MP Board Class 11th Maths Important Questions Chapter 15 Statistics

MP Board Class 11th Maths Important Questions Chapter 15 Statistics

Statistics Important Questions

Statistics Long Answer Type Questions

Question 1.
Find the mean deviation about the mean for the data in following table: (NCERT)
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 1
Solution:
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 2
Mean \(\bar { x }\) = \(\frac{\sum f_{i} x_{i}}{\sum f_{i}}\) = \(\frac { 4000 }{ 80 }\) = 50
Mean deviation about mean = \(\frac{\sum f_{i}\left|x_{i}-\bar{x}\right|}{\sum f_{i}}\)
= \(\frac { 1280 }{ 80 }\) = 16.

Question 2.
Find the mean deviation about the median for the data in following table: (NCERT)
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 3
Solution:
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 4
\(\frac { N }{ 2 }\) = \(\frac { 26 }{ 2 }\) = 13
Which lies on cumulative frequency (C.F.) 14.
∴ Median = Md = 7.
Mean deviation about mean = \(\frac{\sum f_{i}\left|x_{i}-M_{d}\right|}{\sum f_{i}}\)
= \(\frac { 1128.8 }{ 100 }\)

Question 3.
Find the mean deviation about the mean for the data in following table: (NCERT)
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 5
Solution:
Let A = 130
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 6
Mean \(\bar { x }\) = A + \(\frac { Σfd }{ Σf }\) × i = 130 + \(\frac { (- 47) }{ 10 }\) = 10
\(\bar { x }\) = 130 – 4.7 = 125.3
Mean deviation about mean = \(\frac{\sum f_{i}\left|x_{i}-\bar{x}\right|}{\sum f_{i}}\)
= \(\frac { 1128.8 }{ 100 }\) = 11.228.

Question 4.
Find the mean deviation about the median for the data in following table: (NCERT)
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 7
Solution:
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 8
\(\frac { N }{ 2 }\) = \(\frac { 50 }{ 2 }\) = 25
20 – 30 is median class.
Here F = 14, f = 14, l = 20, i = 10
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 9

Question 5.
Calculate the mean deviation about median for the age distribution of 100 persons given below :
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 10
Solution:
First make the class interval uniform:
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 11
\(\frac { N }{ 2 }\) = \(\frac { 100 }{ 2 }\) = 50
Median class = 35.5 – 40.5
where l = 35.5, F = 37, f = 26, i = 5
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 12

Question 6.
Find the mean and variance for the following frequency distribution in following table:
(A)
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 13
Solution:
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 14
Let assumed mean A = 105
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 15

(B)
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 16
Solution:
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 17
Let assumed mean A = 25
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 18

Question 7.
Find the mean and variance and standard deviation using short cut method:
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 19
Solution:
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 20
Let assumed mean A = 92.5
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 21
Standard deviation σ = \(\sqrt {variance}\)
= \(\sqrt {105.58}\) = 10.27

Question 8.
The diameter of circle (in mm) drawn in design are given below :
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 22
Calculate the standard deviation and mean diameter of the circles. (NCERT)
Solution:
First make the class interval uniform :
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 23
Let assume mean A = 42.5
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 24

Question 9.
From the prices of shares X and Y below, find out which is more stable in value:
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 25
Solution:
For first share X : Let assumed mean A = 58, n = 10
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 26
Foe second share Y : Let assumed mean A = 107, n = 10
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 27
Coefficient of variation for X :
Mean M = A + \(\frac { Σd }{ n }\)
= 58 + \(\frac { ( – 70) }{ 10 }\) = 58 – 7
= 51
∴ Coefficient of variance = \(\frac { σ }{ M }\) x 100
= \(\frac { 6 }{ 51 }\) x 100 = 11.8
Coefficient of variance of Y :
Mean M = A + \(\frac { Σd }{ n }\)
= 107 + \(\frac { – 20 }{ 10 }\) = 107 – 2
= 105
∴ Coefficient of variation = \(\frac { σ }{ M }\) x 100
= \(\frac { 2 }{ 105 }\) x 100 = 1.9
Coefficient of variance of Y is less than the coefficient of variance of X.
Share Y is more stable than share X.

Question 10.
An analysis of monthly wages paid to the workers in two firms A and B belonging to the same industry. Give the following results: (NCERT)
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 28

  1. Which firm A or B pays out larger amount as monthly wages.
  2. Which firm A or B shows greater variability in individual wages.

Solution:
For firm A :
Number of wages earners = 586
Mean of monthly wages x = Rs. 5,253
Amount paid by firm A = 586 x 5,253 = Rs. 30, 78, 258
Variance of distribution of wages = 100
Standard deviation σ = \(\sqrt {100}\) = 10
Coefficient of variation = \(\frac{\sigma}{\bar{x}}\) x 100 = \(\frac { 10 }{ 5253 }\) = 0.19

For firm B :
Number of wages earners = 648
Mean \(\bar { x }\) = Rs. 5253
Amount paid by firm B = 648 x 5253 = Rs. 34, 03, 944
Coefficient of variation = 121
Standard deviation σ = \(\sqrt {121}\) = 11
Coefficient of variation = \(\frac{\sigma}{\bar{x}}\) x 100 = \(\frac { 11 }{ 5253 }\) x 100 = 0.21
Thus, firm B pays out larger amount as monthly wages.
∵ C.V. of firm B > C.V. of firm A.
∴ Firm B shows greater variability in individual wages.

Question 11.
The following is the record of goals scored by team A in a football session:
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 29
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? (NCERT)
Solution:
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 30
For team B :
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 31
C.V of team A < C.V of team B. Hence team A is more consistent.

Question 12.
The mean and standard deviation of marks obtained by 50 students of three subjects Mathematics, Physics and Chemistry are given below :
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 32
Which of these three subjects shows the highest variability in marks and which shows lowest? (NCERT)
Solution:
C.V. of Mathematics (C.V.) = \(\frac{\sigma}{\bar{x}}\) x 100
Where σ = 12, \(\bar { x }\) = 42
∴ C.V. of Mathematics = \(\frac { 12 }{ 42 }\) x 100 = 28.57
C.V. of Physics = \(\frac{\sigma}{\bar{x}}\) x 100
Where σ = 15, \(\bar { x }\) = 32
∴ C.V. of Physics = \(\frac { 15 }{ 32 }\) x 100 = 46.87
C.V. of Chemistry = \(\frac{\sigma}{\bar{x}}\) x 100
Where σ = 20, \(\bar { x }\) = 40.9
∴ C.V. of Chemistry = \(\frac { 20 }{ 40.9 }\) x 100
= 48.89
∵ C.V. of Chemistry > C.V. of Physics > C.V. of Mathematics.
∴ Chemistry shows the highest variability and Mathematics shows the least variability.

Question 13.
The mean and standard deviation of a group of 100 observations were found to be 20 and 3 respectively. Later on it was found that three observations were incorrect Which are recorded by 21,21, and 18. Find the mean and standard deviation, if the incorrect observations are omitted. (NCERT)
Solution:
Given: n = 100, \(\bar { x }\) = 20, σ = 3
\(\bar { x }\) = \(\frac{\sum x}{n}\) ⇒ 20 = \(\frac{\sum x}{n}\)
Σx = 20 x 100 = 2000
If incorrect observations 21,21 and 18 are omitted, then correct sum
Σx = 2000 – 21 – 21 – 18 = 2000 – 60 = 1940
Now correct mean of remaining 97 observations are
\(\bar { x }\) = \(\frac{\sum x}{n}\) = \(\frac { 1940 }{ 97 }\) = 20
Given : σ = 3
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 33
correct Σx2 = 40900 – (21)2 – (21)2 – (18)2
= 40900 – 441 – 441 – 324 = 39694
Now correct S.D. of remaining 97 observations are
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 34

Question 14.
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, then And the remaining two observations. (NCERT)
Solution:
Let the two remaining observations are x1 and x2
Mean M = \(\frac{6 + 7 + 10 + 12 + 12 + 13 + x_{1} + x_{2}}{8}\)
9 = \(\frac{60 + x_{1} + x_{2}}{8}\)
⇒ x1 + x2 + 60 = 72
⇒ x1 + x2 = 12 …. (1)
MP Board Class 11th Maths Important Questions Chapter 15 Statistics 35
⇒ x21 + x22 = 722 – 642
⇒ x21 + x22 = 80 …. (2)
From eqn. (1),
x2 = 12 – x1
Put the value of x2 in equation (2),
x21 + (12 – x1)2 = 80
⇒ x2 + 144 + x21 – 24x1 = 80
⇒ 2x21 – 24x1 + 64 = 0
⇒ x21 – 12x1 + 32 = 0
⇒ x21 – 4x1 – 8x1 + 32 = 0
⇒ x1(x1 – 4) – 8(x1 – 4) = 0
⇒ (x1 – 8)(x1 – 4) = 0
⇒ x1 = 4, 8
When x1 = 4, then x2 = 12 – 4 = 8
When x1 = 8, then x2 = 12 – 8 = 4
Hence remaining observations are 4 and 8.

MP Board Class 11th Maths Important Questions

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