Z-score calculator
Convert a value to a Z-score or a Z-score back to x. Read cumulative probability from the highlighted Z table, view the normal curve and calculate areas between bounds.
Your numbers stay in this browser. No account or upload to the server is needed.
The 68–95–99.7 rule
±1σ
About 68.27% of a normal distribution.
±2σ
About 95.45% of a normal distribution.
±3σ
About 99.73% of a normal distribution.
This empirical rule applies to a normal distribution. It is not a test that the data are normal.
How do I use this page?
- Choose value standardisation, reverse conversion or normal distribution areas.
- Enter your data and select “Calculate”. Standard deviation must be greater than zero.
- Review the formula or shaded interval. Use normal probabilities only when the normal model is justified.
Method and worked example
Z = (x − μ) / σ. A positive result is above the mean, a negative result below it, and zero is the mean itself.
Standardisation does not establish normality. The area mode applies only to the standard normal distribution with mean 0 and standard deviation 1.
Density φ(z) = exp(−z²/2) / √(2π). Areas use numerical Simpson integration with a step no larger than 0.005. Input bounds range from −10 to 10.
Tail areas are integrated directly to 12 to reduce cancellation near 1. Results are approximate and rounded for display.
Example: x=85, μ=70, σ=10 gives Z=(85−70)/10=1.5. Under the standard normal model, the interval from −1.96 to 1.96 has probability approximately 0.950004.