Question

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:
A. $\dfrac{{11}}{2}$
B. $6$
C. $\dfrac{{13}}{2}$
D. $\dfrac{5}{2}$

Answer 1

Hint: According to the question we have to determine the variance of the combined data set when for two data sets, each of the size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. So, first of all we have to use the given variances which are 4 and 5 and then their corresponding means which are 2 and 4.
Now, we have to use the formula as mentioned below for the first data as we know for the size of 5.

Formula used: \[ \Rightarrow {\sigma ^2} = \dfrac{1}{n}\sum\limits_{i = 1}^n {{x_i}^2 - {{\left( {\sum\limits_{i = 1}^n {\dfrac{{{x_i}}}{n}} } \right)}^2}} ........................(A)\]
Hence, with the help of the formula (A) above, we can determine the variance and where n is the size if the data set and $\sigma $is the variance.
Now, to determine the mean for the given data set we have to use the formula to determine the mean which is as mentioned below:
$ \Rightarrow \overline x = \sum\limits_{i = 1}^n {\dfrac{{{x_i}}}{n}..............(B)} $
Where, n is the size of the given data set and for the second data set we have to use both of the formulas (A) and (B) above,
Now, after obtaining the required data set we have to determine the variance of the combined data set which can be obtained with the help of the formula as mentioned below:
$ \Rightarrow {\sigma ^2} = \dfrac{1}{n}{\sum\limits_{i = 1}^5 {({x_i}^2 + {y_i}^2) - \left( {\sum\limits_{i = 1}^5 {\dfrac{{{x_i} + {y_i}}}{n}} } \right)} ^2}..............(C)$
Hence, on substituting all the obtained values from the formulas (A) and (B) with the help of both of the data sets we have to substitute the values in the formula (C) above to determine the combined variance.

Complete step-by-step solution:
Step 1: First of all we have to use the formula (A) to determine the variance for the first data set which is as below:
\[ \Rightarrow {\sigma _1}^2 = \dfrac{1}{5}\sum\limits_{i = 1}^5 {{x_i}^2 - {2^2}} \]
Now, on substituting the value of variance as mentioned in the question,
\[
   \Rightarrow 4 = \dfrac{1}{5}\sum\limits_{i = 1}^5 {{x_i}^2 - {2^2}} \\
   \Rightarrow \sum\limits_{i = 1}^5 {{x_i}^2 = 40}
 \]
Step 2: Now, we have to use the formula (B) to determine the mean for the first data set which is as below:
$ \Rightarrow \overline x = \sum\limits_{i = 1}^5 {\dfrac{{{x_i}}}{5}} $
Now, on substituting the value of mean as mentioned in the question,
\[
   \Rightarrow 2 = \sum\limits_{i = 1}^5 {\dfrac{{{x_i}}}{5}} \\
   \Rightarrow \sum\limits_{i = 1}^5 {{x_i} = 10}
 \]
Step 3: Now, we have to use the formula (A) to determine the variance for the second data set which is as below:
\[ \Rightarrow {\sigma _2}^2 = \dfrac{1}{5}\sum\limits_{i = 1}^5 {{y_i}^2 - {4^2}} \]
Now, on substituting the value of variance as mentioned in the question,
\[
   \Rightarrow 5 = \dfrac{1}{5}\sum\limits_{i = 1}^5 {{y_i}^2 - {4^2}} \\
   \Rightarrow \sum\limits_{i = 1}^5 {{y_i}^2 = 105}
 \]
Step 4: Now, we have to use the formula (B) to determine the mean for the first data set which is as below:
$ \Rightarrow \overline x = \sum\limits_{i = 1}^5 {\dfrac{{{y_i}}}{5}} $
Now, on substituting the value of mean as mentioned in the question,
\[
   \Rightarrow 4 = \sum\limits_{i = 1}^5 {\dfrac{{{y_i}}}{5}} \\
   \Rightarrow \sum\limits_{i = 1}^5 {{y_i} = 20}
 \]
Step 5: Now, we have to find the variance for the combined data set with the help of the formula (C) as mentioned in the solution hint. Hence, on substituting all the values as obtained with the help of the formula (A) and (B) in the previous steps in the formula (C),
$
   \Rightarrow {\sigma ^2} = \dfrac{1}{{10}} \times (40 + 105) - \dfrac{1}{{10}} \times 90 \\
   \Rightarrow {\sigma ^2} = \dfrac{{145}}{{10}} - 9 \\
   \Rightarrow {\sigma ^2} = \dfrac{{145 - 90}}{{10}} \\
   \Rightarrow {\sigma ^2} = \dfrac{{11}}{2}
 $
Hence, with the help of the formula (A), (B), and (C) we have determined the variance for the combined data set which is ${\sigma ^2} = \dfrac{{11}}{2}$.

Therefore option (A) is correct.

Note: The term variance refers to statistical measurements of the spread between numbers in a data set and with the help of variance we can measure how far each number in the set is from the mean and thus form every other number in the set.
The mean is simply known as the average, which can be determined by the sum of the collection of the numbers divided by the count of the number in the data set.