Question

Calculate the correlation coefficient between the corresponding values of X and Y in the following table:

X	$2$	$4$	$5$	$6$	$8$	$11$
Y	$18$	$12$	$10$	$8$	$7$	$5$

(A) $ - 0.65$
(B) $ - 0.82$
(C) $ - 0.92$
(D) $ - 0.48$

Answer 1

Hint: Start with writing down the formula for the correlation coefficient, i.e. $\dfrac{{\sum {\left( {{x_i} - \bar x} \right)\left( {{y_i} - \bar y} \right)} }}{{\sqrt {\sum {{{\left( {{x_i} - \bar x} \right)}^2} \times \sum {{{\left( {{y_i} - \bar y} \right)}^2}} } } }}$ . where $\bar x$ represents the mean of values of x-variable. Take the given values in the table and step by step calculate $\left( {x - \bar x} \right){\text{ and }}\left( {y - \bar y} \right)$ by adding separate columns to the table. Now find $\sum {\left( {{x_i} - \bar x} \right)\left( {{y_i} - \bar y} \right)} $ , $\sum {{{\left( {{x_i} - \bar x} \right)}^2}} $ and \[\sum {{{\left( {{y_i} - \bar y} \right)}^2}} \] using the table and substitute these values in the formula to find the answer.

Complete step-by-step answer:
Here in this problem, we are given a table of values of x-variable and corresponding y-variable. There are six values for both variables and with this information, we need to the correlation coefficient of these values. There are four options given, among which the only one is correct.
Before starting with the solution it is very important to understand the concept of the correlation coefficient. A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The Pearson product-moment correlation coefficient is a measure of the strength and direction of the linear relationship between two variables that is defined as the covariance of the variables divided by the product of their standard deviations.
The formula for the Pearson correlation coefficient is:
$ \Rightarrow r = \dfrac{{\sum {\left( {{x_i} - \bar x} \right)\left( {{y_i} - \bar y} \right)} }}{{\sqrt {\sum {{{\left( {{x_i} - \bar x} \right)}^2} \times \sum {{{\left( {{y_i} - \bar y} \right)}^2}} } } }}$ , where ${x_i}{\text{ and }}{y_i}$ are the values of x-variable and y-variable respectively
And $\bar x{\text{ and }}\bar y$ are the mean of the values of x-variable and y-variable respectively
So for calculating the ‘r’ we need to calculate the values of $\left( {{x_i} - \bar x} \right){\text{ and }}\left( {{y_i} - \bar y} \right)$
Let’s first start with finding the mean of both x-variable and y-variable
$ \Rightarrow $ Mean of values of x-variable, i.e. $\bar x = \dfrac{{{\text{Sum of all the values}}}}{{{\text{Number of values}}}} = \dfrac{{\sum {{x_i}} }}{6} = \dfrac{{2 + 4 + 5 + 6 + 8 + 11}}{6} = \dfrac{{36}}{6} = 6$
$ \Rightarrow $ Mean of values of y-variable, i.e. $\bar y = \dfrac{{{\text{Sum of all the values}}}}{{{\text{Number of values}}}} = \dfrac{{\sum {{y_i}} }}{6} = \dfrac{{18 + 12 + 10 + 8 + 7 + 5}}{6} = \dfrac{{60}}{6} = 10$
Now let’s calculate the value for $\left( {x - \bar x} \right){\text{ and }}\left( {y - \bar y} \right)$ in separate columns using the above-calculated value of means.

$x$	$y$	$x - \bar x$	$y - \bar y$
$2$	$18$	$2 - 6 = - 4$	$18 - 10 = 8$
$4$	$12$	$4 - 6 = - 2$	$12 - 10 = 2$
$5$	$10$	$5 - 6 = - 1$	$10 - 10 = 0$
$6$	$8$	$6 - 6 = 0$	$8 - 10 = - 2$
$8$	$7$	$8 - 6 = 2$	$7 - 10 = - 3$
$11$	$5$	$11 - 6 = 5$	$5 - 10 = - 5$

Now we need to calculate the square of $\left( {{x_i} - \bar x} \right){\text{ and }}\left( {{y_i} - \bar y} \right)$ and also the product of these two columns.

$x$	$y$	$x - \bar x$	$y - \bar y$	${\left( {x - \bar x} \right)^2}$	${\left( {y - \bar y} \right)^2}$	$\left( {x - \bar x} \right) \times \left( {y - \bar y} \right)$
$2$	$18$	$ - 4$	$8$	${\left( { - 4} \right)^2} = - 4 \times - 4 = 16$	${\left( 8 \right)^2} = 8 \times 8 = 64$	$ - 4 \times 8 = - 32$
$4$	$12$	$ - 2$	$2$	${\left( { - 2} \right)^2} = - 2 \times - 2 = 4$	${\left( 2 \right)^2} = 2 \times 2 = 4$	$ - 2 \times 2 = - 4$
$5$	$10$	$ - 1$	$0$	${\left( { - 1} \right)^2} = - 1 \times - 1 = 1$	${\left( 0 \right)^2} = 0$	$ - 1 \times 0 = 0$
$6$	$8$	$0$	$ - 2$	${\left( 0 \right)^2} = 0$	${\left( { - 2} \right)^2} = - 2 \times - 2 = 4$	$0 \times - 2 = 0$
$8$	$7$	$2$	$ - 3$	${\left( 2 \right)^2} = 2 \times 2 = 4$	${\left( { - 3} \right)^2} = - 3 \times - 3 = 9$	$2 \times - 3 = - 6$
$11$	$5$	$5$	$ - 5$	${\left( 5 \right)^2} = 5 \times 5 = 25$	${\left( { - 5} \right)^2} = - 5 \times - 5 = 25$	\[5 \times - 5 = - 25\]

Therefore, we get the required values as:

$x$	$y$	$x - \bar x$	$y - \bar y$	${\left( {x - \bar x} \right)^2}$	${\left( {y - \bar y} \right)^2}$	$\left( {x - \bar x} \right) \times \left( {y - \bar y} \right)$
$2$	$18$	$ - 4$	$8$	$16$	$64$	$ - 32$
$4$	$12$	$ - 2$	$2$	$4$	$4$	$ - 4$
$5$	$10$	$ - 1$	$0$	$1$	$0$	$0$
$6$	$8$	$0$	$ - 2$	$0$	$4$	$0$
$8$	$7$	$2$	$ - 3$	$4$	$9$	$ - 6$
$11$	$5$	$5$	$ - 5$	$25$	$25$	\[ - 25\]

Now according to the required formula, we need to find $\sum {\left( {{x_i} - \bar x} \right)\left( {{y_i} - \bar y} \right)} $ , $\sum {{{\left( {{x_i} - \bar x} \right)}^2}} $ and \[\sum {{{\left( {{y_i} - \bar y} \right)}^2}} \]
$ \Rightarrow \sum {\left( {{x_i} - \bar x} \right)\left( {{y_i} - \bar y} \right)} = - 32 + \left( { - 4} \right) + 0 + 0 + \left( { - 6} \right) + \left( { - 25} \right) = - 32 - 4 - 6 - 25 = - 67$
Then from another column, we get:
$ \Rightarrow \sum {{{\left( {{x_i} - \bar x} \right)}^2}} = 16 + 4 + 1 + 0 + 4 + 25 = 16 + 4 + 1 + 4 + 25 = 50$
Similarly, we get:
\[ \Rightarrow \sum {{{\left( {{y_i} - \bar y} \right)}^2}} = 64 + 4 + 0 + 4 + 9 + 25 = 64 + 4 + 4 + 9 + 25 = 106\]
Therefore, now we can use the above formula to find the correlation coefficient as:
$ \Rightarrow r = \dfrac{{\sum {\left( {{x_i} - \bar x} \right)\left( {{y_i} - \bar y} \right)} }}{{\sqrt {\sum {{{\left( {{x_i} - \bar x} \right)}^2} \times \sum {{{\left( {{y_i} - \bar y} \right)}^2}} } } }} = \dfrac{{ - 67}}{{\sqrt {50 \times 106} }} = \dfrac{{ - 67}}{{10\sqrt {53} }} = \dfrac{{ - 6.7}}{{\sqrt {53} }}$
This can be further simplified by calculating the square root of $53$ , we get:
$ \Rightarrow r = - 0.92$
Hence, the option (C) is the correct answer
Note: The correlation coefficient is a statistical measure of the strength of the relationship between the relative movements of two variables. The values range between $ - 1$ and $1$ . A calculated number is greater than $1$ or less than $ - 1$ means that there was 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 zero shows no linear relationship between the movement of the two variables.