# 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 Option 1) $\frac{5}{2}$ Option 2) $\frac{11}{2}$ Option 3) $6$ Option 4) $\frac{13}{2}$

As we learnt in

ARITHMETIC Mean -

The arithmetic mean is defined as the sum of items divided by the number of items.

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and

Variance -

In case of discrete data

$\dpi{100} \sigma ^{2}= \left ( \frac{\sum x_{i}^{2}}{n} \right )-\left ( \frac{\sum x_{i}}{n} \right )^{2}$

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Given v1=4 and v2=5

$\frac{1}{5}\sum x_{i}^{2}-2^{2}=4$

$\sum x_{i}^{2}=40$

and $\frac{1}{5}\sum y_{i}^{2}-4^{2}=5$

$\sum y_{i}^{2}=105$

$\sum( x_{i}^{2}+ y_{i}^{2})=145$

$\sum x_{i}=5\times 2=10$

$\sum y_{i}=5\times 4=20;\:\sum (x_{i}+y_{i})=30$

Variance =$\frac{1}{10}(145)-\left(\frac{30}{10} \right )^{2}$

= 14.5-9=5.5=$\frac{11}{2}$

Option 1)

$\frac{5}{2}$

This is incorrect option

Option 2)

$\frac{11}{2}$

This is correct option

Option 3)

$6$

This is incorrect option

Option 4)

$\frac{13}{2}$

This is incorrect option

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