Question

Calculate the coefficient of correlation from the following data:
X : 9, 8, 7, 6, 5, 4, 3, 2, 1.
Y: 15, 16, 14, 13, 11, 12, 10, 8, 9.​

Answer 1

Given:

The following data is given :

X-samples : 9, 8, 7, 6, 5, 4, 3, 2,1

Y-samples : 15, 16, 14, 13, 11, 12, 10, 8, 9

To Find:

We have to find the coefficient of correlation,r.

Solution:

The statistical measure of the strength of the relationship between the relative movement of two variables is known as the coefficient of correlation. The value ranges from 1 to -1.

The equation is given by,

                                $\[\begin{array}{c}r = \frac{{Cov\left( {x,y} \right)}}{{{\sigma _x}{\sigma _y}}}\\\\ = \frac{{\sum {\left( {{x_i} - \mathop x\limits^ - } \right)\left( {{y_i} - \mathop y\limits^ - } \right)} }}{{\sqrt {\sum {{{\left( {{x_i} - \mathop x\limits^ - } \right)}^2}\sum {{{\left( {{y_i} - \mathop y\limits^ - } \right)}^2}} } } }}\end{array}\]$

where,

r → correlation coefficient

$\[{{x_i}}\]$ → values of the x variable in a sample

$\[{\mathop x\limits^ - }\]$ → mean of the values of the x variable

$\[{{y_i}}\]$ → values of the y variable in a sample

$\[{\mathop y\limits^ - }\]$ → mean of the values of the y variable

                                  $\[\begin{array}{c}{r_{\left( {x,y} \right)}} = \frac{{Cov\left( {u,v} \right)}}{{{\sigma _u}{\sigma _v}}}\\\\ = \frac{{\sum {u * v - \frac{1}{n}\sum {u * \sum v } } }}{{\sqrt {\sum {{u^2} - \frac{{{{\left( {\sum u } \right)}^2}}}{n}} } * \sqrt {\sum {{v^2} - \frac{{{{\left( {\sum v } \right)}^2}}}{n}} } }}\end{array}\]$

The tabular column is shown below.

                          $\[\begin{array}{c}{r_{\left( {x,y} \right)}} = \frac{{\sum {\left( {u * v} \right) - \frac{1}{n}\sum {u * \sum v } } }}{{\sqrt {\sum {{u^2} - \frac{{{{\left( {\sum u } \right)}^2}}}{n}} } * \sqrt {\sum {{v^2} - \frac{{{{\left( {\sum v } \right)}^2}}}{n}} } }}\\\\ = \frac{{57 - \frac{1}{9} * 0 * 0}}{{\sqrt {60 - \frac{0}{9}} * \sqrt {60 - \frac{0}{9}} }}\\\\ = \frac{{57}}{{\sqrt {60} * \sqrt {60} }}\\\\ = \frac{{57}}{{60}}\\\\ = 0 \cdot 95\end{array}\]$

Hence, the coefficient of correlation is given as 0.95