quality:corruption, linguistic:paraphrasing
Unbiased variance estimate

Let x 1, …, x n ∼ N (μ, σ 2) {\displaystyle \ x_{1},\ldots,x_{n}\sim {\mathcal {N}}(\mu,\sigma ^{2})\ } be independent and identically distributed samples from a normal distribution with mean μ {\displaystyle \mu } and variance σ 2. {\displaystyle \ \sigma ^{2}~.} The sample mean and unbiased sample variance are given by:

The resulting (one sample) t statistic is given by

and is distributed according to a Student's t distribution with n − 1 {\displaystyle \ n-1\ } degrees of freedom.

Thus for inference purposes the t statistic is a useful " pivotal quantity " in the case when the mean and variance (μ, σ 2) {\displaystyle (\mu,\sigma ^{2})} are unknown population parameters, in the sense that the t statistic has then a probability distribution that depends on neither μ {\displaystyle \mu } nor σ 2. {\displaystyle \ \sigma ^{2}~.}