Z-Score & Normal Distribution Calculator
Calculate standard z-values, cumulative p-values, tail probabilities, and percentile standings with step-by-step statistical proofs.
What is a Standard Z-Score?
A Z-score (also referred to as a standard normal score) transforms any raw observation from a normal distribution into standard deviations relative to the distribution's mean. By standardizing data, Z-scores allow analysts and scientists to compare observations measured across entirely different scales (e.g., comparing an SAT exam score to an IQ score).
The 68-95-99.7 Empirical Rule Benchmarks
| Standard Deviation Range | Enclosed Population (%) | Outside Tails (%) | Common Applications |
|---|---|---|---|
| ± 1.00 σ | 68.27% | 31.73% | Average normal variation |
| ± 1.645 σ | 90.00% | 10.00% | 90% Confidence Interval |
| ± 1.960 σ | 95.00% | 5.00% | Standard 95% Scientific Alpha (p < 0.05) |
| ± 2.00 σ | 95.45% | 4.55% | Two standard deviations |
| ± 2.576 σ | 99.00% | 1.00% | 99% Confidence Interval |
| ± 3.00 σ | 99.73% | 0.27% | Statistical outlier boundary |
Frequently Asked Questions
What does a Z-score represent?
A Z-score indicates how many standard deviations an observation lies from the average. A positive Z-score indicates a value above the mean, while a negative score indicates a value below the mean.
What is a statistically significant Z-score?
In two-tailed hypothesis testing at an alpha level of α = 0.05, a Z-score greater than +1.96 or less than -1.96 is considered statistically significant (p < 0.05).
How do you convert a Z-score to a percentile?
A Z-score is converted to a percentile by calculating the cumulative area under the standard normal curve to the left of Z using the Gaussian error function [Φ(z) = 0.5 × (1 + erf(z / √2))].