One-Sample T-Test -Running, Interpreting, and Reporting One Sample T-Test

A one-sample t-test is the go-to method when researchers need to check whether a sample’s mean matches a known population mean—useful for judging whether collected data is representative of a target population. The test compares the sample mean to a specified “test value” (the population mean). In practice, it’s applied when a population mean is already known and the question is whether the sample differs significantly from that benchmark.

Common scenarios include economic representativeness checks—such as comparing a city’s per capita income against the country’s per capita income—and quality control—such as determining whether product dimensions have shifted from original specifications during manufacturing. In both cases, the researcher treats the population mean as fixed, collects sample data, and then tests whether the sample mean is statistically different from that population mean.

Running the analysis in SPSS follows a straightforward workflow: go to Analyze → Compare Means → One-Sample T Test. The sample variable (e.g., per capita income data from 100 people) is entered as the “Test Variable,” while the known population mean is entered as the “Test Value.” Once executed, SPSS produces two key tables. The first, “One-Sample Statistics,” reports the sample size (n = 100), the sample mean (reported as 1107 in the example), and the sample standard deviation (271). The second table provides the inferential results needed to decide whether the difference from the population mean is statistically meaningful.

Interpretation hinges on the p-value (labeled as “Sig.” in the output). In the example, the significance is two-tailed and is less than 0.05, which indicates a statistically significant difference between the sample mean and the population mean. That outcome leads to the conclusion that the sample does not adequately represent the population mean.

Reporting the results requires translating the statistics into a clear statement tied to hypotheses. The typical setup is: the null hypothesis claims no significant difference between the city’s per capita income and the country’s per capita income, while the alternative hypothesis claims the opposite. In the provided reporting example, the descriptive statistics are used to contextualize the direction of the difference, and the t-statistic is referenced as exceeding a common critical threshold (t value greater than 1.96). The write-up concludes that the city’s average per capita income is significantly higher than the country’s average, so the null hypothesis is not supported and the sample mean differs from the population mean.

Finally, the transcript suggests practical formatting tips for presenting results: combine relevant tables where possible, include the test value as a note when needed, and present the mean and standard deviation clearly alongside the inferential statistics. The overall takeaway is that the one-sample t-test provides both a statistical decision (via p-value) and a reporting-ready narrative linking sample behavior to a known population benchmark.