Have you ever collected data from a single group and wondered whether its average is meaningfully different from a number you already have in mind? Maybe you run a training institute and want to know whether your students’ average exam score is really different from the national benchmark of 75. Maybe your company promises customers a 10-minute average wait time, and you need to check whether your actual average differs from that target. Or maybe a customer-satisfaction survey came back at 4.1 out of 5, and you want to know if that is statistically different from the industry average of 3.8.
The one-sample t-test is exactly the tool for questions like these. It compares the mean of a single sample to a known or hypothesized population value — called the “test value” — and tells you whether the difference you observe is statistically significant or could plausibly be due to random sampling variation.
In this tutorial, you will learn when to use it, how to check its assumptions, how to run it in SPSS click by click, the paste-ready syntax, how to read the output, and how to report results in APA style (7th edition) — all with a worked example.
What is a one-sample t-test?
A one-sample t-test compares a sample mean to a single known value. The test value is not estimated from your data — it comes from somewhere else: a published norm, a company target, a passing mark, a historical average, or a theoretically meaningful benchmark.
Every one-sample t-test is built around two competing hypotheses:
- Null hypothesis (H₀): the population mean equals the test value. Written formally: H₀: μ = μ₀.
- Alternative hypothesis (H₁): the population mean differs from the test value. Written formally: H₁: μ ≠ μ₀.
The two-tailed version above (μ ≠ μ₀) is the default in SPSS and the version most researchers use.
Real-world examples include:
- Checking whether patients’ average pain scores after a treatment differ from the accepted clinical threshold.
- Determining whether students’ mean scores differ from the passing mark set by an examination board.
- Comparing the average monthly electricity consumption of households in a neighborhood to the city’s reported average.
Assumptions of the one-sample t-test
Like every statistical test, the one-sample t-test works best when its assumptions are reasonably met. Check these four before running the test:
1. Your dependent variable is continuous
The variable you are testing should be measured at the interval or ratio level — things like test scores, reaction times, weights, prices, or temperatures. A one-sample t-test is not appropriate for categorical (nominal) variables; for those you would use a chi-square test instead.
2. Observations are independent
Each case in your data should contribute one independent measurement. The same person appearing twice in the dataset, or scores from students who copied each other’s answers, would violate this assumption. In practice, independent data collection and sensible study design take care of it.
3. The data are approximately normally distributed
The one-sample t-test assumes that the sample mean follows a normal sampling distribution. If your sample is large (roughly 30 or more cases), the central limit theorem makes this a safe assumption even when the raw data are somewhat skewed. With a small sample, check normality with a Shapiro–Wilk test or a Q–Q plot. If the data are clearly non-normal and the sample is small, consider the non-parametric alternative: the Wilcoxon signed-rank test.
4. There are no extreme outliers
A single extreme value can drag the sample mean (and inflate the standard deviation), distorting the test. A quick boxplot of your variable in SPSS (Graphs > Chart Builder, or Analyze > Descriptive Statistics > Explore) will show you whether any extreme cases need attention before you test.
Running a one-sample t-test in SPSS: click by click
Let us run the test on our worked example. A training institute claims that its students average 75 on a certification exam. You collect the scores of 25 students and want to test whether their true average differs from 75. (All numbers in this example are illustrative, chosen to show you how the output works.)
Step 1 — Open your data file
Open your dataset in SPSS Data View. Your test variable (for example, ExamScore) should sit in a single column, with one row per case.
Step 2 — Open the one-sample t-test dialog
From the menu bar, click:
Analyze > Compare Means > One-Sample T Test
This opens the One-Sample T Test dialog box.
Step 3 — Move your variable into the test box
In the left-hand variable list, click your test variable (e.g., ExamScore), then click the arrow button to move it into the Test Variable(s) box on the right. You can test several variables at once, but start with one.
Step 4 — Enter the test value
In the Test Value field, type the benchmark you are comparing against — in our example, 75. This is μ₀, the known value from your hypothesis.
Step 5 — (Optional) Adjust the options
Click the Options button if you want to change the confidence interval (the default is 95%, which suits most analyses) or how missing values are handled. For a single test variable, the defaults are fine. Click Continue to return to the main dialog.
Step 6 — Click OK and read the output
Click OK. SPSS runs the test and displays the results in the Output Viewer. You will see two tables — we will read them both in the next section.
The test works by computing a t-statistic:
t = (sample mean − test value) ÷ (standard deviation ÷ √n)
SPSS then compares this t-value against the t-distribution with n − 1 degrees of freedom to obtain the p-value.
SPSS syntax for the one-sample t-test
If you prefer syntax — or want a reproducible record of your analysis — paste the following into a new syntax window (File > New > Syntax), edit the variable name and test value, select all, and click Run > All (or the green play button):
T-TEST /TESTVAL=75 /MISSING=ANALYSIS /VARIABLES=ExamScore /CRITERIA=CI(.95).
Replace ExamScore with your own variable name and 75 with your benchmark. The /CRITERIA=CI(.95) subcommand sets the confidence interval to 95%. Save the syntax with your project so you can re-run the analysis on updated data in seconds.
Interpreting the SPSS output
SPSS produces two tables for a one-sample t-test. Here is how to read them, using our training-institute example (n = 25 students, benchmark = 75; illustrative numbers).
The One-Sample Statistics table
This table describes your sample:
- N — the number of valid cases (25 in our example).
- Mean — the sample mean (78.40 in our example).
- Std. Deviation — the sample standard deviation (8.60).
- Std. Error Mean — the standard error of the mean (1.72). This is the standard deviation divided by the square root of n, and it measures how precisely the sample mean estimates the population mean.
At a glance, the sample mean of 78.40 is 3.40 points above the benchmark of 75. The question the test answers is whether a difference of 3.40 points, given this sample size and variability, is larger than what random sampling could plausibly produce.
The One-Sample Test table
This is the table with your answer. Its key columns:
- t — the test statistic. In our example, t = (78.40 − 75) ÷ 1.72 = 1.98.
- df — degrees of freedom, equal to n − 1 (24 in our example).
- Sig. (2-tailed) — the p-value. In our example, p = .059.
- Mean Difference — the sample mean minus the test value (3.40).
- 95% Confidence Interval of the Difference — the range of plausible values for the true mean difference. In our example: −0.15 to 6.95.
Making the decision: compare the p-value to your significance level (α = .05 is standard). Because .059 is greater than .05, we fail to reject the null hypothesis. The data do not provide convincing evidence that the true average differs from 75. Notice the careful wording: a non-significant result does not prove the mean equals 75 — it only means we cannot rule it out. Notice also that the 95% confidence interval includes zero, which tells the same story: zero difference is a plausible value.
Don’t stop at the p-value: check the effect size
A p-value tells you whether a difference is statistically significant, not whether it is practically important. For a one-sample t-test, the standard effect size is Cohen’s d, which you compute by hand as:
d = (sample mean − test value) ÷ standard deviation
In our example: d = 3.40 ÷ 8.60 = 0.40. By the conventional benchmarks, 0.20 is a small effect, 0.50 is medium, and 0.80 is large — so 0.40 is a small-to-medium effect. Even if our p-value had been significant, d = 0.40 would tell a reader that the difference, while real, is modest in size. Always report an effect size alongside your p-value: significance answers “is there an effect?”, effect size answers “how big is it?”
Reporting a one-sample t-test in APA style (7th ed.)
A complete APA-style report includes: the test name, the sample mean and SD, the t-statistic with degrees of freedom, the exact p-value, the mean difference, the 95% confidence interval, and the effect size. Report exact p-values (e.g., p = .059); when SPSS prints .000, write p < .001.
Here is a model sentence based on our example:
A one-sample t-test showed that students’ mean exam score (M = 78.40, SD = 8.60) did not differ significantly from the national benchmark of 75, t(24) = 1.98, p = .059, mean difference = 3.40, 95% CI [−0.15, 6.95], d = 0.40.
If the result had been significant, the wording would change only slightly: “differed significantly from the benchmark of 75, t(24) = 2.61, p = .015 …”. Keep the same structure either way — readers of research expect the t-value, degrees of freedom, and p-value in exactly this format.
Key Takeaways
Key Takeaways
- Use a one-sample t-test when you want to compare the mean of one group to a known benchmark value (the test value).
- Check the assumptions first: a continuous dependent variable, independent observations, approximately normal data (or n ≥ 30), and no extreme outliers.
- In SPSS, the path is Analyze > Compare Means > One-Sample T Test — move your variable in, type the test value, click OK.
- Read the One-Sample Test table: the t-value, df, and Sig. (2-tailed) p-value give you the decision; the confidence interval gives you the precision.
- Report t(df) = value, p = value, the mean difference, the 95% CI, and Cohen’s d in APA style — significance plus effect size, every time.
Frequently asked questions
When should I use a one-sample t-test instead of an independent-samples t-test?
Use a one-sample t-test when you have one group and you are comparing its mean to a fixed benchmark that you did not estimate from data — a target, a norm, or a passing mark. Use an independent-samples t-test when you have two separate groups (for example, men vs. women, or treatment vs. control) and you want to compare their means to each other.
What if my data are not normally distributed?
With about 30 or more cases, the one-sample t-test is robust to moderate departures from normality, thanks to the central limit theorem. With a small sample and clearly skewed data, check normality first (Shapiro–Wilk test or Q–Q plot in SPSS). If the assumption fails, use the non-parametric Wilcoxon signed-rank test instead (Analyze > Nonparametric Tests > One Sample).
Can I run a one-sample t-test on Likert scale responses?
Single Likert items are ordinal, so strictly the t-test is not designed for them. In practice, researchers routinely test scale scores — the mean of several Likert items — and the test behaves well with five or more response options and a roughly symmetric distribution. Be transparent about what you did.
My result was not significant. Does that mean there is no difference?
No. A non-significant result means the data do not provide enough evidence to reject the null hypothesis — it does not prove the null is true. The difference might be real but small, or your sample might be too small to detect it (low statistical power). Report the effect size and confidence interval so readers can judge for themselves.
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