A z-score — also called a standard score — converts a raw value into "how many standard deviations from the mean", which makes numbers from completely different scales comparable. Enter the value, the mean and the standard deviation below.
The value 82 sits 1.5 standard deviations above the mean of 70. The shaded area is 93.3% of a normal distribution.
Show the working, step by step
The formula
z = (x − μ) / σ
Subtracting the mean re-centres the data on zero; dividing by the standard deviation re-scales it so that one unit of z is one standard deviation. Together those two steps are called standardising, and they are why a test score, a height and a share price can all be placed on the same axis and compared.
What the sign and size mean
| z | Percentile | Interpretation |
|---|---|---|
| −3.0 | 0.1% | Far below the mean; rare |
| −2.0 | 2.3% | Well below; unusual |
| −1.0 | 15.9% | Below average |
| 0.0 | 50% | Exactly at the mean |
| +1.0 | 84.1% | Above average |
| +1.96 | 97.5% | The 95% two-tailed cutoff |
| +2.0 | 97.7% | Well above; unusual |
| +3.0 | 99.9% | Far above; rare |
Those percentiles come from the standard normal distribution. They are only meaningful if your data is roughly normal — the z-score itself is valid for any data, but its translation into a percentile is not.
A worked example
A student scores 82 on a test with a mean of 70 and a standard deviation of 8.
z = (82 − 70) / 8 = 12 / 8 = 1.5
The score is 1.5 standard deviations above the mean, which places it at roughly the 93rd percentile — better than about 93% of the group, assuming the scores are approximately normally distributed. Note that this is a stronger statement than "12 points above average": whether 12 points is a lot depends entirely on σ. On a test with σ = 20, the same 12 points would give z = 0.6 and a percentile of only about 73.
Where z-scores get used
- Outlier screening — flag any observation with |z| above 3. Useful on roughly normal data; unreliable on skewed data, where the IQR rule is safer.
- Comparing across scales — a z of 1.2 on the SAT and a z of 1.2 on the ACT represent the same relative standing, even though the raw scores are nothing alike.
- Standardising model inputs — regression and machine-learning features are routinely converted to z-scores so that variables measured in different units contribute on equal footing.
- Process control — control limits are conventionally drawn at ±3σ, which is a z-score threshold in different clothing.
A caution on the percentile
The z-score is a pure rescaling and is valid whatever shape your data has. The percentile this calculator reports is not — it reads the z against a normal distribution. For strongly skewed data such as incomes or waiting times, a z of 2 might correspond to the 90th percentile rather than the 97.7th. Check the shape of your distribution before quoting the percentile.
Related calculators
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Standard deviation calculator
Get the σ that a z-score needs, from raw data.
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Confidence interval
The interval built from the same standard error machinery.
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Standard error
For z-scores of a mean rather than of a single value.
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Symbols and notation
What μ, σ and z stand for.
Common questions
What is a z-score?
A z-score says how many standard deviations a value sits from the mean:
z = (x − μ) / σ. A z of +1.5 means the value is one and a half standard
deviations above the mean; −0.4 means it is slightly below.
What counts as a high or unusual z-score?
Common rules of thumb put |z| > 2 at "unusual" (outside roughly 95% of a normal distribution) and |z| > 3 at "rare" (outside roughly 99.7%). Many outlier screens use |z| > 3 as a flag.
Those thresholds assume normality. On skewed data they mislead badly, and the interquartile-range rule is a more robust screen.
Can a z-score be negative?
Yes, and it simply means the value is below the mean. The sign carries direction; the magnitude carries distance. A z of −2.0 and a z of +2.0 are equally unusual.
Should I use the sample or population standard deviation for z?
Strictly, the z-score is defined with the population parameters μ and σ. When you only have sample estimates x̄ and s, the resulting quantity follows a t distribution rather than a normal one — which matters for small n. Above roughly n = 30 the difference is minor and the two are used interchangeably in practice.
How do I convert a z-score back to a raw value?
Rearrange: x = μ + zσ. So with a mean of 100 and an SD of 15, a z of 1.5
corresponds to 100 + 1.5 × 15 = 122.5.