R Squared | Coefficient of Determination

The coefficient of determination, also called R-squared (R² or r²), is a statistical measure that shows how well the independent variable explains the variation in the dependent variable. It represents the proportion of variance in the dependent variable that can be predicted from the independent variable in a regression model.

The coefficient of determination, which is represented by R,² is determined using the following formula:

R² = 1 – (RSS/TSS)

• R² represents the requrired R Squared value,
• RSS represents the residual sum of squares, and
• TSS represents the total sum of squares.

The coefficient of determination can also be calculated using another formula, which is given by: R² = r²

Where,

• n = Number of scores given
• ∑XY = Sum of paired products
• ∑X = Sum of X scores
• ∑Y = Sum of Y scores
• ∑X² = Sum of squares of X scores
• ∑Y² = Sum of squares of Y scores

R-Squared Value Interpretation

The R-squared value shows how well a regression model explains the variation in the dependent variable. A higher R² value indicates a better fit of the model, while a lower R² value indicates a weaker relationship between the variables. However, R² alone is not always sufficient to evaluate the overall performance of a regression model.

Adjusted R Squared

As R squared formula takes into account only 2 variables. If more variables are to be added, then the value of R square never decreases but increases. Thus, we need to adjust the R square in order to compensate for the added variables. By adjusting the R-squared value the model becomes resistant to overfitting and underfitting.

Adjusted R-square formula is given as follows:

\bold{R^2_{adj}=1-\frac{(1-R^2)(N-1)}{N-p-1}}

• R² is the Normal R square value,
• N is the Size of sample, and
• p is the no. of predictors.

R-Squared vs Adjusted R-Squared

The key differences between R-Squared and Adjusted R-Squared, are listed as follows:

Solved Examples

Problem 1: Calculate the coefficient of determination from the following data:

Solution:

To calculate the coefficient of determination from above data we need to calculate ∑x, ∑y, ∑(xy), ∑x², ∑y², (∑x)², (∑y)².

X	Y	XY	X2	Y2
1.2	0	0	1.44	0
1	5	5	1	25
2	2	4	4	4
3	0	0	9	0
∑x = 7.2	∑y = 7	∑xy = 9	∑x 2 = 15.44	∑y 2 = 29

(∑x)² = 51.84 and (∑y)² = 49 and n = 4

Using r = \frac{n\Sigma(xy)-\Sigma x \Sigma y}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}}

⇒ r = \frac{4(9)-(7.2*7)}{\sqrt{[4(15.44) - 51.84][4(29) - 49]}}

⇒ r = \frac{36-50.4}{\sqrt{[61.76 - 51.84][116 - 49]}}

⇒ r = \frac{-14.4}{\sqrt{[9.92][67]}}

⇒ r = \frac{-14.4}{\sqrt{664.64}}

⇒ r = \frac{-14.4}{25.78}

⇒ r = 0.558

Thus, R^2 = r^2 = (0.558)^2

⇒ R^2 = 0.3120 = 31.2 \%

Problem 2: Calculate the coefficient of determination from the following data:

Solution: 

To calculate the coefficient of determination from above data we need to calculate ∑x, ∑y, ∑(xy), ∑x², ∑y², (∑x)², (∑y)².

X	Y	XY	X2	Y2
1	1	1	1	1
2	2	4	4	4
3	3	9	9	9
∑x = 6	∑y = 6	∑xy = 14	∑x2 = 14	∑y2 = 14

(∑x)² = 36 and (∑y)² = 36 and n = 3

Using r = \frac{n\Sigma(xy)-\Sigma x \Sigma y}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}}

⇒ r = \frac{3(14)-(6*6)}{\sqrt{[3(14) - 36][3(14) - 36]}}

⇒ r = \frac{42-36}{\sqrt{[6][6]}}

⇒ r = \frac{6}{\sqrt{36}}

⇒ r = \frac{6}{6}

⇒ r = 1

Thus, R^2 = r^2 = (1)^2

⇒ R^2 = 1 = 100 \%

Problem 3: Calculate the coefficient of determination from the following data:

Solution: 

To calculate the coefficient of determination from above data we need to calculate ∑x, ∑y, ∑(xy), ∑x², ∑y², (∑x)², (∑y)².

X	Y	XY	X2	Y2
1	1	1	1	1
2	4	8	4	16
3	6	18	9	36
∑x = 6	∑y = 11	∑xy = 27	∑x2 = 14	∑y2 = 53

(∑x)² = 36 and (∑y)² = 121 and n = 3

Using r = \frac{n\Sigma(xy)-\Sigma x \Sigma y}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}}

⇒ r = \frac{3(27)-(6*11)}{\sqrt{[3(14) - 36][3(53) - 121]}}

⇒ r = \frac{81-66}{\sqrt{[42-36][159-121]}}

⇒ r = \frac{15}{\sqrt{6*38}}

⇒ r = \frac{15}{15.1}

⇒ r = 0.993

Thus, R^2 = r^2 = (0.993)^2

\Rightarrow R^2 = 0.9867 = 98.67 \%

Problem 4: Calculate the coefficient of determination if RSS = 1.5 and TSS = 1.9.

Solution:

Given RSS = 1.5, TSS = 1.9

Using R² = 1 - (RSS/TSS)

⇒ R² = 1 - (1.5/1.9)

⇒ R² = 1-0.7894 ≈ 79%

⇒ R2 = 0.21 ≈ 21%

Problem 5: Calculate the coefficient of determination if RSS = 1.479 and TSS = 1.89734.

Solution:

Given RSS = 1.479, TSS = 1.89734

Using R² = 1 - (RSS/TSS)

R² = 1 - (1.479/1.89734)

⇒ R² = 1 - 0.7795

⇒ R2 = 0.22 ≈ 22%