 Sample mean and covariance

For observations $X_1,\dots ,X_n\in {\mathbb{R}}^p$, the sample mean and empirical covariance matrix describe average location and joint variation:Each summand is an outer product, so $S_n$ is positive semidefinite. If the observations are independent and identically distributed random variables with covariance matrix $\Sigma$, then $\mathbb{E}{S}_n=(n-1)\Sigma /n$. For $n\ge 2$, replacing the divisor $n$ by $n-1$ gives an unbiased estimator.

For the three observations $(0,0),(1,2),(2,1)$, the sample mean is $(1,1)$. The sum of centered outer products isso the $n$-divisor sample covariance matrix is $\left(\begin{array}{cc}2/3& 1/3\\ 1/3& 2/3\end{array}\right)$, and the $n-1$ version is $\left(\begin{array}{cc}1& 1/2\\ 1/2& 1\end{array}\right)$. Stating the divisor avoids ambiguity when comparing formulas. The rank of a centered sample covariance matrix also explains why too few observations force singularity in high dimension.

Sample mean and covariance are statistical measures that help describe the properties of a dataset. ### Sample Mean The **sample mean** is a measure of central tendency that represents the average of a set of observations. It is calculated by summing all the values in the sample and then dividing by the number of observations in that sample.