Two groups, no links between them
The independent-samples t-test compares two unrelated groups: different people, different classes, no pairing. Its statistic is the mean difference divided by the standard error of that difference, and the modern default form is Welch's, which lets each group keep its own variance instead of pooling them into one. Welch costs almost nothing when variances are equal and rescues the test when they are not, which is why this site, R's t.test and most current teaching default to it.
Class A (n = 8) averages 84.625; class B (n = 8) averages 75.750; the difference is 8.875 marks. Welch's standard error combines the two group variances separately and yields t = 4.484 on about 11.3 degrees of freedom (the fractional df is Welch's signature: it interpolates between the two variance estimates). Two-tailed p = 0.0009, and the 95% interval for the true difference runs 4.531 to 13.219 marks: class A is ahead, by somewhere between a solid and a large margin.
Choosing between paired, Welch and rank tests
The design decides, not the data's mood. Same units twice: paired. Two separate groups with roughly normal data: Welch. Two separate groups with heavy skew or ordinal scores at small n: the Mann-Whitney U test, which compares ranks and asks a slightly different question (stochastic ordering rather than mean difference). Equal-variance pooling (Student's original) survives mostly in textbooks; when a report uses it, Levene's test is the customary justification, and when in doubt Welch is the answer that rarely embarrasses anyone.
Splitting one group at the median and t-testing the halves to "discover" a difference. The split manufactures groups from the same variable being tested, guaranteeing artefacts. Group membership must exist before the outcome is measured.
Frequently asked questions
What is the difference between Welch and Student's t-test?
Student's pools both groups into one variance estimate; Welch keeps separate variances and adjusts the degrees of freedom. Welch is the safer default and is what R runs unless told otherwise.
Why are the degrees of freedom fractional?
Welch's df formula blends the two groups' variance information, and the blend rarely lands on a whole number. Software reports it as is, 11.3 in the example.
Can I use this test on paired data by mistake?
You can compute it, but you would throw away the pairing and its precision. The design, same units measured twice or not, picks the test before any numbers are seen.