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Linear Regression Coefficient of Determination

TopFor Linear Regression Coefficient of determination, we use following steps-
Step 1: First of all, we evaluate value of ‘N’, where ‘N’ is equal to all data entries which is present in a Set and all summations values means,
∑xy, ∑x, ∑y, ∑x2 and ∑y2.

Step 2: After evaluation of all summation values, now we use following formula to evaluate Linear Correlation coefficient –
Correlation coefficient r = N ∑xy - (∑x)(∑y) / √([N ∑x2 - (∑x)2]) . ([N ∑y2 - (∑y)2]),

Step 3: After evaluation of Linear Correlation Coefficient, we know determine the strength and direction between the given variables ‘x’ and y by r2.
Now we take an example to understand the process of linear correlation coefficient-

Example: Find the Linear Regression coefficient of determination for following variables -
X: 26 14 18 10 26 21 7 26 13 19 17 13 16 28 23
Y: 20 10 13 9 19 17 8 15 9 13 12 7 9 17 14?

Solution: We use following steps for linear correlation coefficient-
Step 1: First of all, we find out all values-
N = 15,
∑xy = 3882,
∑x = 277,
∑y = 192,
∑x2 = 5695,
∑y2 = 2698.

Step 2: Now we calculate linear correlation coefficient-
Correlation coefficient r = N ∑xy - (∑x)(∑y) / √([N ∑x2 - (∑x)2]) . ([N ∑y2 - (∑y)2]),
=> r = 15(3882) – (277)(192) / √([15 (5695) - (277)2]) . ([15 (2698) - (192)2]),
=> r = 0.901.

Step 3: Now we calculate r2 means Square of correlation coefficient-
r2 = (0.901)2 = 0.81.
So, this linear correlation coefficient have 81% strong relation between ‘x’ and ‘y’ variable and 19% weak relation between ‘x’ and ‘y’ variable.