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Coefficient of Determination
The coefficient of determination provides a measure of how well observed outcomes are replicated by a model.
Learning Objectives

Compute the coefficient of determination.

Interpret the properties of the coefficient of determination in regard to correlation.
Key Points

The coefficient of determination, r^{2}, is a statistic whose main purpose is either the prediction of future outcomes or the testing of hypotheses on the basis of other related information.

The most general definition of the coefficient of determination is illustrated in , where SS_{err} is the residual sum of squares and SS_{tot} is the total sum of squares.

r^{2}, when expressed as a percent, represents the percent of variation in the dependent variable y that can be explained by variation in the independent variable x using the regression (best fit) line.

1  r^{2} when expressed as a percent, represents the percent of variation in y that is NOT explained by variation in x using the regression line. This can be seen as the scattering of the observed data points about the regression line.
Terms

correlation coefficient
Any of the several measures indicating the strength and direction of a linear relationship between two random variables.

regression
An analytic method to measure the association of one or more independent variables with a dependent variable.
Full Text
The coefficient of determination (denoted r^{2} and pronounced r squared) is a statistic used in the context of statistical models. Its main purpose is either the prediction of future outcomes or the testing of hypotheses on the basis of other related information. It provides a measure of how well observed outcomes are replicated by the model, as the proportion of total variation of outcomes explained by the model. Values for r^{2} can be calculated for any type of predictive model, which need not have a statistical basis.
The Math
A data set will have observed values and modelled values, sometimes known as predicted values. The "variability" of the data set is measured through different sums of squares, such as:
 the total sum of squares (proportional to the sample variance);
 the regression sum of squares (also called the explained sum of squares); and
 the sum of squares of residuals, also called the residual sum of squares.
The most general definition of the coefficient of determination is illustrated in , where SS_{err} is the residual sum of squares and SS_{tot}is the total sum of squares.
Properties and Interpretation of r^{2}
The coefficient of determination is actually the square of the correlation coefficient. It is is usually stated as a percent, rather than in decimal form. In context of data, r^{2 }can be interpreted as follows:
 r^{2}, when expressed as a percent, represents the percent of variation in the dependent variable y that can be explained by variation in the independent variable x using the regression (best fit) line.
 1  r^{2}when expressed as a percent, represents the percent of variation in y that is NOT explained by variation in x using the regression line. This can be seen as the scattering of the observed data points about the regression line.
So r^{2}is a statistic that will give some information about the goodness of fit of a model. In regression, the r^{2} coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An r^{2} of 1 indicates that the regression line perfectly fits the data.
In many (but not all) instances where r^{2} is used, the predictors are calculated by ordinary leastsquares regression: that is, by minimizing SS_{err}. In this case, r^{2} increases as we increase the number of variables in the model. This illustrates a drawback to one possible use of r^{2}, where one might keep adding variables to increase the r^{2} value. For example, if one is trying to predict the sales of a car model from the car's gas mileage, price, and engine power, one can include such irrelevant factors as the first letter of the model's name or the height of the lead engineer designing the car because the r^{2} will never decrease as variables are added and will probably experience an increase due to chance alone. This leads to the alternative approach of looking at the adjusted r^{2}. The explanation of this statistic is almost the same as r^{2} but it penalizes the statistic as extra variables are included in the model.
Note that r^{2 }does not indicate whether:
 the independent variables are a cause of the changes in the dependent variable;
 omittedvariable bias exists;
 the correct regression was used;
 the most appropriate set of independent variables has been chosen;
 there is collinearity present in the data on the explanatory variables; or
 the model might be improved by using transformed versions of the existing set of independent variables.
Example
Consider the third exam/final exam example introduced in the previous section. The correlation coefficient is r = 0.6631. Therefore, the coefficient of determination is r^{2} = 0.6631^{2} = 0.4397.
The interpretation of r^{2} in the context of this example is as follows. Approximately 44% of the variation (0.4397 is approximately 0.44) in the final exam grades can be explained by the variation in the grades on the third exam. Therefore approximately 56% of the variation (1  0.44 = 0.56) in the final exam grades can NOT be explained by the variation in the grades on the third exam.
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Key Term Reference
 bias
 Appears in this related concepts: Defining the Sample and Collecting Data, Generate Alternatives, and Diversity Bias
 collinearity
 Appears in this related concepts: Some Pitfalls: Estimability, Multicollinearity, and Extrapolation, Including and assessing many variables in a model, and Two model selection strategies
 correlation
 Appears in this related concepts: Coefficient of Correlation, Analyzing Data and Growing Conclusions, and Methods for Researching Human Development
 datum
 Appears in this related concepts: Change of Scale, Comparing Nested Models, and Controlling for a Variable
 dependent variable
 Appears in this related concepts: Experimental Design, Evaluating Model Utility, and Experimental Research
 factor
 Appears in this related concepts: Randomized Design: SingleFactor, The Perceptual Process, and Solving Quadratic Equations by Factoring
 goodness of fit
 Appears in this related concepts: Goodness of Fit, Introduction to testing for goodness of fit using chisquare, and Evaluating goodness of fit for a distribution
 independent
 Appears in this related concepts: Regression Analysis for Forecast Improvement, The Rise of Independents, and Unions and Intersections
 independent variable
 Appears in this related concepts: Graphing Functions, Formulating the Hypothesis, and Converting between Exponential and Logarithmic Equations
 line
 Appears in this related concepts: Plotting Lines, Line, and Varieties of Line
 regression line
 Appears in this related concepts: Two Regression Lines, Outliers, and Interpreting regression line parameter estimates
 residual
 Appears in this related concepts: Plotting the Residuals, Models with Both Quantitative and Qualitative Variables, and Degrees of Freedom
 residuals
 Appears in this related concepts: The Correction Factor, Inferences of Correlation and Regression, and Introduction to line fitting, residuals, and correlation
 sample
 Appears in this related concepts: Applications of Statistics, Sampling, and Identify Product Benefits
 statistics
 Appears in this related concepts: What Is Statistics?, Understanding Statistics, and Population Demography
 variable
 Appears in this related concepts: Related Rates, Calculating the NPV, and Fundamentals of Statistics
Sources
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Cite This Source
Source: Boundless. “Coefficient of Determination.” Boundless Statistics. Boundless, 02 Jul. 2014. Retrieved 20 Mar. 2015 from https://www.boundless.com/statistics/textbooks/boundlessstatisticstextbook/correlationandregression11/correlation44/coefficientofdetermination2092661/