Coefficient of Determination Formula: R-Squared Solved Examples

Ans. The correlation coefficient ( R² ) is a measurable statistic of how near the data is to the regression line that has been fitted. For multiple regression, the correlation coefficient is also known as the coefficient of multiple determination. R² of 0% shows that the model does not explain any of the variability in the response data around its mean.

Ans. The most typical application of ( R²) is to see how well the regression model fits the data. R² of 80%, for example, indicates that 80% of the data "accommodate" the regression model. A bigger coefficient usually indicates a better fit for the model. A big r-squared is superlative for the regression model, albeit this is not generally true. However, the coefficient's quality is influenced by a number of factors, including the variables' units, the characteristics of the variables utilised in the model, and the data transformation employed. As a result, even a large coefficient might cause issues with the regression model.

Ans. The required answer is (D) 0.80

Explanation: The reason for this is that R² represents the coefficient of determination. Because the range for the coefficient of correlation is -1 to +1, and therefore the range for the coefficient of determination is 0 to +1, a coefficient of determination of 0.64 suggests that the coefficient of correlation will be 0.8.

Ans. The following are the Properties of Coefficient of Determination

• It aids in calculating the ratio of how a variable that may be predicted from another fluctuates.
• This assessment may be used to judge how clear it is to make predictions based on the facts provided.
• It aids in the search for Total Variation / Explained Variation
• It also tells us how strong the relationship (linear) between the variables is.
• It aids in identifying the degree to which distinct variables are linked.

Ques. The relationship between two variables, m and n, is seen here, with m + 3n = 10. The relationship between the remaining two variables, p and q, is the same, with 2p + 5q = 25. The p/m coefficient is equal to 0.80. Calculate the q on m coefficient. (3 Marks)

Ans. We have, m + 3n = 10

n= m-103-13M -103 -13

Furthermore, 2p + 5q = 25

q= p-252-52p-252 -52

Since, we are familiar of the formula,

Then, 0.80 = -2.5-0.33-2.5-0.33× bqnbqn

0.80= 7.5 bqnbqn

Thus, Bqnbqn= 0.1333 × 0.80 = 8/75

Hence, the required answer is 8/75

Ans. The most typical application of ( R² ) is to see how well the regression model fits the data. R2 of 80%, for example, indicates that 80% of the data "accommodate" the regression model. A bigger coefficient usually indicates a better fit for the model. A big r-squared is superlative for the regression model, albeit this is not generally true. However, the coefficient's quality is influenced by a number of factors, including the variables' units, the characteristics of the variables utilised in the model, and the data transformation employed. As a result, even a large coefficient might cause issues with the regression model.

Ans. 

r = n(∑xy)−(∑x)(∑y) / √[n∑x²−(∑x)²][n∑y²−(∑y)²]

= 4×38−10×16 / √4×30−10²][4×74−16²]

= 4×38−10×16 / √[4×30−100][4×74−256]

= 4×38−10×16[4×30−100][4×74−256]

 = 152−160 / √ [120−100[296−256]

 = −8 / √[20][40]

 = −8 / √800

 = −8 / 28.28

 = -0.28

Ans. 

r = n(∑xy)−(∑x)(∑y) / √[n∑x²−(∑x)2][n∑y²−(∑y)²]

= 4×20−10×10 / √4×30−10²][4×30−10²]

= 4×20−10×10 / √[4×30−100][4×30−100]

= 80 − 100 / √ [120−100[120−100]

 = −20 / √[20][20]

 = −20 / √400

 = −20 / 20

 = −1

Ans. 

r = n(∑xy)−(∑x)(∑y) / √[n∑x²−(∑x)²][n∑y²−(∑y)²]

= 4×48−14×14 / √4×50−14²][4×50−14²]

= 4×48−14×14 / √[4×50−196][4×50−196]

= 192 − 196 / √ [200−196[200−196]

 = −4 / √[4][4]

 = −4 / √16

 = −4 / 4

 = −1

Ans. 

r = n(∑xy)−(∑x)(∑y) / √[n∑x²−(∑x)²][n∑y²−(∑y)²]

= 4×154−19×28 / √[4×113−19²][4×216−28²]

= 4×154−19×28 / √[4×113−361][4×216−784]

= 616 − 532 / √ [452−361[864−784]

 = 84 / √[91][80]

 = 84 / √7280

 = 84 / 85.32

 = 0.98

Ans. R-squared is calculated using the following formula R-squared = 1 - (SSR/SST).

Ans. A good coefficient of determination (R-squared) depends on the context and the specific application. However, in general, a value of 0.7 or higher is considered to be good. This means that the model explains at least 70% of the variation in the dependent variable.

Here is a table of R-squared values and their interpretations:

Just because a model has a high R-squared value does not mean that the independent variables cause the dependent variable.

Ans. Some of the applications of the Coefficient of Correlation Formula are as follows – 

• It is a statistical measure of the strength and direction of the linear relationship between two variables.
• It ranges from -1 to 1, with higher values indicating a stronger relationship.
• It can be used to identify relationships between variables and predict the value of one variable based on the value of the other.
• It also compares the relationships between different variables and the same variable.
• It is used in a wide variety of fields, including business, finance, and science.

Ans. The coefficient of determination (R-squared) and correlation coefficient (r) are both statistical measures of the relationship between two variables. 

• R-squared is a measure of model fit, while r is a measure of the strength and direction of the linear relationship.
• R-squared ranges from 0 to 1, while r ranges from -1 to 1.
• A higher R-squared value indicates a better fit of the model to the data, while a higher r value indicates a stronger linear relationship between the two variables.

Ans. If r = 0.8, then the coefficient of determination (R-squared) is 0.64. This means that the model explains 64% of the variation in the dependent variable.

R-squared = r² = 0.8² = 0.64