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Question
Find correlation coefficient from the following data. `["Given:" sqrt(3) = 1.732]`
| x | 3 | 6 | 2 | 9 | 5 |
| y | 4 | 5 | 8 | 6 | 7 |
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Solution
| xi | yi | xi2 | yi2 | xiyi |
| 3 | 4 | 9 | 16 | 12 |
| 6 | 5 | 36 | 25 | 30 |
| 2 | 8 | 4 | 64 | 16 |
| 9 | 6 | 81 | 36 | 54 |
| 5 | 7 | 25 | 49 | 35 |
| 25 | 30 | 155 | 190 | 147 |
From the table, we have
n = 5, `sum"x"_"i"` = 25, `sum"y"_"i"` = 30, `sum"x"_"i"^2` = 155, `sum"y"_"i"^2` = 190, `sum"x"_"i""y"_"i"` = 147
`bar"x"=(sum"x"_"i")/"n"`
= `25/5`
= 5
`bar"y"=(sum"y"_"i")/"n"`
= `30/5`
= 6
Since, Cov (x, y) = `1/"n"sum"x"_"i""y"_"i"-bar"x"bar"y"`
∴ Cov (x, y) = `1/5xx147-(5xx6)`
= 29.4 − 30
= − 0.6
`sigma_"x"^2=(sum"x"_"i"^2)/"n"-(bar"x")^2`
= `155/5-(5)^2`
= 31 − 25
∴ `sigma_"x"^2` = 6
∴ σx = `sqrt6`
`sigma_"y"^2=(sum"y"_"i"^2)/"n"-(bar"y")^2`
= `190/5-(6)^2`
= 38 − 36
∴ `sigma_"y"^2` = 2
∴ σy = `sqrt2`
∴ σxσy = `sqrt6sqrt2=sqrt12`
= `2sqrt3`
= 2(1.732) = 3.464
Thus, the correlation coefficient between x and y is
r = `("Cov"("x,y"))/(sigma_"x"sigma_"y")`
= `(-0.6)/3.464`
= − 0.1732
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