Z Table (Normal CDF)

Standard normal Z table

The Z table returns the cumulative probability P(Z ≤ z) for the standard normal distribution Z ~ N(0, 1). Reference values: z = 0 → 0.5; z = 1 → 0.8413; z = 1.645 → 0.95 (one-sided 5%); z = 1.96 → 0.975 (two-sided 95%); z = 2.576 → 0.995 (two-sided 99%). How to use: standardize the raw variable with z = (x − μ) / σ and look up the value. Printed tables usually list the right tail; for negative z apply symmetry: P(Z ≤ −z) = 1 − P(Z ≤ z). Today software computes the value directly from the error function (erf), with no tabular interpolation needed.

Applications

Hypothesis testing (z-test for proportions and large means), confidence intervals (95% CI ≈ x̄ ± 1.96·σ/√n), statistical process control (Six Sigma: 3σ defect rate = 0.27%; 6σ ≈ 3.4 ppm), psychometrics (IQ with μ = 100, σ = 15), and financial risk (Value at Risk).

FAQ

How to read negative z? Use symmetry: P(Z ≤ −1.96) = 1 − P(Z ≤ 1.96) = 1 − 0.975 = 0.025.

Z table or t table — when? Z assumes known σ (or large n). With small samples and σ estimated from the data, use the Student t table.

Why use 1.96 for 95% confidence? Because P(−1.96 ≤ Z ≤ 1.96) ≈ 0.95 leaves 2.5% in each tail. People often round to 2, but the exact value 1.96 is the convention in scientific reports.