For a one-sample test about a population mean, use z when the population standard deviation σ is known; use t when σ is unknown and you estimate it with the sample standard deviation s. The sample size alone does not decide between them. A separate rule applies to proportion tests, where z relies on a normal approximation to a binomial distribution.
The one-picture decision rule
Scope: inference about a population mean.
| Question | Choice | Statistic and standard error | Reference distribution |
|---|---|---|---|
| Is population σ known? | Yes | (x̄ − μ0)/(σ/√n) | Standard normal (z) |
| Is σ unknown and estimated by sample s? | Yes | (x̄ − μ0)/(s/√n) | Student’s t, df = n − 1 |
In the second row, replacing σ with s adds uncertainty. The t distribution accounts for that extra estimation uncertainty; as its degrees of freedom increase, its heavier tails shrink and it approaches the normal distribution. The distinction and assumptions are summarized by OpenStax, Introductory Statistics 2e and The Open University’s OpenLearn material.
What changes between z and t?
Known population standard deviation: z
When the population spread σ is genuinely known, the one-sample mean statistic uses σ in its standard error, σ/√n, and is compared with the standard normal distribution. “Known” means the population parameter is available from a reliable source or design—not merely that you calculated a standard deviation from your sample.
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Estimated population standard deviation: t
When σ is not known, calculate the sample standard deviation s and use (x̄ − μ0)/(s/√n). For the ordinary one-sample test, the t distribution has n − 1 degrees of freedom. As OpenStax states, “You use the sample standard deviation to approximate the population standard deviation.” See the full assumptions and notation in OpenStax.
Why t has heavier tails
Estimating σ makes the denominator variable. With fewer degrees of freedom, t therefore places more probability in its tails than the normal distribution, producing more conservative critical values. Increasing n increases the degrees of freedom and makes t increasingly similar to z, but does not change the underlying choice when σ remains unknown.
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Do not confuse mean tests with proportion tests
A z procedure is also common for a population proportion, but that is a different decision. The data follow a binomial model, and a normal approximation is used only when its conditions are adequate. The cited OpenStax section gives np > 5 and nq > 5, with q = 1 − p, together with independence and a common success-probability setup, as conditions for that approximation: OpenStax.
Those np and nq checks do not determine whether a mean test uses z or t. For means, the defining question is whether the population σ is known. For proportions, the defining issue is whether the binomial sampling distribution can be approximated by a normal distribution under the stated sampling conditions.
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Assumptions still matter
Selecting the reference distribution does not make a study valid. Check the design and data before calculating a test statistic.
- Sampling: The one-mean cases described by OpenStax assume a simple random sample.
- Independence: Observations should be independent; for proportions, independence is also part of the normal-approximation conditions.
- Distribution shape: Inspect whether the population or sampling distribution is compatible with the procedure, especially for small samples and skewed or outlier-prone data.
- Parameter and data type: Confirm that the target is a mean, not a proportion or another parameter requiring a different procedure.
See the complete conditions in OpenStax’s hypothesis-testing section.
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A practical selection checklist
- Identify the parameter in the null hypothesis: a population mean μ or a population proportion p.
- If it is a mean, ask whether the population standard deviation σ is known independently of this sample.
- Use z with σ in the standard error when σ is known.
- Use t with s in the standard error when σ is unknown; set df = n − 1 for the ordinary one-sample test.
- If it is a proportion, verify the binomial/normal-approximation conditions, including np > 5 and nq > 5 where that criterion is being used.
- Check sampling, independence and distribution-shape assumptions, then choose the appropriate one- or two-sided alternative and significance level.
Common mistakes to avoid
- “Small n means t, large n means z.” False as a general mean-test rule. Unknown σ points to t; larger n only makes t closer to normal.
- “n ≥ 30 automatically means z.” There is no universal n = 30 cutoff.
- “I know s, so σ is known.” A sample standard deviation is an estimate of σ, not the population parameter itself.
- “Every z-score is a z-test.” A standardized score can describe a value without being a hypothesis test; a test also requires a null hypothesis, sampling model and decision rule.
- Using the mean decision tree for a proportion. Proportion z procedures depend on binomial and normal-approximation conditions instead.
- Ignoring study design. Neither z nor t repairs biased sampling, dependence or a severely unsuitable distribution.
Bottom line for choosing the test
For a population mean, known σ means normal (z); unknown σ estimated by s means t with n − 1 degrees of freedom. The t distribution converges toward normal as the sample grows, so sample size affects tail shape and approximation quality—not the fundamental known-versus-unknown σ decision. Proportion tests are a separate case governed by binomial sampling and normal-approximation conditions.
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