Question

Calculate the correlation coefficient of the following data: x 45 46 46 47 48 49 50 y 44 48 45 48 52 51 49​

Answer 1

Step-by-step explanation:

x

i

​

 y

i

​

 u

i

​

 v

i

​

 u

i

2

​

 v

i

2

​

 u

i

​

v

i

​

5 12 -8 -13 64 169 104

9 20 -4 -5 16 25 20

13 25 0 0 00 00 00

17 33 4 8 16 64 32

21 35 8 10 64 100 80

∑x

i

​

=65 ∑y

i

​

=125 ∑u

i

​

=0 ∑v

i

​

=0 ∑u

i

2

​

=160 ∑v

i

2

​

=358 ∑u

i

​

v

i

​

=236

x

=

n

1

​

∑x

i

​

=

5

1

​

×65=13

y

​

=

n

1

​

∑y

i

​

=

5

1

​

×125=25

ρ(x,y)=ρ(u,v)=

n∑u

i

2

​

−(∑u

i

2

​

)

2

​

 

n∑v

i

2

​

−(∑v

i

2

​

)

2

​

n∑u

i

​

v

i

​

−∑u

i

​

∑v

i

​

​

=

5×160−0

2

​

 

5×358−0

2

​

5×236−0×0

​

=

239.33

236

​

=0.98

Answer 2

Given,

x = 45 46 46 47 48 49 50

y= 44 48 45 48 52 51 49

To find,

Correlation coefficient of given data (r)

Solution,

As it is given, x=45 46 46 47 48 49 50

Using the formula, Mean = $\frac{Sumof all samples}{Number of samples}$

Hence, the mean of x (${\frac{}{X} }$) = $\frac{45+46+46+47+48+49+50}{7}$

⇒${\frac{}{X} }$=$\frac{331}{7}$

⇒${\frac{}{X} }$=47.28

Similarly, the mean of y (${\frac{}{Y} }$)= $\frac{44+48+45+48+52+51+49​}{7}$

⇒${\frac{}{Y} }$=$\frac{337}{7}$

⇒${\frac{}{Y} }$=48.14

Now, Let us find the value of x-${\frac{}{X} }$

x-${\frac{}{X} }$=(45-47.28) +(46-47.28) +(46-47.28) +(47-47.28) +(48-47.28)+(49-47.28) +(50-47.28)

⇒x-${\frac{}{X} }$=(-2.28)+(-1.28)+(-1.28)+(-0.28)+(0.72)+(1.72)+(2.72 )

⇒x-${\frac{}{X} }$=0.04

∴(x-${\frac{}{X} }$)²=((-2.28)²+(-1.28)²+(-1.28)²+(-0.28)²+(0.72)²+(1.72)²+(2.72 )²)

⇒(x-${\frac{}{X} }$)²=19.4288

Similarly,

y-${\frac{}{Y} }$=(44-48.14)+(48-48.14)+(45-48.14)+(48-48.14)+(52-48.14)+(51-48.14)+(49-48.14)

⇒y-${\frac{}{Y} }$=(-4.14)+(-0.14)+(-3.14)+(-0.14)+(3.86)+(2.86)+(0.86)

⇒y-${\frac{}{Y} }$=0.02

∴(y-${\frac{}{Y} }$)²=((-4.14)²+(-0.14)²+(-3.14)²+(-0.14)²+(3.86)²+(2.86)²+(0.86)²)

⇒(y-${\frac{}{Y} }$)²=50.8572

Now, Let us find the value of (x-${\frac{}{X} }$)(y-${\frac{}{Y} }$)

⇒(x-${\frac{}{X} }$)(y-${\frac{}{Y} }$) =(-2.28)(-4.14)+(-1.28)(-0.14)+(-1.28)(-3.14)+(-0.28)(-0.14)+(0.72)(3.86)+(1.72)(2.86)+(2.72 )(0.86)

⇒(x-${\frac{}{X} }$)(y-${\frac{}{Y} }$) = 23.7144

Now, Using the formula,

r =  ∑ (X  -${\frac{}{X} }$)  )(Y - ${\frac{}{Y} }$ ) / √∑(X  -${\frac{}{X} }$)  )²∑(Y  -${\frac{}{Y} }$  )²

⇒r= $\frac{23.7144}{\sqrt{(19.4288)(50.8572)} }$

⇒r=$\frac{23.7144}{31.43}$

⇒r=0.75

Hence the Correlation coefficient of given data (r) is 0.75