Z-Test Online: One-Sample and Two-Sample with Known Variance
What is the One-Sample Z-Test?
The one-sample Z-test is used when the population standard deviation σ is known, testing whether a sample mean equals a target value with the statistic Z = (x̄ − μ0)/(σ/√n), which follows the standard normal distribution. Unlike the t-test it does not consume degrees of freedom, but it requires σ to be known, usually from long-term process history, equipment accuracy or industry standards. The tool returns the p-value for the Z-value and can also compute critical values at a given significance level.
When to Use It
Use it for comparing means when the population variance is known, or when both sample sizes are large (generally n ≥ 30 per group) so the central limit theorem makes the sample means approximately normal. Typical cases include comparing two processes or supplier batch means in high-volume production. When both variances are unknown but both samples are large, substituting sample standard deviations for σ is an acceptable approximation.
How to Use It (Step by Step)
Enter the sample mean or means, the known standard deviation(s), sample size(s) and significance level. The tool computes the Z-value, p-value and confidence interval, and also provides power and required sample size calculations. Use the power function before the study to confirm the design is sensitive enough to avoid false negatives.
Key Formulas / Example
One-sample Z = (x̄ − μ0)/(σ/√n); two-sample Z = (x̄1 − x̄2)/√(σ1²/n1 + σ2²/n2). Power = 1 − β depends on α, effect size and sample size. For example, the two-sided 95% critical Z-value is 1.96, so |Z| > 1.96 rejects H0 at α = 0.05.