Question

A data set of $n$ observations has mean $2\stackrel{-}{x}$ while another data set of $2n$ observations has mean $\stackrel{-}{x}$. The mean of the combined data set of $3n$ observations will be equal to:

• A. $\stackrel{-}{x}$
• B. $\frac{3}{2}\stackrel{-}{x}$
• C. $\frac{2}{3}\stackrel{-}{x}$
• D. $\frac{4}{3}\stackrel{-}{x}$

Answer 1

The correct option is D $\frac{4}{3}\stackrel{-}{x}$

Mean is the sum of the data values divided by the total number of values.

Given that, mean of $n$ observations is $2\overline{x}$

$⟹2\overline{x}=\frac{\text{sum of n observations}}{n}$

Therefore, sum of $n$ observations is $2\overline{x}n$ ------(1)

Given that, mean of $2n$ observations is $\overline{x}$

$⟹\overline{x}=\frac{\text{sum of 2n observations}}{2n}$

Therefore, sum of $2n$ observations is $2\overline{x}n$ ------(2)

Mean of the combined data set is $\frac{\text{sum of n observations + sum of 2n observations}}{n+2n}$

$⟹mean=\frac{2\overline{x}n+2\overline{x}n}{n+2n}$ (from (1) and (2))

$⟹mean=\frac{4\overline{x}n}{3n}$

$⟹mean=\frac{4}{3}\overline{x}$