how do you calculate r2

What is R² in plain language?

R² (read as “R-squared”) is the coefficient of determination. It tells you how much of the variation in your outcome variable (Y) is explained by your model. In simple terms, R² answers this question:

“How well does my regression line explain the ups and downs in the data?”

• R² = 0: the model explains none of the variation.
• R² = 1: the model explains all of the variation perfectly.
• R² = 0.72: about 72% of variation is explained by the model.

For most real-world data, R² is somewhere between 0 and 1, and usually not very close to 1 unless the relationship is very tight.

How do you calculate R²?

The most common formula is:

R² = 1 − (SSE / SST)

Where:

• SSE = Sum of Squared Errors (also called residual sum of squares): Σ(yi − ŷi)²
• SST = Total Sum of Squares: Σ(yi − ȳ)²
• yi = actual observed values
• ŷi = predicted values from the model
• ȳ = mean of observed Y values

If your model’s predictions are much better than just using the mean, SSE becomes small, so R² gets larger.

Step-by-step process

1. Fit your regression model and get predicted values (ŷ).
2. Compute SST = Σ(yi − ȳ)².
3. Compute SSE = Σ(yi − ŷi)².
4. Plug into R² = 1 − (SSE / SST).

Alternative method (simple linear regression only)

If you only have one predictor (simple linear regression), R² is also the square of the Pearson correlation between X and Y:

R² = r²

This shortcut is convenient, but once you have multiple predictors, you should use the sums-of-squares definition from regression output.

Worked mini example

Suppose we have 5 points:

• X: 1, 2, 3, 4, 5
• Y: 2, 4, 5, 4, 5

After fitting the best-fit line, you might get something like:

• SST = 6.8
• SSE = 2.4

Then:

R² = 1 − (2.4 / 6.8) = 0.6471

Interpretation: about 64.7% of variation in Y is explained by the linear trend with X.

How to interpret R² correctly

R² is useful, but easy to misuse. Keep these points in mind:

• High R² does not prove causation. It only shows fit, not cause-and-effect.
• Low R² is not always bad. In noisy domains (human behavior, markets, medicine), even modest R² can be meaningful.
• R² can increase when you add variables, even weak ones. That is why adjusted R² matters.

Adjusted R²: when you have multiple predictors

When you add more independent variables, regular R² never decreases. That can reward overfitting. Adjusted R² penalizes unnecessary complexity.

Formula:

Adjusted R² = 1 − (1 − R²) * ((n − 1) / (n − p − 1))

• n = number of observations
• p = number of predictors

Use adjusted R² when comparing models with different numbers of predictors.

Common mistakes to avoid

1) Treating R² as the only performance metric

Always check residual plots, RMSE/MAE, and out-of-sample performance. A model can have decent R² and still be practically poor.

2) Ignoring nonlinearity

If the true relationship is curved, a straight line may produce a weak R² even though a nonlinear model would fit well.

3) Using R² with no domain context

What is “good” depends on your field. In some physical systems, 0.95 may be expected. In many social systems, 0.20 can still be informative.

Quick recap

• Compute R² with 1 − SSE/SST.
• In simple linear regression, R² = r².
• Interpret as proportion of explained variation.
• Use adjusted R² for multi-variable model comparison.
• Pair R² with residual diagnostics and validation.

If you want a fast answer, use the calculator above. If you want a trustworthy model, pair that number with good statistical judgment.