JEE MAIN - Mathematics (2010 - No. 1)
For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding
means are given to be 2 and 4, respectively. The variance of the combined data set is
$${5 \over 2}$$
$${11 \over 2}$$
6
$${13 \over 2}$$
Explanation
Given that,
$${\sigma _1}^2 = 4$$
and $${\sigma _2}^2 = 5$$
And also given,
$$\overline x = 2\,\,$$ and $$\overline y = 4\,\,$$
So, $$\,\,\,$$ $${{\sum {{x_i}} } \over 5} = 2$$
$$ \Rightarrow \sum {{x_i}} = 10$$ v
and $${{\sum {{y_i}} } \over 5} = 4$$
$$ \Rightarrow \,\,\,\sum {{y_i}} = 20$$
$$\therefore\,\,\,$$ $${\sigma _1}^2 = {{\sum {x_i^2} } \over 5} - {\left( {\overline x } \right)^2}$$
$$ \Rightarrow \,\,\,4 = {{\sum {x_i^2} } \over 5} - 4$$
$$ \Rightarrow \,\,\,\sum {x_i^2} = 40$$
and $${\sigma _2}^2 = {{\sum {y_i^2} } \over 5} - {\left( {\overline y } \right)^2}$$
$$ \Rightarrow \,\,\,\,5 = {{\sum {y_i^2} } \over 5} - 16$$
$$ \Rightarrow \,\,\,\sum {y_i^2} = 105$$
Variance of combined data set
$${\sigma ^2} = {1 \over {10}}\left( {\sum {x_i^2 + \sum {y_i^2} } } \right) - {\left( {{{\overline x + \overline y } \over 2}} \right)^2}$$
$$ = {1 \over {10}}\left( {40 + 105} \right) - 9$$
$$ = {{145 - 90} \over {10}}$$
$$ = {{55} \over {10}}$$
$$ = {{11} \over {2}}$$
$${\sigma _1}^2 = 4$$
and $${\sigma _2}^2 = 5$$
And also given,
$$\overline x = 2\,\,$$ and $$\overline y = 4\,\,$$
So, $$\,\,\,$$ $${{\sum {{x_i}} } \over 5} = 2$$
$$ \Rightarrow \sum {{x_i}} = 10$$ v
and $${{\sum {{y_i}} } \over 5} = 4$$
$$ \Rightarrow \,\,\,\sum {{y_i}} = 20$$
$$\therefore\,\,\,$$ $${\sigma _1}^2 = {{\sum {x_i^2} } \over 5} - {\left( {\overline x } \right)^2}$$
$$ \Rightarrow \,\,\,4 = {{\sum {x_i^2} } \over 5} - 4$$
$$ \Rightarrow \,\,\,\sum {x_i^2} = 40$$
and $${\sigma _2}^2 = {{\sum {y_i^2} } \over 5} - {\left( {\overline y } \right)^2}$$
$$ \Rightarrow \,\,\,\,5 = {{\sum {y_i^2} } \over 5} - 16$$
$$ \Rightarrow \,\,\,\sum {y_i^2} = 105$$
Variance of combined data set
$${\sigma ^2} = {1 \over {10}}\left( {\sum {x_i^2 + \sum {y_i^2} } } \right) - {\left( {{{\overline x + \overline y } \over 2}} \right)^2}$$
$$ = {1 \over {10}}\left( {40 + 105} \right) - 9$$
$$ = {{145 - 90} \over {10}}$$
$$ = {{55} \over {10}}$$
$$ = {{11} \over {2}}$$
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