How to calculate coefficient of determination correctly

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WHAT IS THE CO-EFFICIENT OF DETERMINATION?

Coefficient of determination is also called R-SQUARED (r²). it is used to explain the relationship between independent variable and dependent variables. Business investors use coefficient of determination to conduct trend analysis. 

Coefficient of determination is a further statistical treatment of Regression analysis. In regression analysis there are three types of variation namely;

  • total variation
  • explained variation
  • unexplained variation
the purpose of regression analysis is to try to allocate as much of the total variation in Y in the variation of X variable. This variation should there be as high as possible. NOW THEREFORE, co-efficient of variation expresses the proportion of varience in Y  determined in X. We can now define coefficient of determination as the ratio of explained variation to the total variation.

It is used to measure the strength of the linear relationship.The stronger the linear relation the high the value of the coefficient.

The coefficient of determination is a positive interger between 0 and 1. It cannot be negative because r-squared mean that a negative squared should result to a positive.

INTERPRETATION OF THE VALUE OF THE COEFFICIENT OF DETERMINATION

  • If the coefficient is 1 (unity); this shows that the model is reliable for future forecasting.
  • If the coefficient is 0 ; this means that the model is unreliable for forecasting.
  • Any value between 0 and 1 shows the proportion that the model can explain or forecast. for instannce,if r-squared is  0.64, this means that 64% of the total variation in the dependent variable has been explained by the independent variable.
  • The maximum value of r²is one, which is translated to 100% explained variation,beyond which the is no variation to be explained.
HOW TO CALCULATE COEFFICIENT OF DETERMINATION.

Since it is an application of regresion analysis, coefficient of determination is calculated as under:




                      n (ΣXY) - ΣX ΣY         
  R² = √[(nΣx²-(Σx)²] [(nΣy² -(Σy)²]
EXAMPLE

Covey Limited hopes to achieve a stock market quotation for its shares. A profit forecat is necessary and in order to achieve such a forecast , the company has experimeted with a number of approaches.The following are deatails from a linear regression on the last 13 years profit figures. 
X = years expressed as 1 to l3
Y = annual profit figures.

Σx = 66, Σy= 212.10 ,Σx² = 506, Σxy = 1406.70, Σy²=4254.08.

Required.
calculate the coefficient of determination.

SOLUTION


     R² =          n (ΣXY) - ΣX ΣY       
            √[(nΣx²-(Σx)²] [(nΣy² -(Σy)²]

n is the number of pairs of data items.
n=13

   ∴  R²  =               13 (1406.7) - 66 * 212.1                     
                    √[(13*506-(66)²] [(13*4254.08 -(212.10)²]

THUS, ∴  R²  = 0.896
This means that 89.6% of the variation in dependent variable  can be explained by independent variable.

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