Correlation and Regression Formulas

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1. Co-variance

If two variables x and y takes the values x₁, x₂, x₃….x_n and y₁, y₂, y₃….y_n then covariance is defined as
Cov(x, y) = \(\frac{\Sigma(x-\bar{x})(y-\bar{y})}{n}\)
Where \(\overline{\mathrm{x}} \text { and } \overline{\mathrm{y}}\) are the means of x and y series respectively.

2. Coefficient of Correlation

Karl Pearson gave the following formula for the calculation of correlation coefficient between two variables x and y
r_xy = \(\frac{\Sigma(x-\bar{x})(y-\bar{y})}{\sqrt{\Sigma(x-\bar{x})^{2} \Sigma(y-\bar{y})^{2}}}\) and r_xy = \(\frac { Cov(x,y) }{ \sigma _{ { x } }\sigma _{ { y } } } =\frac { Cov(x,y) }{ \sqrt { Var(x).Var(y) } } \)

3. Rank Correlation

Rank correlation is the correlation between different ranks or grades of the two characteristics. It is given by
1 – \(\frac{6 \Sigma \mathrm{d}^{2}}{\mathrm{n}\left(\mathrm{n}^{2}-1\right)}\); Here d² = \(\sum_{i=1}^{n}\left\{\left(x_{i}-\bar{x}\right)-\left(y_{i}-\bar{y}\right)\right\}^{2}\)
where Σ d² = sum of the squares of the difference of two ranks and n is the number of pairs of observations.

4. Properties of Correlation Coefficient (r)

(a) r lies between – 1 and + 1
(b) The correlation is

• perfect and positive if r = + 1
• perfect and negative if r = – 1
• not correlated if r = 0
• positive if r > 0
• negative if r < 0

(c) It is independent of the change of origin and scale.
(d) It is a pure number and hence unitless
(e) If x and y are independent then r = 0

5. Line of Regression

(i) Line of regression of y on x
\((y-\overline { y } )=\frac { Cov.(x,y) }{ \sigma _{ x }^{ 2 } } (x-\overline { x } )or(y-\overline { y } )=r.\frac { \sigma _{ y } }{ \sigma _{ x } } (x-\overline { x } )\)

(ii) Line of regression of x on y
\((x-\overline { x } )=\frac { Cov.(x,y) }{ \sigma _{ y }^{ 2 } } (y-\overline { y } )or(x-\overline { x } )=r\frac { \sigma _{ x } }{ \sigma _{ y } } (y-\overline { y } )\)

6. Regression Coefficient

(i) The Regression Coefficient of y on x is denoted by b_yx and is given by
\(b_{ yx }-r\cdot \frac { \sigma _{ y } }{ \sigma _{ x } } =\frac { { cov }\cdot (x,y) }{ \sigma _{ x }^{ 2 } } \)
This represents the change in the values of y corresponding to a unit change in x.

(ii) The Coefficient of Regression of x on y is denoted by bxy and is given by
\(b_{ xy }=r\frac { \sigma _{ x } }{ \sigma _{ y } } =\frac { { cov }\cdot (x,y) }{ \sigma _{ y }^{ 2 } } \)
This represents the change in the value of x corresponding to a unit change in y.

7. Properties of Regression Coefficient

(i) r = \(\sqrt{b_{y x} \cdot b_{x y}}\) i.e. the coefficient of correlation is the Geometric Mean between the two Regression Coefficients.

(ii) If byx > 1, then bxy < 1, i.e. If one of the Regression Coefficient is greater then unity then the other will be less than unity.

(iii) b_yx is called the slope of regression line y on x and b_xy is called the slope of regression line x on y.

(iv) b_yx + b_xy > 2 \(\sqrt{b_{y x} \cdot b_{x y}}\) or b_yx + b_xy > 2r i.e the Arithmetic Mean of the regression coefficient is greater than the Correlation Coefficient.

(v) The product of lines of regression’s gradients is given by \(\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}\)

(vi) If the angle between lines of regression is θ then tan θ = \(\left(\frac{1-r^{2}}{r}\right) \cdot\left(\frac{\sigma_{x} \sigma_{y}}{\sigma_{x}^{2}+\sigma_{y}^{2}}\right)\)

(vii) If both b_yx and b_xy are positive, then r will be positive and if both b_yx & b_xy are negative then r will be negative

8. Important points on Regression lines

• If r = 0, then tan 0 is not defined i.e. θ = π/2. Thus, if two variables are not correlated, then the lines of regression are perpendicular to each other.
• If r = ± 1, then tan θ = 0 i.e. θ = 0. Thus the regression lines are coincident
• If regression lines arey = ax + b& x = cy + d then \(\bar{x}=\frac{b c+d}{1-a c}\) and \(\bar{y}=\frac{a d+b}{1-a c}\)

9. Standard error of Prediction

The deviation of the predicted value from the observed value is known as the standard error of prediction and is defined as
S_y = \(\sqrt{\left\{\frac{\Sigma\left(\mathrm{y}-\mathrm{y}_{\mathrm{p}}\right)^{2}}{\mathrm{n}}\right\}}\); where y is actual value and y_p is predicted value.
In relation to coefficient of correlation, it is given by

• Standard error of estimate of x is S_x = \(\sigma_{x} \sqrt{1-r^{2}}\)
• Standard error of estimate of y is S_y = \(\sigma_{y} \sqrt{1-r^{2}}\)

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