# What is a t Statistic?

The *t* Distribution is a continuous random sampling distribution, similar to the z distribution, but also reflects sampling error regarding the standard deviation. The *t* distribution is used to test numerous forms of hypotheses. If we let Z be distributed as a Normal distribution, and Q be Distributed as a Chi-Square distribution with v degrees of freedom, and Q and Z be independent, then

has a *t* distribution with v degrees of freedom.

If we let *X*, be a random variable from a Normal population, with mean, μ, and standard deviation, σ. Then

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

The *t* Distribution has a mean of zero and a variance of v/(v-2). The equation defining the *t* distribution curve is:

The *t* distribution has the following shape, which will be a function of the degrees of freedom, *v*.

As the degrees of freedom increase, the *t* distribution becomes closer and closer to that of a *z* distribution.