The coefficient of determination is a number between 0 and 1 that measures how well a statistical model predicts an outcome. Show
The coefficient of determination is often written as R2, which is pronounced as “r squared.” For simple linear regressions, a lowercase r is usually used instead (r2). Table of contentsWhat is the coefficient of determination?The coefficient of determination (R²) measures how well a statistical model predicts an outcome. The outcome is represented by the model’s . The lowest possible value of R² is 0 and the highest possible value is 1. Put simply, the better a model is at making predictions, the closer its R² will be to 1. Example: Coefficient of determinationImagine that you perform a simple linear regression that predicts students’ exam scores (dependent variable) from their time spent studying ().
More technically, R2 is a measure of goodness of fit. It is the proportion of variance in the dependent variable that is explained by the model. Graphing your linear regression data usually gives you a good clue as to whether its R2 is high or low. For example, the graphs below show two sets of simulated data:
You can see in the first dataset that when the R2 is high, the observations are close to the model’s predictions. In other words, most points are close to the line of best fit: Note: The coefficient of determination is always positive, even when the correlation is negative.In contrast, you can see in the second dataset that when the R2 is low, the observations are far from the model’s predictions. In other words, when the R2 is low, many points are far from the line of best fit: Calculating the coefficient of determinationYou can choose between two formulas to calculate the coefficient of determination (R²) of a simple linear regression. The first formula is specific to simple linear regressions, and the second formula can be used to calculate the R² of many types of statistical models. Formula 1: Using the correlation coefficientFormula 1:
Where r = Pearson correlation coefficient Example: Calculating R² using the correlation coefficientYou are studying the relationship between heart rate and age in children, and you find that the two variables have a negative Pearson correlation:
This value can be used to calculate the coefficient of determination (R²) using Formula 1:
Formula 2: Using the regression outputsFormula 2:
Where:
These values can be used to calculate the coefficient of determination (R²) using Formula 2:
Here's why students love Scribbr's proofreading servicesTrustpilot Discover proofreading & editing Interpreting the coefficient of determinationYou can interpret the coefficient of determination (R²) as the proportion of variance in the that is predicted by the statistical model. Another way of thinking of it is that the R² is the proportion of variance that is shared between the independent and dependent variables. You can also say that the R² is the proportion of variance “explained” or “accounted for” by the model. The proportion that remains (1 − R²) is the variance that is not predicted by the model. If you prefer, you can write the R² as a percentage instead of a proportion. Simply multiply the proportion by 100. R² as an effect sizeLastly, you can also interpret the R² as an effect size: a measure of the strength of the relationship between the dependent and independent variables. Psychologist and statistician Jacob Cohen (1988) suggested the following rules of thumb for simple linear regressions: R² as an effect sizeMinimum coefficient of determination (R²) valueEffect size interpretation.01Small.09Medium.25LargeBe careful: the R² on its own can’t tell you anything about causation. Example: Interpreting R²A simple linear regression that predicts students’ exam scores (dependent variable) from their study time (independent variable) has an R² of .71. From this R² value, we know that:
Studying longer may or may not cause an improvement in the students’ scores. Although this causal relationship is very plausible, the R² alone can’t tell us why there’s a relationship between students’ study time and exam scores. For example, students might find studying less frustrating when they understand the course material well, so they study longer. Reporting the coefficient of determinationIf you decide to include a coefficient of determination (R²) in your research paper, dissertation or thesis, you should report it in your results section. You can follow these rules if you want to report statistics in APA Style:
Practice questionspowered by Typeform Frequently asked questions about the coefficient of determinationWhat is the definition of the coefficient of determination (R²)? The coefficient of determination (R²) is a number between 0 and 1 that measures how well a statistical model predicts an outcome. You can interpret the R² as the proportion of variation in the dependent variable that is predicted by the statistical model. What is the formula for the coefficient of determination (R²)? There are two formulas you can use to calculate the coefficient of determination (R²) of a simple linear regression. Formula 1: Formula 2: How do I calculate the coefficient of determination (R²) in R? You can use the summary() function to view the R² of a linear model in R. You will see the “R-squared” near the bottom of the output. How do I calculate the coefficient of determination (R²) in Excel? You can use the RSQ() function to calculate R² in Excel. If your dependent variable is in column A and your independent variable is in column B, then click any blank cell and type “RSQ(A:A,B:B)”. Cite this Scribbr articleIf you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.
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