R2 Is Calculated in The Following Way

R-squared (R2) is a statistical measure that represents the proportion of the variance in the dependent variable that's explained by the independent variable(s) in a regression model. It ranges from 0 to 1, with higher values indicating better fit. This guide explains how R2 is calculated, how to interpret it, and provides practical examples.

What is R-squared (R2)?

R-squared (R2) is a key metric in regression analysis that measures how well a statistical model predicts an outcome. It shows the proportion of the variance in the dependent variable that is predictable from the independent variable(s).

R2 is calculated by comparing the total sum of squares (SST) to the sum of squares of residuals (SSR). The formula is:

R-squared formula

R² = 1 - (SSR / SST)

Where:

SSR = Sum of Squares of Residuals (difference between actual and predicted values)
SST = Total Sum of Squares (total variation in the dependent variable)

R2 values range from 0 to 1:

0 indicates the model explains none of the variability
0.5 indicates the model explains 50% of the variability
1 indicates the model explains all the variability

How to calculate R2

Calculating R2 involves these steps:

Fit a regression model to your data
Calculate the predicted values for each data point
Compute the residuals (actual - predicted values)
Calculate the sum of squares of residuals (SSR)
Calculate the total sum of squares (SST)
Apply the R2 formula: R² = 1 - (SSR / SST)

Note

R2 can be adjusted for the number of predictors in the model (adjusted R2) to account for overfitting.

Interpreting R2 values

R2 values have specific interpretations:

0.7-1: Excellent fit (70-100% of variance explained)
0.5-0.69: Good fit (50-69% of variance explained)
0.3-0.49: Moderate fit (30-49% of variance explained)
0-0.29: Weak or no fit (0-29% of variance explained)

However, R2 alone doesn't indicate whether the relationship is causal. Correlation doesn't imply causation.

Worked example

Let's calculate R2 for a simple linear regression with these data points:

X (Independent)	Y (Dependent)
1	2
2	3
3	5
4	4
5	6

After fitting a regression line (y = 0.8x + 0.6), we calculate:

SSR = 1.2
SST = 8.4
R² = 1 - (1.2 / 8.4) = 0.857 (or 85.7%)

This indicates the model explains 85.7% of the variance in Y.

Limitations of R2

While R2 is useful, it has several limitations:

It only measures linear relationships
It can be misleading with small sample sizes
It doesn't indicate causation
It can be inflated by adding more predictors
It doesn't account for outliers

For these reasons, R2 should be used alongside other statistical measures and domain knowledge.

FAQ

What does an R2 value of 0.8 mean?: An R2 value of 0.8 means the model explains 80% of the variance in the dependent variable. This indicates a strong fit between the model and the data.
Can R2 be negative?: No, R2 cannot be negative. The minimum value is 0, which indicates the model explains none of the variance. Negative values would imply the model performs worse than a horizontal line, which is impossible.
Is a higher R2 always better?: Not necessarily. While a higher R2 indicates a better fit, it's important to consider other factors like model complexity, overfitting, and practical significance. Sometimes a slightly lower R2 with a simpler model may be preferable.
How does R2 compare to correlation coefficient?: The correlation coefficient (r) measures the strength and direction of a linear relationship, while R2 measures the proportion of variance explained. For simple linear regression, R2 is simply the square of the correlation coefficient (R² = r²).
What's the difference between R2 and adjusted R2?: Adjusted R2 accounts for the number of predictors in the model, penalizing unnecessary complexity. It's calculated as: Adjusted R² = 1 - [(1-R²)(n-1)/(n-k-1)], where n is the sample size and k is the number of predictors. Adjusted R2 will always be less than or equal to R2.