### Chapter 11: Regression Analysis: Simple Linear Regression

### The Coefficient of Determination

Just because a regression line is the *best-fitting* straight line, does not necessarily mean it is a *good* prediction line. To determine how good of a fit the best-fitting straight line is, the *coefficient of determination *can be calculated.

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Coefficient of Determination

The **coefficient of determination #R^2# **measures how much *better *the regression line is able to predict the value of the outcome variable in comparison to the mean of the outcome variable.

Specifically, the coefficient of determination measures how much percent of the variance in the outcome variable #Y# is explained by the predictor variable #X#.

The coefficient of determination ranges from #0# to #1#:

- A coefficient of #0# indicates that the outcome variable cannot be predicted whatsoever by the predictor variable.
- A coefficient of #1# indicates that the outcome variable can be predicted without error by the predictor variable.
- A coefficient between #0# and #1# indicates the extent to which the outcome variable can be predicted by the predictor variable.

For example, a coefficient of determination of #R^2=.72# indicates that #72\%# of the variance in the outcome variable can be explained by the predictor variable.

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