Calculate Mode using Grouping Method from the following grouped data

Class	Frequency
0 - 10	4
10 - 20	16
20 - 30	15
30 - 40	20
40 - 50	7
50 - 60	5

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)
0-10	4
		4+16=20
10-20	16			4+16+15=35
			16+15=31
20-30	15				16+15+20=51
		15+20=35
30-40	20					15+20+7=42
			20+7=27
40-50	7			20+7+5=32
		7+5=12
50-60	5

1) 20 is the maximum value in the column-I and it is an individual frequency of Class 30-40. Therefore, we have tick(✓) this Class.
2) 35 is the maximum value in the column-II and it is the sum of 15 and 20; i.e., of Class 20-30 and 30-40. Therefore, we have tick(✓) this both Class.
3) 31 is the maximum value in the column-III and it is the sum of 16 and 15; i.e., of Class 10-20 and 20-30. Therefore, we have tick(✓) this both Class.
4) 35 is the maximum value in the column-IV and it is the sum of 4, 16, and 15; i.e, of Class 0-10, 10-20 and 20-30. Therefore, we have tick(✓) this three Class.
5) 51 is the maximum value in the column-V and it is the sum of 16, 15, and 20; i.e, of Class 10-20, 20-30 and 30-40. Therefore, we have tick(✓) this three Class.
6) 42 is the maximum value in the column-VI and it is the sum of 15, 20, and 7; i.e, of Class 20-30, 30-40 and 40-50. Therefore, we have tick(✓) this three Class.

Analysis Table :

Column	I	II	III	IV	V	VI	Total
Max Frequency	20	35	31	35	51	42
0-10				✓			1
10-20			✓	✓	✓		3
20-30		✓	✓	✓	✓	✓	5
30-40	✓	✓			✓	✓	4
40-50						✓	1
50-60							-

Since 20-30 has the maximum ticks 5. So mode class is 20-30.
`:.` The mode class is `20 - 30`.

`:. L = `lower boundary point of mode class `=20`

`:. f_1 = ` frequency of the mode class `=15`

`:. f_0 = ` frequency of the preceding class `=16`

`:. f_2 = ` frequency of the succedding class `=20`

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

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

Here `f_1-f_0=-1 < 0` and `f_1-f_2=-5 < 0`

So we can not apply the above formula

Since, `f_2=20 > f_0=16`

Thus, Mode `Z=L+f_2/(f_2+f_0) * c`

`=20+20/(20+16) * 10`

`=20+20/36 * 10`

`=20+5.5556`

`=25.5556`

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