Question

Calculate co-efficient of correlation from the

following data.

x 12 9 8 10 11 13 7

y 14 8 6 9 11 12 3​

Answer 1

Answer:

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Answer 2

Answer:

The correlation coefficient of the given data calculated is $0.689$.

Step-by-step explanation:

Given data:

$x$                  $y$

$12$                $14$

$9$                  $8$

$8$                  $6$

$10$                $9$

$11$                $11$

$13$                $12$

$7$                  $3$

To find: The coefficient of correlation of the given data =?

Firstly, we have to find the mean of $x$ and the mean of $y$.

As we know,

• Mean = $\frac{Sum of terms}{Number of terms}$

For $x$:

• Mean, $\bar{x}$ = $\frac{12+9+8+10+11+13+7}{7}$ = $\frac{70}{7}$ = $10$

For $y$:

• Mean, $\bar{y}$ = $\frac{14+8+6+9+11+12+3}{7}$ = $\frac{63}{7}$ = $9$

As we know,

• Correlation coefficient = $\frac{\sum(x_{i}-\bar{x} )(y_{i}-\bar{y} ) }{\sqrt{\sum(x_{i}-\bar{x} )^{2}\sum(y_{i}-\bar{y} ) } }$

For this, we have to make a table to find the terms in the formula:

$x$            $y$         $(x-\bar{x})$      $(y-\bar{y})$     $(x-\bar{x})(y-\bar{y})$    $(x-\bar{x})^{2}$       $(y-\bar{y})^{2}$

$12$          $14$          $-2$              $5$               $-10$                   $4$                $25$

$9$            $8$           $-1$            $-1$                  $1$                     $1$                 $1$

$8$            $6$           $-2$           $-3$                   $6$                    $4$                 $9$

$10$          $9$              $0$              $0$                   $0$                    $0$                 $0$

$11$          $11$             $1$              $2$                   $2$                    $1$                  $4$

$13$          $12$            $3$              $3$                   $9$                    $9$                  $9$

$7$            $3$           $-3$           $-6$                  $18$                   $9$                  $36$

                                         

Now,

• $\sum (x-\bar{x})(y-\bar{y})$ = $-10 +1 +6 +0 +2+9+18$ = $26$
• $\sum (x-\bar{x})^{2}$ = $4+1+4+0+1+9+9$ = $28$
• $\sum (y-\bar{y})^{2}$ = $25+1+9+0+4+9+36$ = $51$

After putting these calculated values in the formula of the correlation coefficient mentioned above, we get:

• Correlation coefficient = $\frac{\sum(x_{i}-\bar{x} )(y_{i}-\bar{y} ) }{\sqrt{\sum(x_{i}-\bar{x} )^{2}\sum(y_{i}-\bar{y} ) } }$
• Correlation coefficient = $\frac{26}{\sqrt{28\times51 } }$
• Correlation coefficient = $\frac{26}{\sqrt{1428} }$
• Correlation coefficient = $\frac{26}{37.7}$ = $0.689$

Hence, the correlation coefficient of the given data calculated is $0.689$.