Properties of r

Overview of Pearson's Correlation Coefficient

• Definition: Pearson's r is a statistical measure that calculates the strength and direction of a linear relationship between two variables.

Range of Pearson's r

• Possible Range:

  • The correlation coefficient can range from -1 to 1.

  • Negative One (-1): Indicates a perfect negative linear relationship.

  • Zero (0): Indicates no linear relationship.

  • Positive One (1): Indicates a perfect positive linear relationship.

Properties of Pearson's r

• Symmetry:

  • The correlation is symmetric. The correlation of x with y is identical to the correlation of y with x.

  • Example: The correlation between weight and height remains the same whether you calculate weight with respect to height or height with respect to weight.

• Unchanged by Linear Transformations:

  • Pearson's r is invariant to linear transformations of the variables involved:

    • Multiplication by a Constant: The correlation does not change if a variable is multiplied by any constant.

    • Addition of a Constant: The correlation remains the same regardless of adding any constant to a variable.

  • Illustration:

  • The correlation between weight and height is consistent regardless of whether height is measured in inches, feet, or even miles.

  • Adding five points to every student's test score does not affect the correlation between the test score and other variables, such as grade point average (GPA).