Question

For data sets, each of size 5, the variance is given to be 4 and 5, and the corresponding means are given to be 2 and 4, respectively. The double of the variance of the combined data sets is-
A. 13
B. 12
C. 5.5
D. 10

Answer 1

In order to solve this question the given information is variance if x and variance and mean of x and y then we will apply the formula of the radiance of n numbers and put the value of mean in both of them in x as well as in y. From here we can find the summation of ${x^2}$ and the summation of ${y^2}$ individually then we will apply the variance formula.

Complete step by step Solution:
For solving this we have given the data variance is given to us individually of x and y:
${\sigma ^2}x = 4$
And
${\sigma ^2}y = 5$
The mean of each of them is also given to us are:
$\overline x = 2$
And
$\overline y = 4$
Then by applying the formula of variance:
${\sigma ^2}x = \sum {\dfrac{{x_i^2}}{n} - {{\left( {\dfrac{{\sum {{x_i}} }}{n}} \right)}^2}}$
As the value of variance is given in question is equal to 4:
$4 = \sum {\dfrac{{x_i^2}}{n} - {{\left( {\dfrac{{\sum {{x_i}} }}{n}} \right)}^2}}$
Now, $\left( {\dfrac{{\sum {{x_i}} }}{n}} \right)$ is the mean of the x which is equal to 2:
$4 = {\sum {\dfrac{{x_i^2}}{n} - \left( 2 \right)} ^2}$
On putting the value of n=5 and further solving:
$8 = \dfrac{{\sum {{x_i}^2} }}{5}$
The value of $\sum {{x_i}^2} = 40$
Now similarly the variance of y will be:
${\sigma ^2}y = \dfrac{{\sum {y_i^2} }}{n} - {\left( {\dfrac{{\sum {{y_i}} }}{n}} \right)^2}$
And in question it is given to us which is equal to 5:
$5 = \dfrac{{\sum {y_i^2} }}{n} - {\left( {\dfrac{{\sum {{y_i}} }}{n}} \right)^2}$
Now $\left( {\dfrac{{\sum {{y_i}} }}{n}} \right)$ is the mean which is given to us is equal to 4.
$5 = \dfrac{{\sum {y_i^2} }}{n} - {\left( 4 \right)^2}$
Now on putting the value of n=5 and on further solving:
$\dfrac{{\sum {y_i^2} }}{5} = 21$
On further solving we get:
$\sum {y_i^2} = 105$
Now double of the variance of the combined data:
${\sigma ^2} = \dfrac{{\sum {x_i^2} + \sum {y_i^2} }}{{10}} - {\left( {\dfrac{{\sum {{x_i}} + \sum {{y_i}} }}{{10}}} \right)^2}$
Since the total data size of x and y are 5 so for combined it will be equal to 10.
${\sigma ^2} = \dfrac{{145}}{{10}} - {\left( {\dfrac{{10 + 20}}{{10}}} \right)^2}$
On further solution:
${\sigma ^2} = \dfrac{{145}}{{10}} - 3$
On further solving:
${\sigma ^2} = 5.5$

So the correct option is C.

Note:
While solving these types of problems we should apply the correct formula because there are complications in these while we apply, and as the mean is given to us so we should analyze the formula that is mean and apply it in place of the formula.
One important thing is we will have to put the data size of combined sets as we are finding the variance of combined sets.