# What are the necessary and sufficient criteria for a distribution to be a Student's t-distribution?

Wikipedia states

In probability and statistics, Student's t-distribution (or simply the t-distribution) is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown.

And Britannica also states similarly

Student’s t-test, in statistics, a method of testing hypotheses about the mean of a small sample drawn from a normally distributed population when the population standard deviation is unknown.

Is the small sample size really a necessary condition for the distribution to be a t-distribution? Isn't a sample drawn from a normal population with unknown variance a necessary and sufficient criterion?

• You’re right about necessary and sufficient. Their phrasing only describes a sufficient condition, though it would be nice if they didn’t make it sound necessary.
– Dave
Jul 4 '20 at 14:57