Z-Score Calculator
Find how many standard deviations a value lies above or below the mean
Z-score
How we calculated it
About the Z-Score Calculator
A z-score shows how far an observation lies from a mean in units of standard deviation. Positive values are above the mean, negative values are below it and zero lies exactly at the mean.
Z-scores are useful for standardizing values measured on the same distribution and for connecting normally distributed observations with probabilities.
How to Use the Z-Score Calculator
- Enter the observation value.
- Enter the mean of the relevant data set or distribution.
- Enter the standard deviation using the same units as the observation.
- Calculate the z-score and interpret its sign and magnitude rather than treating it as a percentage.
How the Calculation Works
The formula is z = (x − μ) ÷ σ for a population mean μ and standard deviation σ.
Subtracting the mean centers the observation; dividing by standard deviation expresses that distance in standardized units.
Worked example
Z-Score Calculator Worked Example
Value 85, mean 70, standard deviation 10
z = (85−70) ÷ 10 = 15 ÷ 10 = 1.5. The observation is 1.5 standard deviations above the mean.
Result guide
Understanding Your Z-Score Calculator Result
A z-score describes relative position. Whether a value of 1.5 is unusual depends on the distribution and the purpose of the analysis. A value several standard deviations from the mean deserves closer inspection, but whether it is statistically or practically important depends on the field and the quality of the underlying distribution model.
Before you rely on the answer
Assumptions and Limitations
- Observation, mean and standard deviation refer to the same variable.
- Standard deviation is positive.
- Normal-distribution probability interpretations require the distributional assumption to be reasonable.
Practical tip
Using This Result in Practice
Do not compare raw z-scores from unrelated populations without considering how each mean and standard deviation was estimated.
Related calculations
Useful Calculations to Continue With
If you want the probability associated with a z-score under a normal model, use the Normal Distribution Calculator.
If you first need to calculate the spread of the underlying data, use the Standard Deviation Calculator.
Common questions
Z-Score Calculator FAQ
What does a z-score of 0 mean?
The observation equals the mean.
Can a z-score be negative?
Yes. A negative z-score means the observation is below the mean.
Is a z-score a percentile?
No. A z-score is a standardized distance. Under a specified distribution it can be converted to a percentile or probability.