One-Sample t-test
One-Sample Student t-test · Also known as: single-sample t-test, one-group t-test, one-sample t, Student one-sample t-test
The one-sample t-test is a parametric hypothesis test that determines whether the mean of a single sample differs significantly from a known or hypothesized population value. Derived from Student's (Gosset's) 1908 t-distribution, it assumes continuous, approximately normally distributed data and is one of the most fundamental tests in applied statistics.
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When to use it
Use the one-sample t-test when you have one group of continuous observations and want to compare the group mean against a fixed reference value — a legal limit, an established norm, a theoretical prediction, or a historical average. The data should be approximately normally distributed (verify with Shapiro-Wilk or a Q-Q plot), and observations must be independent of each other. Do not use it when the comparison value itself is estimated from data (use independent t-test or paired t-test instead), when the data are ordinal or heavily skewed in small samples (use the Wilcoxon signed-rank test instead), or when comparing more than one group.
Strengths & limitations
- Conceptually simple: requires only one sample and a reference value.
- Powerful for detecting mean shifts when normality holds, especially with moderate-to-large samples.
- Produces an interpretable effect size (Cohen's d) and a confidence interval for the mean difference.
- Directly available in every major statistical package (SPSS, R, Python, Stata, Excel).
- Robust to mild non-normality when sample size exceeds about 30, by the central limit theorem.
- Assumes approximate normality; unreliable with small samples from strongly skewed or heavy-tailed distributions.
- The reference value must be fixed and known a priori, not estimated from another sample.
- Sensitive to outliers, which can inflate the standard deviation and mask or exaggerate the true effect.
- Provides no information about variability or distribution shape beyond what the standard deviation captures.
Frequently asked
What is the 'test value' and how do I choose it?
The test value (also called mu-zero) is the hypothesized population mean — a fixed number derived from theory, regulation, a published norm, or a historical record. It must be specified before data collection. Common examples are a national test-score average, a legal threshold, or a manufacturing specification.
When should I switch to the Wilcoxon signed-rank test?
Switch when the sample is small (n < 30) and the Shapiro-Wilk test rejects normality, or when the data contain strong outliers or come from an ordinal scale. The Wilcoxon signed-rank test is the nonparametric alternative that compares the median (or the distribution more broadly) to the reference value without assuming normality.
Can I use the one-sample t-test for a one-tailed hypothesis?
Yes. If you predict a directional difference before seeing the data (e.g., the mean will be greater than the norm), a one-tailed test focuses all rejection power on that direction, making it easier to reject the null in that direction but impossible to detect a significant effect in the opposite direction. Use one-tailed tests only when the direction is theoretically justified a priori.
How is this different from the paired samples t-test?
In the paired t-test you compute the difference score for each participant (e.g., post minus pre) and then test whether that difference mean equals zero — which is essentially a one-sample t-test on difference scores. The paired t-test is appropriate when each observation has two related measurements; the one-sample t-test is used when you compare one set of observations against a fixed external value.
What effect size should I report?
Cohen's d is the standard: divide the difference between the sample mean and the test value by the sample standard deviation. Values around 0.2 are small, 0.5 medium, and 0.8 large by Cohen's (1988) benchmarks. Also report the 95% confidence interval for the mean difference to convey estimation precision.
Sources
- Student (1908). The probable error of a mean. Biometrika, 6(1), 1–25. DOI: 10.1093/biomet/6.1.1 ↗
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). SAGE. ISBN: 978-1446249185
How to cite this page
ScholarGate. (2026, June 3). One-Sample Student t-test. ScholarGate. https://scholargate.app/en/statistics/one-sample-t-test
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Independent samples t-testStatistics↔ compare
- One-way ANOVAStatistics↔ compare
- Paired samples t-testStatistics↔ compare