Properties of the Linear Correlation Coefficient

Exploring Relationships Between Two Variables

Properties of the Linear Correlation Coefficient

The linear correlation coefficient, commonly denoted as r, is a statistical measure that quantifies the strength and direction of a linear relationship between two quantitative variables. Understanding its properties is essential for interpreting data and making informed conclusions about variable associations.

• Definition: The linear correlation coefficient r measures the degree to which two variables are linearly related.

• Range: The value of r is always between -1 and 1, inclusive. That is, $-1 \leq r \leq 1$.

• Perfect Positive Linear Relation: If $r = 1$, there is a perfect positive linear relationship between the two variables. All data points lie exactly on a line with positive slope.

• Perfect Negative Linear Relation: If $r = -1$, there is a perfect negative linear relationship between the two variables. All data points lie exactly on a line with negative slope.

• No Linear Relation: If $r$ is close to 0, there is little to no evidence of a linear relationship between the two variables. However, this does not imply that there is no relationship at all—only that it is not linear.

• Strength of Association: The closer $r$ is to +1 or -1, the stronger the evidence of a linear association (positive or negative, respectively) between the variables.

• Unitless Measure: The value of r does not depend on the units of measurement for the variables x and y.

Interpreting the Strength of r

The strength of the linear relationship can be loosely categorized based on the value of r:

r Value Range	Strength of Relationship	Direction
-1 to -0.75	Strong	Negative
-0.75 to -0.5	Moderate	Negative
-0.5 to -0.25	Weak	Negative
-0.25 to 0.25	None or Very Weak	None
0.25 to 0.5	Weak	Positive
0.5 to 0.75	Moderate	Positive
0.75 to 1	Strong	Positive

Additional info: These categories are guidelines; interpretation may vary by context and field of study.

Examples of Linear Correlation

• Perfect Positive Linear Correlation: $r = 1$ All points lie exactly on a straight line with positive slope. For example, the relationship between Celsius and Fahrenheit temperature scales.

• Perfect Negative Linear Correlation: $r = -1$ All points lie exactly on a straight line with negative slope. For example, the relationship between the amount of time spent on a task and the time remaining to complete it (if total time is fixed).

• No Linear Correlation: $r = 0$ There is no apparent linear relationship between the variables. For example, the relationship between shoe size and intelligence.

Formula for the Linear Correlation Coefficient

The formula for the sample linear correlation coefficient is:

$r = \frac{\sum_{i=1}^{n} (x_i - \overline{x})(y_i - \overline{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \overline{x})^2} \sqrt{\sum_{i=1}^{n} (y_i - \overline{y})^2}}$

• $x_i, y_i$ are the individual sample values

• $\overline{x}, \overline{y}$ are the sample means of $x$ and $y$

• $n$ is the number of paired observations

Additional info: The correlation coefficient is sensitive to outliers, which can greatly affect its value. Always examine scatterplots before interpreting r.