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

To find the class marks, the following relation is used.

Taking 135 as assumed mean (a), d_i, u_i, f_iu_i are calculated according to step deviation method as follows.

Monthly consumption (in units)	Number of consumers (f i)	xi class mark	di= xi− 135
65 − 85	4	75	− 60	− 3	− 12
85 − 105	5	95	− 40	− 2	− 10
105 − 125	13	115	− 20	− 1	− 13
125 − 145	20	135	0	0	0
145 − 165	14	155	20	1	14
165 − 185	8	175	40	2	16
185 − 205	4	195	60	3	12
Total	68				7

From the table, we obtain

From the table, it can be observed that the maximum class frequency is 20, belonging to class interval 125 − 145.

Modal class = 125 − 145

Lower limit (l) of modal class = 125

Class size (h) = 20

Frequency (f₁) of modal class = 20

Frequency (f₀) of class preceding modal class = 13

Frequency (f₂) of class succeeding the modal class = 14

To find the median of the given data, cumulative frequency is calculated as follows.

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

From the table, we obtain

n = 68

Cumulative frequency (cf) just greater than is 42, belonging to interval 125 − 145.

Therefore, median class = 125 − 145

Lower limit (l) of median class = 125

Class size (h) = 20

Frequency (f) of median class = 20

Cumulative frequency (cf) of class preceding median class = 22

Therefore, median, mode, mean of the given data is 137, 135.76, and 137.05 respectively.

The three measures are approximately the same in this case.