Calculate Mode using Grouping Method from the following grouped data

Class	Frequency
10 - 20	4
20 - 30	6
30 - 40	5
40 - 50	10
50 - 60	20
60 - 70	22
70 - 80	24
80 - 90	6
90 - 100	3

Solution:
Mode using Grouping Method :
1) In column-I, write the original frequency
2) In column-II, combine the frequency two by two, starting from top
3) In column-III, combine the frequency two by two, starting from second
4) In column-IV, combine the frequency three by three, starting from top
5) In column-V, combine the frequency three by three, starting from second
6) In column-VI, combine the frequency three by three, starting from third

Grouping Table :

Class	I Frequency	II (1+2)	III (2+3)	IV (1+2+3)	V (2+3+4)	VI (3+4+5)
10-20	4
		4+6=10
20-30	6			4+6+5=15
			6+5=11
30-40	5				6+5+10=21
		5+10=15
40-50	10					5+10+20=35
			10+20=30
50-60	20			10+20+22=52
		20+22=42
60-70	22				20+22+24=66
			22+24=46
70-80	24					22+24+6=52
		24+6=30
80-90	6			24+6+3=33
			6+3=9
90-100	3

1) 24 is the maximum value in the column-I and it is an individual frequency of Class 70-80. Therefore, we have tick(✓) this Class.
2) 42 is the maximum value in the column-II and it is the sum of 20 and 22; i.e., of Class 50-60 and 60-70. Therefore, we have tick(✓) this both Class.
3) 46 is the maximum value in the column-III and it is the sum of 22 and 24; i.e., of Class 60-70 and 70-80. Therefore, we have tick(✓) this both Class.
4) 52 is the maximum value in the column-IV and it is the sum of 10, 20, and 22; i.e, of Class 40-50, 50-60 and 60-70. Therefore, we have tick(✓) this three Class.
5) 66 is the maximum value in the column-V and it is the sum of 20, 22, and 24; i.e, of Class 50-60, 60-70 and 70-80. Therefore, we have tick(✓) this three Class.
6) 52 is the maximum value in the column-VI and it is the sum of 22, 24, and 6; i.e, of Class 60-70, 70-80 and 80-90. Therefore, we have tick(✓) this three Class.

Analysis Table :

Column	I	II	III	IV	V	VI	Total
Max Frequency	24	42	46	52	66	52
10-20							-
20-30							-
30-40							-
40-50				✓			1
50-60		✓		✓	✓		3
60-70		✓	✓	✓	✓	✓	5
70-80	✓		✓		✓	✓	4
80-90						✓	1
90-100							-

Since 60-70 has the maximum ticks 5. So mode class is 60-70.
`:.` The mode class is `60 - 70`.

`:. L =`lower boundary point of mode class `=60`

`:. f_1 =` frequency of the mode class `=22`

`:. f_0 =` frequency of the preceding class `=20`

`:. f_2 =` frequency of the succedding class `=24`

`:. c =` class length of mode class `=10`

`Z=L+(|f_1-f_0|)/(|f_1-f_0|+|f_1-f_2|) * c`

`=60+(|22-20|)/(|22-20| + |22-24|) * 10`

`=60+2/4 * 10`

`=60+5`

`=65`

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