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Page 1
NCERT
SOLUTIONS
CLASS - 11th
aglase .co
SOLUTIONS
CLASS - 11th
aglase .co
Page 2
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
Class : 11th
Subject : Statistics
Chapter : 6
Chapter Name : Measures of Dispersion
Q1 A measure of dispersion is a good supplement to the central value in
understanding a frequency distribution. Comment.
Answer. In order to have a better understanding of the central value and also of the
entire frequency distribution table, we need to study the dispersion of data among the
various observations. For example, if there is a huge difference between the low income
group and the high income group, then we can conclude that there is a large income
disparity in the region.
Page : 89 , Block Name : Exercises
Q2 Which measure of dispersion is the best and how?
Answer. The best measure of dispersion is the Standard Deviation. It is because:
1. It can compare multiple quantities easily.
2. It takes into account every aspect and hence provides a comprehensive
assessment.
3. It is least affected by the change of samples.
4. It is independent of the origin.
5. It is algebraic in nature as it takes into account the positive and negative
deviations.
Page : 89 , Block Name : Exercises
Q3 Some measures of dispersion depend upon the spread of values whereas
some are estimated on the basis of the variation of values from a central
value. Do you agree?
Answer. Yes, some measures of dispersion depend upon the spread of values whereas
some are estimated on the basis of the variation of values from a central
Value. This is the reason the measures are classified into absolute and relative
measures. The absolute measures are the range, standard deviation, etc. whereas the
relative measures are coefficient of variation, coefficient of standard deviation, etc.
Page 1 of 21
Class : 11th
Subject : Statistics
Chapter : 6
Chapter Name : Measures of Dispersion
Q1 A measure of dispersion is a good supplement to the central value in
understanding a frequency distribution. Comment.
Answer. In order to have a better understanding of the central value and also of the
entire frequency distribution table, we need to study the dispersion of data among the
various observations. For example, if there is a huge difference between the low income
group and the high income group, then we can conclude that there is a large income
disparity in the region.
Page : 89 , Block Name : Exercises
Q2 Which measure of dispersion is the best and how?
Answer. The best measure of dispersion is the Standard Deviation. It is because:
1. It can compare multiple quantities easily.
2. It takes into account every aspect and hence provides a comprehensive
assessment.
3. It is least affected by the change of samples.
4. It is independent of the origin.
5. It is algebraic in nature as it takes into account the positive and negative
deviations.
Page : 89 , Block Name : Exercises
Q3 Some measures of dispersion depend upon the spread of values whereas
some are estimated on the basis of the variation of values from a central
value. Do you agree?
Answer. Yes, some measures of dispersion depend upon the spread of values whereas
some are estimated on the basis of the variation of values from a central
Value. This is the reason the measures are classified into absolute and relative
measures. The absolute measures are the range, standard deviation, etc. whereas the
relative measures are coefficient of variation, coefficient of standard deviation, etc.
Page 1 of 21
Page 3
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
Page : 89 , Block Name : Exercises
Q4 In a town, 25% of the persons earned more than Rs 45,000 whereas
75% earned more than 18,000. Calculate the absolute and relative values
of dispersion.
Answer. The absolute value of dispersion will be quartile deviation.
Quartile Deviation =
=
= 13500
The relative value of dispersion will be Coefficient of Quartile Deviation.
Coefficient of Quartile Deviation =
=
= 0.478
Page : 89 , Block Name : Exercises
Q5 The yield of wheat and rice per acre for 10 districts of a state is as
under :
Distri 1 2 3 4 5 6 7 8 9 10
ct
Whea 12 10 15 19 21 16 18 9 25 10
r
Rice 22 29 13 23 18 15 12 34 18 12
Calculate for each crop,
(i) Range
Page 2 of 21
Page : 89 , Block Name : Exercises
Q4 In a town, 25% of the persons earned more than Rs 45,000 whereas
75% earned more than 18,000. Calculate the absolute and relative values
of dispersion.
Answer. The absolute value of dispersion will be quartile deviation.
Quartile Deviation =
=
= 13500
The relative value of dispersion will be Coefficient of Quartile Deviation.
Coefficient of Quartile Deviation =
=
= 0.478
Page : 89 , Block Name : Exercises
Q5 The yield of wheat and rice per acre for 10 districts of a state is as
under :
Distri 1 2 3 4 5 6 7 8 9 10
ct
Whea 12 10 15 19 21 16 18 9 25 10
r
Rice 22 29 13 23 18 15 12 34 18 12
Calculate for each crop,
(i) Range
Page 2 of 21
Page 4
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
(ii) Q.D.
(iii) Mean deviation about Mean
(iv) Mean deviation about Median
(v) Standard deviation
(vi) Which crop has greater variation?
(vii) Compare the values of different measures for each crop.
Answer.
(i) The range is the difference between the highest and the lowest values.
Wheat Range = 25-9
= 16
Rice Range = 34-12
= 22
(ii) To calculate Q.D., we need to calculate and .
For Wheat:
= th item
= 2.75th item
= 10
=3 th item
= 8.25th item
= 19.50
Quartile Deviation =
=
Page 3 of 21
(ii) Q.D.
(iii) Mean deviation about Mean
(iv) Mean deviation about Median
(v) Standard deviation
(vi) Which crop has greater variation?
(vii) Compare the values of different measures for each crop.
Answer.
(i) The range is the difference between the highest and the lowest values.
Wheat Range = 25-9
= 16
Rice Range = 34-12
= 22
(ii) To calculate Q.D., we need to calculate and .
For Wheat:
= th item
= 2.75th item
= 10
=3 th item
= 8.25th item
= 19.50
Quartile Deviation =
=
Page 3 of 21
Page 5
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
= 4.75
For Rice :
= th item
= 2.75th item
= 12
=3 th item
= 8.25th item
= 24.50
Quartile Deviation =
=
= 6.75
(iii) Mean of Wheat =
= 15.5
Tabulation of Values:
Quantity (X) Deviation
9 6.5
10 5.5
10 5.5
12 3.5
15 0.5
16 0.5
18 2.5
19 3.5
Page 4 of 21
= 4.75
For Rice :
= th item
= 2.75th item
= 12
=3 th item
= 8.25th item
= 24.50
Quartile Deviation =
=
= 6.75
(iii) Mean of Wheat =
= 15.5
Tabulation of Values:
Quantity (X) Deviation
9 6.5
10 5.5
10 5.5
12 3.5
15 0.5
16 0.5
18 2.5
19 3.5
Page 4 of 21
Page 6
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
21 5.5
25 9.5
Sum = 43
Deviation about Mean =
=
= 4.3
Mean of Rice=
= 19.5
Tabulation of Values:
Quantity (X) Deviation (d=X-X’)
12 7.5
12 7.5
12 7.5
15 4.5
18 1.5
18 1.5
22 2.5
23 3.5
29 9.5
34 14.5
Sum = 60
Deviation about Mean =
Page 5 of 21
21 5.5
25 9.5
Sum = 43
Deviation about Mean =
=
= 4.3
Mean of Rice=
= 19.5
Tabulation of Values:
Quantity (X) Deviation (d=X-X’)
12 7.5
12 7.5
12 7.5
15 4.5
18 1.5
18 1.5
22 2.5
23 3.5
29 9.5
34 14.5
Sum = 60
Deviation about Mean =
Page 5 of 21
Page 7
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
=
=6
(iv) Median of Wheat =
= 15.5
Tabulation of Values:
Quantity (X) Deviation (d=X-X’)
9 6.5
10 5.5
10 5.5
12 3.5
15 0.5
16 0.5
18 2.5
19 3.5
21 5.5
25 9.5
Sum = 43
Deviation about Median =
=
= 4.3
Median of Rice =
= 18
Page 6 of 21
=
=6
(iv) Median of Wheat =
= 15.5
Tabulation of Values:
Quantity (X) Deviation (d=X-X’)
9 6.5
10 5.5
10 5.5
12 3.5
15 0.5
16 0.5
18 2.5
19 3.5
21 5.5
25 9.5
Sum = 43
Deviation about Median =
=
= 4.3
Median of Rice =
= 18
Page 6 of 21
Page 8
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
Tabulation of Values:
Quantity (X) Deviation (d=X-X’)
12 6
12 6
12 6
15 3
18 0
18 0
22 4
23 5
29 11
34 16
Sum = 57
Deviation about Mean =
=
= 5.7
(v) For Wheat, the standard deviation can be calculated as
Wheat (X) d = X – X'
(X' = 15.5)
9 -6.5 42.25
10 -5.5 30.25
10 -5.5 30.25
Page 7 of 21
Tabulation of Values:
Quantity (X) Deviation (d=X-X’)
12 6
12 6
12 6
15 3
18 0
18 0
22 4
23 5
29 11
34 16
Sum = 57
Deviation about Mean =
=
= 5.7
(v) For Wheat, the standard deviation can be calculated as
Wheat (X) d = X – X'
(X' = 15.5)
9 -6.5 42.25
10 -5.5 30.25
10 -5.5 30.25
Page 7 of 21
Page 9
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
12 -3.5 12.25
15 -0.5 0.25
16 0.5 0.25
18 2.5 6.25
19 3.5 12.25
21 5.5 30.25
25 9.5 90.25
Therefore, =
=
= 5.04
For Rice:
as
Rice (Y) d = Y – Y'
(X' = 15.5)
12 -7.5 56.25
12 -7.5 56.25
12 -7.5 56.25
15 -4.5 20.25
18 -1.5 2.25
18 -1.5 2.25
Page 8 of 21
12 -3.5 12.25
15 -0.5 0.25
16 0.5 0.25
18 2.5 6.25
19 3.5 12.25
21 5.5 30.25
25 9.5 90.25
Therefore, =
=
= 5.04
For Rice:
as
Rice (Y) d = Y – Y'
(X' = 15.5)
12 -7.5 56.25
12 -7.5 56.25
12 -7.5 56.25
15 -4.5 20.25
18 -1.5 2.25
18 -1.5 2.25
Page 8 of 21
Page 10
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
22 2.5 6.25
23 3.5 12.25
29 9.5 90.25
34 14.5 210.25
Therefore, =
=
= 7.16
(vi) The Coefficient of Variation of Wheat can be calculated as:
CV =
CV =
CV = 32.51
The Coefficient of Variation of Rice can be calculated as:
0
CV =
CV =
CV = 36.71
Therefore, rice has greater variation.
(vii) The comparison can be made using the following table.
Page 9 of 21
22 2.5 6.25
23 3.5 12.25
29 9.5 90.25
34 14.5 210.25
Therefore, =
=
= 7.16
(vi) The Coefficient of Variation of Wheat can be calculated as:
CV =
CV =
CV = 32.51
The Coefficient of Variation of Rice can be calculated as:
0
CV =
CV =
CV = 36.71
Therefore, rice has greater variation.
(vii) The comparison can be made using the following table.
Page 9 of 21
Page 11
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
Parameter Wheat Rice
Range 16 22
QD 4.75 6.75
MD about Mean 4.3 6
MD about Median 4.3 5.7
SD 5.04 7.16
CV 32.51 36.71
Page : 89 , Block Name : Exercises
Q6 In the previous question, calculate the relative measures of variation
and indicate the value which, in your opinion, is more reliable.
Answer. The coefficient of range of wheat can be calculated as:
Coefficient of Range =
=
=
= 0.47
The coefficient of range of rice can be calculated as:
Coefficient of Range =
Page 10 of 21
Parameter Wheat Rice
Range 16 22
QD 4.75 6.75
MD about Mean 4.3 6
MD about Median 4.3 5.7
SD 5.04 7.16
CV 32.51 36.71
Page : 89 , Block Name : Exercises
Q6 In the previous question, calculate the relative measures of variation
and indicate the value which, in your opinion, is more reliable.
Answer. The coefficient of range of wheat can be calculated as:
Coefficient of Range =
=
=
= 0.47
The coefficient of range of rice can be calculated as:
Coefficient of Range =
Page 10 of 21
Page 12
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
=
=
= 0.48
The Coefficient of QD of wheat can be calculated as:
Coefficient of QD =
=
= 0.32
The Coefficient of QD of rice can be calculated as:
Coefficient of QD =
=
= 0.48
The Coefficient of mean deviation of wheat:
Coefficient of =
=
= 0.28
Coefficient of =
=
= 0.29
The Coefficient of mean deviation of rice:
Page 11 of 21
=
=
= 0.48
The Coefficient of QD of wheat can be calculated as:
Coefficient of QD =
=
= 0.32
The Coefficient of QD of rice can be calculated as:
Coefficient of QD =
=
= 0.48
The Coefficient of mean deviation of wheat:
Coefficient of =
=
= 0.28
Coefficient of =
=
= 0.29
The Coefficient of mean deviation of rice:
Page 11 of 21
Page 13
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
Coefficient of =
=
= 0.31
Coefficient of =
=
= 0.32
Page : 89 , Block Name : Exercises
Q7 A batsman is to be selected for a cricket team. The choice is between X
and Y on the basis of their scores in five previous tests which are :
X 25 85 40 80 120
Y 50 70 65 45 80
Which batsman should be selected if we want,
(i) a higher run getter, or
(ii) a more reliable batsman in the team?
Answer. The values of dispersion for batsman X are:
X x = X - X' where, X =
70
25 -45 2025
85 15 225
40 -30 900
80 10 100
120 50 2500
Page 12 of 21
Coefficient of =
=
= 0.31
Coefficient of =
=
= 0.32
Page : 89 , Block Name : Exercises
Q7 A batsman is to be selected for a cricket team. The choice is between X
and Y on the basis of their scores in five previous tests which are :
X 25 85 40 80 120
Y 50 70 65 45 80
Which batsman should be selected if we want,
(i) a higher run getter, or
(ii) a more reliable batsman in the team?
Answer. The values of dispersion for batsman X are:
X x = X - X' where, X =
70
25 -45 2025
85 15 225
40 -30 900
80 10 100
120 50 2500
Page 12 of 21
Page 14
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
= 5750
X' =
=
= 70
X X = X - X' where,
X' = 70
25 -45 2025
85 15 225
40 -30 900
80 10 100
120 50 2500
= 350 = 5750
CV =
The values of dispersion for batsman Y are:
Y y = Y - Y' where, Y' =
70
50 -12 144
70 8 64
Page 13 of 21
= 5750
X' =
=
= 70
X X = X - X' where,
X' = 70
25 -45 2025
85 15 225
40 -30 900
80 10 100
120 50 2500
= 350 = 5750
CV =
The values of dispersion for batsman Y are:
Y y = Y - Y' where, Y' =
70
50 -12 144
70 8 64
Page 13 of 21
Page 15
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
65 3 9
45 -17 289
80 18 324
= 310 = 830
CV =
Therefore, to get a higher run getter, we should select batsman X due to high mean.
(ii) As calculated from the values in part (i), we should select batsman Y for reliability
due to his lower variation.
Page : 89 , Block Name : Exercises
Q8 To check the quality of two brands of lightbulbs, their life in burning
hours was estimated as under for 100 bulbs of each brand.
Life (in hours) Brand A Brand B
0-50 15 2
50-100 20 8
100-150 18 60
150-200 25 25
200-250 22 5
Page 14 of 21
65 3 9
45 -17 289
80 18 324
= 310 = 830
CV =
Therefore, to get a higher run getter, we should select batsman X due to high mean.
(ii) As calculated from the values in part (i), we should select batsman Y for reliability
due to his lower variation.
Page : 89 , Block Name : Exercises
Q8 To check the quality of two brands of lightbulbs, their life in burning
hours was estimated as under for 100 bulbs of each brand.
Life (in hours) Brand A Brand B
0-50 15 2
50-100 20 8
100-150 18 60
150-200 25 25
200-250 22 5
Page 14 of 21
Page 16
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
100 100
(i) Which brand gives higher life?
(ii) Which brand is more dependable
Answer. The dispersion parameters can be calculated as:
For Brand A:
Life No. of Mid d=X–A fd'
Bulbs value
(f) (m)
0-50 15 25 –100 –2 4 –30 60
50-100 20 75 –50 –1 1 –20 20
100-150 18 125 = A 0 0 0 0 0
150-200 25 175 50 1 1 25 25
200-250 22 225 100 2 4 44 88
= = =
100 19 193
= 69
CV = =51.3
For Brand B:
Page 15 of 21
100 100
(i) Which brand gives higher life?
(ii) Which brand is more dependable
Answer. The dispersion parameters can be calculated as:
For Brand A:
Life No. of Mid d=X–A fd'
Bulbs value
(f) (m)
0-50 15 25 –100 –2 4 –30 60
50-100 20 75 –50 –1 1 –20 20
100-150 18 125 = A 0 0 0 0 0
150-200 25 175 50 1 1 25 25
200-250 22 225 100 2 4 44 88
= = =
100 19 193
= 69
CV = =51.3
For Brand B:
Page 15 of 21
Page 17
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
Life No. of Mid d=X–A fd'
Bulbs (f) value d' =
(m)
0-50 2 25 –100 –2 4 –4 8
50-100 8 75 –50 –1 1 –8 8
100-150 60 125 = A 0 0 0 0 0
150-200 25 175 50 1 1 25 25
200-250 5 225 100 2 4 10 10
= = =
100 23 61
= 37.5
CV = =27.47
(i) Brand B, due to higher mean.
(ii) Brand A, due to lower variation.
Page : 89 , Block Name : Exercises
Q9 Average daily wage of 50 workers of a factory was Rs 200 with a standard
deviation of Rs 40. Each worker is given a raise of Rs 20. What is the
new average daily wage and standard deviation? Have the wages become
more or less uniform?
Answer. The new average daily wage can be calculated as:
Page 16 of 21
Life No. of Mid d=X–A fd'
Bulbs (f) value d' =
(m)
0-50 2 25 –100 –2 4 –4 8
50-100 8 75 –50 –1 1 –8 8
100-150 60 125 = A 0 0 0 0 0
150-200 25 175 50 1 1 25 25
200-250 5 225 100 2 4 10 10
= = =
100 23 61
= 37.5
CV = =27.47
(i) Brand B, due to higher mean.
(ii) Brand A, due to lower variation.
Page : 89 , Block Name : Exercises
Q9 Average daily wage of 50 workers of a factory was Rs 200 with a standard
deviation of Rs 40. Each worker is given a raise of Rs 20. What is the
new average daily wage and standard deviation? Have the wages become
more or less uniform?
Answer. The new average daily wage can be calculated as:
Page 16 of 21
Page 18
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
N = 50
X = 50
= 40
X' =
200 =
50
= ₹10000
Now, increased wage rate = ₹20
Now,
= ₹11000
New X' =
=
= ₹220
N=50
=200
=40
200 =
Thus, = 200 50 = Rs 10,000
Now, increased wage rate = Rs 20
New =10,000+1,000 =Rs 11,000
New =Rs 220
Since standard deviation is independent of origin, it will not change at all.
Page 17 of 21
N = 50
X = 50
= 40
X' =
200 =
50
= ₹10000
Now, increased wage rate = ₹20
Now,
= ₹11000
New X' =
=
= ₹220
N=50
=200
=40
200 =
Thus, = 200 50 = Rs 10,000
Now, increased wage rate = Rs 20
New =10,000+1,000 =Rs 11,000
New =Rs 220
Since standard deviation is independent of origin, it will not change at all.
Page 17 of 21
Page 19
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
Page : 90 , Block Name : Exercises
Q10 If in the previous question, each worker is given a hike of 10% in wages,
how are the mean and standard deviation values affected?
Answer. If each worker is given a hike of 10%, the new values will be:
N = 50
X' = 200
= 40
Now,
New X' = 200 + 10% of ₹200
New X' = ₹220
Initial SD = ₹40
New SD = ₹40 + ₹4
New SD = ₹44
Page : 90 , Block Name : Exercises
Q11 Calculate the mean deviation using mean and Standard Deviation for
the following distribution.
Classes Frequencies
20-40 3
40-80 6
80-100 20
100-120 12
120-140 9
50
Answer. The tabulation of values can be done as:
Page 18 of 21
Page : 90 , Block Name : Exercises
Q10 If in the previous question, each worker is given a hike of 10% in wages,
how are the mean and standard deviation values affected?
Answer. If each worker is given a hike of 10%, the new values will be:
N = 50
X' = 200
= 40
Now,
New X' = 200 + 10% of ₹200
New X' = ₹220
Initial SD = ₹40
New SD = ₹40 + ₹4
New SD = ₹44
Page : 90 , Block Name : Exercises
Q11 Calculate the mean deviation using mean and Standard Deviation for
the following distribution.
Classes Frequencies
20-40 3
40-80 6
80-100 20
100-120 12
120-140 9
50
Answer. The tabulation of values can be done as:
Page 18 of 21
Page 20
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
Classes (f) Mid d=X–A d' = fd'
value
(m)
20-40 3 30 –60 –6 36 –18 108
40-80 6 60 –30 –3 9 –18 54
80-100 20 90 = A 0 0 0 0 0
100-120 12 110 20 2 2 24 48
120-140 9 130 40 4 16 36 144
= = 24 = 354
50
= 26.17
= =19.2
Page : 90 , Block Name : Exercises
Q12 The sum of 10 values is 100 and the sum of their squares is 1090. Find out the
coefficient of variation.
Answer. The coefficient of variation can be calculated as:
X = 100
N = 10
So, X' =
Page 19 of 21
Classes (f) Mid d=X–A d' = fd'
value
(m)
20-40 3 30 –60 –6 36 –18 108
40-80 6 60 –30 –3 9 –18 54
80-100 20 90 = A 0 0 0 0 0
100-120 12 110 20 2 2 24 48
120-140 9 130 40 4 16 36 144
= = 24 = 354
50
= 26.17
= =19.2
Page : 90 , Block Name : Exercises
Q12 The sum of 10 values is 100 and the sum of their squares is 1090. Find out the
coefficient of variation.
Answer. The coefficient of variation can be calculated as:
X = 100
N = 10
So, X' =
Page 19 of 21
Page 21
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
=
= 10
= 1090
=
=3
CV =
=100
N=10
So, =10
=1090
=
=
=3
CV =
=
=30
Page : 90 , Block Name : Exercises
Page 20 of 21
=
= 10
= 1090
=
=3
CV =
=100
N=10
So, =10
=1090
=
=
=3
CV =
=
=30
Page : 90 , Block Name : Exercises
Page 20 of 21
Page 22
Book : Statistics for Economics Ncert Solutions | Chapter - 6 Statistics
Page 21 of 21
Page 21 of 21
Document Details
| Board / Org | NCERT |
| Exam | Class 11 |
| Type | Solution |
| Pages | 22 |
| Updated | 30 Apr 2026 |