One-Sample z-Test

z = (x̄ − μ) / (σ/√n). Compare sample mean to known population.

Inputs

Result

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How to use this calculator

  • Enter sample mean + population μ + σ + n.

About this calculator

One-sample z-test compares sample mean to a known/hypothesized population mean (with known σ). Used when population SD is known and sample is large (n ≥ 30). Otherwise prefer t-test. Foundational in quality control (process drift detection), educational testing (group vs. norm), epidemiology (incidence comparison). Source: Wolfram MathWorld - Z-Test.

Frequently asked

When use z vs. t?+
z: σ known, n large. t: σ estimated from sample. Practically: use t (more conservative, generalizes).
One-tail vs. two-tail?+
Two-tail: H₁ "different" (most common). One-tail: H₁ specifies direction (a priori). One-tail gives smaller p but only valid if pre-specified.
Critical z values?+
α = 0.05 two-tail: |z| > 1.96. α = 0.01: |z| > 2.576. α = 0.05 one-tail: z > 1.645.
Effect size?+
Cohen's d = (x̄ − μ) / σ. Same scale as t-test.
Confidence interval?+
x̄ ± z_α/2 × σ/√n. 95% CI: x̄ ± 1.96 × SE.

Related calculators

Two-Sample t-Test (Welch's)
Test whether two means differ. t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂).
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χ² = Σ(O−E)²/E. Test if observed frequencies match expected.
Sample Size for Margin of Error
n = (z × σ / E)². Required sample size for given margin E at confidence z.

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