TITLE: What is the T-distribution, and what is it used for?
QUESTION [8 upvotes]: (I'll post my own answer to this, but don't hesitate to post your own!)
Student's t-distribution, or T-distribution, was introduced in 1908 by William Sealey Gossett writing under the pseudonym "Student".
What is it, and what is it for?

REPLY [3 votes]: Oh, that's much more than I was going to say:  $T$ is the distribution of
$${\bar x - \mu \over s / \sqrt{n}}$$
where $x$ is a normally distributed variable with mean $\mu$, and $\bar x$ and $s$ are
the sample mean and standard deviation (resp.) of a random sample of size $n$.  (I'm assuming everyone understands that a random sample meets the same conditions as Michael Hardy's $X_1,\dots,X_n$.)
I can add two things:  Gossett worked for Guiness, which did not allow its researchers to publish; hence he resorted to the pseudonym.  The t-distribution is developed in The Probable Error of the Mean, Biometrika, 6(1), (Mar. 1908) 1-25.