Complete guide
Z score calculator: standard deviations from the mean
Enter a value, mean and positive standard deviation. Calculator24 reports the z-score, raw difference and whether the observation lies above or below the mean.
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 interpretation
| z | Location |
|---|---|
| 0 | At the mean |
| +1 | One SD above |
| −1 | One SD below |
| +2 | Two SD above |
| −2 | Two SD below |
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.
What to keep in mind
A z-score describes standardized distance. Distribution probabilities require additional assumptions.
Results display up to 8 decimal places; exported numbers preserve calculation precision. This is a calculation summary, not an official certificate.