# [[Linear Correlation Coefficient]] (Mathematics > Statistics) ## Definition The linear correlation coefficient, denoted $\rho$ (Greek letter "rho"), quantifies the strength and direction of the linear relationship between two sets of paired observations $(x, y)$. It ranges from $-1$ to $1$, where: - $\rho = 1$ indicates perfect positive linear correlation. - $\rho = -1$ indicates perfect negative linear correlation. - $\rho = 0$ indicates no linear correlation. The constraint on $\rho$ is: $ -1 \leq \rho \leq 1 $ ## Key Concepts - $\rho = 1$: Perfect positive linear relationship (data points lie on a line with positive slope). - $\rho = -1$: Perfect negative linear relationship (data points lie on a line with negative slope). - $\rho = 0$: No linear correlation (random or no specific pattern). ## Important Properties 1. The sign of $\rho$ indicates the direction of the relationship (positive or negative). 2. The magnitude of $\rho$ reflects the strength of the linear relationship. 3. $\rho$ is dimensionless and remains unchanged by scaling or shifting the data. ## Essential Formulas - For a sample, the linear correlation coefficient $r$ is calculated as: $ r = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^n (x_i - \bar{x})^2} \sqrt{\sum_{i=1}^n (y_i - \bar{y})^2}} $ ## Core Examples 1. **Perfect positive correlation**: $x = \{1, 2, 3\}$ and $y = \{2, 4, 6\}$ give $\rho = 1$, indicating a perfect positive linear relationship. 2. **No linear correlation**: $x = \{1, 2, 3\}$ and $y = \{5, 5, 5\}$ give $\rho = 0$, indicating no linear relationship. ## Related Theorems/Rules - **Cauchy-Schwarz inequality** ensures that $\rho$ lies within the interval $[-1, 1]$. - **Covariance** is related to correlation: $ \rho = \frac{\text{Cov}(x, y)}{\sigma_x \sigma_y} $ where $\sigma_x$ and $\sigma_y$ are the standard deviations of $x$ and $y$. ## Common Pitfalls - Confusing correlation with causation; correlation does not imply one variable causes the other to change. - Misinterpreting $\rho = 0$ as indicating independence; it only means no linear relationship. ## Related Topics - [[Covariance]] - [[Regression (Mathematics > Statistics)]] ## Quick Review Questions 1. What does a correlation coefficient of $\rho = -1$ imply about the relationship between $x$ and $y$? 2. How does correlation differ from covariance in terms of interpretation and scale? *** ![[Linear_Correlation_Coefficient_visualization]]