Question

If the median of the data $6,7,x-2,x,18,21$ written in ascending order is $16$, then the variance of the data is

• A. $30\frac{1}{5}$
• B. $31\frac{1}{3}$
• C. $32\frac{1}{2}$
• D. $33\frac{1}{3}$

Answer 1

The correct option is B $31\frac{1}{3}$

The given observations are,

$6,7,x-2,x,18,21$

Number of observations $\left(n\right)=6$ [ even ]

$\Rightarrow$ $Median=\frac{3^{rd}obs+4^{th}obs}{2}$

$\Rightarrow$ $16=\frac{x-2+x}{2}$

$\Rightarrow$ $32=x-2+x$

$\Rightarrow$ $32=2x-2$

$\Rightarrow$ $34=2x$

$\therefore$ $x=17$

$\therefore$ The observations becomes,

$6,7,15,17,18,21$

$\overline{x}=\frac{6+7+15+17+18+21}{6}=\frac{84}{6}=14$

$Variance\left({\sigma }^2\right)=\frac{(6-14{)}^2+(7-14{)}^2+(15-14{)}^2+(17-14{)}^2+(18-14{)}^2+(21-14{)}^2}{6}$

$=\frac{(-8{)}^2+(-7{)}^2+(1{)}^2+(3{)}^2+(4{)}^2+(7{)}^2}{6}$

$=\frac{64+49+1+9+16+49}{6}$

$=\frac{188}{6}$

$=\frac{94}{3}$

$=31\frac{1}{3}$