Pearson correlation coefficient

\(\rho_{X, Y}=\frac{\operatorname{cov}(X, Y)}{\sigma_X \sigma_Y}=\frac{E\left[\left(X-\mu_X\right)\left(Y-\mu_Y\right)\right]}{\sigma_X \sigma_Y}\)

So \(\sigma_X\) or \(\sigma_Y\) coudn't be equal to 0.

Spearman’s Rank Correlation Coefficient

For a sample of size n, the n raw scores \(X_{i},Y_{i}\) are converted to ranks R⁡(\(X_{i}\)) R⁡(\(Y_{i}\)) , and \(r_s\) is computed as

\(r_s=\rho_{\mathrm{R}(X), \mathrm{R}(Y)}=\frac{\operatorname{cov}(\mathrm{R}(X), \mathrm{R}(Y))}{\sigma_{\mathrm{R}(X)} \sigma_{\mathrm{R}(Y)}}\)

Only if all n ranks are distinct integers, it can be computed using the popular formula

\(r_s=1-\frac{6 \sum d_i^2}{n\left(n^2-1\right)}\)

Value X Value Y X Order Y Order X Rank Y Rank \(d_i\) \(d_i^2\)
10 12 1 1 1 1 0 0
9 11 2 2 (2+3)/2=2.5 2 0.5 0.25
9 10 3 3 (2+3)/2=2.5 (3+4)/2=3.5 -1 1
8 10 4 4 4 (3+4)/2=3.5 -0..5 0.25

\(\sum d_i^2= 1.5\)

\(\rho=1-\frac{6 \sum d_i^2}{n\left(n^2-1\right)}\)

\(\rho = 0.25\)

So, we can see that there must have same number X and Y.

Reference