NCERT solution class 11 chapter 15 Statistics exercise 15.2 mathematics

EXERCISE 15.2

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Page No 371:

Question 1:

Find the mean and variance for the data 6, 7, 10, 12, 13, 4, 8, 12

Answer:

6, 7, 10, 12, 13, 4, 8, 12
Mean, 
The following table is obtained.

xi
6	–3	9
7	–2	4
10	–1	1
12	3	9
13	4	16
4	–5	25
8	–1	1
12	3	9
		74

Question 2:

Find the mean and variance for the first n natural numbers

Answer:

The mean of first n natural numbers is calculated as follows.

Question 3:

Find the mean and variance for the first 10 multiples of 3

Answer:

The first 10 multiples of 3 are
3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Here, number of observations, n = 10

The following table is obtained.

xi
3	–13.5	182.25
6	–10.5	110.25
9	–7.5	56.25
12	–4.5	20.25
15	–1.5	2.25
18	1.5	2.25
21	4.5	20.25
24	7.5	56.25
27	10.5	110.25
30	13.5	182.25
		742.5

Question 4:

Find the mean and variance for the data

xi	6	10	14	18	24	28	30
f i	2	4	7	12	8	4	3

Answer:

The data is obtained in tabular form as follows.

xi	f i	fixi
6	2	12	–13	169	338
10	4	40	–9	81	324
14	7	98	–5	25	175
18	12	216	–1	1	12
24	8	192	5	25	200
28	4	112	9	81	324
30	3	90	11	121	363
	40	760			1736

Here, N = 40, 

Question 5:

Find the mean and variance for the data

xi	92	93	97	98	102	104	109
f i	3	2	3	2	6	3	3

Answer:

The data is obtained in tabular form as follows.

xi	f i	fixi
92	3	276	–8	64	192
93	2	186	–7	49	98
97	3	291	–3	9	27
98	2	196	–2	4	8
102	6	612	2	4	24
104	3	312	4	16	48
109	3	327	9	81	243
	22	2200			640

Here, N = 22, 

Question 6:

Find the mean and standard deviation using short-cut method.

xi	60	61	62	63	64	65	66	67	68
fi	2	1	12	29	25	12	10	4	5

Answer:

The data is obtained in tabular form as follows.

xi	fi		yi2	fiyi	fiyi2
60	2	–4	16	–8	32
61	1	–3	9	–3	9
62	12	–2	4	–24	48
63	29	–1	1	–29	29
64	25	0	0	0	0
65	12	1	1	12	12
66	10	2	4	20	40
67	4	3	9	12	36
68	5	4	16	20	80
	100	220		0	286

Mean, 

Question 7:

Find the mean and variance for the following frequency distribution.

Classes	0-30	30-60	60-90	90-120	120-150	150-180	180-210
Frequencies	2	3	5	10	3	5	2

Answer:

Class	Frequency fi	Mid-point xi		yi2	fiyi	fiyi2
0-30	2	15	–3	9	–6	18
30-60	3	45	–2	4	–6	12
60-90	5	75	–1	1	–5	5
90-120	10	105	0	0	0	0
120-150	3	135	1	1	3	3
150-180	5	165	2	4	10	20
180-210	2	195	3	9	6	18
	30				2	76

Mean, 

Page No 372:

Question 8:

Find the mean and variance for the following frequency distribution.

Classes	0-10	10-20	20-30	30-40	40-50
Frequencies	5	8	15	16	6

Answer:

Class	Frequency fi	Mid-point xi		yi2	fiyi	fiyi2
0-10	5	5	–2	4	–10	20
10-20	8	15	–1	1	–8	8
20-30	15	25	0	0	0	0
30-40	16	35	1	1	16	16
40-50	6	45	2	4	12	24
	50				10	68

Mean, 

Question 9:

Find the mean, variance and standard deviation using short-cut method

Heightin cms	No. of children
70-75	3
75-80	4
80-85	7
85-90	7
90-95	15
95-100	9
100-105	6
105-110	6
110-115	3

Answer:

Class Interval	Frequency fi	Mid-point xi		yi2	fiyi	fiyi2
70-75	3	72.5	–4	16	–12	48
75-80	4	77.5	–3	9	–12	36
80-85	7	82.5	–2	4	–14	28
85-90	7	87.5	–1	1	–7	7
90-95	15	92.5	0	0	0	0
95-100	9	97.5	1	1	9	9
100-105	6	102.5	2	4	12	24
105-110	6	107.5	3	9	18	54
110-115	3	112.5	4	16	12	48
	60				6	254

Mean, 

Question 10:

The diameters of circles (in mm) drawn in a design are given below:

Diameters	No. of children
33-36	15
37-40	17
41-44	21
45-48	22
49-52	25

Answer:

Class Interval	Frequency fi	Mid-point xi		fi2	fiyi	fiyi2
32.5-36.5	15	34.5	–2	4	–30	60
36.5-40.5	17	38.5	–1	1	–17	17
40.5-44.5	21	42.5	0	0	0	0
44.5-48.5	22	46.5	1	1	22	22
48.5-52.5	25	50.5	2	4	50	100
	100				25	199

Here, N = 100, h = 4
Let the assumed mean, A, be 42.5.
Mean, 

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