triggers investigation — the origin of the phrase "six sigma".
- Hypothesis testing. A z-test statistic is a z-score of a sample mean, standardised by the standard error rather than the standard deviation.
- Machine learning. Standardising features to zero mean and unit variance is routine preprocessing, because algorithms that use distance are otherwise dominated by whichever variable happens to have the largest scale.
Standard deviation or standard error?
A frequent confusion. When standardising a single observation against a distribution, divide by the standard deviation — that is what this calculator does. When standardising a sample mean for a hypothesis test, divide by the standard error instead:
The second is a much larger number for the same difference, because the standard error is smaller than the standard deviation. Using the wrong one is a reliable way to reach a wrong conclusion about significance.
Frequently asked questions
Can a z-score be negative?
What is a good z-score?
Do I use the sample or population standard deviation?
Why is my percentile different from another calculator?
Can I use z-scores on skewed data?
What z-score corresponds to the top 5%?
Is my data stored?
Next steps
If you do not yet have a standard deviation for your data, compute one with the standard deviation calculator, which shows every deviation and squared deviation. The critical z-values on this page (±1.96, ±1.645) are the same ones used by the confidence interval calculator. For a first look at the shape of your data — which determines whether percentile conversion is safe — use the descriptive summary.