Example Background: (This is a extension of the previous Temperature/Water example.)  Typically, in the summer time as the temperature increases people are thirstier.  In the previous simple correlation example, we considered the two numerical variables, temperature and water consumption.  For that example, we found that temperature and water consumption are positively correlated (i.e. the higher the temperature, the more water a given person consumes).  Though this linear relationship was strong (r= 0.963), it was not a perfect linear relationship.  Thus we might look for another variable which would be related to the thirst that occurs in the summer while people are outside.  We will use the example below to demonstrate multiple regression.

       Specific Multiple Correlation Example: Assume that during a three-hour period spent outside, a person recorded the temperature and their water consumption and the time they spent mowing the grass. The experiment was conducted on 7 randomly selected days during the summer.  The data is shown in the table below with the temperature placed in increasing order.

To see a 3-D graph of the data, please click on Corresponding 3-D Scatter Plot

• To see the relationship for multiple correlation instead of looking for a line of best fit, you now need to look for a curve (3-D surface) of best fit.
• R, called the Multiple Correlation Coefficient, is a measure of the strength of the association between the independent (explanatory) variables and the dependent (prediction) variable.
• Note:  At this time we are not using R-sq, the Coefficient of Multiple Determination.  R-sq  tells the amount of variation in the dependent variable that is explained by the model and the independent variables.  At this point we have not specified a multiple regression model yet.
  

Next let's learn more about the Multiple Correlation Coefficient, R.

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