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Karnataka 1st PUC Maths Question Bank Chapter 15 Statistics

Question 1.
Find the mean deviation about the mean for the following data
(i) 6, 7, 10, 12, 13, 4, 8, 12
(ii) 4, 7, 8, 9, 10, 12,13,17
(iii) 38, 70, 48, 40, 42, 55, 63, 46, 54, 44
Answer:
1st PUC Maths Question Bank Chapter 15 Statistics 1
1st PUC Maths Question Bank Chapter 15 Statistics 2

Question 2.
Find the mean deviation about the mean for the data.
12, 3, 18, 17, 4, 9, 17, 19, 20, 15, 8, 17, 2, 3, 16, 11, 3, 1, 0, 5.
Answer:
Try your self.
\(N=20, \bar{x}=10, \Sigma\left|x_{i}-\bar{x}\right|=124, \mathrm{M} \cdot \mathrm{D} \cdot(\bar{x})=6 \cdot 2\)

KSEEB Solutions

Question 3.
Find the mean deviation about the median for the data:
(i) 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17
(ii) 36, 72, 46, 60, 45, 53, 46, 51, 49, 42
(iii) 3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21
Answer:
(i) Arrange the data in ascending order as
1st PUC Maths Question Bank Chapter 15 Statistics 3
1st PUC Maths Question Bank Chapter 15 Statistics 4

Question 4.
Find the mean deviation about the mean for the following data.
1st PUC Maths Question Bank Chapter 15 Statistics 5
Answer:
1st PUC Maths Question Bank Chapter 15 Statistics 6
1st PUC Maths Question Bank Chapter 15 Statistics 7

1st PUC Maths Question Bank Chapter 15 Statistics 8
KSEEB Solutions

Question 5.
Find the mean deviation about the median for the data,
1st PUC Maths Question Bank Chapter 15 Statistics 9
1st PUC Maths Question Bank Chapter 15 Statistics 10
Answer:
1st PUC Maths Question Bank Chapter 15 Statistics 11
1st PUC Maths Question Bank Chapter 15 Statistics 12
1st PUC Maths Question Bank Chapter 15 Statistics 13
Question 6.
Find the mean deviation about the mean for the following data.
(i)

Income per day Number of persons
0 – 100 4
100 – 200 8
200 – 300 9
300 – 400 10
400 – 500 7
500 – 600 5
600 – 700 4
700 – 800 3

(ii)

Height (cms) Number of boys
95 – 105 9
105 – 115 13
115 – 125 26
125 – 135 30
135 – 145 12
145 – 155 10

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(iii)

Marks obtained Number of students
0 – 10 12
10 – 20 18
20 – 30 27
30 – 40 20
40 – 50 17
50 – 60 6

(iv)

Marks obtained Number of students
10 – 20 2
20 – 30 3
30 – 40 8
40 – 50 14
50 – 60 8
60 – 70 3
70 – 80 2

Answer:
1st PUC Maths Question Bank Chapter 15 Statistics 14

Method (II)
1st PUC Maths Question Bank Chapter 15 Statistics 15

Method (I)
1st PUC Maths Question Bank Chapter 15 Statistics 16

Method (ii)
Here a = 130, h = 10
1st PUC Maths Question Bank Chapter 15 Statistics 17

(iii) Method (I)
1st PUC Maths Question Bank Chapter 15 Statistics 18

Question 7.
Find the mean deviation about the median for the data:
(i)

Class Frequency
0 – 10 6
10 – 20 7
20 – 30 15
30 – 40 16
40 – 50 4
50 – 60 2

(ii)

Marks Number of Girls
0 – 10 6
10 – 20 8
20 – 30 14
30 – 40 16
40 – 50 4
50 – 60 2

Answer:
1st PUC Maths Question Bank Chapter 15 Statistics 19
(ii)
1st PUC Maths Question Bank Chapter 15 Statistics 20
1st PUC Maths Question Bank Chapter 15 Statistics 21
KSEEB Solutions

Question 8.
Calculate the mean deviation about median age for the age distribution of 100 persons given below.

Age Number
16 – 20 5
21 – 25 6
26 – 30 12
31 – 35 14
36 – 40 26
41- 45 12
46 – 50 16
51 – 55 9

Answer:
First, modify the classes to make the data. continuous by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit.
1st PUC Maths Question Bank Chapter 15 Statistics 22
1st PUC Maths Question Bank Chapter 15 Statistics 23

Question 9.
Find Mean and variance for each data:
(i) 6, 8, 10, 12, 14, 16, 18, 20, 22, 24.
(ii) 6, 7, 10, 12, 13, 4, 8, 12
(iii) First n natural numbers.
(iv) First 10 multiples of 3.
Answer:
1st PUC Maths Question Bank Chapter 15 Statistics 24
1st PUC Maths Question Bank Chapter 15 Statistics 25
1st PUC Maths Question Bank Chapter 15 Statistics 26
1st PUC Maths Question Bank Chapter 15 Statistics 27
Question 10.
Find mean and variance.
(i)

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

(ii)

xi 92 93 97 98 102 104 109
fi 3 2 3 2 6 3 3

Answer:
(i)
1st PUC Maths Question Bank Chapter 15 Statistics 28
(ii)
1st PUC Maths Question Bank Chapter 15 Statistics 29

KSEEB Solutions

Question 11.
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:
1st PUC Maths Question Bank Chapter 15 Statistics 30
1st PUC Maths Question Bank Chapter 15 Statistics 31

Question 12.
Find the mean and variance
(i)

Class Frequency
0 – 30 2
30 – 60 3
60 – 90 5
90 – 120 10
120 – 150 3
150 – 180 5
180 – 210 2

(ii)

Class Frequency
0 – 10 5
10 – 20 8
20 – 30 15
30 – 40 16
40 – 50 6

Answer:
1st PUC Maths Question Bank Chapter 15 Statistics 32
1st PUC Maths Question Bank Chapter 15 Statistics 33

Question 13.
Find the mean, variance and standard deviation using short-cut method.

Height (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:
1st PUC Maths Question Bank Chapter 15 Statistics 34
1st PUC Maths Question Bank Chapter 15 Statistics 35

Question 14.
The diameters of circles (in mm ) drawn in a design are given below calculate S.D and mean diameter.

Diameter Circles
70 – 75 15
75 – 80 17
80 – 85 21
85 – 90 22
90 – 95 25

Answer:
1st PUC Maths Question Bank Chapter 15 Statistics 36
1st PUC Maths Question Bank Chapter 15 Statistics 37

KSEEB Solutions

Question 15.
From the data given below state which group is more variable, A or B?

Marks Group A Group B
10 – 20 9 10
20 – 30 17 20
30 – 40 32 30
40 – 50 33 25
50 – 60 40 143
60 – 70 10 15
70 – 80 9 7

Answer:
We know that, the group having greater C.V.(co-efficient of variation) is said w be more variable than other. So, we compute mean and S.D. for groups A and B.
First we compute mean and S.D. for A.
1st PUC Maths Question Bank Chapter 15 Statistics 38
1st PUC Maths Question Bank Chapter 15 Statistics 39
1st PUC Maths Question Bank Chapter 15 Statistics 40
Clearly, B having greater C. V. than A. Therefore B is more variable.

Note:
For two frequency distributions with equal means, the series with greater S.D. is more variable than the other.

KSEEB Solutions

Question 16.
From the prices of shares X and Y below, find out which is more stable in value.
1st PUC Maths Question Bank Chapter 15 Statistics 41
Answer:
Here, number of items for both = 10.
First we compute mean and S.D. for x, taking a = 56, N = 10.
1st PUC Maths Question Bank Chapter 15 Statistics 42
1st PUC Maths Question Bank Chapter 15 Statistics 43
1st PUC Maths Question Bank Chapter 15 Statistics 44
Question 17.
An analysis of monthly wages paid to workers in two firms A and B, belonging to the same industry, gives the following results.

A B
No. of wage earners 586 648
Mean of monthly wages (Rs ) 5253 5253
Variance of the distribution of wages 100 121

(i) Which firm A or B pays larger amount as monthly wages?
(ii) Which firm A or B shows greater variability in individual wages?
Answer:
(i) Amount of monthly wages paid by firm A.
= 586 x mean wages
= 586 x 5253 = Rs.3078258
And amount of monthly wages paid by form B
= 648 x mean wages
= 648 x 5253 = Rs. 3403944.
Clearly, firm B pays more wages.
Alternatively,
The variance of wages in A is 100.
∴ S.D. of distribution of wage in A = 10
Also the variance of distribution of wages in firm B is 121.
∴ S.D. of distribution of wages in B = 11.
since the average monthly wages in both firms is same i.e., Rs. 5253, therefore the firm with greater S.D. will have more variability.
‍∴ Firm B pays more wages.

(ii) Firm B with greater S.D. shows greater variability in individual wages.

KSEEB Solutions

Question 18.
Two plants A and B of a factory show following results about the number of workers and the wages paid to them.

A B
No – of workers 5000 6000
Average monthly wages 2500 2500
Variance distribution of wages 81 100

In which plant A or B is there greater variability in individual wages?
Answer:
B. (Try yourself) (Ex. 13 Text book)

Question 19.
The following is the record of goals scored by team A in a football session:
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:
First, compute mean number of goals and S.D. for team A.
1st PUC Maths Question Bank Chapter 15 Statistics 45
Given : Mean number of goals for the team B is 2 and S.D. is 1.25
Since S.D. For A is less than S.D for B.
∴ A is more consistent.

Question 20.
The sum and sum of squares corresponding to length x (in cm) and weight y (in gm) of 50 plant products are given below:
1st PUC Maths Question Bank Chapter 15 Statistics 46
Which is more varying, the length or weight?
Answer:
1st PUC Maths Question Bank Chapter 15 Statistics 47

KSEEB Solutions

Question 21.
Co-efficient of variation of two distributions are 60 and 70, and the standard deviations are 21 and 16, respectively. What are their arithmetic means.
Answer:
Let \(\bar{x}_{1} \text { and } \bar{x}_{2} \) be arithmetic means of first and second distribution respectively. Then
1st PUC Maths Question Bank Chapter 15 Statistics 48

Question 22.
The following values are calculated in respect of heights and weight of the students of a section of class XI.

Height Weight
Mean 162 – 6 cm 52-36 kg
Variance 127 – 69 cm2 23. 1361kg2

Can we say that the weight show greater variation than the height.
Answer:
1st PUC Maths Question Bank Chapter 15 Statistics 49
Clearly C.V. (weights) is greater than C.V. (heights). Weights show more variability than heights.

Miscellaneous Examples

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:
Let the remaining two observation be x and y.
1st PUC Maths Question Bank Chapter 15 Statistics 50

1st PUC Maths Question Bank Chapter 15 Statistics 51

Question 2.
The mean of 5 observations is 4.4 and their variance is 8.24. If three observations are 1, 2 and 6, find the remaining two observations.
Answer:
Let the remaining two observations be x and
∴ Five observations are 1, 2, 6, x, y.
1st PUC Maths Question Bank Chapter 15 Statistics 52
1st PUC Maths Question Bank Chapter 15 Statistics 53

Question 3.
The mean and variance of 7 observations are 8 and 16 respectively. If five of the observations are 2, 4, 10, 12, 14. Find the remaining two observations.
Answer:
6 and 8 (Try yourself)
Here, x + y = 14, x2 + y2 = 100

Question 4.
The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation.
Answer:
Let six observations be
1st PUC Maths Question Bank Chapter 15 Statistics 54

KSEEB Solutions

Question 5.
Given that \(\bar{x} \)is the mean and σ2 is the variance of n observations x1 , x2,……….. ,xn . Prove that the mean and variance of the observations ax1,ax2,………………,axn are \(a \bar{x} \) and a2a1 respectively (a ≠ 0)
Answer:
1st PUC Maths Question Bank Chapter 15 Statistics 55

Question 6.
If each of the observation x1,x2,………. ,xn is increased by ‘a’, where a is positive or negative number, show that the variance remains unchanged.
Answer:
1st PUC Maths Question Bank Chapter 15 Statistics 56
1st PUC Maths Question Bank Chapter 15 Statistics 57

Question 7.
The mean and standard deviation 20 observations are found to be 10 and 2 respectively. On rechecking, it was fond that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases :
(i) If wrong item is omitted,
(ii) If it is replaced by 12.
Answer:
1st PUC Maths Question Bank Chapter 15 Statistics 58
1st PUC Maths Question Bank Chapter 15 Statistics 59
1st PUC Maths Question Bank Chapter 15 Statistics 60

Question 8.
The mean and standard deviation of 100 observations were calculated as 40 and 5.1 respectively by a student who took by mistake 50 instead of 40 for one observation. What are the correct mean and standard deviation?
Answer:
Given: N = 100
Incorrect mean = 40 = \(a \bar{x} \)
Incorrect S.D. = 5.1
We have, \(\bar{x}=\frac{1}{N} \Sigma x_{i}\)
⇒\(\Sigma x_{i}=N \bar{x}=100 \times 40=4000\)
∴ Incorrect sum of observations = 4000.
∴ Correct sum of observations = 4000 – 50 + 40 = 3990.
1st PUC Maths Question Bank Chapter 15 Statistics 61

KSEEB Solutions

Question 9.
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 were recorded as 21, 21 and 18. Find and standard deviation if the incorrect observations are omitted.
Answer:
Given N = 100,New N = 97
1st PUC Maths Question Bank Chapter 15 Statistics 62
1st PUC Maths Question Bank Chapter 15 Statistics 63
Question 10.
The mean and standard deviation of marks obtained by 50 students of a class in three subjects, Mathematics, Physics and Chemistry are given below.

Subject Mathematics Physics Chemistry
Mean 42 32 40 – 9
S D 12 15 20

Which of three subjects shows the highest variability in marks and which shows the lowest?
Answer:
1st PUC Maths Question Bank Chapter 15 Statistics 64
Chemistry shows high variability and Mathematics shows lowest variability.