One-Sample t-Test — StatReason

Sample mean
Difference from benchmark
95% CI of the mean
t
Degrees of freedom
p-value
Cohen's d

What this answers

This test answers "is my sample's average meaningfully different from a specific known benchmark?" Common uses include checking whether a manufacturing process's average output matches a target specification, or whether a sample's average score differs from a known population value.

When to use it, and when not to

Use this when you have one sample and a single fixed number to compare it against, not another sample. If you have two groups to compare against each other, use the Independent t-Test (unrelated groups) or Paired t-Test (matched measurements) instead.

How it is calculated

The t statistic is the difference between your sample mean and the benchmark, divided by the standard error of the mean (sample standard deviation divided by the square root of n). Degrees of freedom are n minus 1. Cohen's d, the effect size reported here, is that same difference divided by the sample standard deviation rather than the standard error, giving a scale-free measure of how large the gap is in standard-deviation units.

Worked example

For the sample 1, 2, 3 tested against a benchmark of 0: the sample mean is 2, the standard error is 1 divided by the square root of 3 (about .577), giving a t statistic of about 3.46 on 2 degrees of freedom, a fairly large t for such a small sample.

Assumption audit

Calculated from your data: sample size, mean, standard deviation, and whether the sample is constant (zero variance).

Evidence to review: whether your sample looks roughly symmetric, which affects how reliable the p-value is, especially for a small sample.

You must verify: that your sample was drawn independently and randomly with respect to the question being asked, and that the benchmark value is a genuinely meaningful, pre-specified target, not chosen after seeing the data.

What this result does not mean

A significant result means the data provide evidence the true mean differs from the benchmark at your chosen threshold, not that the difference is practically important; check Cohen's d for that. A nonsignificant result means the test was inconclusive at that threshold, not proof the true mean equals the benchmark exactly.

Limitations

With a small sample, this test has limited power to detect a real but modest difference from the benchmark, and the confidence interval will be correspondingly wide.