May 9, 2009 - Correlation Notes

A. Correlation

A correlation is a relationship between two variables.

The correlated variables can be represented by an ordered pair (x, y) where x is the
independent, or explanatory, variable, and y is the dependent, or response, variable.
A scatter plot is the graphical representation of the ordered pairs in the form of points in a coordinate plane.

The correlation coefficient is a mathematical measure of the strength and the direction of a linear relationship between two variables. The symbol r represents the sample correlation coefficient. The range of r is –1 to 1.
Types of correlation:

• Negative linear correlation: A negative linear correlation implies that when x
increases, y tends to decrease. When r approaches –1, x and y are said to have a strong negative, or inverse, linear correlation.


• Positive linear correlation: A positive linear correlation implies that when x
increases, y tends to increase. When r approaches 1, x and y are said to have a strong positive, or direct, linear correlation.

• No correlation or weak correlation: No correlation or a weak linear correlation
implies that the magnitude of x has little or no effect on the magnitude of y. When r is near 0, x and y are said to have no correlation or a weak linear correlation.

• Nonlinear correlation: A scatter plot of the data shows a pattern, but it is not that of a line. It might resemble a U, an arc, or some other shape.
Cause and effect relations refer to situations when two variables, x and y, are related in such a way that changing one variable causes the other to change. Statisticians are careful to avoid claiming that because x and y are correlated, x causes y. In other words, you want to emphasize that correlation does not imply causation.

B. Linear Regression

The technique of fitting a linear equation to real data points gives a line called a regression line. This line is used to predict the value y—the response variable—for a given x, often called the predictor variable.

The equation of a regression line for an independent variable x and a dependent variable y is written as ŷ = mx + b, where m is the slope of the equation and ŷ is the predicted y-value for a given x-value.