How to Find the Sample Coefficient of Determination Given Adjusted Sample Coefficient of Determination?

Sample coefficient of determination can be defined as measuring source of dependent variables whatever changes occur in them. This measurement tells that how regression line looks. Sample coefficient of determination always exists as 0 ≤ r² ≤ 1. r² is the symbol used to represent sample coefficient of determination.

When two variables are not identical then there is always some variation. Total variation in a variable or Set is made up of two parts. One of them is explained by regression equation and other which is not described by regression equation.
A formula of r² is given as
r² = (1 / N) ∑ (a i − aˉ) (b i − b) / σ a σ b ²
Where 'N' is number of observation, aˉ and bˉ is Mean of ‘a’ and ‘b’ values, σ a and σ b is the Standard Deviation of ‘a’ and ‘b’ values.

Let’s see How to Find the Sample Coefficient of Determination Given Adjusted Sample Coefficient of Determination
Adjusted coefficient of determination of a Linear Regression model is given in form of sample coefficient of determination as follows,
r² (adj) = 1 – (1 – r²) (x- 1) / (x – y – 1),
From above formula, sample coefficient of determination can be determined by rearranging the above expression as shown below.
r² (adj) = 1 – (1 – r2) (x - 1) / (x – y – 1) = (x – y – 1) - (1 – r2) (x - 1) / (x – y – 1)
Then,
(x – y – 1) r² (adj) = (x – y – 1) - (x - 1 – xr² + r²) = x – y – 1 - x + 1 + xr² - r² = x r² - r² - y,
Here 'x' is the number of observations in data set and 'y' is the number of independent variables.
On solving above equation, we get the sample coefficient of determination.