Question
For the data $$3,\,5,\,1,\,6,\,5,\,9,\,5,\,2,\,8,\,6$$ the mean, median and mode are $$x,\,y$$ and $$z$$ respectively. Which one of the following is correct ?
A.
$$x = y \ne z$$
B.
$$x \ne y = z$$
C.
$$x \ne y \ne z$$
D.
$$x = y = z$$
Answer :
$$x = y = z$$
Solution :
Given data $$3,\,5,\,1,\,6,\,5,\,9,\,5,\,2,\,8,\,6$$ and mean, median and mode are $$x,\,y,\,z$$ respectively.
Rearranging data $$1,\,2,\,3,\,5,\,5,\,5,\,6,\,6,\,8,\,9$$
$$\eqalign{
& {\text{Mean}} = x \cr
& = \frac{{1 + 2 + 3 + 5 + 5 + 5 + 6 + 6 + 8 + 9}}{{10}} \cr
& = \frac{{50}}{{10}} \cr
& = 5 \cr
& {\text{Median}} = y \cr
& = \frac{{\frac{{{n^{th}}}}{2}{\text{term}} + {{\left( {\frac{n}{2} + 1} \right)}^{th}}{\text{term}}}}{2} \cr
& = \frac{{5 + 5}}{2} \cr
& = 5 \cr} $$
Mode $$\left( z \right) = $$ most frequently occurring value $$= 5$$
Hence $$x = y = z.$$