## What is the Coefficient of Determination?

Coefficient of determination is the measure of accuracy of project that how project is working and its future outcome and is calculated by-product of one divided by number of variables into sum of product of the first variable minus mean of the first variable into the second variable minus mean of second variable whole divided by standard deviation of the first variable into standard deviation of the second variable.

It is also known as R Squared determines the extent of the variance of the dependent variable which can be explained by the independent variable. By looking at R^2 value one can judge whether the regression equation is good enough to be used. Higher the coefficient better the regression equation as it implies that the independent variable chosen in order to determine the dependent variable is chosen properly.

### Detailed Explanation of the Coefficient of Determination

Where

- R= Correlation
- R^2= Coefficient of determination 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

The value of the coefficient ranges from 0 to 1, where a value of 0 indicates that the independent variable does not explain the variation of the dependent variable and a value of 1 indicates that the independent variable perfectly explains the variation in the dependent variable.

### Examples

#### Example #1

Let us try and understand the coefficient of determination formula 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 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. We can find the correlation with the help of the formula and square that to get the coefficient of the regression equation. The data set and the variables are presented in the excel sheet attached.

**Solution:**

Below is given data for calculation of the coefficient of determination.

Therefore, the calculation of the coefficient of determination formula is as follows,

R = -424520/√(683696*81071100)

R will be –

**R = -0.057020839**

R^2 will be –

**R^2 = 0.325%**

#### Example #2

Let us try and understand the concept of coefficient of determination 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. We can find the correlation with the help of the formula and square that to get the R^2 of the regression equation. The data set and the variables are presented in the excel sheet attached.

**Solution:**

Below is given data for calculation of the coefficient of determination.

Therefore, the calculation of the coefficient of determination formula is as follows,

R = 34.62/√(169204*3245)

**R = 0.000467045**

**R^2 = 0.000000218**

### Interpretation

The coefficient of determination 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 analysis to validate whether what he thinks of the relationship between two variables, is also validated by the regression equation. Higher the coefficient 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%.

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