FOLLOW US:

Regression :A mathematical relation-ship between two corelated variables is called regression . (one variable will be independent and another will be dependent).

---

Regression Line : Regression lines, gives us a relation-ship between dependent and independent variables, that is very helpful to forecast .

---

There are two types of regression lines

(1) . line y on x , y = a + bx
y = dependent variable.
x = independent variable.
a and b are unknown parameters.

(2) . line x on y , x = a + by
x = dependent variable.
y = independent variable.
a and b are unknown parameters.

---

There are two methods to find regression lines

(1) . Least square method or Normal equations method .

(2) . Regression coefficient method.

To find regression equations using .

---

---

(1) Least square method

(i) Line y on x .

y = a+bx

∑ y = na + b∑ x ----------(i)

∑ xy = a ∑ x + b∑x² --------------(ii)

On solving these two Normal equations , we get the value of a and b .

---

(ii) Line x on y

x = a + by

∑ x = na + b∑ y ----------(i)

∑ xy = a ∑ y + b∑y² --------------(ii)

On solving these two Normal equations , we get the value of a and b .

---

(2) To find regression equations using
regression coefficient method .

(i) Line y on x .

y = a+bx

y - y̅ = b_yx (x - x̅)

where y̅ = mean of y , x̅ = mean of x

b_yx = regression coefficient y on x .

b_yx = r

σ_y/σ _x

=

cov(xy)/σ_x . σ_y

.

σ_y/σ _x

=

cov(xy)/σ _x²

=

1/n ∑(x-x̅)(y-y̅)/ 1/n ∑(x-x̅)²

=

∑(x-x̅)(y-y̅)/ ∑(x-x̅)²

=

1/n ∑xy - x̅y̅)/ 1/n ∑x² - (x̅)²

=

1/n∑xy - (∑x/n).(∑y/n)/1/n∑x² - (∑x/n)²

=

∑xy - (∑x . ∑y/n)/∑x² - (∑x)²/n

---

(ii) Line x on y .

x = a+by

x - x̅ = b_xy (y - y̅)

---

where y̅ = mean of y , x̅ = mean of x

b_xy = regression coefficient x on y .

b_xy = r

σ_x/σ _y

=

cov(xy)/σ_x . σ_y

.

σ_x/σ _y

=

cov(xy)/σ _y²

=

1/n ∑(x-x̅)(y-y̅)/ 1/n ∑(y-y̅)²

=

∑(x-x̅)(y-y̅)/ ∑(y-y̅)²

=

1/n ∑xy - x̅y̅)/ 1/n ∑y² - (y̅)²

=

1/n∑xy - (∑x/n).(∑y/n)/1/n∑y² - (∑y/n)²

=

∑xy - (∑x . ∑y/n)/∑y² - (∑y)²/n

---

Note:Regression coefficient are independent of change of origin but not of scale.