What is Adjusted R Squared?

Adjusted R Squared or Modified R^2 determines the extent of the variance of the dependent variable, which the independent variable can explain. The specialty of the modified R^2 is that it does not consider the impact of all independent variables but only those which impact the variation of the dependent variable. Therefore, the value of the modified R^2 can also be negative, though it is not always negative.

Adjusted R squared Formula

The formula to calculate the adjusted R square of regression is below:

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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 moreN= Number of observations in the regression equationXi= Independent variable of the regression equationX= Mean of the independent variable of the regression equationYi= Dependent variable of the regression equationY= 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 equationσx = Standard deviation of the independent variableσy = Standard deviation of the dependent variable.

Please Note

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

Interpretation

The value of the modified R^2 can also be negative, though it is not always negative. 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. It 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. First, let us try to find out the relation between the distance covered by the truck driver and the age of the truck driver. Then, someone does a regression equation to validate whether what he thinks of the relationship between two variables is also validated by the regression equation.

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 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 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 student’s GPA, and the independent variable is the height of the students.

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 closest to 100%.

Adjusted R Square is a significant output to determine whether the data set is a good fit. Someone does a regression equation to validate whether what he thinks of the relationship between two variables is also validated by the regression equation. The higher the value, the better the regression equation, which implies that the independent variable chosen to determine the dependent variable is chosen appropriately. Ideally, a researcher will look for the coefficient of determination closest to 100%.

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

  • Coefficient of DeterminationCoefficient Of DeterminationCoefficient of determination, also known as R Squared determines the extent of the variance of the dependent variable which can be explained by the independent variable. Therefore, the higher the coefficient, the better the regression equation is, as it implies that the independent variable is chosen wisely.read moreGini Coefficient FormulaGini Coefficient FormulaGini Coefficient or Gini Index is statistical dispersion depicting the income dispersions amongst the population of a country i.e. it represents the wealth inequalities of the citizens of a particular country. read moreR SquaredRegression vs ANOVAExpected Value FormulaExpected Value FormulaThe expected value formula depicts the possible value of an investment or asset in a future period. It is evaluated as the sum of the occurrence probabilities of all the random variables.read more