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Facts About the Correlation Coefficient for Linear Regression

Module by: Kathy Chu. E-mail the author

Based on: Linear Regression and Correlation: Facts About the Correlation Coefficient for Linear Regression by Susan Dean, Barbara Illowsky, Ph.D.

Summary: This module provides an overview of Facts About the Correlation Coefficient for Linear Regression as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

  • A positive rr means that when xx increases, yy increases and when xx decreases, yy decreases (positive correlation).
  • A negative rr means that when xx increases, yy decreases and when xx decreases, yy increases (negative correlation).
  • An rr of zero means there is absolutely no linear relationship between xx and yy (no correlation).
  • High correlation does not suggest that xx causes yy or yy causes xx. We say "correlation does not imply causation." For example, every person who learned math in the 17th century is dead. However, learning math does not necessarily cause death!

Figure 1
Positive CorrelationNegative CorrelationZero Correlation
(a) A scatter plot showing data with a positive correlation.(b) A scatter plot showing data with a negative correlation.(c) A scatter plot showing data with zero correlation.
Scatterplot of points ascending from the lower left to the upper right.Scatterplot of points descending from the upper left to the lower right.Scatterplot of points in a horizontal configuration.

If r=-1r=-1 or r=+1r=+1, then all the data points lie exactly on a straight line.

If the linear correlation is strong, then the line can be used to predict a yy value.

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