Adjusted R Squared

What is Adjusted R Squared?

Adjusted R Squared refers to the statistical tool which helps the investors in measuring the extent of the variance of the variable which is dependent that can be explained with the independent variable and it considers the impact of only those independent variables which have an impact on the variation of the dependent variable.

Adjusted R Squared or Modified R^2 determines the extent of the variance of the dependent variable, which can be explained by the independent variable. The specialty of the modified R^2 is it does not take into count the impact of all independent variables rather only those which impact the variation of the dependent variable. The value of the modified R^2 can be negative also, though it is not negative most of the time.

Adjusted R squared Formula

The formula to calculate the adjusted R square of regression is represented as below,

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For eg:
Source: Adjusted R Squared (wallstreetmojo.com)

Where

• R^2= adjusted R square of the regression equationRegression EquationThe regression formula is used to evaluate the relationship between the dependent and independent variables and to determine how the change in the independent variable affects the dependent variable. Y = a + b X +read more
• N= Number of observations in the regression equation
• Xi= Independent variable of the regression equation
• X= Mean of the independent variable of the regression equation
• Yi= Dependent variable of the regression equation
• Y= MeanMeanMean refers to the mathematical average calculated for two or more values. There are primarily two ways: arithmetic mean, where all the numbers are added and divided by their weight, and in geometric mean, we multiply the numbers together, take the Nth root and subtract it with one.read more of the dependent variable of the regression equation
• σx = Standard deviation of the independent variable
• σy = Standard deviation of the dependent variable.

Interpretation

Adjusted R square determines the extent of the variance of the dependent variable, which can be explained by the independent variable. By looking at the adjusted R^2 value, one can judge whether the data in the regression equation is a good fit. The higher the adjusted R^2 better the regression equation as it implies that the independent variable is chosen to determine the dependent variable can explain the variation in the dependent variable.

The value of the modified R^2 can be negative also, though it is not negative most of the time. In the adjusted R square, the value of the adjusted R square will go up with the addition of an independent variable only when the variation of the independent variableIndependent VariableIndependent variable is an object or a time period or a input value, changes to which are used to assess the impact on an output value (i.e. the end objective) that is measured in mathematical or statistical or financial modeling.read more impacts the variation in the dependent variable. This is not applicable in the case of R^2, only relevant to the value of adjusted R^2.

Examples

Example #1

Let us try and understand the concept of adjusted R^2 with the help of an example. Let us try to find out what is the relation between the distance covered by the truck driver and the age of the truck driver. Someone does a regression equation to validate whether what he thinks of the relationship between two variables is also validated by the regression equation.

In this particular example, we will see which variable is the dependent variable and which variable is the independent variable. The dependent variable in this regression equation is the distance covered by the truck driver, and the independent variable is the age of the truck driver. By running a regression with the variables, we got the adjusted R square to be 65%. The snapshot below depicts the regression output for the variables. The data set and the variables are presented in the excel sheet attached.

The adjusted R^2 value of 65% for this regression implies that the independent variable explains 65% of the variation in the dependent variable. Ideally, a researcher will look for the coefficient of determination, which is closest to 100%.

Example #2

Let us try and understand the concept of adjusted R square with the help of another example. Let us try to find out what is the relation between the height of the students of a class and the GPA grade of those students. In this particular example, we will see which variable is the dependent variable and which variable is the independent variable. The dependent variable in this regression equation is the GPA of the students, and the independent variable is the height of the students.

By running a regression with the variables, we got the adjusted R^2 to be negligible or negative. The snapshot below depicts the regression output for the variables. The data set and the variables are presented in the excel sheet attached.

The adjusted R^2 value is negligible for this regression, which implies that the independent variable does not explain the variation in the dependent variable. Ideally, a researcher will look for the coefficient of determination, which is closest to 100%.

Interpretation

Adjusted R square is a significant output to find out whether the data set is a good fit or not. Someone does a regression equation to validate whether what he thinks of the relationship between two variables is also validated by the regression equation. Higher the value, the better the regression equation as it implies that the independent variable chosen to determine the dependent variable is appropriately chosen. Ideally, a researcher will look for the coefficient of determination, which is closest to 100%.

Recommended Articles

This has been a guide to what is Adjusted R Squared and its meaning. Here we discuss how to perform Adjusted R Square using its formula and examples and a downloadable excel template. You can learn more about statistical modeling from the following articles –

• Coefficient of Determination
• Gini Coefficient Formula
• R Squared
• Regression vs ANOVA
• Expected Value Formula