# When should I search for the covariance matrix instead of the variance?

Suppose I have a random variable $X$ and $n$ realizations of this variable: $x_1, ..., x_n$. It seems clear to me in that case that if I am interested in knowing the variability I have in my data (realizations) then I should calculate the variance of $X$ i.e. $var(X)=E[(X-E(X))^2]$ where if I'm interested in its value I can compute its estimate (unbiased sample variance for example): $\frac{1}{n-1}\sum_{i=1}^n(x_i-\hat{\mu})^2$ where $\hat{\mu}=\frac{1}{n}\sum_{i=1}^n x_i$ is the sample mean.

Suppose now that I just take the realizations $x_1, ..., x_n$ and put them inside a vector $\mathbf{x}$. Then if I'm again interested in knowing the variability I have in my data then what should I compute?

• Is it $var(\mathbf{x})$ and what does it give in that case?
• Is it a covariance matrix I need to compute i.e. $E[(\mathbf{x}-E(\mathbf{x}))(\mathbf{x}-E(\mathbf{x}))^H]$? If so why?

Let $X$ be univariate random variable, say height of an individual. Then $x_{1},x_{2}\cdots,x_{n}$ be a realization on the variable height, and variance is what we compute.
In multivariate analysis, the components of the random vector $X$ are different variables. In this case we will e computing to know how the different components related to each other.
Suppose our random vector is $X=\left(\begin{array}{c} X_{1}\\ X_{2} \end{array}\right)$ where, $X_{1}$ is height, and $X_{2}$ is weight of an individual.
Then, a realization of $n$ values on $X$ will be $\left(\begin{array}{c} x_{11}\\ x_{21} \end{array}\right)$ $\left(\begin{array}{c} x_{12}\\ x_{22} \end{array}\right)$ $\left(\begin{array}{c} x_{13}\\ x_{23} \end{array}\right)$ $\cdots$ $\left(\begin{array}{c} x_{1n}\\ x_{2n} \end{array}\right)$ where the first component of each data value corresponds to height and the second component corresponds to weight. For such data, we compute covariance matrix.