When to Use a One-Sample t-Test
A one-sample t-test compares the mean of a single sample to a known or hypothesized population value. It answers one straightforward question: does this group differ from a specific standard?
You would use a one-sample t-test when:
- Testing against a population norm. A researcher wants to know whether students at a particular university score differently on a standardized IQ test compared to the national average of 100.
- Comparing to a benchmark or criterion. A quality control engineer measures whether the average weight of cereal boxes differs from the labeled weight of 500 g.
- Evaluating change from a baseline. A psychologist assesses whether average reaction times in a treatment group differ from a known baseline of 250 ms.
The key requirement is that you have a single continuous variable measured on one group, and a fixed reference value to compare it against. The data should be approximately normally distributed, especially with small samples.
The APA Reporting Template
APA 7th edition requires a specific format for reporting t-test results. For a one-sample t-test, the standard template is:
t(df) = X.XX, p = .XXX, d = X.XX
Where:
| Symbol | Meaning | |--------|---------| | t | The t-statistic (italicized) | | df | Degrees of freedom, calculated as N - 1 | | p | The p-value (no leading zero) | | d | Cohen's d effect size (with leading zero) |
Formatting rules to remember:
- Use italics for statistical symbols: t, p, d, M, SD, N
- Report p-values without a leading zero (.034, not 0.034) because p cannot exceed 1.0
- Report effect sizes with a leading zero (0.75, not .75) because d can exceed 1.0
- Use two decimal places for t and d, three for p
- For very small p-values, write p < .001 rather than the exact value
Step 1: Report Descriptive Statistics
Before presenting inferential results, APA style requires you to report the descriptive statistics of your sample and clearly state the test value.
The sample (N = 45) had a mean IQ score of M = 105.30 (SD = 12.40). Scores were compared to the population mean of 100.
Key elements to include:
| Element | What to report | Example | |---------|---------------|---------| | Sample size | N = | N = 45 | | Sample mean | M = | M = 105.30 | | Standard deviation | SD = | SD = 12.40 | | Test value | The known/hypothesized value | Population mean of 100 |
Always specify what the test value represents. Simply writing "compared to 100" is insufficient. State the source: a population parameter, a published norm, a regulatory standard, or a theoretical expectation.
Step 2: Report the t-Test Results
After the descriptives, report the inferential statistics. Include the t-statistic, degrees of freedom, exact p-value, and a confidence interval around the mean difference.
A one-sample t-test revealed that participants' IQ scores were significantly higher than the population mean of 100, t(44) = 2.87, p = .006, 95% CI [1.58, 9.02].
Breaking this down:
- df = 44 because N - 1 = 45 - 1 = 44
- 95% CI [1.58, 9.02] is the confidence interval of the mean difference (sample mean minus test value). Since the interval does not include zero, this confirms the significant result.
- The direction of the effect is stated in words ("significantly higher than") rather than relying solely on the sign of the t-statistic.
If the p-value is extremely small:
t(44) = 4.52, p < .001
Step 3: Report Effect Size (Cohen's d)
APA 7th edition strongly recommends reporting effect sizes alongside significance tests. For a one-sample t-test, Cohen's d is calculated as:
d = (M - mu) / SD
Where M is the sample mean, mu is the test value, and SD is the sample standard deviation.
Interpretation guidelines (Cohen, 1988):
| Cohen's d | Interpretation | |-------------|----------------| | 0.20 | Small effect | | 0.50 | Medium effect | | 0.80 | Large effect |
For the IQ example: d = (105.30 - 100) / 12.40 = 0.43, indicating a small-to-medium effect.
The effect size was moderate, d = 0.43.
Always report the effect size even when the result is not significant. A non-significant p-value with a medium effect size tells a different story than a non-significant p-value with a negligible effect.
Complete APA Reporting Example
Here is a full paragraph combining all elements for a one-sample t-test, suitable for the Results section of a manuscript.
Scenario: A researcher measured IQ scores in a sample of 45 university students to determine whether their cognitive ability differed from the general population mean of 100.
A one-sample t-test was conducted to determine whether the mean IQ score of the sample differed from the population mean of 100. The sample mean (M = 105.30, SD = 12.40) was significantly higher than the test value, t(44) = 2.87, p = .006, d = 0.43, 95% CI of the difference [1.58, 9.02]. The effect size indicated a small-to-medium difference between the sample and the population norm.
This paragraph includes every element a reviewer expects: the purpose, descriptive statistics, test results, effect size, confidence interval, and a brief interpretation.
Reporting Non-Significant Results
When the one-sample t-test is not significant, you still report all of the same statistics. The key difference is in the language: avoid saying the groups "are equal" or that "there was no difference." Instead, state that no statistically significant difference was found.
Scenario: A nutritionist measured the daily caloric intake of 30 participants and compared it to the recommended 2,000 calories.
A one-sample t-test indicated that the mean daily caloric intake (M = 2,045.00, SD = 180.50) did not differ significantly from the recommended value of 2,000 calories, t(29) = 1.37, p = .182, d = 0.25, 95% CI [-22.40, 112.40]. The small effect size suggests that any deviation from the recommendation was minimal.
Notice that the confidence interval includes zero, which is consistent with the non-significant result. Also note that the effect size is still reported and interpreted.
One-Tailed vs Two-Tailed One-Sample t-Tests
By default, a one-sample t-test is two-tailed, testing whether the sample mean differs from the test value in either direction. A one-tailed test is appropriate only when you have a strong directional hypothesis established before data collection.
Two-tailed (default):
A one-sample t-test was conducted to determine whether scores differed from the national average of 75.
One-tailed:
A one-sample t-test was conducted to determine whether scores exceeded the national average of 75.
When reporting a one-tailed test in APA format, you must:
- Justify the directionality in the introduction or method section
- State the direction clearly in the results (e.g., "exceeded," "was lower than")
- Label the p-value to avoid ambiguity:
t(39) = 1.92, p = .031, one-tailed, d = 0.30
Some journals require you to report the one-tailed p-value explicitly. Others prefer that you report the two-tailed p-value and note that the test was one-tailed. Check your target journal's guidelines.
One-Sample t-Test vs Other Tests
vs Independent-Samples t-Test
These tests answer fundamentally different questions. A one-sample t-test compares one group to a fixed value. An independent-samples t-test compares two separate groups to each other. If you are comparing exam scores from two different classes, that is an independent-samples test. If you are comparing one class against a national standard, that is a one-sample test.
vs One-Sample Wilcoxon Signed-Rank Test
When the normality assumption is violated and the sample is small (typically N < 30), the one-sample Wilcoxon signed-rank test is the non-parametric alternative. It tests whether the median (rather than the mean) differs from the test value.
A one-sample Wilcoxon signed-rank test indicated that median response times (Mdn = 260.50 ms) were significantly higher than the baseline of 250 ms, T = 312, z = 2.15, p = .032, r = .34.
Use the Wilcoxon alternative when your data are heavily skewed, contain outliers, or are measured on an ordinal scale.
vs Paired-Samples t-Test
A common confusion: the paired-samples t-test also involves a single group, but it compares two related measurements (e.g., pre-test vs post-test). The one-sample t-test compares one measurement to a fixed constant, not to another measurement from the same participants.
Common Mistakes in One-Sample t-Test Reporting
1. Not specifying the test value. Every one-sample t-test report must state what value the sample was compared to and where that value comes from. Writing "a one-sample t-test was significant" without mentioning the comparison value is incomplete.
2. Omitting effect size. Reporting t and p alone is no longer considered sufficient. APA 7th edition requires or strongly recommends an effect size measure. Cohen's d takes minimal effort to calculate and adds meaningful context.
3. Confusing one-sample with paired-samples t-test. If you are comparing pre-treatment to post-treatment scores within the same participants, that is a paired test, not a one-sample test. The one-sample t-test requires a fixed, known comparison value that does not come from your data.
4. Not checking the normality assumption. The one-sample t-test assumes approximately normal data. For small samples, run a Shapiro-Wilk test or inspect a Q-Q plot. If normality is violated, consider the Wilcoxon signed-rank test or note the violation and rely on the t-test's robustness for larger samples.
5. Using a leading zero for p-values. Write p = .034, not p = 0.034. This is a specific APA convention because p-values are bounded between 0 and 1.
6. Failing to state the direction. Always describe whether the sample mean was above or below the test value. The sign of the t-statistic alone is not enough for readers to understand the practical meaning of the result.
One-Sample t-Test APA Checklist
Before submitting your manuscript, verify that your one-sample t-test reporting includes all required elements:
- [ ] Purpose of the test clearly stated
- [ ] Test value specified with its source
- [ ] Sample size (N) reported
- [ ] Descriptive statistics: M and SD
- [ ] t-statistic with degrees of freedom: t(df) = X.XX
- [ ] Exact p-value (or p < .001): p = .XXX
- [ ] Effect size with interpretation: d = X.XX
- [ ] 95% confidence interval of the mean difference
- [ ] Direction of effect described in words
- [ ] Normality assumption addressed
- [ ] Italics used for all statistical symbols
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