Z-score Calculator
Use a mean and SD as a ruler for this return's relative position.
Returns entered in % · SD in percentage points · no event probabilities
Results
Review or adjust your inputs, then calculate to see your results and interpretation.
Understand this calculation
A Z-score divides the observation's distance from the mean by the SD. Positive is above the mean, negative is below, and zero is at the mean. The SD must be positive.
See calculation method and assumptions
Formula
Z-score
Z is the observation minus its mean, divided by a positive SD in the same units.
Variables, rates and periods
- xᵢ / yᵢ
- Observations from the same period. A return input of 2 means 2%, calculated directly in percentage units.
- μ / σ
- Reference mean and a positive SD in the same units as the observation; Z is undefined at σ=0.
- Z
- A dimensionless relative position, not a normality assumption or event probability.
Standardizing is not normalizing a distribution
- Computing a Z-score does not require normality, but it does not remove heavy tails, skewness or time dependence.
- The diagram's marks show SD distances, not particular probabilities.
Compare a return with its mean and measure its signed distance in standard deviations.
Worked example
- Enter an observed return of 6 and a mean of 2, both in %.
- Enter an SD of 2, meaning two percentage points.
- Calculate (6−2)/2=2 and read the position two SDs above the mean.
Frequently asked questions
Does a Z-score have units?
No. Observation, mean and SD must use matching input units; division produces a unitless relative distance.
Does Z=2 give an event probability?
Not by itself. Standardizing does not make data normally distributed. Tail probabilities require an appropriate, validated distribution model.
What if the SD is zero?
Division by the SD is undefined, so there is no Z-score. Even an observation equal to the mean must not be reported as zero in this case.
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