Question

Observations of a raw data are $5,28,15,10,15,8,24$. Add four more numbers so that mean and median of the data remain the same, but mode increases by $1$

Answer 1

$Mean$ $\overline{x}=\frac{\sum x}{n}.$

$\overline{x}=\frac{5+8+10+15+15+24+28}{7}=\frac{105}{7}=15.$

$n=7$ [$n$ is odd.]

$median=15.$ [the $mid-value.$]

$mode=15$ [It occurs often.]

Since, $four$ more numbers has to be added, where $mode$ has to be increased by $1.$

$\therefore$ new mode$=16.$ [Has to occur atleast $thrice$.]

$\therefore$ Mean has to be same.

$\therefore 16,16,16,12$ has to be the numbers.

$verifying:$

$\overline{x}=\frac{\sum x}{n}=\frac{105+16+16+16+12}{11}=\frac{165}{11}=15$ $\left(verified\right).$

$median⟹$ $n=11$ [$n$ is odd.]

$median=15.$ [median is the mid value.]

$mode⟹$ $16$ occuring $3$ times.

$mode=16.$ $\left(verified\right).$