It was introduced by Karl Pearson in 1908.

In Statistics, Regression analysis is a method for the prediction of future events.

The relationship between several independent variables or predictors and a dependent variable or criterion is known as Multiple Regression.

The value of a house depends on the location where it is situated and the condition of the house i.e. Rooms, East Open or West Open and Proper Water Supply etc.

The Salary of an Employee depends on many variables like his education, his experience, his hard work and his skills etc.

We depend on our Parents. If one will die than it will definitely affects our life.

Y = a + b X1 + c X2

Where,Y = Dependent VariableX1 & X2 = Independent Variablea, b & c = Constants

The Coefficient of Multiple Correlation measures the relationship between a dependent variable and the whole group of independent variables.

The Coefficient of Multiple Correlation between “Y” and the two independent variables “X1 & X2” is denoted as Rx1.y.x2 .

The Coefficient of Multiple Correlation is computed by the following formula.

Rx1.y.x2 = r²y.x1 + r²x1.x2 – 2 ry.x1 ry.x2 rx1.x2

1 - r²y.x2

Where,

r = Simple Coefficient of Correlation having the general formula,

r = n ∑ XY – ( ∑ X ) ( ∑ Y )

n ∑ X² - ( ∑ X )² n ∑ Y² - ( ∑ Y )²

Following is the data of the Assessed Value

(in thousands of dollars) of 15 houses in a certain

locality that depends on the Heating Area of

Dwelling (thousands of square feet) and Age

(in years). Fit a Multiple Regression Equation.

Y = a + b X1 + c X2

Data of Assessed Value, Heating Area of Dwelling & Age:

HOUSEASSESSED

VALUE ($000)HEATING AREA OF DWELLING

(THOUSANDS OF SQUARE FEET)

AGE (YEARS)

01 84.4 2.00 3.42

02 77.4 1.71 11.50

03 75.7 1.45 8.33

04 85.9 1.76 0.00

05 79.1 1.93 7.42

06 70.4 1.20 32.00

07 75.8 1.55 16.00

08 85.9 1.93 2.00

09 78.5 1.59 1.75

10 79.2 1.50 2.75

11 86.7 1.39 0.00

12 79.3 1.90 0.00

13 74.5 1.54 12.58

14 83.8 1.89 2.75

15 76.8 1.59 7.17

Plot a Graph among all the three variables.

Find the Multiple Regression Equation.

Predict the Assessed Value if the Heating area of dwelling is 1.10 and Age is 34.00 years.

Calculate Multiple Correlation.

AS

SES

SED

VA

LY

E (

Y)

HEATING AREA OF DWELLING (X1)

AGE (X2)

Y X1 X2 Y² X²1 X²2 X1Y X2Y X1X2

84.4 2.00 3.42 7123.36 4 11.6964 168.8 288.648 6.84

77.4 1.71 11.50 5990.76 2.9241 132.25 132.354 890.1 19.665

75.7 1.45 8.33 5730.49 2.1025 69.3889 109.765 630.581 12.0785

85.9 1.76 0.00 7378.81 3.0976 0.00 151.184 0.00 0.00

79.1 1.93 7.42 6256.81 3.7249 55.0564 152.663 586.922 14.3206

70.4 1.20 32.00 4956.16 1.44 1024 84.48 2252.8 38.4

75.8 1.55 16.00 5745.64 2.4025 256 117.49 1212.8 24.8

85.9 1.93 2.00 7378.81 3.7249 4.00 165.787 171.8 3.86

78.5 1.59 1.75 6162.25 2.5281 3.0625 124.815 137.375 2.7825

79.2 1.50 2.75 6272.64 2.25 7.5625 118.8 217.8 4.125

86.7 1.39 0.00 7516.89 1.9321 0.00 120.51 0.00 0.00

79.3 1.90 0.00 6288.49 3.61 0.00 150.67 0.00 0.00

74.5 1.54 12.58 5550.25 2.3716 158.2564 114.73 937.21 19.3732

83.8 1.89 2.75 7022.44 3.5721 7.5625 158.382 230.45 5.1975

76.8 1.59 7.17 5898.24 2.5281 51.4089 122.112 550.656 11.4003

∑Y = 1193.4

∑X1 = 24.93

∑X2 = 107.67

∑Y² = 95272.04

∑X²1 = 42.2085

∑X²2 = 1780.245

∑X1Y = 1992.545

∑X2Y = 8107.142

∑X1X2 = 162.8426

Now eq. 1 , 2 & 3 becomes,

1 => 15a + 24.93b + 107.67c = 1193.4

2 => 24.93a + 42.2085b + 162.8426c = 1992.545

3 => 107.67a + 162.8426b + 1780.245c = 8107.142

Solving equations 1 and 2 ,

Multiplying eq. 1 by 24.93 and eq. 2 by 15, we get

1 => 373.95a + 621.5049b + 2684.2131c = 29751.462

2 => 373.95a + 633.1275b + 2442.639c = 29888.175

Now subtracting eq. 1 and eq. 2 ,

+ 373.95a + 621.5049b + 2684.2131c = + 29751.462

+ 373.95a + 633.1275b + 2442.639c = + 29888.175

- - - -

- 11.6226 b + 241.5741 c = - 136.713 eq. 4

Solving equations 2 and 3 ,

Multiplying eq. 2 by 107.67 and eq. 3 by 24.93, we get

2 => 2684.2131 a + 4544.589195 b + 17533.26274 c = 214537.32023 => 2684.2131 a + 4059.666018 b + 44381.50785 c = 202111.0501

Now subtracting eq. 2 and 3 ,

+ 2684.2131 a + 4544.589195 b + 17533.26274 c = + 214537.3202 + 2684.2131 a + 4059.666018 b + 44381.50785 c = + 202111.0501 - - - -

+ 484.923177 b - 26848.24511 c = 12426.2701 eq. 5

Solving equations 4 and 5 ,

Multiplying eq. 4 by 484.923177 and eq. 5 by 11.6226, we get

4 => - 5636.068117 b + 117144.8801 c = - 66295.3023

5 => + 5636.068117 b – 312046.4136 c = + 144425.5669

Now Adding eq. 4 and 5 ,

4 => - 5636.068117 b + 117144.8801 c = - 66295.3023

5 => +5636.068117 b – 312046.4136 c = + 144425.5669

- 194901.5335 c = + 78130.2646

c = - 78130.2646

194901.5335

c = - 0.400870445

Now Put (c = - 0.400870445) in eq. 4 to get the value of b,

- 11.6226 b + 241.5741 (- 0.400870445) = - 136.713 - 11.6226 b – 96.83991697 = - 136.713 - 11.6226 b = -136.713 + 96.83991697 - 11.6226 b = - 39.87308303

b = 39.87308303

11.6226

b = 3.43065089

Now Put (b = 3.43065089) and (c = - 0.400870445) in eq. 1 to

get the value of a,

15a + 24.93 (3.43065089) + 107.67 (- 0.400870445) = 1193.4

15a + 85.52612669 – 43.16172081 = 1193.4

15a +42.36440588 = 1193.4 15a = 1193.4 – 42.36440588 a = 1151.035594

15

a = 76.73570627

We have,

a = 76.73570627

b = 3.43065089

c = - 0.400870445

Now eq. A becomes,

Y = 76.73570627 + 3.43065089 X1 – 0.400870445 X2

By using Regression Equation, Y = 76.73570627 + 3.43065089 (1.10) – 0.400870445 (34.00)

Y = 66.879 (in thousands of dollars)

The Coefficient of Multiple Correlation is computed by the following formula.

Rx1.y.x2 = r²y.x1 + r²x1.x2 – 2 ry.x1 ry.x2 rx1.x2

1 - r²y.x2

First we find ry.x1 , rx1.x2 and ry.x2.

Now the Coefficient of Multiple Correlation

will be ,

Rx1.y.x2 = r²y.x1 + r²x1.x2 – 2 ry.x1 ry.x2 rx1.x2

1 - r²y.x2

Rx1.y.x2 = (0.57)² + (- 0.57)² – 2 (0.57)(- 0.80)(- 0.57)

1 – (- 0.80)²

Rx1.y.x2 = 0.60 (Moderate Correlation)

We can now conclude that the value of each house depends on the time period since it was built and the heating area that affects it.

We have many dependent variables in our life and many independent also. Regression is the best method to know or to measure the relationship between those variables.

Thank you so much

Our honorable teacher Sir Zafar Ali.

The students of BS Commerce (3rd Semester) to cooperate with us. We

wish you all the very best for your future.

Please pardon us if we hurt you throughout the Presentation.