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The coefficient of determination is a statistical measurement that examines how differences in one variable can be explained by the difference in a second variable, when predicting the outcome of a given event.
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Published on April 22, 2022 by Shaun Turney. Revised on September 14, 2022. The coefficient of determination is a number between 0 and 1 that measures how well a statistical
model predicts an outcome.
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). What 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 dependent variable. 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 (independent variable).
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: 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:
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See an example Interpreting the coefficient of determinationYou can interpret the coefficient of determination (R²) as the proportion of variance in the dependent variable 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 size
Be 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 questionsFrequently asked questions about the coefficient of determinationCite 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.
Is this article helpful?You have already voted. Thanks :-) Your vote is saved :-) Processing your vote... What happens if coefficient of determination is equal to 1?A value of 1.0 indicates a perfect fit, and is thus a highly reliable model for future forecasts, while a value of 0.0 would indicate that the calculation fails to accurately model the data at all.
Is the coefficient of determination is equal to 1 then the correlation coefficient?The coefficient of determination is the square of the correlation(r), thus it ranges from 0 to 1. With linear regression, the coefficient of determination is equal to the square of the correlation between the x and y variables.
What is a correlation coefficient of 1?Correlation coefficients are expressed as values between +1 and -1. A coefficient of +1 indicates a perfect positive correlation: A change in the value of one variable will predict a change in the same direction in the second variable.
Can the coefficient of determination be 1?The coefficient of determination is a number between 0 and 1 that measures how well a statistical model predicts an outcome.
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