Calculating person product moment correlation coefficient on a 3 X 3 table

Question

Usually we are given problems that only involve 2 rows (x and y), but recently saw a problem asking how to compute the correlation coefficient on a table of data that has 3 rows and am not sure how to go about it.

Answer 1

Probably, it means that you have to calculate a correlation matrix. I.e., for $X_1, X_2, X_3$ you have to calculate $R_{3\times3}$ symmetric matrix, where the $ij$ entry is the Pearson's correlation coefficients between $X_i$ and $X_j$, namely $\rho_ij=\hat{Cov}(X_i, X_j)/(\hat{\sigma}_{X_i}\hat{\sigma}_{X_j})$, and the main diagonal is $1$.