1. Calculate Mode using Grouping Method from the following grouped data

X	Frequency
10	8
15	12
20	36
25	35
30	28
35	18
40	9

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 :

x	I Frequency	II (1+2)	III (2+3)	IV (1+2+3)	V (2+3+4)	VI (3+4+5)
10	8
		8+12=20
15	12			8+12+36=56
			12+36=48
20	36				12+36+35=83
		36+35=71
25	35					36+35+28=99
			35+28=63
30	28			35+28+18=81
		28+18=46
35	18				28+18+9=55
			18+9=27
40	9

1) 36 is the maximum value in the column-I and it is an individual frequency of x 20. Therefore, we have tick(✓) this x.
2) 71 is the maximum value in the column-II and it is the sum of 36 and 35; i.e., of x 20 and 25. Therefore, we have tick(✓) this both x.
3) 63 is the maximum value in the column-III and it is the sum of 35 and 28; i.e., of x 25 and 30. Therefore, we have tick(✓) this both x.
4) 81 is the maximum value in the column-IV and it is the sum of 35, 28, and 18; i.e, of x 25, 30 and 35. Therefore, we have tick(✓) this three x.
5) 83 is the maximum value in the column-V and it is the sum of 12, 36, and 35; i.e, of x 15, 20 and 25. Therefore, we have tick(✓) this three x.
6) 99 is the maximum value in the column-VI and it is the sum of 36, 35, and 28; i.e, of x 20, 25 and 30. Therefore, we have tick(✓) this three x.

Analysis Table :

Column	I	II	III	IV	V	VI	Total
Max Frequency	36	71	63	81	83	99
10							-
15					✓		1
20	✓	✓			✓	✓	4
25		✓	✓	✓	✓	✓	5
30			✓	✓		✓	3
35				✓			1
40							-

Since 25 has the maximum ticks 5. So mode is 25.