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
$65-85$ $85-105$ $105-125$ $125-145$ $145-165$ $165-185$ $185-205$	$4$ $5$ $13$ $20$ $14$ $8$ $4$

Answer 1

Calculation of median:

Preparing the table to compute median :

Monthly consumption (in units)	Number of consumers (Frequency)	Cumulative frequency
$65-85$	$4$	$4$
$85-105$	$5$	$9$
$105-125$	$13$	$22$
$125-145$	$20$	$42$
$145-165$	$14$	$56$
$165-185$	$8$	$64$
$185-205$	$4$	$68$
Total	$n=68$

We have, $n=68$

$\frac{n}{2}=34$

The cumulative frequency just greater than $\frac{n}{2}$ is $42$ and the corresponding class is $125-145$.

Thus,125-145$ is the median class such that

$\frac{n}{2}=34,l=125,f=20,cf=22,$ and $h=20$

Substituting these values in the formula

Median, $M=l+\left(\frac{\frac{n}{2}-cf}{f}\right)\times h$

$M=125+\left(\frac{34-22}{20}\right)\times 20$

$M=125+12=137$ units

For calculation of mean:

Class- Interval	Mid-Value $\left(x_i\right)$	Frequency $\left(f_i\right)$	$u_i=\frac{x_i-135}{20}$	$f_i{u}_i$
$65-85$	$75$	$4$	$-3$	$-12$
$85-105$	$95$	$5$	$-2$	$-10$
$105-125$	$105$	$13$	$-1$	$-13$
$125-145$	$135$	$20$	$0$	$0$
$145-165$	$155$	$14$	$1$	$14$
$165-185$	$175$	$8$	$2$	$16$
$185-205$	$195$	$4$	$3$	$12$
Total		$68$		$7$

Let the assumed mean, $A=135$ and class interval, $h=20$

So, $u_i=\frac{x_i-A}{h}=\frac{x_i-135}{20}$

Mean, $\overline{x}=A+h\times \frac{\sum f_i{u}_i}{\sum f_i}$

$\overline{x}=135+20\times \frac{7}{68}$

$\Rightarrow \overline{x}=135+2.05=137.05$

Therefore, Mean $=137.05$ units

Calculation of Mode:

The class $125–145$ has the maximum frequency,therefore, this is the modal class.

Here,

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

Now, let us substitute these values in the formula

Mode $=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\times h$

$=125+\frac{20-13}{40-13-14}\times 20$

$=125+\frac{7}{13}\times 20=125+10.76=135.76$ units

It can be seen that the three measures are approximately the same in this case.