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NCERT · Class 11 · Maths

Chapter 15: Statistics Exercise 15.3

Practice each question, then open the answer or worked solution.

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Questions

5 questions
Q1Written answer

From the data given below, the 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
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Worked solution

Solution:-

To compare the variability or dispersion of two series, we calculate the coefficient of variance for each series. The series having greater C.V. is said to be more variable than the other. The series having lesser C.V. is said to be more consistent than the other.

Co-efficient of variation (C.V.) = (σ/ x̅) × 100

Where, σ = standard deviation, x̅ = mean

For Group A,

Marks Group A

fi

 Mid-point

Xi

 Yi = (xi – A)/h (Yi)2 fiyi fi(yi)2
10 – 20 9 15 ((15 – 45)/10) = -3 (-3)2

= 9

 – 27 81
20 – 30 17 25 ((25 – 45)/10) = -2 (-2)2

= 4

 – 34 68
30 – 40 32 35 ((35 – 45)/10) = – 1 (-1)2

= 1

 – 32 32
40 – 50 33 45 ((45 – 45)/10) = 0 02 0 0
50 – 60 40 55 ((55 – 45)/10) = 1 12

= 1

 40 40
60 – 70 10 65 ((65 – 45)/10) = 2 22

= 4

 20 40
70 – 80 9 75 ((75 – 45)/10) = 3 32

= 9

 27 81
Total 150 -6 342

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 37

Where A = 45,

and yi = (xi – A)/h

Here, h = class size = 20 – 10

h = 10

So, x̅ = 45 + ((-6/150) × 10)

= 45 – 0.4

= 44.6

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 38

σ2 = (102/1502) [150(342) – (-6)2]

= (100/22500) [51,300 – 36]

= (100/22500) × 51264

= 227.84

Hence, standard deviation = σ = √227.84

= 15.09

∴ C.V for group A = (σ/ x̅) × 100

= (15.09/44.6) × 100

= 33.83

Now, for group B.

Marks Group B

fi

 Mid-point

Xi

 Yi = (xi – A)/h (Yi)2 fiyi fi(yi)2
10 – 20 10 15 ((15 – 45)/10) = -3 (-3)2

= 9

 – 30 90
20 – 30 20 25 ((25 – 45)/10) = -2 (-2)2

= 4

 – 40 80
30 – 40 30 35 ((35 – 45)/10) = – 1 (-1)2

= 1

 – 30 30
40 – 50 25 45 ((45 – 45)/10) = 0 02 0 0
50 – 60 43 55 ((55 – 45)/10) = 1 12

= 1

 43 43
60 – 70 15 65 ((65 – 45)/10) = 2 22

= 4

 30 60
70 – 80 7 75 ((75 – 45)/10) = 3 32

= 9

 21 63
Total 150 -6 366

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 39

Where A = 45

h = 10

So, x̅ = 45 + ((-6/150) × 10)

= 45 – 0.4

= 44.6

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 40

σ2 = (102/1502) [150(366) – (-6)2]

= (100/22500) [54,900 – 36]

= (100/22500) × 54,864

= 243.84

Hence, standard deviation = σ = √243.84

= 15.61

∴ C.V for group B = (σ/ x̅) × 100

= (15.61/44.6) × 100

= 35

By comparing the C.V. of group A and group B,

C.V. of Group B > C.V. of Group A.

So, Group B is more variable.

Q2Written answer

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
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Worked solution

Solution:-

From the given data,

Let us make the table of the given data and append other columns after calculations.

X (xi) Y (yi) Xi2 Yi2
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
Total = 510 1050 26360 110290

We have to calculate the Mean for x.

Mean x̅ = ∑xi/n

Where n = Number of terms

= 510/10

= 51

Then, Variance for x =NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 41

= (1/102)[(10 × 26360) – 5102]

= (1/100) (263600 – 260100)

= 3500/100

= 35

WKT Standard deviation = √variance

= √35

= 5.91

So, the co-efficient of variation = (σ/ x̅) × 100

= (5.91/51) × 100

= 11.58

Now, we have to calculate the Mean for y.

Mean ȳ = ∑yi/n

Where, n = Number of terms

= 1050/10

= 105

Then, Variance for y =NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 42

= (1/102)[(10 × 110290) – 10502]

= (1/100) (1102900 – 1102500)

= 400/100

= 4

WKT Standard deviation = √variance

= √4

= 2

So, the co-efficient of variation = (σ/ x̅) × 100

= (2/105) × 100

= 1.904

By comparing the C.V. of X and Y,

C.V. of X > C.V. of Y.

So, Y is more stable than X.

Q3Written answer

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

Firm A Firm B
No. of wages earners 586 648
Mean of monthly wages Rs 5253 Rs 5253
The variance of the distribution of wages 100 121

(i) Which firm, A or B, pays a larger amount as monthly wages?

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

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Worked solution

Solution:-

(i) From the given table,

Mean monthly wages of firm A = Rs 5253

and Number of wage earners = 586

Then,

Total amount paid = 586 × 5253

= Rs 3078258

Mean monthly wages of firm B = Rs 5253

Number of wage earners = 648

Then,

Total amount paid = 648 × 5253

= Rs 34,03,944

So, firm B pays a larger amount as monthly wages.

(ii) Variance of firm A = 100

We know that standard deviation (σ)= √100

=10

Variance of firm B = 121

Then,

Standard deviation (σ)=√(121 )

=11

Hence, the standard deviation is more in the case of Firm B, which that means in firm B, there is greater variability in individual wages.

Q4Written answer

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

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Worked solution

Solution:-

From the given data,

Let us make the table of the given data and append other columns after calculations.

Number of goals scored xi Number of matches fi fixi Xi2 fixi2
0 1 0 0 0
1 9 9 1 9
2 7 14 4 28
3 5 15 9 45
4 3 12 16 48
Total 25 50 130

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 43

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 44

The C.V. of firm B is greater.

∴ Team A is more consistent.

Q5Written answer

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

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 45

Which is more varying, the length or weight?

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Worked solution

Solution:-

First, we have to calculate the Mean for Length x.

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 46

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 47