Let our data be:

X = x_{1, 1} \dots x_{n, 1} x_{1, 2} x_{n, 2} x_{1, 3} x_{n, 3} \dots \dots x_{1, p} x_{n, p}

Sample mean

\overset{ˉ}{X} = \overset{x_{1}}{ˉ} \overset{x_{2}}{ˉ} ⋮ \overset{x_{p}}{ˉ}_{p \times 1}

where each $x_{i}$ is another random variable. Expected Value.

\overset{x_{i}}{ˉ} = \frac{1}{n} j = 1 \sum n x_{ji} for i = 1, 2, \dots, p

Variance-Covariance Matrix

We denote sample Variance & Covariance in a matrix of size $p \times p$ .

S_{n} = s_{11} ⋮ s_{p 1} s_{12} ⋱ s_{p 2} \dots \dots s_{1 p} s_{pp}

Here any $s_{ii}$ denote the variance of variable $i$ and any $s_{ij}$ denote the covariance between variable $i$ and $j$ .

s_{ij} = \frac{1}{n} k = 1 \sum n [(s_{ki} - \overset{x_{k}}{ˉ}) (s_{kj} - \overset{x_{j}}{ˉ})] for i, j \in 1, 2, \dots, p

R = 1 r_{21} ⋮ r_{p 1} r_{12} 1 ⋮ \dots \dots \dots ⋱ \dots r_{1 p} r_{2 p} r_{3 p} 1_{p \times p}

where,

r_{ij} = \frac{C o v ( x _{i} , x _{j} )}{V ( x _{i} ) V ( x _{j} )} = \frac{s _{ij}}{s _{ii} s _{jj}}