[Maths Class Notes] on What are the Discrete Facets of Coefficient of Determination? Pdf for Exam

The coefficient of determination method in statistical analysis is used to forecast and describe the future outcomes of a model. This method is also referred to as R squared, which acts as a guideline to measure the accuracy of the model.

The coefficient of determination or R squared method is the variance of the dependent variable in the proportion that is predicted through an independent variable. It denotes the level of variation in the provided data.

• The coefficient of determination is nothing but the square of the correlation(r), hence it ranges from 0 to 1.

• As per linear regression, the coefficient of determination is the same as the square of the correlation between the x and y variables.

• If R² is 0, the independent variable cannot predict the dependent variable.

• If R² is 1, the independent variable without any error cannot predict the dependent variable.

• If R² ranges between 0 and 1, then it denotes the extent that can be predicted by the dependent variable. If R² is 0.10, then the x variable has predicted 10 percent of the variance in the y variable. If R² of 0.20, then the x variable has predicted 10 percent of the variance in the y variable, and so on.

• Whether the model would fit the given data set or not, can be determined using the R² value. For any percentage of the variation, the good fit would be different. 

Coefficient of Determination Formula

The formula to find the coefficient of determination is classified into two types- The correlation coefficient and the sum of squares.

Formula 1:

[ r = frac{n(sum xy) – (sum x) (sum y)}{sqrt{[nsum x^{2} – (sum x)^{2}] [nsum y^{2} – (sum y)^{2}]}}]

Where,

n is the total number of observations.

Σx is the total of the first variable value.

Σy is the total of the second variable value.

Σxy is the sum of the product of the first & the second value.

Σx2 is the sum of the squares of the first value.

Σy2 is the sum of the squares of the second value.

Thus, the coefficient of determination = (correlation coefficient)2 = r2.

Formula 2:

[R^{2} = 1 – frac{RSS}{TSS}]

Where,

R2 is the coefficient of determination.

RSS is the residual sum of squares.

TSS is the total sum of squares.

Properties of Coefficient of Determination

• It helps to determine the variation in the ratio of how a variable is predicted from the other.

• It helps to determine how to make predictions from the given data by using this measurement.

• It helps to determine the explained variation / total variation

• It helps to understand the strength of the association (linear) between the variables.

• If the value of R² is close to 1, The values of y will be close to the regression line, and similarly, if it is nearing 0, the values move away from the regression line.

• It helps to determine the strength of association between different variables.

Steps to Find the Coefficient of Determination

1. Find r, Correlation Coefficient

2. Square ‘r’.

3. Change the value to a percentage.

Conclusion:

This article focuses on the Coefficient of Determination and its application. You need to be thorough with the topic to be able to apply it in practice. Mathematics requires you to repeatedly do sums to perfect the concepts.