1. **Stating the problem:** We want to understand the formulas for Pearson's correlation coefficient $r$, the test value $t$ for significance testing of $r$, and Spearman's rank-order correlation coefficient $r$. 2. **Pearson's correlation coefficient formula:** $$r = \frac{n \sum xy - \sum x \sum y}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}$$ This formula measures the linear relationship between two variables $x$ and $y$. 3. **Explanation of terms:** - $n$ is the number of paired observations. - $\sum xy$ is the sum of the product of paired scores. - $\sum x$ and $\sum y$ are sums of the individual variables. - $\sum x^2$ and $\sum y^2$ are sums of squares of the variables. 4. **Test value for significance of $r$:** $$t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}}$$ This $t$-value follows a $t$-distribution with $n-2$ degrees of freedom and is used to test if the correlation is significantly different from zero. 5. **Spearman Rank-order correlation coefficient formula:** $$r = 1 - \frac{6 \sum d^2}{N(N^2 - 1)}$$ Where: - $d$ is the difference between the ranks of each pair. - $N$ is the number of pairs. 6. **Summary:** - Pearson's $r$ measures linear correlation between raw scores. - Spearman's $r$ measures correlation between ranks, useful for non-parametric data. - The test value $t$ helps determine if Pearson's $r$ is statistically significant. These formulas are fundamental in statistics for measuring and testing relationships between variables.