Solved – Obtain z test statistic for testing Spearman rho with z-score formula

Let $R^x_1, R^x_2,ldots, R^x_{n_x}$ be the ranks of $x$ and $R^y_1, R^y_2,ldots,R^y_{n_y}$ be the ranks of $y$. Spearman correlation is determined as begin{equation} r_s = 1 – frac{6D}{n^3-n} end{equation} where begin{equation} D = sum_{i=1}^n d_i^2 = sum_{i=1}^n (R^x_{(i)}-R^y_{(i)})^2 end{equation}

Significance testing of $r_s$ is based on the standard normal distribution. Using the expected value of $D$ as begin{equation} E(D) = frac{n^3-n}{6} = frac{n(n-1)(n+1)}{6} end{equation} begin{equation} V(D) = frac{n^2(n+1)^2(n-1)}{36} end{equation} begin{equation} Z = frac{D – E(D)}{sqrt{V(D)}} Longrightarrow frac{x-mu}{sigma} end{equation}