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)
(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.