The *t* distributions were discovered by William S. Gosset in 1908.
Gosset was a statistician employed by the Guinness brewing company which
had stipulated that he not publish under his own name. He therefore
wrote under the pen name ``Student.'' These distributions arise in
the following situation.

Suppose we have a simple random sample of size *n* drawn from a Normal
population with mean and standard deviation . Let
denote the sample mean and *s*, the sample standard
deviation. Then the quantity

has a *t* distribution with *n*-1 degrees of freedom.

Note that there is a different *t* distribution for each sample size,
in other words, it is a class of distributions. When we speak of a
specific *t* distribution, we have to specify the *degrees of
freedom*. The degrees of freedom for this *t* statistics comes from
the sample standard deviation *s* in the denominator of
equation 1.

The *t* density curves are symmetric and bell-shaped like the normal
distribution and have their peak at 0. However, the spread is more
than that of the standard normal distribution. This is due to the fact
that in formula 1, the denominator is *s* rather than
. Since *s* is a random quantity varying with various samples,
the variability in *t* is more, resulting in a larger spread.

The larger the degrees of freedom, the closer the *t*-density is to
the normal density. This reflects the fact that the standard deviation
*s* approaches for large sample size *n*. You can
visualize this in the applet below by moving the sliders.

The stationary curve is the standard normal density.

Mon Jul 22 01:00:41 PDT 1996