# Kendal's tau correlation if only one variable is ordinal

I have a vector with 3000 data points consisting of values on a likert scale from 1 to 9. It is ordinal data without normal distribution. Now I would like to calculate the correlation of this vector to three other vectors each of size 3000 as follows:

1. Correlation to a vector with continuous values.

2. Correlation to a vector containing 21 different continuous values between 0.25 and 0.8.

3. Correlation to a vector containing only 4 different values (-2,2,-0.2,0.2).

Is it correct if I'm using for all three cases the Kendal's tau correlation?

Kendal tau is not just for ordinal variables, but rather for the case where the correlation is ordinal. For instance, imagine that the true relationship is $y=\ln x+\varepsilon$, but you don't know it. You got your sample $(x_i,y_i)$, and run a Pearson correlation analysis $cor[x_i,y_i]$. The problem is that although both variables are not just ordinal but cardinal, their relationship is not linear. That's why Pearson correlation is not appropriate here. If you'd know the true relationship, you do look at Pearson correlation of $\ln x$ and $y$, but in this case you don't know this. In such a case, you could run Kendall $\tau$ correlation because it's ordinal and $\ln x$ happens to be a monotonic function, which is often the case.
Hence, Kendall $\tau$ analysis will work in a wider set of circumstances compared to Pearson correlation. Particularly, when the relationship between variables is monotonic it will work. Linear is monotonic, so it will work wherever Pearson work, but for linear relationship Pearson would be a better test.