Z Table (Normal CDF)
Compute P(Z ≤ z) and P(|Z| ≤ z) for the standard normal distribution.
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.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.