---
title: "One-sample z-test (known sigma)"
method_id: SM.DIF.PAR.1SAM.ZTEST
family: Difference
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/one_sample_z_test.md
source: Clarus method library
---

# One-sample z-test (known sigma)

> A classical statistical method in the Clarus library (Difference family).
> Clarus selects, assumption-checks and runs this method on your data with a
> deterministic rule engine and real numerical libraries (scipy / statsmodels);
> it never generates numbers, citations or results.

**Also known as:** single-sample z-test; one-sample location z-test; z-test against mu0 with known population SD; large-sample mean test.

## Hypotheses

- **Null (H0):** mu_{dependent} = {mu0}
- **Alternative (H1):** mu_{dependent} != {mu0}

## When to use it

Use this method when your goal is to:
- compare one mean to reference

## Data it expects

- **dependent:** 1 continuous
- **independent:** none
- **pairing:** single_sample_vs_constant
- **population_sd:** known

## Assumptions Clarus checks

- **known population sd** (severity: error; on violation: reroute)
- **normality or large n** (gate: shapiro_wilk; severity: warning; on violation: warn_only)
- **independence of observations** (severity: warning; on violation: warn_only)

## Effect size reported

- **cohen d** (Cohen (1988))

## Honest limitations

- The z-test assumes the population standard deviation sigma is genuinely KNOWN; if it is estimated from the sample, the p-value and CI are too narrow and a t-test should be used.
- Large n can pair statistical significance with a trivial difference from mu0 — read Cohen's d and the confidence interval.
- A one-sample z-test is observational; a difference from the reference value cannot support a causal claim.
- Results generalize only to the population from which the sample was drawn, and assume the reference value mu0 is meaningful on the same scale.
- If p >= alpha, this is absence of evidence for a difference, not evidence that the mean equals mu0.

## Primary sources

- Moore, McCabe & Craig (2017). Introduction to the Practice of Statistics (9th ed.). W.H. Freeman, Ch.6 (z procedures, sigma known).
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum.
- Field, A. (2013). Discovering Statistics Using IBM SPSS (4th ed.). Sage, Ch.5.

## How to cite this

To cite this Clarus method page (the page itself — for the method's own primary sources, see above):

**Plain text**

Clarus, "One-sample z-test (known sigma)", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/one_sample_z_test.md

**APA**

Clarus. (2026). One-sample z-test (known sigma) (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/one_sample_z_test.md

**BibTeX**

```bibtex
@misc{clarus-one-sample-z-test,
  author       = {Clarus},
  title        = {One-sample z-test (known sigma)},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/one_sample_z_test.md}
}
```

_No DOI is minted for Clarus method pages yet; this citation uses the canonical URL, version and date. Cite the primary sources above for the method's scientific provenance._

---

METHOD SM.DIF.PAR.1SAM.ZTEST · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library