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

The cumulative frequencies for each class interval are presented below.

Monthly consumption (in units)	Number of consumers	Cumulative frequency
60 - 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

From the table, n = 68. The cumulative frequency immediately greater than \( \frac{n}{2} = \frac{68}{2} = 34 \) is 42, which corresponds to the class interval 125 - 145.
Median class: 125 - 145
Lower limit (l) of the median class: 125
Frequency (f) of the median class: 20
Cumulative frequency (cf) of the median class: 22
Class size (h): 20

Median = \(l + (\frac{\frac{n}{2} - cf}{f} \times h)\)

Median = \(125 + (\frac{34 - 22}{20} \times 20)\)

Median = 125 + 12
Median = 137

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The class mark (xi) for each interval is calculated using the formula:

Class mark (xi) = \( \frac{\text{Upper \,limit + Lower \,limit}}{2} \)

Using an assumed mean (a) of 11.5, di, ui, and fiui are calculated via the step deviation method.

Monthly consumption  (in units)	Number of consumers	\( \bf{x_i}\)	\( \bf{d_i = x_i -11.5}\)	\( \bf{u_i = \frac{d_i}{3}}\)	\( \bf{f_iu_i}\)
60 - 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, \( \sum f_i = 68 \) and \( \sum f_iu_i = 7 \).

Mean, \( \overset{-}{x} = a + (\frac{\sum f_iu_i}{\sum f_i})\times h \)

\( \overset{-}{x} \) = \( 135 + (\frac{7 }{68})\times 20 \)

\( \overset{-}{x} \) = 135 + \( \frac{140}{68} \)
Mean, \( \overset{-}{x} \) = 137.058

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The data can be represented as follows:

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

The maximum class frequency is 20, corresponding to the class interval 125 - 145.

Modal class: 125 - 145
Lower limit (l) of the modal class: 125
Frequency (f1) of the modal class: 20
Frequency (f0) of the preceding class: 13
Frequency (f2) of the succeeding class: 14
Class size (h): 20

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

Mode = \(125 + (\frac{20 - 13 }{ 2(20) - 13 - 14}) \times 20\)

Mode = \(125+ [\frac{7}{13}] \times 20\)

Mode = \(125 +( \frac{ 140}{ 13})\)
Mode = 135.76

The median, mode, and mean of the given data are 137, 135.76, and 137.058, respectively. These values are approximately equal.