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R-squared is a measure of how well a regression model fits the data. It is calculated by taking the sum of the squared errors of the regression model and dividing it by the total sum of squared errors of the mean. The final result is the R-squared value, which can range from 0 to 1, with 1 being a perfect fit.
In statistics, R-squared (R2) measures the proportion of the variance in the that can be explained by the predictor variable in a regression model.
We use the following formula to calculate R-squared:
R2 = [ (nΣxy – (Σx)(Σy)) / (√nΣx2-(Σx)2 * √nΣy2-(Σy)2) ]2
The following step-by-step example shows how to calculate R-squared by hand for a given regression model.
Step 1: Create a Dataset
First, let’s create a dataset:
Step 2: Calculate Necessary Metrics
Next, let’s calculate each metric that we need to use in the R2 formula:
Step 3: Calculate R-Squared
Lastly, we’ll plug in each metric into the formula for R2:
- R2 = [ (nΣxy – (Σx)(Σy)) / (√nΣx2-(Σx)2 * √nΣy2-(Σy)2) ]2
- R2 = [ (8*(2169) – (72)(223)) / (√8*(818)-(72)2 * √8*(6447)-(223)2) ]2
- R2 = 0.6686
Note: The n in the formula represents the number of observations in the dataset and turns out to be n = 8 observations in this example.
Assuming x is the predictor variable and y is the response variable in this regression model, the R-squared for the model is 0.6686.
This tells us that 66.86% of the variation in the variable y can be explained by variable x.