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R squared formula or the correlation coefficient of determination is sometimes referred to as goodness of fit. It is the number that indicates variance in the dependent variable that is to be predicted from the independent variable. These statistics measure the correlation between investment performance and specific benchmark indexes. It is explained as an independent variable or variables in a regression model. This correlation explains the strength of the relationship between two independent and dependent variables.
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Key Takeaways: Investment, value, outcome, coefficient, square, sum, value, correlation, determination.
What is the R Squared?
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R2 or r2 is pronounced as r squared is the correlation between the anticipated scores and the actual scores. This model is used to predict the outcome and is regarded as a testing hypothesis. The trigonometric value of r squared varies from -1 to +1.
Read more : cardinal numbers
Formula for R Squared
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R squared coefficient helps to summarise the relationship between the two variables in a single number.
The formula for r squared is as follows:
N = No of scores given
∑XY = the sum of paired product
x = x score sum
y= y score sum
∑ x2 = the square of X score sum
∑x2 = the square of Y score sum
Note:
Coefficient of determination = Maths Processing Error
Correlation coefficient = Maths Processing Error
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Interpretation of R2
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As mentioned earlier the significance of r2 varies from -1 to +1.
- An R2 of 0 indicates- the dependent variable is unable to be anticipated from the independent variable.
- An R2 of 1 indicates- the dependent variable can be anticipated error-free from the independent variable.
- An R2 between 0 and 1 indicates- the magnitude to which the dependent variable is foreseeable
- An R2 of 0.10- indicates that 10% of the variance in y is foreseeable from x.
- An R2 of 0.20- indicates that 20% is foreseeable.
A value of 1.0 indicates a perfect fit, thus showing a highly reliable model for future forecasts. A value of 0 indicates the model fails to accurately model the data.
Also Read: Principle of Mathematical Induction
Adequacy of R2
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- More the number of samples added, the coefficient would exhibit the probability of a new point falling on the line.
- A strong relationship between the linear equation in two variables doesn't mean that determination is not evidence of causality.
- Although interpretations of fit depend on the analysis, a high R2 value indicates the model is a good fit for the data.
Also Read: Prism Formula
Points to Remember
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Following are some important points:
- R squared is also known as the coefficient of determination.
- This theory focuses on the statistical assessment of future data.
- Coefficient of Determination formula = (Correlation Coefficient)2.
- The value of r squared varies from -1 to +1.
- A value of 0 shows the failure to correctly model the data.
- A value of 1 shows a perfect fit for future anticipation.
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Sample Questions
Ques: What will be the coefficient of correlation for the following?
x= 4,8,12,16
y=5, 10, 15, 20
Construct a table to get the required values of the formula. (4 Marks)
Ans: Following is the table given below:
| x | y | x2 | y2 | XY |
|---|---|---|---|---|
| 4 | 5 | 16 | 25 | 20 |
| 8 | 10 | 64 | 100 | 80 |
| 12 | 15 | 144 | 225 | 180 |
| 16 | 20 | 256 | 400 | 320 |
| ∑ x=40 | ∑ y=50 | ∑ x2=480 | ∑ y2=750 | ∑ xy= 600 |
We know that,
R2=\(\frac{N\sum xy \sum x \sum y }{\sqrt{N\sum x^2-(\sum x)2 N\sum y2-(\\sum y)2}}\)
Inserting the values,
R2= \(\frac{4X600 - (40X50)}{\sqrt 4X480-(40)2 4X750-(50)2}\)
R2 = \(\frac{400}{17.89X22.36}\)
= \(\frac{400}{400}\)
= 1
Therefore, 1 is the coefficient correlation.
Ques: Find the coefficient correlation of the following data?
x= 2,5,6,7
y=2,5,4,3
Construct a table to the required values of the formula. (4 Marks)
Ans: Following is the table given below:
| x | y | x2 | y2 | XY |
|---|---|---|---|---|
| 2 | 2 | 4 | 4 | 4 |
| 5 | 5 | 25 | 25 | 25 |
| 6 | 4 | 36 | 16 | 24 |
| 7 | 3 | 49 | 9 | 21 |
| ∑x=20 | ∑y=14 | ∑x2=114 | ∑y2= 54 | ∑ xy= 74 |
In this,
N= 4,
R2=\(\frac{N\sum xy \sum x \sum y }{\sqrt{N\sum x^2-(\sum x)2 N\sum y2-(\\sum y)2}}\)
Inserting the values that we acquired earlier,
R2= \(\frac{4X74 - (20X14)}{\sqrt 4X114-(20)2 4X54-(14)2}\)
= \(\frac{296-280}{7.48X4.472}\)
= 0.478
R2= ( 0.478)2
= 0.22848
Therefore, the coefficient of determination is 0.22848.
Ques: Find the conjugate of (3-2i)( 2+3i )( 1+2i)( 2-i ). (2 Marks)
Ans: We have,
(3-2i)( 2+3i )( 1+2i)( 2-i )
= 6+9i-4i+62-i+4i+2
= 12+5i4+3i x 4-3i4-3i
= 48-36i+20i + 1516+9
= 63-16i25
=6325 - 1625i
Therefore the conjugate is 6325 - 1625i.
Ques: Solve x2 + x + 1= 0. (2 Marks)
Ans: We know that,
b2-4ac= 12 - 4*1*1
= 1 - 4
= -3
Therefore the solution is given by,
x= \(\frac{-1+- (\sqrt-3)}{2X1}\)
= \(x= \frac{-1+- (\sqrt-3)}{2}\)
Ques: If x+ iy =\(\frac{a+ib}{a-ib}\), prove that x2+ y2 = 1. (3 Marks)
Ans: We have ,
x+ iy = \(\frac{(a+ib)(a+ib)}{(a-ib)(a+ib)}\)
=\(\frac{a^2-b^2+2abi}{a^2+b^2}\)
=\(\frac{a^2- b^2}{a^2+b^2}=\frac{2ab}{a^2+b^2}\)
So that,
x-iy=a2-b2a2+b2- 2aba2+b2i
Therefore,
x2+y2= ( x+ iy ) ( x-iy )
=\(\frac{(a^2- b^2)2}{(a^2+b^2)2}=\frac{4a^2b^2}{a^2+b^2}\)
= \(\frac{(a^2- b^2)2}{(a^2+b^2)2}\)
= 1
Ques: Find the conjugate of (3-2i)( 2+3i )( 1+2i)( 2-i ). (2 Marks)
Ans: We have,
(3-2i)( 2+3i )( 1+2i)( 2-i )
= 6+9i-4i+62-i+4i+2
= 12+5i4+3i x 4-3i4-3i
= 48-36i+20i + 1516+9
= 63-16i25
=6325 - 1625i.
Therefore the conjugate is 6325 - 1625i.
Ques: Solve \(\sqrt5x2\) + x+ \(\sqrt 5\) = 0. (2 Marks)
Ans: Here the discriminant of the equation is,
12 - 4 x\(\sqrt 5\) X\(\sqrt 5\)
= 1- 20
= -19
Therefore the solution,
= \(\frac{-1-+\sqrt -19}{2 \sqrt 5}\)
= \(\frac{-1-+\sqrt -19i}{2 \sqrt 5}\)
Ques: Express ( 5- 3i ) 3 in the form a+ ib. (1 Mark)
Ans: ( 5- 3i )3 = 53 - 3 X 5 2 X 3i + 3 X 5 ( 3i ) 2 - ( 3i )
125 - 225i - 135 + 27i
= -10 - 198i
Ques: Express the following in the form a + ib. (3 Marks)
a.\(\frac{5+\sqrt 2i}{1- \sqrt 2i}\)
b.i-35
Ans:
- We have,
\(\frac{5+\sqrt 2i}{1- \sqrt 2i}\)
=\(\frac{5+\sqrt 2i}{1- \sqrt 2i} X \frac{5+\sqrt 2i}{1+\sqrt 2i}\)
= \(\frac{5+5\sqrt 2i + \sqrt 2i-2}{1- (\sqrt 2i)2}\)
= \(\frac{3+6\sqrt2i}{1+2}\)
=\(\frac{3+(1+2\sqrt2i)}{3}\)
= 1 + 2 \(\sqrt2i\)
- i-35
= \( \frac{1}{i-35}\)
=\( \frac{i}{i2(17i)}\)
=\(\frac{1}{-i}=\frac{1}{+i}\)
= \(\frac{1}{i2}\)
=i
Ques: Solve \(\sqrt6\) x2 + x + \(\sqrt6\) =0 . (2 Marks)
Ans: Here the discriminant of the equation is ,
12- 4x \(\sqrt6\) X \(\sqrt6\)
= 1- 24
= -23.
Therefore the solutions are
\(\frac{-1+-\sqrt -23}{2\sqrt6}\)
=\(\frac{-1+-\sqrt -23i}{2\sqrt6}\)
Ques: If 4x + i ( 3x- y ) = 3+ i ( -6 ) , where x and y are real numbers , find the values of x and y. (3 Marks)
Ans: Here we have,
4 x+ i ( 3x-y)
= 3 + i ( -6 ) ………………( 1 )
Equating the real and imaginary parts of ( 1 ) we get ,
4x =3, 3 x-y= -6,
Which on solving simultaneously give,
x= \(\frac{3}{4}\)
y= \(\frac{33}{4}\)
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