DIY Cost Tools
Z-Score Calculator
Calculate how many standard deviations a value lies above or below a specified mean. The prefilled example shows a useful result immediately.
All fields are required. Standard deviation must be positive. Input magnitudes may not exceed 1 trillion.
How to use
Enter the required numbers, then select Calculate. Invalid inputs clear previous results and show an explanation. Answers display up to ten significant digits unless an exact integer factorization is shown.
Formula and explanation
z = (x − μ)/σ. Subtract the reference mean μ from the observed value x and divide by the reference standard deviation σ. A positive z-score lies above the mean; a negative z-score lies below it.
Worked example
A value of 85 with mean 70 and standard deviation 10 has z = (85 − 70)/10 = 1.5.
Important limitations
A z-score can be calculated without a normal distribution, but normal percentiles and tail probabilities require additional assumptions. This tool does not calculate p-values or diagnose health conditions. Use the standard deviation appropriate to your reference population or sample.
Frequently asked questions
What does zero mean?
The observed value equals the reference mean.
Can a z-score be negative?
Yes. A value below the mean has a negative z-score.
Does 1.5 mean the 1.5th percentile?
No. It means 1.5 standard deviations above the mean.
Why is zero standard deviation invalid?
The formula would divide by zero.
Review notes
Reviewed October 1, 2026. Default outputs and boundary cases were independently checked. Blank, nonfinite, excessive, and calculator-specific invalid inputs were tested before publication. This is an educational tool; verify precision and domain assumptions for professional use.