R-squared in Simple Linear Regression Analysis

1. R-squared in Regression Analysis

The r-squared is also called coefficient of determination. It shows the proportion of the total variation of the dependent variable can be explained by independent variables. However, this value can be inflated as the number of independent variables increases. Thus, some people prefer to use the adjusted r-squared. This concept is likely to be introduce during the university statistics lessons.

2. Formula for calculating R-squared

R-squared = Sum of square of residual / Sum of square of total

Exercise

Calcuate the R-squared for the following regression lines.

Question 1

Y1	X1
10	12
15	15
13	10
14	15
10	15
14	15
13	15
13	11
10	10
10	15
10	10
14	11
10	14
11	11
10	11
13	15
14	12
10	14
10	11
13	10
10	10

Question 2

Y2	X2
10	14
11	13
12	15
15	14
13	15
12	15
13	11
10	10
15	10
12	15
11	12
15	10
12	15
13	15
15	14
14	14
13	10
12	10
15	13
10	15
11	12

Question 3

Y3	X3
10	14
10	13
12	13
10	11
14	13
13	12
14	14
15	12
13	13
12	10
12	10
12	14
15	15
10	15
14	11
14	13
15	14
12	10
10	11
10	15
14	10

Question 4

Y4	X4
11	11
14	12
14	10
12	14
14	10
12	10
12	11
13	12
10	15
11	10
14	15
12	11
12	13
11	11
11	14
12	12
14	13
12	13
14	15
12	13
10	15

Question 5

Y5	X5
12	12
10	12
11	10
15	14
10	10
11	14
10	11
12	10
10	15
12	14
14	14
10	14
11	15
13	15
14	14
13	15
10	13
14	11
13	13
15	11
15	10

Question 6

Y6	X6
11	11
14	12
14	13
14	10
10	12
15	12
13	15
15	10
12	12
12	10
15	11
10	10
12	10
15	10
12	10
14	13
14	10
12	11
11	12
11	12
12	11

Answer

0.059
0.014
0.000
0.015
0.000
0.000

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Y1	X1
10	12
15	15
13	10
14	15
10	15
14	15
13	15
13	11
10	10
10	15
10	10
14	11
10	14
11	11
10	11
13	15
14	12
10	14
10	11
13	10
10	10

Y2	X2
10	14
11	13
12	15
15	14
13	15
12	15
13	11
10	10
15	10
12	15
11	12
15	10
12	15
13	15
15	14
14	14
13	10
12	10
15	13
10	15
11	12

Y3	X3
10	14
10	13
12	13
10	11
14	13
13	12
14	14
15	12
13	13
12	10
12	10
12	14
15	15
10	15
14	11
14	13
15	14
12	10
10	11
10	15
14	10

Y4	X4
11	11
14	12
14	10
12	14
14	10
12	10
12	11
13	12
10	15
11	10
14	15
12	11
12	13
11	11
11	14
12	12
14	13
12	13
14	15
12	13
10	15

Y5	X5
12	12
10	12
11	10
15	14
10	10
11	14
10	11
12	10
10	15
12	14
14	14
10	14
11	15
13	15
14	14
13	15
10	13
14	11
13	13
15	11
15	10

Y6	X6
11	11
14	12
14	13
14	10
10	12
15	12
13	15
15	10
12	12
12	10
15	11
10	10
12	10
15	10
12	10
14	13
14	10
12	11
11	12
11	12
12	11

Y1	X1
10	12
15	15
13	10
14	15
10	15
14	15
13	15
13	11
10	10
10	15
10	10
14	11
10	14
11	11
10	11
13	15
14	12
10	14
10	11
13	10
10	10

Y2	X2
10	14
11	13
12	15
15	14
13	15
12	15
13	11
10	10
15	10
12	15
11	12
15	10
12	15
13	15
15	14
14	14
13	10
12	10
15	13
10	15
11	12

Y3	X3
10	14
10	13
12	13
10	11
14	13
13	12
14	14
15	12
13	13
12	10
12	10
12	14
15	15
10	15
14	11
14	13
15	14
12	10
10	11
10	15
14	10

Y4	X4
11	11
14	12
14	10
12	14
14	10
12	10
12	11
13	12
10	15
11	10
14	15
12	11
12	13
11	11
11	14
12	12
14	13
12	13
14	15
12	13
10	15

Y5	X5
12	12
10	12
11	10
15	14
10	10
11	14
10	11
12	10
10	15
12	14
14	14
10	14
11	15
13	15
14	14
13	15
10	13
14	11
13	13
15	11
15	10

Y6	X6
11	11
14	12
14	13
14	10
10	12
15	12
13	15
15	10
12	12
12	10
15	11
10	10
12	10
15	10
12	10
14	13
14	10
12	11
11	12
11	12
12	11

Y1	X1
10	12
15	15
13	10
14	15
10	15
14	15
13	15
13	11
10	10
10	15
10	10
14	11
10	14
11	11
10	11
13	15
14	12
10	14
10	11
13	10
10	10

Y2	X2
10	14
11	13
12	15
15	14
13	15
12	15
13	11
10	10
15	10
12	15
11	12
15	10
12	15
13	15
15	14
14	14
13	10
12	10
15	13
10	15
11	12

Y3	X3
10	14
10	13
12	13
10	11
14	13
13	12
14	14
15	12
13	13
12	10
12	10
12	14
15	15
10	15
14	11
14	13
15	14
12	10
10	11
10	15
14	10

Y4	X4
11	11
14	12
14	10
12	14
14	10
12	10
12	11
13	12
10	15
11	10
14	15
12	11
12	13
11	11
11	14
12	12
14	13
12	13
14	15
12	13
10	15

Y5	X5
12	12
10	12
11	10
15	14
10	10
11	14
10	11
12	10
10	15
12	14
14	14
10	14
11	15
13	15
14	14
13	15
10	13
14	11
13	13
15	11
15	10

Y6	X6
11	11
14	12
14	13
14	10
10	12
15	12
13	15
15	10
12	12
12	10
15	11
10	10
12	10
15	10
12	10
14	13
14	10
12	11
11	12
11	12
12	11