Higher Engineering Mathematics

LINEAR CORRELATION 569

J

LetXbe the expenditure in thousands of pounds and
Ybe the days lost.

The coefficient of correlation,

r=

∑

xy

√{(∑

x^2

)(∑

y^2

)}

wherex=(X−X) andy=(Y−Y),XandYbeing
the mean values ofXandYrespectively. Using a
tabular approach:

X Y x=(X−X) y=(Y−Y)

3.5 241 −6.57 69.9

5.0 318 −5.07 146.9

7.0 174 −3.07 2.9

10 110 −0.07 −61.1

12 147 1.93 −24.1

15 122 4.93 −49.1

18 86 7.93 −85.1

∑

X= 70 .5, X=

70. 5

7

= 10. 07

∑

Y=1198, Y=

1198

7

= 171. 1

xy x^2 y^2

−459.2 43.2 4886

−744.8 25.7 21580

−8.9 9.4 8

4.3 0 3733

−46.5 3.7 581

−242.1 24.3 2411

−674.8 62.9 7242

∑

xy=− 2172

∑

x^2 = 169. 2

∑

y^2 = 40441

Thus

r=

− 2172

√

[169. 2 ×40441]

=− 0. 830

This shows that there isfairly good inverse corre-
lationbetween the expenditure on welfare and days
lost due to absenteeism.

Problem 3. The relationship between monthly

car sales and income from the sale of petrol for

a garage is as shown:

Cars sold 2 5 3 12 14 7 3 28 14 7 3 13

Income from

petrol sales

(£′000) 12 9 13 21 17 22 31 47 17 10 9 11

Determine the linear coefficient of correlation

between these quantities.

LetXrepresent the number of cars sold andYthe

income, in thousands of pounds, from petrol sales.

Using the tabular approach:

X Y x=(X−X) y=(Y−Y)

2 12 −7.25 −6.25

5 9 −4.25 −9.25

3 13 −6.25 −5.25

12 21 2.75 2.75

14 17 4.75 −1.25

7 22 −2.25 3.75

3 31 −6.25 12.75

28 47 18.75 28.75

14 17 4.75 −1.25

7 10 −2.25 −8.25

3 9 −6.25 −9.25

13 11 3.75 −7.25

∑

X=111, X=

111

12

= 9. 25

∑

Y=219, Y=

219

12

= 18. 25

xy x^2 y^2

45.3 52.6 39.1

39.3 18.1 85.6

32.8 39.1 27.6

7.6 7.6 7.6

−5.9 22.6 1.6

−8.4 5.1 14.1

−79.7 39.1 162.6

539.1 351.6 826.6

−5.9 22.6 1.6

18.6 5.1 68.1

57.8 39.1 85.6

−27.2 14.1 52.6

∑

xy= 613. 4

∑

x^2 = 616. 7

∑

y^2 = 1372. 7