1. **Problem Statement:** You want to understand the Pearson correlation coefficient formula and how to calculate it. 2. **Formula:** The Pearson correlation coefficient $r_{xy}$ is given by: $$r_{xy} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$$ 3. **Explanation of terms:** - $x_i$ and $y_i$ are individual data points in samples $x$ and $y$ respectively. - $\bar{x}$ and $\bar{y}$ are the means (averages) of the $x$ and $y$ samples. 4. **Step-by-step calculation:** 1. Calculate the mean of $x$ values: $\bar{x} = \frac{1}{n} \sum x_i$ 2. Calculate the mean of $y$ values: $\bar{y} = \frac{1}{n} \sum y_i$ 3. For each pair $(x_i, y_i)$, find the difference from the mean: $(x_i - \bar{x})$ and $(y_i - \bar{y})$ 4. Multiply these differences for each pair: $(x_i - \bar{x})(y_i - \bar{y})$ 5. Sum all these products: $\sum (x_i - \bar{x})(y_i - \bar{y})$ 6. Calculate the sum of squares for $x$: $\sum (x_i - \bar{x})^2$ 7. Calculate the sum of squares for $y$: $\sum (y_i - \bar{y})^2$ 8. Multiply the sums of squares: $\sum (x_i - \bar{x})^2 \times \sum (y_i - \bar{y})^2$ 9. Take the square root of the product: $\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}$ 10. Divide the sum of products by the square root from step 9: $$r_{xy} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$$ 5. **Interpretation:** - $r_{xy}$ ranges from -1 to 1. - $r_{xy} = 1$ means perfect positive linear correlation. - $r_{xy} = -1$ means perfect negative linear correlation. - $r_{xy} = 0$ means no linear correlation. This formula measures how strongly $x$ and $y$ are linearly related.