Linear Regression Formula Class 12 ISC Mathematics

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Linear Regression

Formula

Regression Coefficients

The Regression Coefficient of y on x is denoted by $\mathbf{{\color{Purple} b_{yx}}}$

$b_{yx}=\frac{\sum xy-\frac{1}{n}\sum x\sum y}{\sum x^{2}-\frac{1}{n}(\sum x)^{2}}$ (we use this formula when x, y are small numbers)

$b_{yx}=\frac{\sum (x-x\bar{})(y-y\bar{})}{\sum (x-x\bar{})^{2}}$ (when $x - x\bar{}, y-y\bar{}$ are small fraction less numbers)

$b_{yx}=\frac{\sum uv-\frac{1}{n}\sum u\sum v}{\sum u^{2}-\frac{1}{n}(\sum u)^{2}}$ (when $u=x-A,\, v=y-B,\, A \, and\, B$ are assumed means)

$b_{yx}= r.\, \frac{\sigma _{y}}{\sigma _{x}\, }$ (where $\sigma _{x}$ is the standard deviation of x-variate, $\sigma _{y}$ is the standard deviation of y-variate and r is the coefficient of correlation)

The Regression Coefficient of x on y is denoted by $\mathbf{{\color{Purple} b_{xy}}}$

$b_{xy}=\frac{\sum xy-\frac{1}{n}\sum x\sum y}{\sum y^{2}-\frac{1}{n}(\sum y)^{2}}$ (we use this formula when x, y are small numbers)

$b_{xy}=\frac{\sum (x-x\bar{})(y-y\bar{})}{\sum (y-y\bar{})^{2}}$ (when $x-x\bar{}, \, y-y\bar{}$ are small fraction less numbers)

$b_{xy}=\frac{\sum uv-\frac{1}{n}\sum u\sum v}{\sum v^{2}-\frac{1}{n}(\sum v)^{2}}$ (when $u=x-A,\, v=y-B$ , $A\, \, and\, \, B$ are assumed means)

$b_{xy}= r.\, \frac{\sigma _{x}}{\sigma _{y}}$ (where $\sigma _{x}$ is the standard deviation of x-variate, $\sigma _{y}$ is the standard deviation of y-variate and r is the coefficient of correlation)

Coefficient of correlation: $\mathbf{{\color{Purple} r(x,y)\, or\, \rho (x,y)}}$

$r^{2}=b_{yx}.b_{xy}\, \, \, \, and\, \, \, 0\leq r^{2}\leq 1$

$r=\sqrt{b_{yx}.b_{xy}}\, \, \, and\, \, \, -1\leq r\leq 1$

$b_{xy},\, b_{yx}\, \, and\, \, \rho (x,y)$ are of the same sign.

Equations of two lines of regression

The regression equation of y on x is $y-y\bar{}=b_{yx}(x-x\bar{})$

The regression equation of x on y is $x-x\bar{}=b_{xy}(y-y\bar{})$

The two regression lines intersect at $(x\bar{},y\bar{})$

The acute angle θ between two regression lines is given by

$tan\, \theta =\left | \frac{1-r^{2}}{b_{xy}+b_{yx}} \right |$

If two lines coincide then θ = 0 . $\mathrm{So}\, \, \, 1-r^{2}=0\, \, \, and\, \, \, r=\pm 1$

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ISC Class 12 Mathematics

Previous Years Board Questions	Solutions of Previous years Board Questions
Inverse Trigonometric Functions (2000 to 2020)	Inverse Trigonometric Functions (2000 to 2020)
Matrices (2000 to 2020)	Matrices (2000 to 2020)
Determinants (2000 to 2020)	Determinants (2000 to 2020)
Differentiation (2000 to 2020)	Differentiation (2000 to 2020)
Rolle’s Theorem and Lagrange’s Mean Value Theorem (1998 to 2020)	Rolle’s Theorem and Lagrange’s Mean Value Theorems (1998 to 2020)
Maxima and Minima (2000 to 2020)	Maxima and Minima (2000 to 2020)
Integration (1990 – 2011)	Integration (1990 – 2011)
Integration (2012 to 2020)	Integration (2012 to 2020)
Differential Equations (1991 to 2005)	Differential Equations (1991 to 2005)

Differential Equations (2006 to 2020)	Differential Equations (2006 to 2020)
Probability (Addition and Multiplication Theorem-Conditional Probability) (1993 to 2020)	Probability (Addition and Multiplication Theorem, Conditional Probability) (1993 to 2020)
Probability (Baye’s Theorem) (2000 to 2019)	Probability (Baye’s Theorem) (2000 to 2019)
Probability (Probability distribution and Binomial distribution) (1992 to 2020)	Probability (Probability distribution and Binomial distribution) (1992 to 2020)
Application of Calculus in Commerce and Economics (2005 to 2020)	Application of Calculus in Commerce and Economics (2005 to 2020)
Linear Programming (2008 to 2020)	Linear Programming (2008 to 2020)
Linear Regression (2005 to 2020)

ISC – Class 12 – Mathematics – Sem 2 – 2022 – Sample Paper

Sample Paper 1
Sample Paper 2

ISC- Class 12 – Mathematics – Sem 2 – 2022 – Chapterwise Questions

Differential Equations

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