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

Monthly consumption (in units)	Number of consumers $f_i$	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	$n=68$

$n=68$ gives $\frac{n}{2}=34$
So, we have the median class (125-145)
$l=125,n=68,f=20,cf=22,h=20$
$Median=l+\left\{\frac{\frac{n}{2}-cf}{f}\right\}\times h$
$=125+\left\{\frac{34-22}{\cdot 20}\right\}\times 20=137units$.

(ii) Modal class is (25-145) having maximum frequency $f_m=20,f)1-13,f_2=14,l=125$ and $h=20$
$Mode=l+\left\{\frac{f_m-f_1}{2f_m-f_1-f_2}\right\}\times h$
$=125+\left\{\frac{20-13}{40-13-14}\right\}\times 20=125+\frac{7\times 20}{13}$
$=125+\frac{140}{13}=125+10.76=135.76units$

(iii) $n=68,a=135,h=20$ and $\sum f_i{u}_i=7$

Monthly consumption (in units)	Number of consumers	Class mark $x_i$	$u_i=\frac{x_i-135}{20}$	$f_i\times u_i$
65-85	4	75	-3	-12
85-105	5	95	-2	-10
105-125	13	115	-1	-13
125-145	20	135=a	0	0
145-165	14	155	1	14
165-185	8	175	2	16
185-205	4	195	3	12
Total	n=68			7

$n=68,a=135,h=20$ and $\sum f_i{u}_i=7$
By step-deviation method.
$Mean=a+h\times \frac{1}{n}\times \sum f_i{u}_i=135+20\times \frac{1}{68}\times 7$
$=135+\frac{35}{17}=135+2.05=137.05units$.