Question

Calculate the correlation coefficient from the following data, and interpret it.

X	1	3	5	7	9	11	13
Y	12	10	8	6	4	2	0

Answer 1

	xi	yi	xi2	yi2	xiyi
	1	12	1	144	12
	3	10	9	100	30
	5	8	25	64	40
	7	6	49	36	42
	9	4	81	16	36
	11	2	121	4	22
	13	0	169	0	0
Total	49	42	455	364	182

From the table, we have
n = 7, `sum"x"_"i"` = 49, `sum"y"_"i"` = 42, `sum"x"_"i"^2` = 455, `sum"y"_"i"^2` = 364, `sum"x"_"i""y"_"i"^2` = 182

∴ `bar"x" = (sum"x"_"i")/"n" = 49/7` = 7,

`bar"y" = (sum"y"_"i")/"n" = 42/7` = 6

Cov (X, Y) = `1/"n" sum"x"_"i""y"_"i" - bar"x" bar"y"`

= `1/7 xx 182 - (7 xx 6)`

= 26 − 42

∴ Cov (X, Y) = – 16

`sigma_"x"^2 = (sum"x"_"i"^2)/"n" - (bar"x")^2`

= `455/7 - (7)^2`
= 65 – 49
∴ `sigma_"x"^2` = 16

∴ `sigma_"x"` = 4

`sigma_"y"^2 = (sum"y"_"i"^2)/"n" - (bar"y")^2`

= `364/7 - (6)^2`

= 52 – 36
`sigma_"y"^2` = 16
∴ `sigma_"y"` = 4
Thus, the coefficient of correlation between X and Y is

r = `("Cov (X, Y)")/(sigma_"x" sigma_"y")`

= `(-16)/(4 xx 4)`

= – 1
∴ The value of r indicates perfect negative correlation between x and y.