1. Pearson's Correlation Coefficient Example-2 (Class-X & Y)

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1. Pearson's Correlation Coefficient example ( Enter your problem )

( Enter your problem )

Other related methods

Correlation Coefficient r
Covariance - Population Covariance, Sample Covariance

1. Formula & Example-1 (Class-X & Y)
(Previous example)

3. Example-3 (X & Y)
(Next example)

2. Example-2 (Class-X & Y)

Calculate Correlation Coefficient r without cov(x,y), Correlation Coefficient r with population cov(x,y), Correlation Coefficient r with sample cov(x,y) from the following data
Class-X Y
20 - 25 110
25 - 30 170
30 - 35 80
35 - 40 45
40 - 45 40
45 - 50 35

Solution:
Mean `bar x = (sum x_i)/n`

`=(22.5+27.5+32.5+37.5+42.5+47.5)/6`

`=210/6`

`=35`

Mean `bar y = (sum y_i)/n`

`=(110+170+80+45+40+35)/6`

`=480/6`

`=80`

Class-X	Mid value `x`	`y`	`X=(x-35)/1`	`Y=(y-80)/5`	`X^2`	`Y^2`	*`XY`**
20-25	22.5	110	-12.5	6	156.25	36	-75
25-30	27.5	170	-7.5	18	56.25	324	-135
30-35	32.5	80	-2.5	0	6.25	0	0
35-40	37.5	45	2.5	-7	6.25	49	-17.5
40-45	42.5	40	7.5	-8	56.25	64	-60
45-50	47.5	35	12.5	-9	156.25	81	-112.5
---	---	---	---	---	---	---	---
	`210`	`480`	`sum X=0`	`sum Y=0`	`sum X^2=437.5`	`sum Y^2=554`	`sum X*Y=-400`

Correlation Coefficient r :
`r = (sum XY)/(sqrt(sum X^2) * sqrt(sum Y^2))`

`=-400/(sqrt(437.5) * sqrt(554))`

`=-400/(20.9165 * 23.5372)`

`=-0.8125`

Correlation Coefficient r with Population Cov(x,y) :

Population `Cov(x,y) = (sum (x-bar x)(y-bar y))/(n)`

`=-400/6`

`=-66.6667`

Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/(n))`

`=sqrt(437.5/6)`

`=sqrt(72.9167)`

`=8.5391`

Population Standard deviation `sigma = sqrt((sum (y - bar y)^2)/(n))`

`=sqrt(554/6)`

`=sqrt(92.3333)`

`=9.609`

Now, `r = (cov(x,y))/(sigma_x * sigma_y)`

`= (-66.6667)/(8.5391 * 9.609)`

`=-0.8125`

Correlation Coefficient r with Sample Cov(x,y) :

Sample `Cov(x,y) = (sum (x-bar x)(y-bar y))/(n-1)`

`=-400/5`

`=-80`

Sample Standard deviation `sigma = sqrt((sum (x - bar x)^2)/(n-1))`

`=sqrt(437.5/5)`

`=sqrt(87.5)`

`=9.3541`

Sample Standard deviation `sigma = sqrt((sum (y - bar y)^2)/(n-1))`

`=sqrt(554/5)`

`=sqrt(110.8)`

`=10.5262`

Now, `r = (cov(x,y))/(sigma_x * sigma_y)`

`= (-80)/(9.3541 * 10.5262)`

`=-0.8125`

This material is intended as a summary. Use your textbook for detail explanation.
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