Question

The following frequency distribution gives the monthly consumption of electricity of $68$ consumers of a locality. Find the median, mean and mode of the data and compare them.

Monthly consumption (in units)	Number of consumers
$65-85$	$4$
$85-105$	$5$
$105-125$	$13$
$125-145$	$20$
$145-165$	$14$
$165-185$	$8$
$185-205$	$4$

Answer 1

$C.I$	$x_i$	$f_i$	$f_i{x}_i$	$c.f$
$65-85$	$75$	$4$	$300$	$4$
$85-105$	$95$	$5$	$475$	$9$
$105-125$	$115$	$13$	$1495$	$22$
$125-145$	$135$	$20$	$2700$	$42$
$145-165$	$155$	$14$	$2170$	$56$
$165-185$	$175$	$8$	$1400$	$64$
$185-205$	$195$	$4$	$780$	$68$
		$\sum f_i=68$	$\sum f_i{x}_i=9320$

$\Rightarrow$ $Mean=\frac{\sum f_i{x}_i}{\sum f_i}=\frac{9320}{68}=137.05$

Now, $n=68$ and $\frac{n}{2}=34$

$\therefore$ Median class $=125-145$

We have, $l=125,c.f=22,f=20$ and $h=20$

$\Rightarrow$ $Median=l+\frac{\frac{n}{2}-c.f}{f}\times h$

$=125+\frac{34-22}{20}\times 20$

$=125+12$

$\therefore$ $Median=137$

Now, modal class $=125-145$

Then we have $l=125,f_0=13,f_1=20,f_2=14$ and $h=20$

$\Rightarrow$ $Mode=l+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h$

$=125+\frac{20-13}{2\times 20-13-14}\times 20$

$=125+10.77$

$\therefore$ $Mode=135.77$