Editing Student's t-distribution (section)

=====Unbiased variance estimate=====
Let <math>\ x_1, \ldots, x_n \sim {\mathcal N}(\mu, \sigma^2)\ </math> be independent and identically distributed samples from a normal distribution with mean <math>\mu</math> and variance <math>\ \sigma^2 ~.</math> The sample mean and unbiased [[sample variance]] are given by:

: <math>
\begin{align}
 \bar{x} &= \frac{\ x_1+\cdots+x_n\ }{ n }\ , \\[5pt]
 s^2     &= \frac{ 1 }{\ n-1\ }\ \sum_{i=1}^n (x_i - \bar{x})^2 ~.
\end{align}
</math>

The resulting (one sample) {{mvar|t}}&nbsp;statistic is given by

: <math> t = \frac{\bar{x} - \mu}{\ s / \sqrt{n \ }\ } \sim t_{n - 1} ~.</math>

and is distributed according to a Student's {{mvar|t}}&nbsp;distribution with <math>\ n - 1\ </math> degrees of freedom.

Thus for inference purposes the {{mvar|t}}&nbsp;statistic is a useful "[[pivotal quantity]]" in the case when the mean and variance <math>(\mu, \sigma^2)</math> are unknown population parameters, in the sense that the {{mvar|t}}&nbsp;statistic has then a probability distribution that depends on neither <math>\mu</math> nor <math>\ \sigma^2 ~.</math>