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NCERT Solutions for Class 11 Maths Chapter 15 Statistics

NCERT Solutions for Class 11 Maths Chapter 15 Statistics

Edited By Ramraj Saini | Updated on Sep 25, 2023 10:18 PM IST

Statistics Class 11 Questions And Answers

NCERT Solutions for Class 11 Maths Chapter 15 Statistics are provided here. You have learnt the basic statistics like mean, mode and median known as "measures of the central tendency" in the previous Class NCERT syllabus. In this NCERT book chapter, you will learn important topics like measures of dispersion, mean deviation, standard deviation, quartile deviation, range and many more. In class 11 maths chapter 15 question answer, you will get questions related to all the above topics. These NCERT solutions are prepared by expert team at Careers360 keeping in my the latest syllabus of CBSE 2023-24, easy and simple language, step by step comprehensive coverage of the concepts.

This statistics class 11 solutions is important for the CBSE class 11 final examination as well as in the various competitive exams like JEE Main, BITSAT, etc . As statistics deal with the collected data for specific purposes and in the digital world, everything that we do comes in the form of data. This is a very useful technique to interpret, analyze the data, and make a conclusive decision based on the data. Class 11 maths chapter 15 NCERT solutions will build your fundamentals of statistics which will be useful to learn advanced statistical techniques. There are 27 questions in 3 exercises in the NCERT textbook. Interested students can find all NCERT solutions for class 11 at one place which will help to understand the concepts in an easy way.

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Statistics Class 11 Questions And Answers PDF Free Download

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Statistics Class 11 Solutions - Important Formulae

Measure of Dispersion:

Dispersion measures the degree of variation in the values of a variable. It quantifies how scattered observations are around the central value in a distribution.

Range:

Range is the simplest measure of dispersion. It is defined as the difference between the largest and smallest observations in a distribution.

Range of distribution = Largest observation – Smallest observation.

Mean Deviation:

Mean Deviation for Ungrouped Data:

  • For n observations x1, x2, x3, ..., xn, the mean deviation about their mean x¯ is calculated as:

    • MD(\bar x) = (Σ |xᵢ - x¯|) / n

  • The mean deviation about its median M is calculated as:

    • MD(M) = (Σ |xᵢ - M|) / n

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Mean Deviation for Discrete Frequency Distribution:

  • For a discrete frequency distribution with values xᵢ, frequencies fᵢ, mean x¯, and total frequency N, the mean deviation is calculated as:

    • MD(\bar x) = (Σ fᵢ |xᵢ - x¯|) / (Σ fᵢ) = (Σ fᵢ |xᵢ - x¯|) / N

Variance:

  • Variance is the average of the squared deviations from the mean x¯. If x, x, …, xₙ are n observations with mean x¯, the variance denoted by σ² is calculated as:

    • σ² = (Σ(xᵢ - x¯)²) / n

Standard Deviation:

  • Standard deviation, denoted as σ, is the square root of the variance σ². If σ² is the variance, then the standard deviation is given by:

    • σ = √(σ²)

  • For a discrete frequency distribution with values xᵢ, frequencies fᵢ, mean x¯, and total frequency N, the standard deviation is calculated as:

    • σ = √((Σ fᵢ(xᵢ - x¯)²) / N)

Coefficient of Variation:

  • The coefficient of variation (CV) is used to compare two or more frequency distributions. It is defined as:

    • Coefficient of Variation = (Standard Deviation / Mean) × 100

    • CV = (σ / x¯) × 100

Free download NCERT Solutions for Class 11 Maths Chapter 15 Statistics for CBSE Exam.

Statistics Class 11 NCERT Solutions (Intext Questions and Exercise)

Class 11 maths chapter 15 question answer - Exercise: 15.1

Question:1 . Find the mean deviation about the mean for the data.

\small 4,7,8,9,10,12,13,17

Answer:

Mean ( \overline{x} ) of the given data:

\overline{x} = \frac{1}{8}\sum_{i=1}^{8}x_i = \frac{4+ 7+ 8+ 9+ 10+ 12+ 13+ 17}{8} = 10

The respective absolute values of the deviations from mean, |x_i - \overline{x}| are

6, 3, 2, 1, 0, 2, 3, 7

\therefore \sum_{i=1}^{8}|x_i - 10| = 24

\therefore M.D.(\overline{x}) = \frac{1}{n}\sum_{i=1}^{n}|x_i - \overline{x}|

= \frac{24}{8} = 3

Hence, the mean deviation about the mean is 3.

Question:2. Find the mean deviation about the mean for the data.

\small 38,70,48,40,42,55,63,46,54,44

Answer:

Mean ( \overline{x} ) of the given data:

\\ \overline{x} = \frac{1}{8}\sum_{i=1}^{8}x_i = \frac{38+70+48+40+42+55+63+46+54+44}{10} \\ = \frac{500}{10} = 50

The respective absolute values of the deviations from mean, |x_i - \overline{x}| are

12, 20, 2, 10, 8, 5, 13, 4, 4, 6

\therefore \sum_{i=1}^{8}|x_i - 50| = 84

\therefore M.D.(\overline{x}) = \frac{1}{n}\sum_{i=1}^{n}|x_i - \overline{x}|

= \frac{84}{10} = 8.4

Hence, the mean deviation about the mean is 8.4.

Question:3. Find the mean deviation about the median.

\small 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17

Answer:

Number of observations, n = 12, which is even.

Arranging the values in ascending order:

10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18.

Now, Median (M)

\\ = \frac{(\frac{12}{2})^{th} observation + (\frac{12}{2}+ 1)^{th} observation}{2} \\ = \frac{13 + 14}{2} = \frac{27}{2}= 13.5

The respective absolute values of the deviations from median, |x_i - M| are

3.5, 2.5, 2.5, 1.5, 0.5, 0.5, 0.5, 2.5, 2.5, 3.5, 3.5, 4.5

\therefore \sum_{i=1}^{8}|x_i - 13.5| = 28

\therefore M.D.(M) = \frac{1}{12}\sum_{i=1}^{n}|x_i - M|

= \frac{28}{12} = 2.33

Hence, the mean deviation about the median is 2.33.

Question:4. Find the mean deviation about the median.

\small 36, 72, 46, 42, 60, 45, 53, 46, 51, 49

Answer:

Number of observations, n = 10, which is even.

Arranging the values in ascending order:

36, 42, 45, 46, 46, 49, 51, 53, 60, 72

Now, Median (M)

\\ = \frac{(\frac{10}{2})^{th} observation + (\frac{10}{2}+ 1)^{th} observation}{2} \\ = \frac{46 + 49}{2} = \frac{95}{2}= 47.5

The respective absolute values of the deviations from median, |x_i - M| are

11.5, 5.5, 2.5, 1.5, 1.5, 1.5, 3.5, 5.5, 12.5, 24.5

\therefore \sum_{i=1}^{8}|x_i - 47.5| = 70

\therefore M.D.(M) = \frac{1}{10}\sum_{i=1}^{n}|x_i - M|

= \frac{70}{10} = 7

Hence, the mean deviation about the median is 7.

Question:5 Find the mean deviation about the mean.

\small \\x_i\hspace{1cm}5\hspace{1cm}10\hspace{1cm}15\hspace{1cm}20\hspace{1cm}25\\\small f_i\hspace{1cm}7\hspace{1cm}4\hspace{1.15cm}6\hspace{1.22cm}3\hspace{1.3cm}5

Answer:

x_i
f_i
f_ix_i
|x_i - \overline{x}|
f_i|x_i - \overline{x}|
5
7
35
9
63
10
4
40
4
16
15
6
90
1
6
20
3
60
6
18
25
5
125
11
55

\sum{f_i}
= 25
\sum f_ix_i
= 350

\sum f_i|x_i - \overline{x}|
=158

N = \sum_{i=1}^{5}{f_i} = 25 ; \sum_{i=1}^{5}{f_ix_i} = 350

\overline{x} = \frac{1}{N}\sum_{i=1}^{n}f_ix_i = \frac{350}{12} = 14

Now, we calculate the absolute values of the deviations from mean, |x_i - \overline{x}| and

\sum f_i|x_i - \overline{x}| = 158

\therefore M.D.(\overline{x}) = \frac{1}{25}\sum_{i=1}^{n}f_i|x_i - \overline{x}|

= \frac{158}{25} = 6.32

Hence, the mean deviation about the mean is 6.32

Question:6. Find the mean deviation about the mean.

\small \\x_i\hspace{1cm}10\hspace{1cm}30\hspace{1cm}50\hspace{1cm}70\hspace{1cm}90\\\small f_i\hspace{1cm}4\hspace{1.1cm}24\hspace{1.1cm}28\hspace{1.1cm}16\hspace{1.2cm}8

Answer:

x_i
f_i
f_ix_i
|x_i - \overline{x}|
f_i|x_i - \overline{x}|
10
4
40
40
160
30
24
720
20
480
50
28
1400
0
0
70
16
1120
20
320
90
8
720
40
320

\sum{f_i}
= 80
\sum f_ix_i
= 4000

\sum f_i|x_i - \overline{x}|
=1280

N = \sum_{i=1}^{5}{f_i} = 80 ; \sum_{i=1}^{5}{f_ix_i} = 4000

\overline{x} = \frac{1}{N}\sum_{i=1}^{n}f_ix_i = \frac{4000}{80} = 50

Now, we calculate the absolute values of the deviations from mean, |x_i - \overline{x}| and

\sum f_i|x_i - \overline{x}| = 1280

\therefore M.D.(\overline{x}) = \frac{1}{80}\sum_{i=1}^{5}f_i|x_i - \overline{x}|

= \frac{1280}{80} = 16

Hence, the mean deviation about the mean is 16

Question:7. Find the mean deviation about the median.

\small x_i \small 5 \small 7 \small 9 \small 10 \small 12 \small 15

\small f_i \small 8 \small 6 \small 2 \small 2 \small 2 \small 6

Answer:

x_i
f_i
c.f.
|x_i - M|
f_i|x_i - M|
5
8
8
2
16
7
6
14
0
0
9
2
16
2
4
10
2
18
3
6
12
2
20
5
10
15
6
26
8
48

Now, N = 26 which is even.

Median is the mean of 13^{th} and 14^{th} observations.

Both these observations lie in the cumulative frequency 14, for which the corresponding observation is 7.

Therefore, Median, M = \frac{13^{th} observation + 14^{th} observation}{2} = \frac{7 + 7}{2} = \frac{14 }{2} = 7

Now, we calculate the absolute values of the deviations from median, |x_i - M| and

\sum f_i|x_i - M| = 84

\therefore M.D.(M) = \frac{1}{26}\sum_{i=1}^{6}|x_i - M|

= \frac{84}{26} = 3.23

Hence, the mean deviation about the median is 3.23

Question:8 Find the mean deviation about the median.

\small x_i \small 15 \small 21 \small 27 \small 30 \small 35

\small f_i \small 3 \small 5 \small 6 \small 7 \small 8

Answer:

x_i
f_i
c.f.
|x_i - M|
f_i|x_i - M|
15
3
3
13.5
40.5
21
5
8
7.5
37.5
27
6
14
1.5
9
30
7
21
1.5
10.5
35
8
29
6.5
52

Now, N = 30, which is even.

Median is the mean of 15^{th} and 16^{th} observations.

Both these observations lie in the cumulative frequency 21, for which the corresponding observation is 30.

Therefore, Median, M = \frac{15^{th} observation + 16^{th} observation}{2} = \frac{30 + 30}{2} = 30

Now, we calculate the absolute values of the deviations from median, |x_i - M| and

\sum f_i|x_i - M| = 149.5

\therefore M.D.(M) = \frac{1}{29}\sum_{i=1}^{5}|x_i - M|

= \frac{149.5}{29} = 5.1

Hence, the mean deviation about the median is 5.1

Question:9. Find the mean deviation about the mean.

Income per day in Rs
\small 0-100
\small 100-200
\small 200-300
\small 300-400
\small 400-500
\small 500-600
\small 600-700
\small 700-800
Number of persons
\small 4
\small 8
\small 9
\small 10
\small 7
\small 5
\small 4
\small 3






Answer:

Income
per day
Number of
Persons f_i
Mid
Points x_i
f_ix_i
|x_i - \overline{x}|
f_i|x_i - \overline{x}|
0 -100
4
50
200
308
1232
100 -200
8
150
1200
208
1664
200-300
9
250
2250
108
972
300-400
10
350
3500
8
80
400-500
7
450
3150
92
644
500-600
5
550
2750
192
960
600-700
4
650
2600
292
1168
700-800
3
750
2250
392
1176

\sum{f_i}
=50

\sum f_ix_i
=17900

\sum f_i|x_i - \overline{x}|
=7896


N = \sum_{i=1}^{8}{f_i} = 50 ; \sum_{i=1}^{8}{f_ix_i} = 17900

\overline{x} = \frac{1}{N}\sum_{i=1}^{8}f_ix_i = \frac{17900}{50} = 358

Now, we calculate the absolute values of the deviations from mean, |x_i - \overline{x}| and

\sum f_i|x_i - \overline{x}| = 7896

\therefore M.D.(\overline{x}) = \frac{1}{50}\sum_{i=1}^{8}f_i|x_i - \overline{x}|

= \frac{7896}{50} = 157.92

Hence, the mean deviation about the mean is 157.92

Question:10. Find the mean deviation about the mean.

Height in cms
\small 95-105
\small 105-115
\small 115-125
\small 125-135
\small 135-145
\small 145-155
Number of persond
\small 9
\small 13
\small 26
\small 30
\small 12
\small 10

Answer:

Height
in cms
Number of
Persons f_i
Mid
Points x_i
f_ix_i
|x_i - \overline{x}|
f_i|x_i - \overline{x}|
95 -105
9
100
900
25.3
227.7
105 -115
13
110
1430
15.3
198.9
115-125
26
120
3120
5.3
137.8
125-135
30
130
3900
4.7
141
135-145
12
140
1680
14.7
176.4
145-155
10
150
1500
24.7
247

\sum{f_i}
=100

\sum f_ix_i
=12530

\sum f_i|x_i - \overline{x}|
=1128.8


N = \sum_{i=1}^{6}{f_i} = 100 ; \sum_{i=1}^{6}{f_ix_i} = 12530

\overline{x} = \frac{1}{N}\sum_{i=1}^{6}f_ix_i = \frac{12530}{100} = 125.3

Now, we calculate the absolute values of the deviations from mean, |x_i - \overline{x}| and

\sum f_i|x_i - \overline{x}| = 1128.8

\therefore M.D.(\overline{x}) = \frac{1}{100}\sum_{i=1}^{6}f_i|x_i - \overline{x}|

= \frac{1128.8}{100} = 11.29

Hence, the mean deviation about the mean is 11.29

Question:11. Find the mean deviation about median for the following data :

Marks
\small 0-10
\small 10-20
\small 20-30
\small 30-40
\small 40-50
\small 50-60
Number of girls
\small 6
\small 8
\small 14
\small 16
\small 4
\small 2

Answer:

Marks
Number of
Girls f_i
Cumulative
Frequency c.f.
Mid
Points x_i
|x_i - M|
f_i|x_i - M|
0-10
6
6
5
22.85
137.1
10-20
8
14
15
12.85
102.8
20-30
14
28
25
2.85
39.9
30-40
16
44
35
7.15
114.4
40-50
4
48
45
17.15
68.6
50-60
2
50
55
27.15
54.3





\sum f_i|x_i - M|
=517.1

Now, N = 50, which is even.

The class interval containing \left (\frac{N}{2} \right)^{th} or 25^{th} item is 20-30. Therefore, 20-30 is the median class.

We know,

Median = l + \frac{\frac{N}{2}- C}{f}\times h

Here, l = 20, C = 14, f = 14, h = 10 and N = 50

Therefore, Median = 20 + \frac{25 - 14}{14}\times 10 = 20 + 7.85 = 27.85

Now, we calculate the absolute values of the deviations from median, |x_i - M| and

\sum f_i|x_i - M| = 517.1

\therefore M.D.(M) = \frac{1}{50}\sum_{i=1}^{6}f_i|x_i - M|

= \frac{517.1}{50} = 10.34

Hence, the mean deviation about the median is 10.34

Question:12 Calculate the mean deviation about median age for the age distribution of \small 100 persons given below:

Age (in years)
\small 16-20
\small 21-25
\small 26-30
\small 31-35
\small 36-40
\small 41-45
\small 46-50
\small 51-55
\small 5
\small 6
\small 12
\small 14
\small 26
\small 12
\small 16
\small 9

[ Hint Convert the given data into continuous frequency distribution by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of each class interval]

Answer:

Age
(in years)
Number
f_i
Cumulative
Frequency c.f.
Mid
Points x_i
|x_i - M|
f_i|x_i - M|
15.5-20.5
5
5
18
20
100
20.5-25.5
6
11
23
15
90
25.5-30.5
12
23
28
10
120
30.5-35.5
14
37
33
5
70
35.5-40.5
26
63
38
0
0
40.5-45.5
12
75
43
5
60
45.5-50.5
16
91
48
10
160
50.5-55.5
9
100
53
15
135





\sum f_i|x_i - M|
=735

Now, N = 100, which is even.

The class interval containing \left (\frac{N}{2} \right)^{th} or 50^{th} item is 35.5-40.5. Therefore, 35.5-40.5 is the median class.

We know,

Median = l + \frac{\frac{N}{2}- C}{f}\times h

Here, l = 35.5, C = 37, f = 26, h = 5 and N = 100

Therefore, Median = 35.5 + \frac{50 - 37}{26}\times 5 = 35.5 + 2.5 = 38

Now, we calculate the absolute values of the deviations from median, |x_i - M| and

\sum f_i|x_i - M| = 735

\therefore M.D.(M) = \frac{1}{100}\sum_{i=1}^{8}f_i|x_i - M|

= \frac{735}{100} = 7.35

Hence, the mean deviation about the median is 7.35

Class 11 maths chapter 15 ncert solutions - Exercise: 15.2

Question:1. Find the mean and variance for each of the data.

\small 6, 7, 10, 12, 13, 4, 8, 12

Answer:

Mean ( \overline{x} ) of the given data:

\overline{x} = \frac{1}{8}\sum_{i=1}^{8}x_i = \frac{6+ 7+ 10+ 12+ 13+ 4+ 8+ 12}{8} = \frac{72}{8} = 9

The respective values of the deviations from mean, (x_i - \overline{x}) are

-3, -2, 1 3 4 -5 -1 3

\therefore \sum_{i=1}^{8}(x_i - 10)^2 = 74

\therefore \sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \overline{x})^2

\frac{1}{8}\sum_{i=1}^{8}(x_i - \overline{x})^2= \frac{74}{8} = 9.25

Hence, Mean = 9 and Variance = 9.25

Question:2. Find the mean and variance for each of the data.

First n natural numbers.

Answer:

Mean ( \overline{x} ) of first n natural numbers:

\overline{x} = \frac{1}{n}\sum_{i=1}^{n}x_i = \frac{\frac{n(n+1)}{2}}{n} = \frac{n+1}{2}

We know, Variance \sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \overline{x})^2

\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}\left (x_i - \frac{n+1}{2} \right )^2

We know that (a-b)^2 = a^2 - 2ab + b^2

\\ \therefore n\sigma^2 = \sum_{i=1}^{n}x_i^2 + \sum_{i=1}^{n}(\frac{n+1}{2})^2 - 2\sum_{i=1}^{n}x_i\frac{n+1}{2} \\ = \frac{n(n+1)(2n+1)}{6} + \frac{(n+1)^2}{4}\times n - 2.\frac{(n+1)}{2}.\frac{n(n+1)}{2}

\\ \implies \sigma^2 = \frac{(n+1)(2n+1)}{6} + \frac{(n+1)^2}{4} - \frac{(n+1)^2}{2}

\\ = \frac{(n+1)(2n+1)}{6} - \frac{(n+1)^2}{4} \\ = (n+1)\left [\frac{4n+2 - 3n -3}{12} \right ] \\ = (n+1).\frac{(n-1)}{12} \\ = \frac{n^2-1}{12}

Hence, Mean = \frac{n+1}{2} and Variance = \frac{n^2-1}{12}

Question:3. Find the mean and variance for each of the data

First 10 multiples of 3

Answer:

First 10 multiples of 3 are:

3, 6, 9, 12, 15, 18, 21, 24, 27, 30

Mean ( \overline{x} ) of the above values:

\\ \overline{x} = \frac{1}{10}\sum_{i=1}^{10}x_i = \frac{3+ 6+ 9+ 12+ 15+ 18+ 21+ 24+ 27+ 30}{10} \\ = 3.\frac{\frac{10(10+1)}{2}}{10} = 16.5

The respective values of the deviations from mean, (x_i - \overline{x}) are

-13.5, -10.5, -7.5, -4.5, -1.5, 1.5, 4.5, 7.5, 10.5, 13.5

\therefore \sum_{i=1}^{10}(x_i - 16.5)^2 = 742.5

\therefore \sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \overline{x})^2

\frac{1}{10}\sum_{i=1}^{10}(x_i - \overline{x})^2= \frac{742.5}{10} = 74.25

Hence, Mean = 16.5 and Variance = 74.25

Question:4. Find the mean and variance for each of the data.

\small x_i
6
10
14
18
24
28
30
\small f_i
2
4
7
12
8
4
3


Answer:

x_i
f_i
f_ix_i
(x_i - \overline{x})
(x_i - \overline{x})^2
f_i(x_i - \overline{x})^2
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
13
169
363

\sum{f_i}
= 40
\sum f_ix_i
= 760


\sum f_i(x_i - \overline{x})^2
=1736

N = \sum_{i=1}^{7}{f_i} = 40 ; \sum_{i=1}^{7}{f_ix_i} = 760

\overline{x} = \frac{1}{N}\sum_{i=1}^{n}f_ix_i = \frac{760}{40} = 19

We know, Variance, \sigma^2 = \frac{1}{N}\sum_{i=1}^{n}(x_i - \overline{x})^2

\implies \sigma^2 = \frac{1736}{40} = 43.4

Hence, Mean = 19 and Variance = 43.4

Question:5. Find the mean and variance for each of the data.

\small x_i
92
93
97
98
102
104
109
\small f_i
3
2
3
2
6
3
3


Answer:

x_i
f_i
f_ix_i
(x_i - \overline{x})
(x_i - \overline{x})^2
f_i(x_i - \overline{x})^2
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

\sum{f_i}
= 22
\sum f_ix_i
= 2200


\sum f_i(x_i - \overline{x})^2
=640

N = \sum_{i=1}^{7}{f_i} = 22 ; \sum_{i=1}^{7}{f_ix_i} = 2200

\overline{x} = \frac{1}{N}\sum_{i=1}^{n}f_ix_i = \frac{2200}{22} = 100

We know, Variance, \sigma^2 = \frac{1}{N}\sum_{i=1}^{n}(x_i - \overline{x})^2

\implies \sigma^2 = \frac{640}{22} = 29.09

Hence, Mean = 100 and Variance = 29.09

Question:6 Find the mean and standard deviation using short-cut method.

\small x_i
60
61
62
63
64
65
66
67
68
\small f_i
2
1
12
29
25
12
10
4
5


Answer:

Let the assumed mean, A = 64 and h = 1

x_i
f_i
y_i = \frac{x_i-A}{h}
y_i^2
f_iy_i
f_iy_i^2
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

\sum{f_i}
=100


\sum f_iy_i
= 0
\sum f_iy_i ^2
=286

N = \sum_{i=1}^{9}{f_i} = 100 ; \sum_{i=1}^{9}{f_iy_i} = 0

Mean,

\overline{x} = A + \frac{1}{N}\sum_{i=1}^{n}f_iy_i\times h =64 + \frac{0}{100} = 64

We know, Variance, \sigma^2 = \frac{1}{N^2}\left [N\sum f_iy_i^2 - (\sum f_iy_i)^2 \right ]\times h^2

\\ \implies \sigma^2 = \frac{1}{(100)^2}\left [100(286) - (0)^2 \right ] \\ = \frac{28600}{10000} = 2.86

We know, Standard Deviation = \sigma = \sqrt{Variance}

\therefore \sigma = \sqrt{2.86} = 1.691

Hence, Mean = 64 and Standard Deviation = 1.691

Question:​​​​​​​7. Find the mean and variance for the following frequency distributions.

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


Answer:

Classes
Frequency
f_i
Mid point
x_i
f_ix_i
(x_i - \overline{x})
(x_i - \overline{x})^2
f_i(x_i - \overline{x})^2
0-30
2
15
30
-92
8464
16928
30-60
3
45
135
-62
3844
11532
60-90
5
75
375
-32
1024
5120
90-120
10
105
1050
2
4
40
120-150
3
135
405
28
784
2352
150-180
5
165
825
58
3364
16820
180-210
2
195
390
88
7744
15488

\sum{f_i} = N
= 30

\sum f_ix_i
= 3210


\sum f_i(x_i - \overline{x})^2
=68280


\overline{x} = \frac{1}{N}\sum_{i=1}^{n}f_ix_i = \frac{3210}{30} = 107

We know, Variance, \sigma^2 = \frac{1}{N}\sum_{i=1}^{n}(x_i - \overline{x})^2

\implies \sigma^2 = \frac{68280}{30} = 2276

Hence, Mean = 107 and Variance = 2276

Question:​​​​​​​8. Find the mean and variance for the following frequency distributions.

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


Answer:

Classes
Frequency
f_i
Mid-point
x_i
f_ix_i
(x_i - \overline{x})
(x_i - \overline{x})^2
f_i(x_i - \overline{x})^2
0-10
5
5
25
-22
484
2420
10-20
8
15
120
-12
144
1152
20-30
15
25
375
-2
4
60
30-40
16
35
560
8
64
1024
40-50
6
45
270
18
324
1944

\sum{f_i} = N
= 50

\sum f_ix_i
= 1350


\sum f_i(x_i - \overline{x})^2
=6600


\overline{x} = \frac{1}{N}\sum_{i=1}^{n}f_ix_i = \frac{1350}{50} = 27

We know, Variance, \sigma^2 = \frac{1}{N}\sum_{i=1}^{n}(x_i - \overline{x})^2

\implies \sigma^2 = \frac{6600}{50} = 132

Hence, Mean = 27 and Variance = 132

Question:9. Find the mean, variance and standard deviation using short-cut method.

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






Answer:

Let the assumed mean, A = 92.5 and h = 5

Height
in cms
Frequency
f_i
Midpoint
x_i
\dpi{100} y_i = \frac{x_i-A}{h}
y_i^2
f_iy_i
f_iy_i^2
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

\sum{f_i} =N = 60



\sum f_iy_i
= 6
\sum f_iy_i ^2
=254

Mean,

\overline{y} = A + \frac{1}{N}\sum_{i=1}^{n}f_iy_i\times h =92.5 + \frac{6}{60}\times5 = 93

We know, Variance, \sigma^2 = \frac{1}{N^2}\left [N\sum f_iy_i^2 - (\sum f_iy_i)^2 \right ]\times h^2

\\ \implies \sigma^2 = \frac{1}{(60)^2}\left [60(254) - (6)^2 \right ] \\ = \frac{1}{(60)^2}\left [15240 - 36 \right ] \\ = \frac{15204}{144} = 105.583

We know, Standard Deviation = \sigma = \sqrt{Variance}

\therefore \sigma = \sqrt{105.583} = 10.275

Hence, Mean = 93, Variance = 105.583 and Standard Deviation = 10.275

Question:10. The diameters of circles (in mm) drawn in a design are given below:

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

Calculate the standard deviation and mean diameter of the circles.
[ Hint First make the data continuous by making the classes as 32.5-36.5,36.5-40.4,40.5-44.5,44.5-48.5,48.5-52.5 and then proceed.]

Answer:

Let the assumed mean, A = 92.5 and h = 5

Diameters
No. of
circles f_i
Midpoint
x_i
\dpi{100} y_i = \frac{x_i-A}{h}
y_i^2
f_iy_i
f_iy_i^2
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

\sum{f_i} =N = 100



\sum f_iy_i
= 25
\sum f_iy_i ^2
=199

Mean,

\overline{x} = A + \frac{1}{N}\sum_{i=1}^{n}f_iy_i\times h =42.5 + \frac{25}{100}\times4 = 43.5

We know, Variance, \sigma^2 = \frac{1}{N^2}\left [N\sum f_iy_i^2 - (\sum f_iy_i)^2 \right ]\times h^2

\\ \implies \sigma^2 = \frac{1}{(100)^2}\left [100(199) - (25)^2 \right ]\times4^2 \\ = \frac{1}{625}\left [19900 - 625 \right ] \\ = \frac{19275}{625} = 30.84

We know, Standard Deviation = \sigma = \sqrt{Variance}

\therefore \sigma = \sqrt{30.84} = 5.553

Hence, Mean = 43.5, Variance = 30.84 and Standard Deviation = 5.553

Statistics class 11 NCERT solutions - Exercise: 15.3

Question:1. From the data given below state which group is more variable, A or B?

Marks
10-20
20-30
30-40
40-50
50-60
60-70
70-80
Group A
9
17
32
33
40
10
9
Group B
10
20
30
25
43
15
7


Answer:

The group having a higher coefficient of variation will be more variable.

Let the assumed mean, A = 45 and h = 10

For Group A

Marks
Group A
f_i
Midpoint
x_i
\dpi{100} y_i = \frac{x_i-A}{h}
= \frac{x_i-45}{10}
y_i^2
f_iy_i
f_iy_i^2
10-20
9
15
-3
9
-27
81
20-30
17
25
-2
4
-34
68
30-40
32
35
-1
1
-32
32
40-50
33
45
0
0
0
0
50-60
40
55
1
1
40
40
60-70
10
65
2
4
20
40
70-80
9
75
3
9
27
81

\sum{f_i} =N = 150



\sum f_iy_i
= -6
\sum f_iy_i ^2
=342

Mean,

\overline{x} = A + \frac{1}{N}\sum_{i=1}^{n}f_iy_i\times h =45 + \frac{-6}{150}\times10 = 44.6

We know, Variance, \sigma^2 = \frac{1}{N^2}\left [N\sum f_iy_i^2 - (\sum f_iy_i)^2 \right ]\times h^2

\\ \implies \sigma^2 = \frac{1}{(150)^2}\left [150(342) - (-6)^2 \right ]\times10^2 \\ = \frac{1}{15^2}\left [51264 \right ] \\ =227.84

We know, Standard Deviation = \sigma = \sqrt{Variance}

\therefore \sigma = \sqrt{227.84} = 15.09

Coefficient of variation = \frac{\sigma}{\overline x}\times100

C.V.(A) = \frac{15.09}{44.6}\times100 = 33.83

Similarly,

For Group B

Marks
Group A
f_i
Midpoint
x_i
\dpi{100} y_i = \frac{x_i-A}{h}
= \frac{x_i-45}{10}
y_i^2
f_iy_i
f_iy_i^2
10-20
10
15
-3
9
-30
90
20-30
20
25
-2
4
-40
80
30-40
30
35
-1
1
-30
30
40-50
25
45
0
0
0
0
50-60
43
55
1
1
43
43
60-70
15
65
2
4
30
60
70-80
7
75
3
9
21
72

\sum{f_i} =N = 150



\sum f_iy_i
= -6
\sum f_iy_i ^2
=375

Mean,

\overline{x} = A + \frac{1}{N}\sum_{i=1}^{n}f_iy_i\times h =45 + \frac{-6}{150}\times10 = 44.6

We know, Variance, \sigma^2 = \frac{1}{N^2}\left [N\sum f_iy_i^2 - (\sum f_iy_i)^2 \right ]\times h^2

\\ \implies \sigma^2 = \frac{1}{(150)^2}\left [150(375) - (-6)^2 \right ]\times10^2 \\ = \frac{1}{15^2}\left [56214 \right ] \\ =249.84

We know, Standard Deviation = \sigma = \sqrt{Variance}

\therefore \sigma = \sqrt{249.84} = 15.80

Coefficient of variation = \frac{\sigma}{\overline x}\times100

C.V.(B) = \frac{15.80}{44.6}\times100 = 35.42

Since C.V.(B) > C.V.(A)

Therefore, Group B is more variable.

Question:2 From the prices of shares X and Y below, find out which is more stable in value:

X
35
54
52
53
56
58
52
50
51
49
Y
108
107
105
105
106
107
104
103
104
101


Answer:

X( x_i )
Y( y_i )
x_i^2
y_i^2
35
108
1225
11664
54
107
2916
11449
52
105
2704
11025
53
105
2809
11025
56
106
8136
11236
58
107
3364
11449
52
104
2704
10816
50
103
2500
10609
51
104
2601
10816
49
101
2401
10201
=510
= 1050
=26360
=110290

For X,

Mean , \overline{x} = \frac{1}{n}\sum_{i=1}^{n}x_i = \frac{510}{10} = 51

Variance, \sigma^2 = \frac{1}{n^2}\left [n\sum x_i^2 - (\sum x_i)^2 \right ]

\\ \implies \sigma^2 = \frac{1}{(10)^2}\left [10(26360) - (510)^2 \right ] \\ = \frac{1}{100}.\left [263600 - 260100 \right ] \\ \\ = 35

We know, Standard Deviation = \sigma = \sqrt{Variance}= \sqrt{35} = 5.91

C.V.(X) = \frac{\sigma}{\overline x}\times100 = \frac{5.91}{51}\times100 = 11.58

Similarly, For Y,

Mean , \overline{y} = \frac{1}{n}\sum_{i=1}^{n}y_i = \frac{1050}{10} = 105

Variance, \sigma^2 = \frac{1}{n^2}\left [n\sum y_i^2 - (\sum y_i)^2 \right ]

\\ \implies \sigma^2 = \frac{1}{(10)^2}\left [10(110290) - (1050)^2 \right ] \\ = \frac{1}{100}.\left [1102900 - 1102500 \right ] \\ \\ = 4

We know, Standard Deviation = \sigma = \sqrt{Variance}= \sqrt{4} = 2

C.V.(Y) = \frac{\sigma}{\overline y}\times100 = \frac{2}{105}\times100 = 1.904

Since C.V.(Y) < C.V.(X)

Hence Y is more stable.

Question:3(i) An analysis of monthly wages paid to workers in two firms A and B, belonging to the same industry, gives the following results:


No. of wage earners
586
648
Mean of monthly wages
Rs\hspace {1mm} 5253
Rs\hspace {1mm} 5253
Variance of the distribution of wages
100
121

Which firm A or B pays larger amount as monthly wages?

Answer:

Given, Mean of monthly wages of firm A = 5253

Number of wage earners = 586

Total amount paid = 586 x 5253 = 30,78,258

Again, Mean of monthly wages of firm B = 5253

Number of wage earners = 648

Total amount paid = 648 x 5253 = 34,03,944

Hence firm B pays larger amount as monthly wages.

Question:3(ii) An analysis of monthly wages paid to workers in two firms A and B, belonging to the same industry, gives the following results:


No. of wage earners
586
648
Mean of monthly wages
Rs\hspace {1mm} 5253
Rs\hspace {1mm} 5253
Variance of the distribution of wages
100
121

Which firm, A or B, shows greater variability in individual wages?

Answer:

Given, Variance of firm A = 100

Standard Deviation = \sigma_A = \sqrt{Variance}= \sqrt{100} = 10

Again, Variance of firm B = 121

Standard Deviation = \sigma_B = \sqrt{Variance}= \sqrt{121} = 11

Since \sigma_B>\sigma_A , firm B has greater variability in individual wages.

Question:4 The following is the record of goals scored by team A in a football session:

No. of goals scored
0
1
2
3
4
No. of matches
1
9
7
5
3

For the team B, mean number of goals scored per match was 2 with a standard deviation 1.25 goals. Find which team may be considered more consistent?

Answer:

No. of goals
scored x_i
Frequency
f_i
x_i^2
f_ix_i
f_ix_i^2
0
1
0
0
0
1
9
1
9
9
2
7
4
14
28
3
5
9
15
45
4
3
16
12
48

\sum{f_i} =N = 25

\sum f_ix_i
= 50
\sum f_ix_i ^2
=130

For Team A,

Mean,

\overline{x} = \frac{1}{N}\sum_{i=1}^{n}f_ix_i =\frac{50}{25}= 2

We know, Variance, \sigma^2 = \frac{1}{N^2}\left [N\sum f_ix_i^2 - (\sum f_ix_i)^2 \right ]

\\ \implies \sigma^2 = \frac{1}{(25)^2}\left [25(130) - (50)^2 \right ] \\ \\ = \frac{750}{625} =1.2

We know, Standard Deviation = \sigma = \sqrt{Variance}

\therefore \sigma = \sqrt{1.2} = 1.09

C.V.(A) = \frac{\sigma}{\overline x}\times100 = \frac{1.09}{2}\times100 = 54.5

For Team B,

Mean = 2

Standard deviation, \sigma = 1.25

C.V.(B) = \frac{\sigma}{\overline x}\times100 = \frac{1.25}{2}\times100 = 62.5

Since C.V. of firm B is more than C.V. of A.

Therefore, Team A is more consistent.

Question:5 The sum and sum of squares corresponding to length x (in cm) and weight y (in gm) of 50 plant products are given below:

\sum_{i=1}^{50}x_i=212, \sum _{i=1}^{50}x_i^2=902.8,\sum _{i=1}^{50}y_i=261,\sum _{i=1}^{50}y_i^2=1457.6
Which is more varying, the length or weight?

Answer:

For lenght x,

Mean, \overline{x} = \frac{1}{n}\sum_{i=1}^{n}x_i =\frac{212}{50}= 4.24

We know, Variance, \sigma^2 = \frac{1}{n^2}\left [n\sum f_ix_i^2 - (\sum f_ix_i)^2 \right ]

\\ \implies \sigma^2 = \frac{1}{(50)^2}\left [50(902.8) - (212)^2 \right ] \\ \\ = \frac{196}{2500} =0.0784

We know, Standard Deviation = \sigma = \sqrt{Variance}= \sqrt{0.0784} = 0.28

C.V.(x) = \frac{\sigma}{\overline x}\times100 = \frac{0.28}{4.24}\times100 = 6.603

For weight y,

Mean,

Mean, \overline{y} = \frac{1}{n}\sum_{i=1}^{n}y_i =\frac{261}{50}= 5.22

We know, Variance, \sigma^2 = \frac{1}{n^2}\left [n\sum f_iy_i^2 - (\sum f_iy_i)^2 \right ]

\\ \implies \sigma^2 = \frac{1}{(50)^2}\left [50(1457.6) - (261)^2 \right ] \\ \\ = \frac{4759}{2500} =1.9036

We know, Standard Deviation = \sigma = \sqrt{Variance}= \sqrt{1.9036} = 1.37

C.V.(y) = \frac{\sigma}{\overline y}\times100 = \frac{1.37}{5.22}\times100 = 26.24

Since C.V.(y) > C.V.(x)

Therefore, weight is more varying.

Statistics maths chapter 15 class 11 - Miscellaneous Exercise

Question:1 The mean and variance of eight observations are 9 and 9.25 , respectively. If six of the observations are 6,7,10,12,12 and 13 , find the remaining two observations.

Answer:

Given,

The mean and variance of 8 observations are 9 and 9.25, respectively

Let the remaining two observations be x and y,

Observations: 6, 7, 10, 12, 12, 13, x, y.

∴ Mean, \overline X = \frac{6+ 7+ 10+ 12+ 12+ 13+ x+ y}{8} = 9

60 + x + y = 72

x + y = 12 -(i)

Now, Variance

= \frac{1}{n}\sum_{i=1}^8(x_i - x)^2 = 9.25

\implies 9.25 = \frac{1}{8}\left[(-3)^2+ (-2)^2+ 1^2+ 3^2+ 4^2+ x^2+ y^2 -18(x+y)+ 2.9^2 \right ]

\implies 9.25 = \frac{1}{8}\left[(-3)^2+ (-2)^2+ 1^2+ 3^2+ 4^2+ x^2+ y^2+ -18(12)+ 2.9^2 \right ] (Using (i))

\implies 9.25 = \frac{1}{8}\left[48+x^2+ y^2 -216+ 162 \right ] = \frac{1}{8}\left[x^2+ y^2 - 6 \right ]

\implies x^2+ y^2 = 80 -(ii)

Squaring (i), we get

x^2+ y^2 +2xy= 144 (iii)

(iii) - (ii) :

2xy = 64 (iv)

Now, (ii) - (iv):

\\ x^2+ y^2 -2xy= 80-64 \\ \implies (x-y)^2 = 16 \\ \implies x-y = \pm 4 -(v)

Hence, From (i) and (v):

x – y = 4 \implies x = 8 and y = 4

x – y = -4 \implies x = 4 and y = 8

Therefore, The remaining observations are 4 and 8. (in no order)

Question:2 The mean and variance of 7 observations are 8 and \small 16 , respectively. If five of the observations are \small 2,4,10,12,14 . Find the remaining two observations.

Answer:

Given,

The mean and variance of 7 observations are 8 and 16, respectively

Let the remaining two observations be x and y,

Observations: 2, 4, 10, 12, 14, x, y

∴ Mean, \overline X = \frac{2+ 4+ 10+ 12+ 14+ x+ y}{7} = 8

42 + x + y = 56

x + y = 14 -(i)

Now, Variance

= \frac{1}{n}\sum_{i=1}^8(x_i - \overline x)^2 = 16

\implies 16 = \frac{1}{7}\left[(-6)^2+ (-4)^2+ 2^2+ 4^2+ 6^2+ x^2+ y^2 -16(x+y)+ 2.8^2 \right ]

\implies 16 = \frac{1}{7}\left[36+16+4+16+36+ x^2+ y^2 -16(14)+ 2(64) \right ] (Using (i))

\implies 16 = \frac{1}{7}\left[108+x^2+ y^2 -96 \right ] = \frac{1}{7}\left[x^2+ y^2 + 12\right ]

\implies x^2+ y^2 = 112- 12 =100 -(ii)

Squaring (i), we get

x^2+ y^2 +2xy= 196 (iii)

(iii) - (ii) :

2xy = 96 (iv)

Now, (ii) - (iv):

\\ x^2+ y^2 -2xy= 100-96 \\ \implies (x-y)^2 = 4 \\ \implies x-y = \pm 2 -(v)

Hence, From (i) and (v):

x – y = 2 \implies x = 8 and y = 6

x – y = -2 \implies x = 6 and y = 8

Therefore, The remaining observations are 6 and 8. (in no order)

Question:3 The mean and standard deviation of six observations are \small 8 and \small 4 , respectively. If each observation is multiplied by \small 3 , find the new mean and new standard deviation of the resulting observations.

Answer:

Given,

Mean = 8 and Standard deviation = 4

Let the observations be x_1, x_2, x_3, x_4, x_5\ and\ x_6

Mean, \overline x = \frac{x_1+ x_2+ x_3+ x_4+ x_5+ x_6}{6} = 8

Now, Let y_i be the the resulting observations if each observation is multiplied by 3:

\\ \overline y_i = 3\overline x_i \\ \implies \overline x_i = \frac{\overline y_i}{3}

New mean, \overline y = \frac{y_1+ y_2+ y_3+ y_4+ y_5+ y_6}{6}

= 3\left [\frac{x_1+ x_2+ x_3+ x_4+ x_5+ x_6}{6} \right] =3\times 8

= 24

We know that,

Standard Deviation = \sigma = \sqrt{Variance}

\dpi{100} =\sqrt{ \frac{1}{n}\sum_{i=1}^n(x_i - \overline x)^2}

\dpi{100}\\ \implies 4^2=\frac{1}{6}\sum_{i=1}^6(x_i - \overline x)^2 \\ \implies \sum_{i=1}^6(x_i - \overline x)^2 = 6\times16 = 96 -(i)

Now, Substituting the values of x_i\ and\ \overline x in (i):

\dpi{100} \\ \implies \sum_{i=1}^6(\frac{y_i}{3} - \frac{\overline y}{3})^2 = 96 \\ \implies \sum_{i=1}^6(y_i - \overline y)^2 = 96\times9 =864

Hence, the variance of the new observations = \frac{1}{6}\times864 = 144

Therefore, Standard Deviation = \sigma = \sqrt{Variance} = \sqrt{144} = 12

Question:4 Given that \small \bar {x} is the mean and \small \sigma^2 is the variance of \small n observations .Prove that the mean and variance of the observations \small ax_1,ax_2,ax_3,....,ax_n , are \small a\bar{x} and \small a^2 \sigma^2 respectively, \small (a \neq 0) .

Answer:

Given, Mean = \small \bar {x} and variance = \small \sigma^2

Now, Let y_i be the the resulting observations if each observation is multiplied by a:

\\ \overline y_i = a\overline x_i \\ \implies \overline x_i = \frac{\overline y_i}{a}

\overline y =\frac{1}{n}\sum_{i=1}^ny_i = \frac{1}{n}\sum_{i=1}^nax_i

\overline y = a\left [\frac{1}{n}\sum_{i=1}^nx_i \right] = a\overline x

Hence the mean of the new observations \small ax_1,ax_2,ax_3,....,ax_n is \small a\bar{x}

We know,

\dpi{100} \sigma^2=\frac{1}{n}\sum_{i=1}^n(x_i - \overline x)^2

Now, Substituting the values of x_i\ and\ \overline x :

\\ \implies \sigma^2= \frac{1}{n}\sum_{i=1}^n(\frac{y_i}{a} - \frac{\overline y}{a})^2 \\ \implies a^2\sigma^2= \frac{1}{n}\sum_{i=1}^n(y_i - \overline y)^2

Hence the variance of the new observations \small ax_1,ax_2,ax_3,....,ax_n is a^2\sigma^2

Hence proved.

Question:5(i) The mean and standard deviation of \small 20 observations are found to be \small 10 and \small 2 , respectively. On rechecking, it was found that an observation \small 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases:

If wrong item is omitted.

Answer:

Given,

Number of observations, n = 20

Also, Incorrect mean = 10

And, Incorrect standard deviation = 2

\overline x =\frac{1}{n}\sum_{i=1}^nx_i

\implies 10 =\frac{1}{20}\sum_{i=1}^{20}x_i \implies \sum_{i=1}^{20}x_i = 200

Thus, incorrect sum = 200

Hence, correct sum of observations = 200 – 8 = 192

Therefore, Correct Mean = (Correct Sum)/19

=\frac{192}{19}

= 10.1

Now, Standard Deviation,

\sigma =\sqrt{\frac{1}{n}\sum_{i=1}^nx_i ^2 - (\frac{1}{n}\sum_{i=1}^n x)^2}

\\ \implies 2^2 =\frac{1}{n}\sum_{i=1}^nx_i ^2 - (\overline x)^2 \\ \implies \frac{1}{n}\sum_{i=1}^nx_i ^2 = 4 + (\overline x)^2 \\ \implies \frac{1}{20}\sum_{i=1}^nx_i ^2 = 4 + 100 = 104 \\ \implies \sum_{i=1}^nx_i ^2 = 2080 ,which is the incorrect sum.

Thus, New sum = Old sum - (8x8)

= 2080 – 64

= 2016

Hence, Correct Standard Deviation =

\sigma' =\sqrt{\frac{1}{n'}\left (\sum_{i=1}^nx_i ^2 \right )' - (\overline x')^2} = \sqrt{\frac{2016}{19} - (10.1)^2}

\dpi{100} = \sqrt{106.1 - 102.01} = \sqrt{4.09}

=2.02

Question:5(ii) The mean and standard deviation of \small 20 observations are found to be \small 10 and \small 2 , respectively. On rechecking, it was found that an observation \small 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases:

If it is replaced by \small 12.

Answer:

Given,

Number of observations, n = 20

Also, Incorrect mean = 10

And, Incorrect standard deviation = 2

\overline x =\frac{1}{n}\sum_{i=1}^nx_i

\implies 10 =\frac{1}{20}\sum_{i=1}^{20}x_i \implies \sum_{i=1}^{20}x_i = 200

Thus, incorrect sum = 200

Hence, correct sum of observations = 200 – 8 + 12 = 204

Therefore, Correct Mean = (Correct Sum)/20

\dpi{100} =\frac{204}{20}

= 10.2

Now, Standard Deviation,

\sigma =\sqrt{\frac{1}{n}\sum_{i=1}^nx_i ^2 - (\frac{1}{n}\sum_{i=1}^n x)^2}

\\ \implies 2^2 =\frac{1}{n}\sum_{i=1}^nx_i ^2 - (\overline x)^2 \\ \implies \frac{1}{n}\sum_{i=1}^nx_i ^2 = 4 + (\overline x)^2 \\ \implies \frac{1}{20}\sum_{i=1}^nx_i ^2 = 4 + 100 = 104 \\ \implies \sum_{i=1}^nx_i ^2 = 2080 ,which is the incorrect sum.

Thus, New sum = Old sum - (8x8) + (12x12)

= 2080 – 64 + 144

= 2160

Hence, Correct Standard Deviation =

\sigma' =\sqrt{\frac{1}{n}\left (\sum_{i=1}^nx_i ^2 \right )' - (\overline x')^2} = \sqrt{\frac{2160}{20} - (10.2)^2}

= \sqrt{108 - 104.04} = \sqrt{3.96}

= 1.98

Question:6 The mean and standard deviation of marks obtained by \small 50 students of a class in three subjects, Mathematics, Physics and Chemistry are given below:

Subject
Mathematics
Physics
Chemistry
Mean
\small 42
\small 32
\small 40.9
Standard deviation
\small 12
\small 15
\small 20

Which of the three subjects shows highest variability in marks and which shows the lowest?

Answer:

Given,

Standard deviation of physics = 15

Standard deviation of chemistry = 20

Standard deviation of mathematics = 12

We know ,

C.V. = \frac{Standard\ Deviation}{Mean}\times 100

The subject with greater C.V will be more variable than others.

C.V. _P= \frac{15}{32}\times 100 = 46.87

C.V. _C= \frac{20}{40.9}\times 100 = 48.89

C.V. _M= \frac{12}{42}\times 100 = 28.57

Hence, Mathematics has lowest variability and Chemistry has highest variability.

Question:7 The mean and standard deviation of a group of \small 100 observations were found to be \small 20 and \small 3 , respectively. Later on it was found that three observations were incorrect, which were recorded as \small 21,21 and \small 18 . Find the mean and standard deviation if the incorrect observations are omitted.

Answer:

Given,

Initial Number of observations, n = 100

\overline x =\frac{1}{n}\sum_{i=1}^nx_i

\dpi{100} \implies 20 =\frac{1}{100}\sum_{i=1}^{100}x_i \implies \sum_{i=1}^{100}x_i = 2000

Thus, incorrect sum = 2000

Hence, New sum of observations = 2000 - 21-21-18 = 1940

New number of observation, n' = 100-3 =97

Therefore, New Mean = New Sum)/100

=\frac{1940}{97}

= 20

Now, Standard Deviation,

\sigma =\sqrt{\frac{1}{n}\sum_{i=1}^nx_i ^2 - (\frac{1}{n}\sum_{i=1}^n x)^2}

\\ \implies 3^2 =\frac{1}{n}\sum_{i=1}^nx_i ^2 - (\overline x)^2 \\ \implies \frac{1}{n}\sum_{i=1}^nx_i ^2 = 9 + (\overline x)^2 \\ \implies \frac{1}{100}\sum_{i=1}^nx_i ^2 = 9 + 400 = 409 \\ \implies \sum_{i=1}^nx_i ^2 = 40900 ,which is the incorrect sum.

Thus, New sum = Old (Incorrect) sum - (21x21) - (21x21) - (18x18)

= 40900 - 441 - 441 - 324

= 39694

Hence, Correct Standard Deviation =

\sigma' =\sqrt{\frac{1}{n'}\left (\sum_{i=1}^nx_i ^2 \right )' - (\overline x')^2} = \sqrt{\frac{39694}{97} - (20)^2}

= \sqrt{108 - 104.04} = \sqrt{3.96}

= 3.036

Summary And Formulae Of Statistics Maths Chapter 15 class 11

  • Measures of dispersion- Range, mean deviation, Quartile deviation, standard deviation, variance are measures of dispersion.
  • Range- The difference between the maximum and the minimum values of each series is called the ‘Range’ of the data. Range = Maximum Value – Minimum Value.
  • Mean- The arithmetic average of a range of values or quantities, found by adding all values and dividing by the number of values. For example, the mean of 4, 10, and 1 is (4+10+1)/3=15/3=5.

\bar{X}=\frac{\Sigma X}{N}
X represents values or scores,
N represents the number of values or scores.

  • Median- If the total number of values(n) is an odd number, then the formula is-

Median= (\frac{n+1}{2})^{th}\:term

  • Median- If the total number of values(n) is an even number, then the formula is-

Median= \frac{(\frac{n}{2})^{th}\:term+(\frac{n+1}{2})^{th}\:term}{2}

  • Mode- It is the most frequently occurring value or number.
  • Mean deviation for ungrouped data-

M.D.(\bar{x})=\frac{\Sigma \:|x_{i}-\bar{x}|}{n}, \:\:\:\:\:\:\:\:\: M.D.(M)=\frac{\Sigma \:|x_{i}-M|}{n}

  • Mean deviation for grouped data-

M.D.(\bar{x})=\frac{\Sigma f_i\:|x_{i}-\bar{x}|}{N}, \:\:\:\:\:\:\:\:\: M.D.(M)=\frac{f_i\Sigma \:|x_{i}-M|}{N}, \: where \:N=\Sigma f_i

interested students can practice class 11 maths ch 15 question answer using the following exercises.

NCERT Solutions For Class 11 Mathematics

Key Features Of Statistics Class 11 NCERT Solutions

Easy to Understand: The ch 15 maths class 11 solutions are written in simple language and are easy to understand, which makes them accessible for students who may not have a strong background in statistics.

Comprehensive Coverage: The statistics NCERT solutions cover all the important concepts and topics in Statistics Class 11 NCERT curriculum. This ensures that students are well-prepared and have a thorough understanding of the subject.

Structured Format: The ch 15 maths class 11 solutions are presented in a structured format that helps students to organize their thoughts and approach problems in a systematic manner.

NCERT Solutions For Class 11 - Subject Wise

This chapter seems very easy but it very useful in the field of science, research reliable predictions, or forecasts for future use, decision making, etc. There are 7 questions are given in the miscellaneous exercise. Try to solve them also to get command in this chapter. In NCERT solutions for class 11 maths chapter 15 statistics, you will get solutions to miscellaneous exercise too.

NCERT Books and NCERT Syllabus

Frequently Asked Questions (FAQs)

1. Explain the term standard deviation, which is discussed in NCERT solutions for class 11 maths chapter 15.

The concept of standard deviation involves measuring the amount of variation or deviation present in a given set of values, and the range is determined by considering the level of standard deviation. For a clearer understanding of this term, students are encouraged to download the PDF solutions provided by Careers360, which are available in both chapter-wise and exercise-wise formats to cater to their specific needs.

2. What are the topics discussed in Chapter 15 of NCERT Solutions for Class 11 Maths?

The topics discussed in class 11 maths statistics are
1. Introduction
2. Methods of Dispersion
3. Range
4. Mean Deviation
5. Variance and Standard Deviation
6. Analysis of Frequency Distributions

3. Are the NCERT Solutions for statistics class 11 maths helpful for the students?

The NCERT Solutions for class 11 chapter 15 maths offer precise and straightforward explanations that aid students in achieving good grades in their exams. The solutions' methodical approach to problem-solving provides students with a clear understanding of the marks allocation as per the latest CBSE Syllabus 2023. By using these class 11th statistics solutions , students can identify their weak areas and work towards improving them, resulting in better academic performance.

4. Where can I find the complete solutions of class 11 statistics solutions ?

Students can find a detailed NCERT solutions for class 11 maths  by clicking on the link. they can practice these solutions to get confidence in the concepts and in-depth understanding that ultimately lead to high score in the exam.

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