Z score calculator: standard deviations from the mean
Enter a value, mean and positive standard deviation. HitEquals reports the z-score, raw difference and whether the observation lies above or below the mean.
Inputs used by this calculator
Z Score Calculator uses the fields below. The starting values reproduce the page example; replace them with values from the same units and time period before using the result.
| Input | Starting example | Accepted value | How it is used |
|---|---|---|---|
| Observed value x | 85 | Any finite number supported by the formula | Supplies the observed value x quantity used by the published equation or result breakdown. |
| Mean μ or x̄ | 70 | Any finite number supported by the formula | Supplies the mean μ or x̄ quantity used by the published equation or result breakdown. |
| Standard deviation σ or s | 10 | 1e-12 to 1000000000000 | Supplies the standard deviation σ or s quantity used by the published equation or result breakdown. |
What a z-score measures
A z-score expresses signed distance from the mean in standard-deviation units. Zero is at the mean; positive values are above it and negative values below it.
Z-score formula
Subtract the mean and divide by a positive standard deviation.
z=(x−mean)/standard deviationInterpret sign and magnitude
The sign gives direction. Magnitude gives distance in standardized units, allowing values on different scales to be compared.
Population versus sample inputs
Use the mean and standard deviation that belong to the same population or sample definition. This calculator does not estimate them from raw data.
Outliers and probability
NIST cautions that z-score rules alone can mislead for small samples or multiple outliers. A probability also depends on an assumed distribution.
Z-score and percentile are different
A z-score is a standardized distance. Converting it to a percentile requires a distribution model. Under a standard normal model, z=0 corresponds to the 50th percentile and z≈1.645 to the 95th percentile, but those probability statements do not automatically apply to skewed or discrete data.
Comparing measurements on different scales
Standardization can compare relative standing across compatible populations. A test score and a laboratory value may share the same z-score even though their raw units differ, provided each uses the correct mean and standard deviation for its own reference group.
Z-score interpretation
| z | Location |
|---|---|
| 0 | At the mean |
| +1 | One SD above |
| −1 | One SD below |
| +2 | Two SD above |
| −2 | Two SD below |
Normal-model landmarks
| z | Approximate percentile | Position |
|---|---|---|
| −1.96 | 2.5th | lower two-sided 95% landmark |
| −1 | 15.9th | one SD below |
| 0 | 50th | at the mean |
| 1 | 84.1st | one SD above |
| 1.96 | 97.5th | upper two-sided 95% landmark |
Frequently asked questions
What does z=0 mean?
The value equals the mean.
Can a z-score be negative?
Yes. It means the value is below the mean.
Is z=2 always an outlier?
No. An outlier decision depends on context, distribution and sample size.
Can standard deviation be zero?
Not for this calculation; division by zero is undefined.
Does a z-score give probability?
Only with a specified probability model, commonly a normal distribution.
Is a z-score the same as a percentile?
No. A percentile additionally requires a distribution or an empirical ranking.
Can I compare z-scores from different units?
Yes when each value is standardized against the correct and comparable reference population.
What to keep in mind
A z-score describes standardized distance. Distribution probabilities require additional assumptions.
Results are rounded for display in US format, with money in dollars and cents; CSV exports keep full calculation precision. This is a calculation summary, not an official certificate.