Calculate Mode using Grouping Method from the following grouped data

X	Frequency
10	8
11	15
12	20
13	100
14	98
15	95
16	90
17	75
18	50
19	30

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+15=23
11	15			8+15+20=43
			15+20=35
12	20				15+20+100=135
		20+100=120
13	100					20+100+98=218
			100+98=198
14	98			100+98+95=293
		98+95=193
15	95				98+95+90=283
			95+90=185
16	90					95+90+75=260
		90+75=165
17	75			90+75+50=215
			75+50=125
18	50				75+50+30=155
		50+30=80
19	30

1) 100 is the maximum value in the column-I and it is an individual frequency of x 13. Therefore, we have tick(✓) this x.
2) 193 is the maximum value in the column-II and it is the sum of 98 and 95; i.e., of x 14 and 15. Therefore, we have tick(✓) this both x.
3) 198 is the maximum value in the column-III and it is the sum of 100 and 98; i.e., of x 13 and 14. Therefore, we have tick(✓) this both x.
4) 293 is the maximum value in the column-IV and it is the sum of 100, 98, and 95; i.e, of x 13, 14 and 15. Therefore, we have tick(✓) this three x.
5) 283 is the maximum value in the column-V and it is the sum of 98, 95, and 90; i.e, of x 14, 15 and 16. Therefore, we have tick(✓) this three x.
6) 260 is the maximum value in the column-VI and it is the sum of 95, 90, and 75; i.e, of x 15, 16 and 17. Therefore, we have tick(✓) this three x.

Analysis Table :

Column	I	II	III	IV	V	VI	Total
Max Frequency	100	193	198	293	283	260
10							-
11							-
12							-
13	✓		✓	✓			3
14		✓	✓	✓	✓		4
15		✓		✓	✓	✓	4
16					✓	✓	2
17						✓	1
18							-
19							-

Since 4 is the maximum ticks repeated 2 times, So grouping method fails to give the modal class.

We use formula Mode = 3 Median - 2 Mean

Find Mean, Median

`x` `(1)`	Frequency `(f)` `(2)`	`f*x` `(3)=(2)xx(1)`	`cf` `(5)`
10	8	80	8
11	15	165	23
12	20	240	43
13	100	1300	143
14	98	1372	241
15	95	1425	336
16	90	1440	426
17	75	1275	501
18	50	900	551
19	30	570	581
---	---	---	---
	`n=581`	`sum f*x=8767`	--

Mean `bar x = (sum fx)/n`

`=8767/581`

`=15.0895`

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Median :
M = value of `((n+1)/2)^(th)` observation

= value of `(582/2)^(th)` observation

= value of `291^(st)` observation

From the column of cumulative frequency `cf`, we find that the `291^(st)` observation is `15`.

Hence, the median of the data is `15`.

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We have given Mean (`bar X`) `=15.0895`, Median(`M`) `=15`, Mode(`Z`) `=?`

`Z=3 M - 2 bar X`

`Z=3*15-2*15.0895`

`Z=45-30.179`

`Z=14.821`

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