Arithmetic Mean (Discrete Series)- Direct and Short-cut Method Solved Examples

By Dr. Swati Gupta January 05, 2022

Arithmetic Mean Discrete Series

Direct Method- To find out the total of items in discrete series, frequency of each value is multiplied with the respective size. The values so obtained are totaled up. This total is then divided by the total number of frequencies to obtain the arithmetic mean.

Question 1(a): Calculate mean from the following data by using direct method.

Value: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Frequency: 21, 30, 28, 40, 26, 34, 40, 9, 15, 57

*Value (x*)	*Frequency (f*)	fx
1	21	21
2	30	60
3	28	84
4	40	160
5	26	130
6	34	204
7	40	280
8	9	72
9	15	135
10	57	570
	N = 300	*∑fx* = 1716

Question 1(b): Calculate mean from the following data by using short cut method.

Value: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Frequency: 21, 30, 28, 40, 26, 34, 40, 9, 15, 57

*Value (x*)	*Frequency (f*)	*A = 6* *d (X - A)*	fd
1	21	-5	-105
2	30	-4	-120
3	28	-3	-84
4	40	-2	-80
5	26	-1	-26
6	34	0	0
7	40	1	40
8	9	2	18
9	15	3	45
10	57	4	228
	N = 300		*∑fd = -84*

Question 2(a): Calculate mean from the following data by using direct method.

Marks: 52 58 60 65 68 70 75

No. of Students: 5 3 2 4 1 1 2

*No. of Students [f]*	*Marks [x]*	fx
5	52	260
3	58	174
2	60	120
4	65	260
1	68	68
1	70	70
2	75	150
*N=18*		*∑fx = 1102*

Question 2(b): Calculate mean from the following data by using short cut method.

Marks: 52 58 60 65 68 70 75

No. of Students: 5 3 2 4 1 1 2

*Marks [x]*	*No. of Students [f]*	*A=60* *d=[X-A]*	fd
52	5	-8	-40
58	3	-2	-6
60	2	0	0
65	4	5	20
68	1	8	8
70	1	10	10
75	2	15	30
	*N=18*		*∑fd =22*

Question 3(a): Calculate mean from the following data by using direct method.

Student’s Age [x]: 20, 23, 25, 28, 30

No. of Students [f]: 3, 5, 10, 6, 1

*Student’s Age [x]*	*No. of Students [f]*	fx
20	3	60
23	5	115
25	10	250
28	6	168
30	1	30
	*N=25*	*∑fx = 623*

Question 3(b): Calculate mean from the following data by using short cut method.

Student’s Age [x]: 20 23 25 28 30

No. of Students [f]: 3 5 10 6 1

*Student’s Age [x]*	*No. of Students [f]*	*A= 25* *D=[x-A]*	fd
20	3	-5	-15
23	5	-2	-10
25	10	0	0
28	6	3	18
30	1	5	5
	*N=25*		*∑fd = -2*

Question 4(a): Calculate mean from the following data by using direct method.

The coins are tossed 1024 times. The theoretical frequencies of 10 heads, 9 heads, …0 heads are given below:

No. of heads: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Frequency: 1 10 45 120 210 252 210 120 45 10 1

*No. of heads [x]*	*Frequency [f]*	fx
0	1	0
1	10	10
2	45	90
3	120	360
4	210	840
5	252	1260
6	210	1260
7	120	840
8	45	360
9	10	90
10	1	10
	*N=1024*	*∑fx* =5120

Question 4(b): Calculate mean from the following data by using direct method.

The coins are tossed 1024 times. The theoretical frequencies of 10 heads, 9 heads, …0 heads are given below:

No. of heads: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Frequency: 1 10 45 120 210 252 210 120 45 10 1

*No. of heads [x]*	*Frequency [f]*	*A=5* *d=[x-A]*	fd
0	1	-5	-5
1	10	-4	-40
2	45	-3	-135
3	120	-2	-240
4	210	-1	-210
5	252	0	0
6	210	1	210
7	120	2	240
8	45	3	135
9	10	4	40
10	1	5	5
	N=1024		*∑fd* = 0

Question 5(a): Calculate mean from the following data by using direct method.

Values: 0 1 2 3 4 5 6 7 8 9 10

Frequency: 4 10 41 131 105 158 211 118 52 7 2

*Values [x]*	*Frequency [f]*	fx
0	4	0
1	10	10
2	41	82
3	131	393
4	105	420
5	158	790
6	211	1266
7	118	826
8	52	416
9	7	63
10	2	20
	*N=839*	*∑fx* =4286

Question 5(b): Calculate mean from the following data by using direct method.

Values: 0 1 2 3 4 5 6 7 8 9 10

Frequency: 4 10 41 131 105 158 211 118 52 7 2

*Values [x]*	*Frequency [f]*	*A-5* *d=[x-A]*	fd
0	4	-5	-20
1	10	-4	-40
2	41	-3	-123
3	131	-2	-262
4	105	-1	-105
5	158	0	0
6	211	1	211
7	118	2	236
8	52	3	156
9	7	4	28
10	2	5	10
	*N=839*		*∑fd*=91