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The standard deviation for variables x and y be $3$ and $4$ respectively and their covariance is $8$, then coefficient of correlation between them is:
A) $\dfrac{2}{3}$
B) $\dfrac{3}{{2\sqrt 2 }}$
C) $\dfrac{{2\sqrt 2 }}{3}$
D) $\dfrac{2}{9}$

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Last updated date: 18th Jun 2024
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Answer
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Hint: The correlation coefficient is a statistical measure of the strength in relationship between the relative movements of two variables, for the given variables x and y the formula become coefficient of correlation = $\dfrac{{Cov(x,y)}}{{{\sigma _X}{\sigma _Y}}}$

Complete step by step solution:
As in the question we have to find coefficient of correlation, for this
The correlation coefficient is a statistical measure of the strength in relationship between the relative movements of two variables. The values range in between $ - 1$ and $1$ if the calculated number is greater than $1$ or less than $ - 1$ means that there is an error in the correlation measurement. A correlation of $ - 1$ shows a perfect negative correlation, while a correlation of $1$ shows a perfect positive correlation. A correlation of $0$ shows no linear relationship between the movement of the two variables.
To calculate the product-moment correlation, one must first determine the covariance of the two variables in question. Next, one must calculate each variable's standard deviation. The correlation coefficient is determined by dividing the covariance by the product of the two variables' standard deviations.
So the relation is
Coefficient of correlation = $\dfrac{{Cov(x,y)}}{{{\sigma _X}{\sigma _Y}}}$
where
$Cov(x,y)$ is covariance of x and y that is \[8\]
${\sigma _X}$ is standard deviation of x that is $3$
${\sigma _y}$ is standard deviation of x that is $4$
Hence on putting the value of these ,
Coefficient of correlation = $\dfrac{8}{{3 \times 4}}$
Hence it is equal to the $\dfrac{2}{3}$

Option A will be the correct answer.

Note:Correlation coefficients are used to measure the strength of the relationship between two variables. Pearson correlation is the one most commonly used in statistics. This measures the strength and direction of a linear relationship between two variables that we use in this question.
Correlation coefficient values less than $0.8$ or greater than $ - 0.8$ are not considered significant.