🧮 One-Sample z-Test Calculator

Evaluate a population mean against a hypothesised value when the population standard deviation is known (or the sample size is large enough to treat it as known). Enter your sample statistics to obtain the z statistic, p-value, decision, effect size, and a narrated interpretation.

Tail Direction

Significance Level (\\(\alpha\\))

Custom α must be between 0.001 and 0.25.

Formula Reference

The one-sample z statistic is

\\[ z = \dfrac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \\]

The standard error is \\(\sigma / \sqrt{n}\\). For two-tailed tests, we reject \\(H_0\\) when \\(|z| \ge z_{1-\alpha/2}\\). For one-tailed tests we reject when \\(z \le z_{\alpha}\\) (left) or \\(z \ge z_{1-\alpha}\\) (right).

How to Use This Calculator

  1. Collect your sample mean \\(\bar{x}\\), population standard deviation \\(\sigma\\), sample size \\(n\\), and the null hypothesis value \\(\mu_0\\).
  2. Select the tail that matches your alternative hypothesis and choose the significance level \\(\alpha\\).
  3. Press “Run z-Test” to compute the statistic, critical value, p-value, and effect size.
  4. Compare the p-value with \\(\alpha\\) or the z statistic with the critical region to conclude whether to reject \\(H_0\\).
  5. Review the interpretation and effect-size narrative to report your findings.

References

  • Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
  • Montgomery, D. C., & Runger, G. C. (2014). Applied Statistics and Probability for Engineers (6th ed.). Wiley.
  • Rice, J. A. (2006). Mathematical Statistics and Data Analysis (3rd ed.). Cengage.

Disclaimer

This calculator assumes independent observations and known population variance (or large-sample approximation). Always review the conditions for inference before using the result in decision-making.