Question

If the variances of two variables \[x\] and $y$ are respectively $9$ and $16$ and their covariance is $8$, then their coefficient of correlation is
A) $\dfrac{2}{3}$
B) $\dfrac{8}{{3\sqrt 2 }}$
C) $\dfrac{9}{{8\sqrt 2 }}$
D) $\dfrac{2}{9}$

Answer 1

Hint: To find the coefficient of correlation, we are to use the formula,
${r_{xy}} = \dfrac{{Cov(xy)}}{{{\sigma _x}{\sigma _y}}}$
Where, ${r_{xy}} = $coefficient of correlation
$Cov(xy) = $ covariance of \[x\] and $y$
${\sigma _x} = $ standard deviation of $x$
${\sigma _y} = $ standard deviation of $y$
Then, we have to convert the variance into standard deviation of the number. This is very simple as, standard deviation is the square root of variance of a variable.

Complete step by step answer:
Given, variance of the two variables,
$\sigma _x^2 = 9$
$\sigma _y^2 = 16$
And, covariance of \[x\] and $y$, $Cov(xy) = 8$
Now, we have to find the standard deviation of the variables \[x\] and $y$.
We know that, standard deviation, $\sigma = \sqrt {{\sigma ^2}} $
Therefore, standard variation of$x$ is, ${\sigma _x} = \sqrt {\sigma _x^2} = \sqrt 9 = 3$
And, standard deviation of $y$ is, ${\sigma _y} = \sqrt {\sigma _y^2} = \sqrt {16} = 4$
Therefore, from the formula of coefficient of correlation we have,
${r_{xy}} = \dfrac{{Cov(xy)}}{{{\sigma _x}{\sigma _y}}}$
Where, ${r_{xy}} = $coefficient of correlation
$Cov(xy) = $ covariance of \[x\] and $y$
${\sigma _x} = $ standard deviation of $x$
${\sigma _y} = $ standard deviation of $y$
Now, substituting the values in the above formula, we get,
${r_{xy}} = \dfrac{8}{{3 \times 4}}$
$ \Rightarrow {r_{xy}} = \dfrac{2}{3}$
Therefore, the coefficient of correlation is $\dfrac{2}{3}$, i.e., option (A).

Note:
Correlation coefficients are used to measure how strong a relationship is between two variables. Correlation coefficients ranges from $ - 1$ to $ + 1$, where $ \pm 1$ indicates the strongest possible agreement and $0$ the strongest possible disagreement. Correlation coefficients are used to assess the strength and direction of the linear relationships between pairs of variables. Correlation coefficients do not communicate information about whether one variable moves in response to another. There is no attempt to establish one variable as dependent and the other as independent.