## 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,

**R^2 = {(1 / N) * Σ [(xi – x) * (yi – y)] / (σx * σy)}^2**

Where

- R^2= adjusted R square of the regression equation
- 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= Mean of the dependent variable of the regression equation
- σx = Standard deviation of the independent variable
- σy = Standard deviation of the dependent variable.

**Please Note**

For calculating it in excel it needs to be provided y and x variables in the excel and the whole output along with Adjusted R^2 is generated by Excel. It is a special case where it is difficult to provide the output in text format, unlike other formulas.

### 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. Higher the adjusted R^2 better the regression equation as it implies that the independent variable chosen in order to determine the dependent variable is able to 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 case of 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 variable impacts the variation in the dependent variable. This is not applicable in the case of R^2, only applicable to the value of adjusted R^2.

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### 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 actually does 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 65% of the variation in the dependent variable is explained by the independent 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 variation in the dependent variable is not explained by the independent variable. Ideally, a researcher will look for the coefficient of determination which is closest to 100%.

### Interpretation

Adjusted R square is a very important output in order to find out whether the data set is a good fit or not. Someone actually 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, better the regression equation as it implies that the independent variable chosen in order to determine the dependent variable is chosen properly. 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 along with examples and 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

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