Question

When the origin is changed, then find the change in the coefficient of correlation.
A. Becomes zero
B. Varies
C. Remains fixed
D. None of these

Answer 1

Hint: First write the correlation coefficient and its dependency to obtain the required result.

Formula Used:
The formula of correlation coefficient is,
$r = \dfrac{{\sum {\left( {{x_i} - \overline x } \right)\left( {{y_i} - \overline y } \right)} }}{{\sqrt {\sum {{{\left( {{x_i} - \overline x } \right)}^2}} } \sum {{{\left( {{y_i} - \overline y } \right)}^2}} }}$ ,
Where ${x_i},{y_i}$ are the x variable and y variable in the sample respectively and $\overline x ,\overline y $ are the means values of x and y respectively.

Complete step by step solution:
The formula of correlation coefficient is,
$r = \dfrac{{\sum {\left( {{x_i} - \overline x } \right)\left( {{y_i} - \overline y } \right)} }}{{\sqrt {\sum {{{\left( {{x_i} - \overline x } \right)}^2}} } \sum {{{\left( {{y_i} - \overline y } \right)}^2}} }}$ ,
Where ${x_i},{y_i}$ are the x variable and y variable in the sample respectively and $\overline x ,\overline y $ are the means values of x and y respectively.
Here we can see that the correlation coefficient is independent of the origin, hence it will remain unchanged when origin is changed.

Option ‘C’ is correct

Additional Information
A statistical concept known as the correlation coefficient aids in establishing a relationship between expected and actual values obtained through statistical experimentation. Changes to the scale and origin have no effect on the coefficient of correlation's value.

Note: Sometimes the students substitute x and y as (0,0) in the formula of correlation coefficient but here we need not to do that, just observe the formula and say whether it's dependent on the origin or not.