(Q3) 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) 65-85 85-105 105-125 125-145 145-165 165-185 185-205
Number of consumers 4 5 13 20 14 8 4

Answer :

Monthly Consumption (in units) Number of Consumers (fi) Cumulative frequency (xi) Class marks ui =
xi - a
h

(h = 20) 		fiui
65-85 4 4 75 -3 -
85-105 5 9 95 -2 -
105-125 13 22 115 -1 -
125-145 20 42 135 (a) 0
145-165 14 56 155 1
165-185 8 64 175 2
185-205 4 68 195 3
∑fi = 68 - - - ∑fixi =

Sum of the frequencies = 68

∴
n
2
 =
68
2
 = 34

Hence median class = 125-145

Lower boundary of the median class, l = 125

cf - cumulative frequency of the class preceding the median class = 22

f - frequency of the median class = 20

h = class size = 20

Median = l +
(
n
2
 - cf )
f
 × h
Median = +
[ - ]
20
 ×

∴ Median = + =

Maximum number of consumers lie in the class 125-145

Modal class is 125-145

l - lower limit of the modal class =125

f1 - frequency of the modal class = 20

f0 - frequency of the class preceding the modal class = 13

f2 - frequency of the class succeeding the modal class = 14

h - size of the class = 20

∴ Mode (Z) = l +
f1 - f0
2f1 - f0 - f2
 × h
Mode (Z) = +
-
( - )+( - )
 × 20

Mode(Z) = .769

Mean = x̄ = a +
∑fiui
∑fi
 × h

a = assumed mean = 135

∴ x = 135 +
68
 = .102

Mean, Median and Mode are approximately same in this case.