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}

 

Answers (1)

As we learnt in

ARITHMETIC Mean -

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

-

 

 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}

-

 

 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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