This calculator finds the Explained Sum of Squares (ESS) and the R-squared value for a linear regression model using values for the predictor and response variables.
To use the calculator, provide a list of values for the predictor and the response, ensuring they are the same length, and then click the "Calculate ESS and R-squared" button.
Explained Sum of Squares (ESS):
R-squared:
Explained Sum of Squares (ESS), Total Sum of Squares (TSS), and R-squared
The Explained Sum of Squares (ESS) measures the variability of the predicted values from a linear regression model compared to the mean of the observed data.
Key Components
- Predictor Variable (\( {X} \)): The independent variable used to predict the response.
- Response Variable (\( {Y} \)): The dependent variable that is being predicted.
- Fitted Value (\( \hat{Y} \)): The predicted value of Y for a given X, based on the linear regression model.
Explained Sum of Squares (ESS) Formula
The ESS is calculated as the sum of the squared differences between the predicted values and the mean of the observed values:
Total Sum of Squares (TSS) and R-squared
The Total Sum of Squares (TSS) represents the total variability in the response variable \(Y\), while R-squared is a normalized measure of how well the model explains the variation in \(Y\). The formula for R-squared is:
Relationship Between ESS, TSS, and RSS
The relationship between ESS, TSS, and RSS (Residual Sum of Squares) is given by:
In this equation, TSS represents the total variability in \(Y\), ESS represents the portion of the variability explained by the model, and RSS represents the unexplained variability.
R-squared Interpretation
The R-squared value ranges from 0 to 1:
- An R-squared value of 1 means the model perfectly explains the variability in the response variable.
- An R-squared value of 0 means the model explains none of the variability in the response variable.
Caveats and Conditions
- Linear Assumption: These calculations assume a linear relationship between the predictor \(X\) and response \(Y\). Non-linear relationships may result in a misleading R-squared value.
- Overfitting: A high R-squared value could indicate overfitting when too many predictors are used.
- Outliers: Outliers can disproportionately affect the ESS, TSS, and RSS, leading to a skewed R-squared value.
Further Reading
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