# 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 - Mathematics and Statistics

Sum

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

#### Solution

 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.

Concept: Interpretation of Value of Correlation Coefficient
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Chapter 5: Correlation - Miscellaneous Exercise 5 [Page 64]

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