The normal distribution — the familiar bell curve — is the single most important distribution in statistics. It describes countless natural and social measurements (height, measurement error, exam scores) and, via the Central Limit Theorem, approximates the sampling distribution of a sample mean even when the underlying data are not themselves normal. Its shape is fixed by only two parameters: the mean μ (center) and the standard deviation σ (spread). Standardizing any normal variable X into a Z-score, Z = (X − μ) / σ, converts it to the standard normal distribution N(0, 1), for which a single reference table works for every possible μ and σ.
Critical Z Values
| Confidence / p | One-tailed z | Two-tailed z |
|---|
| 90% (α = .1) | 1.282 | 1.645 |
| 95% (α = .05) | 1.645 | 1.960 |
| 97.5% (α = .025) | 1.960 | 2.241 |
| 99% (α = .01) | 2.326 | 2.576 |
| 99.5% (α = .005) | 2.576 | 2.807 |
| 99.9% (α = .001) | 3.090 | 3.291 |
Generated directly from src/core/distributions.js (normInv) — never hand-typed.
Frequently Asked Questions
What does z = 1.96 mean?
z = 1.96 is the value that cuts off exactly 2.5% of the area in each tail of the standard normal distribution — so 95% of the area lies between z = −1.96 and z = 1.96. It is the standard critical value for a 95% confidence interval and for a two-tailed test at α = .05.
How do I read a Z-table?
A standard Z-table gives Φ(z) = P(Z ≤ z), the cumulative probability up to z, for the standard normal N(0,1). Find your z-score's row (whole number + first decimal) and column (second decimal) to read off the cumulative probability directly — no interpolation needed with this calculator, since it computes Φ(z) exactly rather than reading a rounded table.
When do I use z instead of t?
Use z when the population standard deviation σ is known (rare in practice) or when the sample is large enough (conventionally n ≥ 30) that the t-distribution has essentially converged to the normal. Otherwise — a small sample with an estimated (not known) standard deviation — use the t-distribution, which has heavier tails to account for the extra uncertainty in estimating σ from the sample itself.
What is the difference between one-tailed and two-tailed p-values?
A one-tailed p-value, P(Z ≥ z), is the area beyond z in one direction only — appropriate when your hypothesis specifies a direction (e.g. 'greater than'). A two-tailed p-value doubles this (or equivalently sums both tails) and is appropriate for a non-directional hypothesis ('different from'), which is the more conservative and far more common default in research.