- Related Topics:
- estimated regression equation
- goodness-of-fit test
- On the Web:
- Open Library Publishing Platform - Introduction to Statistics - Coefficient of Determination (Oct. 30, 2026)
coefficient of determination
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What is the coefficient of determination?
The coefficient of determination, denoted as ( R^2 ) (or ( r^2 )), gauges a model's predictive accuracy within linear regression. Specifically, it indicates the proportion of variance in the dependent variable (Y) that the independent variable (X) predicts or explains.
A high ( R^2 ) value suggests the model fits the data well, though interpretations vary by analysis context. For instance, an ( R^2 ) of 0.35 means the model explains 35% of the outcome's variation using included covariates.
The coefficient of determination can be found with the following formula: (R^2 = \frac{MSS}{TSS} = \frac{(TSS - RSS)}{TSS}). In this equation, MSS represents the model sum of squares, TSS is the total sum of squares, and RSS is the residual sum of squares.
Expressed as a percentage, ( r^2 ) indicates the percentage of the total sum of squares explained by the estimated regression equation.
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How is the coefficient of determination related to correlation?
The coefficient of determination, or (R^2), measures how well a statistical model predicts outcomes in linear regression. It shows the proportion of variance in the dependent variable (Y) that the independent variable (X) predicts or explains.
When there is only one predictor in the model, the coefficient of determination relates mathematically to Pearson's correlation coefficient, r. Squaring the correlation coefficient gives the coefficient of determination. The (R^2) ranges from 0 to 1, where values closer to 1 suggest a better fit.
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What does the coefficient of determination show in a statistical model?
The coefficient of determination, denoted as ( R^2 ), assesses a statistical model's ability to predict or explain an outcome in a linear regression setting. It indicates the proportion of the variance in the dependent variable (Y) that the independent variable (X) predicts or explains.
( R^2 ) ranges from 0 to 1, where a high value suggests the model fits the data well. For instance, an ( R^2 ) of 0.35 means the model explains 35% of the variation in the outcome using the included covariates.
The formula for the coefficient of determination is:
[ R^2 = \frac{MSS}{TSS} = \frac{(TSS - RSS)}{TSS} ]Where:
* MSS is the model sum of squares
* TSS is the total sum of squares
* RSS is the residual sum of squares2 Britannica SourcesCreated from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles. -
How do you interpret a coefficient of determination value?
The coefficient of determination, or (R^2), measures how well a statistical model predicts outcomes in linear regression. It shows the proportion of variance in a dependent variable (Y) that an independent variable (X) predicts or explains.
An (R^2) value ranges from 0 to 1, where a high value indicates a good model fit, but interpretations depend on the context. For instance, an (R^2) of 0.35 means the model explains 35% of the variation in the outcome using the included covariates. Such a percentage may be high in social sciences but not in physical sciences. The coefficient of determination indicates the magnitude of the association, but not its statistical significance or whether one variable causes another.
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How is the coefficient of determination used in linear regression?
The coefficient of determination, or R^2, measures a linear regression model's ability to predict or explain an outcome. It shows the proportion of variance in the dependent variable (Y) that the independent variable (X) predicts.
A high R^2 value suggests the model fits the data well, but interpretations depend on the context. For example, an R^2 of 0.35 means 35% of the outcome variation is explained by the model's covariates. The adjusted R^2 penalizes the number of predictor variables to account for the effect of new variables, providing a more realistic estimate of the variation proportion predicted by the model's covariates.
The formula to find the coefficient of determination is R^2 = MSS/TSS = (TSS − RSS)/TSS. However, R^2 only indicates the magnitude of the association, not its statistical significance or whether one variable causes another.
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What are some limitations of using the coefficient of determination (R-squared)?
The coefficient of determination, or R^2, measures how well a statistical model predicts outcomes in linear regression. A high R^2 suggests a good fit, but interpretation varies by context; for example, 0.35 might be significant in social sciences but less so in physical sciences.
R^2 cannot determine causation, only association, and it only indicates the magnitude of the association, not its statistical significance. Also, R^2 tends to increase with added predictor variables, even if those variables aren't truly associated with the outcome. To counter this, the adjusted R^2 penalizes the inclusion of more predictor variables, offering a more realistic estimate of the variance proportion predicted by the model's covariates.
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coefficient of determination, in statistics, R2 (or r2), a measure that assesses the ability of a model to predict or explain an outcome in the linear regression setting. More specifically, R2 indicates the proportion of the variance in the dependent variable (Y) that is predicted or explained by linear regression and the predictor variable (X, also known as the independent variable).
In general, a high R2 value indicates that the model is a good fit for the data, although interpretations of fit depend on the context of analysis. An R2 of 0.35, for example, indicates that 35 percent of the variation in the outcome has been explained just by predicting the outcome using the covariates included in the model. That percentage might be a very high portion of variation to predict in a field such as the social sciences; in other fields, such as the physical sciences, one would expect R2 to be much closer to 100 percent. The theoretical minimum R2 is 0. However, since linear regression is based on the best possible fit, R2 will always be greater than zero, even when the predictor and outcome variables bear no relationship to one another.
R2 increases when a new predictor variable is added to the model, even if the new predictor is not associated with the outcome. To account for that effect, the adjusted R2 (typically denoted with a bar over the R in R2) incorporates the same information as the usual R2 but then also penalizes for the number of predictor variables included in the model. As a result, R2 increases as new predictors are added to a multiple linear regression model, but the adjusted R2 increases only if the increase in R2 is greater than one would expect from chance alone. In such a model, the adjusted R2 is the most realistic estimate of the proportion of the variation that is predicted by the covariates included in the model.
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When only one predictor is included in the model, the coefficient of determination is mathematically related to the Pearson’s correlation coefficient, r. Squaring the correlation coefficient results in the value of the coefficient of determination. The coefficient of determination can also be found with the following formula: R2 = MSS/TSS = (TSS − RSS)/TSS, where MSS is the model sum of squares (also known as ESS, or explained sum of squares), which is the sum of the squares of the prediction from the linear regression minus the mean for that variable; TSS is the total sum of squares associated with the outcome variable, which is the sum of the squares of the measurements minus their mean; and RSS is the residual sum of squares, which is the sum of the squares of the measurements minus the prediction from the linear regression.
The coefficient of determination shows only association. As with linear regression, it is impossible to use R2 to determine whether one variable causes the other. In addition, the coefficient of determination shows only the magnitude of the association, not whether that association is statistically significant.