How to Calculate R-Squared by Hand

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In statistics, R-squared (R²) measures the proportion of the variance in the response variable 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.

Additional Resources

Introduction to Simple Linear Regression
Introduction to Multiple Linear Regression
R vs. R-Squared: What’s the Difference?
What is a Good R-squared Value?