When you fit a regression model using most statistical software, you’ll often notice the following two values in the output:

**Multiple R:** The multiple correlation coefficient between three or more variables.

**R-Squared:** This is calculated as (Multiple R)^{2} and it represents the proportion of the variance in the response variable of a regression model that can be explained by the predictor variables. This value ranges from 0 to 1.

In practice, we’re often interested in the R-squared value because it tells us how useful the predictor variables are at predicting the value of the response variable.

However, each time we add a new predictor variable to the model the R-squared is guaranteed to increase even if the predictor variable isn’t useful.

The **adjusted R-squared** is a modified version of R-squared that adjusts for the number of predictors in a regression model. It is calculated as:

**Adjusted R ^{2} = 1 – [(1-R^{2})*(n-1)/(n-k-1)]**

where:

**R**: The R^{2}^{2}of the model**n**: The number of observations**k**: The number of predictor variables

Since R-squared always increases as you add more predictors to a model, adjusted R-squared can serve as a metric that tells you how useful a model is, *adjusted for the number of predictors in a model*.

To gain a better understanding of each of these terms, consider the following example.

**Example: Multiple R, R-Squared, & Adjusted R-Squared**

Suppose we have the following dataset that contains the following three variables for 12 different students:

Suppose we fit a multiple linear regression model using *Study Hours* and *Current Grade* as the predictor variables and *Exam Score* as the response variable and get the following output:

We can observe the values for the following three metrics:

**Multiple R: 0.978**. This represents the multiple correlation between the response variable and the two predictor variables.

**R Square: 0.956**. This is calculated as (Multiple R)^{2} = (0.978)^{2} = 0.956. This tells us that 95.6% of the variation in exam scores can be explained by the number of hours spent studying by the student and their current grade in the course.

**Adjusted R-Square: 0.946**. This is calculated as:

**Adjusted R ^{2} **= 1 – [(1-R

^{2})*(n-1)/(n-k-1)] = 1 – [(1-.956)*(12-1)/(12-2-1)] = 0.946.

This represents the R-squared value, *adjusted for the number of predictor variables in the model*.

This metric would be useful if we, say, fit another regression model with 10 predictors and found that the Adjusted R-squared of that model was **0.88**. This would indicate that the regression model with just two predictors is better because it has a higher adjusted R-squared value.

**Additional Resources**

Introduction to Multiple Linear Regression

What is a Good R-squared Value?