gen-define-the-correlation-coefficient-and-give-one-short-worked-9d82

Define the correlation coefficient and give one short worked example.

# Correlation Coefficient ## Definition The **correlation coefficient** (typically Pearson's r) measures the strength and direction of the linear relationship between two variables. It ranges from -1 to +1, where: - **r = +1**: perfect positive linear relationship - **r = -1**: perfect negative linear relationship - **r = 0**: no linear relationship ## Formula $$r = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i - \bar{x})^2 \sum(y_i - \bar{y})^2}}$$ ## Worked Example Find the correlation between hours studied (x) and test scores (y): | Hours (x) | Score (y) | |-----------|-----------| | 2 | 65 | | 3 | 70 | | 4 | 80 | | 5 | 85 | **Solution:** - Mean: $\bar{x} = 3.5$, $\bar{y} = 75$ | x | y | $(x-\bar{x})$ | $(y-\bar{y})$ | $(x-\bar{x})(y-\bar{y})$ | $(x-\bar{x})^2$ | $(y-\bar{y})^2$ | |---|---|---|---|---|---|---| | 2 | 65 | -1.5 | -10 | 15 | 2.25 | 100 | | 3 | 70 | -0.5 | -5 | 2.5 | 0.25 | 25 | | 4 | 80 | 0.5 | 5 | 2.5 | 0.25 | 25 | | 5 | 85 | 1.5 | 10 | 15 | 2.25 | 100 | $$r = \frac{35}{\sqrt{5 \times 250}} = \frac{35}{35.36} \approx 0.99$$ **Interpretation:** Strong positive correlation - more study hours associate with higher scores.