JEE MAIN - Mathematics (2013 (Offline) - No. 2)
All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10
to each of the students. Which of the following statistical measures will not change even after the grace
marks were given?
median
mode
variance
mean
Erläuterung
As we know variance does not change with the change of origin. So, here even after adding grace marks $$10$$, the variance will be same.
Let's see with an example,
Assume initial variance $$ = {{\sum {{{\left( {{x_i} - \overline x } \right)}^2}} } \over N}$$
After adding grace marks $$10$$ with each student,
the final variance $$ = {{{\sum {\left[ {\left( {{x_i} + 10} \right) - \left( {\overline x + 10} \right)} \right]} } \over N}^2}$$
$$ = {{\sum {{{\left( {{x_i} - \overline x } \right)}^2}} } \over N}$$
$$ = $$ Initial variance.
Let's see with an example,
Assume initial variance $$ = {{\sum {{{\left( {{x_i} - \overline x } \right)}^2}} } \over N}$$
After adding grace marks $$10$$ with each student,
the final variance $$ = {{{\sum {\left[ {\left( {{x_i} + 10} \right) - \left( {\overline x + 10} \right)} \right]} } \over N}^2}$$
$$ = {{\sum {{{\left( {{x_i} - \overline x } \right)}^2}} } \over N}$$
$$ = $$ Initial variance.