Question

The frequency distribution table given below shows the monthly consumption of electricity of 68 consumers of a locality.

Monthly Consumption	Number of Consumers
65 – 85	4
85 – 105	5
105 – 125	13
125 – 145	20
145 – 165	14
165 – 185	8
185 – 205	4
Total	68

                        Calculate the following:

1. Mean
2. Mode
3. Median
4. Quartile 1
5. Quartile 3
6. Inter Quartile

step by step please..tq

Answer 1

Solution:

Class (1)	Frequency (f) (2)	Mid value (x) (3)	d=x-Ah=x-13520 A=135,h=20 (4)	f⋅d (5)=(2)×(4)	cf (7)
65 - 85	4	75	-3	-12	4
85 - 105	5	95	-2	-10	9
105 - 125	13	115	-1	-13	22
125 - 145	20	135=A	0	0	42
145 - 165	14	155	1	14	56
165 - 185	8	175	2	16	64
185 - 205	4	195	3	12	68
---	---	---	---	---	---
	n=68	-----	-----	∑f⋅d=7	-----

a ) Mean ˉx=A+∑fdn⋅h

=135+768⋅20

=135+0.1029⋅20

=135+2.0588

=137.0588

Mean = 137

b ) Mode
Here, maximum frequency is 20.

∴ The mode class is 125-145.

∴L=lower boundary point of mode class =125

∴f1= frequency of the mode class =20

∴f0= frequency of the preceding class =13

∴f2= frequency of the succedding class =14

∴c= class length of mode class =20

Z=L+(f1-f02⋅f1-f0-f2)⋅c

=125+(20-132⋅20-13-14)⋅20

=125+(713)⋅20

=125+10.7692

=135.7692
Mode = 135.8

c ) Median
= value of (n2)th observation

= value of (682)th observation

= value of 34th observation

From the column of cumulative frequency cf, we find that the 34th observation lies in the class 125-145.

∴ The median class is 125-145.

Now,
∴L=lower boundary point of median class =125

∴n=Total frequency =68

∴cf=Cumulative frequency of the class preceding the median class =22

∴f=Frequency of the median class =20

∴c=class length of median class =20

Median M=L+n2-cff⋅c

=125+34-2220⋅20

=125+1220⋅20

=125+12

=137

Median = 137


d ) Quartile 1

Class with (n4)th value of the observation in cf column

=(684)th value of the observation in cf column

=(17)th value of the observation in cf column

and it lies in the class 105-125.

∴Q1 class : 105-125

The lower boundary point of 105-125 is 105.

∴L=105

Q1=L+n4-cff⋅c

=105+17-913⋅20

=105+813⋅20

=105+12.3077

=117.3077

Quartile 1 = 117.3

d ) Quartile 3

Q3=L+3n4-cff⋅c

=145+51-4214⋅20

=145+914⋅20

=145+12.8571

=157.8571

Quartile 3 = 157.8

f ) Inter Quartile =Quartile 3 -Quartile 1

= 157.8 - 117.3

= 40.5

Inter Quartile = 40.5