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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 Objective

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
$SS_\text{err}$ is the residual sum of squares and$SS_\text{tot}$ 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.$1r^2$ 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
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:
where
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,
$r^2$ , 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.$1r^2$ 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
In many (but not all) instances where
Note that
 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
The interpretation of
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Key Term Reference
 bias
 Appears in these related concepts: Determine a Course, Biases in Experimental Design: Validity, Reliability, and Other Issues, and Social Psychology
 collinearity
 Appears in this related concept: Some Pitfalls: Estimability, Multicollinearity, and Extrapolation
 correlation
 Appears in these related concepts: Coefficient of Correlation, Descriptive and Correlational Statistics, and Methods for Researching Human Development
 datum
 Appears in these related concepts: Change of Scale, Type I and II Errors, and Lab 1: Confidence Interval (Home Costs)
 dependent variable
 Appears in these related concepts: Experimental Design, Formulating the Hypothesis, and Experimental Research
 factor
 Appears in these related concepts: Randomized Design: SingleFactor, Finding Factors of Polynomials, and Solving Quadratic Equations by Factoring
 goodness of fit
 Appears in these related concepts: Goodness of Fit, Evaluating goodness of fit for a distribution, and The ChiSquare Distribution: Test for Homogeneity
 independent
 Appears in these related concepts: Probability Histograms, The Rise of Independents, and Unions and Intersections
 independent variable
 Appears in these related concepts: Evaluating Model Utility, Graphical Representations of Functions, and Converting between Exponential and Logarithmic Equations
 line
 Appears in these related concepts: Plotting Lines, Line, and Qualities of Line
 regression line
 Appears in these related concepts: Two Regression Lines, Outliers, and Line fitting, residuals, and correlation exercises
 residual
 Appears in these related concepts: Plotting the Residuals, Models with Both Quantitative and Qualitative Variables, and Degrees of Freedom
 residuals
 Appears in these related concepts: Inferences of Correlation and Regression, Midterm elections and unemployment, and Diagnostics for the email classifier
 sample
 Appears in these related concepts: Defining the Sample and Collecting Data, Surveys, and Basic Inferential Statistics
 statistics
 Appears in these related concepts: What Is Statistics?, Communicating Statistics, and Understanding Statistics
 variable
 Appears in these related concepts: Related Rates, Controlling for a Variable, and Math Review
Sources
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Cite This Source
Source: Boundless. “Coefficient of Determination.” Boundless Statistics. Boundless, 26 May. 2016. Retrieved 29 May. 2016 from https://www.boundless.com/statistics/textbooks/boundlessstatisticstextbook/correlationandregression11/correlation44/coefficientofdetermination2092661/