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Page No 71:
Question 1:
Which average would be suitable in the following cases?
(i) Average size of readymade garments.
(ii) Average intelligence of students in a class.
(iii) Average production in a factory per shift.
(iv) Average wage in an industrial concern.
(v) When the sum of absolute deviations from average is least.
(vi) When quantities of the variable are in ratios.
(vii) In case of openended frequency distribution.
Answer:
(i) The demand for the average size of any readymade garment is the maximum. As, the modal value represents the value with the highest frequency, so the number of the average size to be produced is given by the Modal value.
(ii) Median will be the best measure for calculating the average intelligence of students in a class. It is the value that divides the series into two equal parts. So, number of students below and above the average intelligence can easily be estimated by median.
(iii) It is advisable to use mean for calculating the average production in a factory per shift. The average production is best calculated by arithmetic mean.
(iv) Mean will be the most suitable measure. It is calculated by dividing the sum of wages of all the labour by the total number of labours in the industry.
(v) When the sum of absolute deviations from average is the least, then mean could be used to calculate the average. This is an important mathematical property of arithmetic mean. The algebraic sum of the deviations of a set of n values from A.M. is 0.
(vi) Median will be the most suitable measure in case the variables are in ratios. It is least affected by the extreme values.
(vii) In case of open ended frequency distribution, Median is the most suitable measure as it can be easily computed. Moreover, the median value can be estimated even in case of incomplete statistical series.
Page No 71:
Question 2(i):
The most suitable average for qualitative measurement is
(a) arithmetic mean
(b) median
(c) mode
(d) geometric mean
(e) none of the above
Answer:
Median is the most suitable average for qualitative measurement. This is because Median divides a series in two equal parts.
Page No 72:
Question 2(ii):
Which average is affected most by the presence of extreme items?
(a) median
(b) mode
(c) arithmetic mean
(d) none of the above
Answer:
Arithmetic mean is the most affected by the presence of extreme items. It is one of the prime demerits of the arithmetic mean. It is easily distorted by the extreme values, and also the value of arithmetic mean may not figure out at all in the series.
Page No 72:
Question 2(iii):
The algebraic sum of deviation of a set of n values from A.M. is
(a) n
(b) 0
(c) 1
(d) none of the above
Answer:
The algebraic sum of deviation of a set of n values from A.M. is zero. This is one of the mathematical properties of arithmetic mean.
Page No 72:
Question 3:
Comment whether the following statements are true or false.
(i) The sum of deviation of items from median is zero.
(ii) An average alone is not enough to compare series.
(iii) Arithmetic mean is a positional value.
(iv) Upper quartile is the lowest value of top 25% of items.
(v) Median is unduly affected by extreme observations.
Answer:
(i) The sum of deviation of items from median is zero. False
The statement is false. This mathematical property applies to the arithmetic mean that states that the sum of the deviation of all items from the mean is zero.
(ii) An average alone is not enough to compare series. True
An average indicates only the behaviour of a particular series. Therefore, in order to measure the extent of divergence of different items from the central tendency is measured by dispersion. So, average is not enough to compare the series.
(iii) Arithmetic mean is a positional value. False
This statement is false as mean is not a positional average, rather the statement holds true for median and mode. The calculation of median and modal values is based on the position of the items in the series, i.e. why these are also termed as positional averages.
(iv) Upper quartile is the lowest value of top 25% of items. True
The value that divides a statistical series into four equal parts, the end value of each part is called quartile. The third quartile or the upper quartile has 75 % of the items below it and 25 % of items above it,
(v) Median is unduly affected by extreme observations. False
This statement is true for Arithmetic mean. Arithmetic mean is most affected by the presence of extreme items. It is one of the prime demerits of the arithmetic mean. It is easily distorted by the extreme values, and also the value of arithmetic mean may not figure out at all in the series.
Page No 72:
Question 4:
If the arithmetic mean of the data given below is 28, find (a) the missing frequency, and (b) the median of the series:
Profit per retail shop (in Rs) 
0 – 10 
10 – 20 
20 – 30 
30 – 40 
40 – 50 
50 – 60 
Number of retail shops 
12 
18 
27 
– 
17 
6 
Answer:
(i) Let the missing frequency be f_{1 }
Arithmetic Mean = 28
Profit per Retail Shop (in Rs) 
No of Retail Shops 
Mid Value 

Class Interval 
(f) 
(m) 
fm 
0 – 10 
12 
5 
60 
10 – 20 
18 
15 
270 
20 – 30 
27 
25 
675 
30 – 40 
f_{1} 
35 
35f_{1} 
40 – 50 
17 
45 
765 
50 – 60 
6 
55 
330 


or, 2240 + 28f_{1} = 2100 + 35f_{1}
or, 2240 – 2100 = 35f_{1} – 28f_{1}
or, 140 = 7f_{1}
f_{1} = 20
(ii)

Class Interval 
Frequency
(f) 
Cumulative Frequency (CF) 

0 – 10 
12 
12 

10 – 20 
18 
30 

20 – 30 
27 
57 

30 – 40 
20 
77 

40 – 50 
17 
94 

50 – 60 
6 
100 

Total 

So, the Median class = Size of item
= 50^{th} item
50^{th} item lies in the 57^{th} cumulative frequency and the corresponding class interval is 20 – 30.
Page No 72:
Question 5:
The following table gives the daily income of ten workers in a factory. Find the arithmetic mean.
Workers 
A 
B 
C 
D 
E 
F 
G 
H 
I 
J 
Daily Income (in Rs) 
120 
150 
180 
200 
250 
300 
220 
350 
370 
260 
Answer:
Workers 
Daily Income (in Rs) (X) 
A 
120 
B 
150 
C 
180 
D 
200 
E 
250 
F 
300 
G 
220 
H 
350 
I 
370 
J 
260 
Total 
N = 10
Arithmetic mean = Rs 240
Page No 72:
Question 6:
Following information pertains to the daily income of 150 families. Calculate the arithmetic mean.
Income (in Rs) 
Number of families 
More than 75 
150 
More than 85 
140 
More than 95 
115 
More than 105 
95 
More than 115 
70 
More than 125 
60 
More than 135 
40 
More than 145 
25 
Answer:
Income 
No. of families 
Frequency 
Mid Value 
fm 
Class Interval 
(CF) 
(f) 
(m) 

75 – 85 
150 
150 – 140 = 10 
80 
800 
85 – 95 
140 
140 – 115 = 25 
90 
2250 
95 – 105 
115 
115 – 95 = 20 
100 
2000 
105 – 115 
95 
95 – 70 = 25 
110 
2750 
115 – 125 
70 
70 – 60 = 10 
120 
1200 
125 – 135 
60 
60 – 40 = 20 
130 
2600 
135 – 145 
40 
40 – 25 = 15 
140 
2100 
145 – 155 
25 
25 
150 
3750 
Total 


= Rs 116.33
Page No 73:
Question 7:
The size of land holdings of 380 families in a village is given below. Find the median size of land holdings.
Size of Land Holdings (in acres) 
Less than 100 
100 – 200 
200 – 300 
300 – 400 
400 and above 
Number of families 
40 
89 
148 
64 
39 
Answer:
Size of Land Holdings Class Interval 
No. of Families
(f) 
Cumulative Frequency (CF) 
0 – 100 
40 
40 
100 – 200 
89 
129 
200 – 300 
148 
277 
300 – 400 
64 
341 
400 – 500 
39 
380 
Total 

So, the Median class = Size of item = 190^{th} item
190^{th} item lies in the 129^{th} cumulative frequency and the corresponding class interval is 200 – 300.
Median size of land holdings = 241.22 acres
Page No 73:
Question 8:
The following series relates to the daily income of workers employed in a firm. Compute (a) highest income of lowest 50% workers (b) minimum income earned by the top 25% workers and (c) maximum income earned by lowest 25% workers.
Daily Income (in Rs) 
10 – 14 
15 – 19 
20 – 24 
25 – 29 
30 – 34 
35 – 39 
Number of workers 
5 
10 
15 
20 
10 
5 
(Hint: Compute median, lower quartile and upper quartile)
Answer:
Daily Income (in Rs) Class Interval 
No. of Workers (f) 
Cumulative frequency (CF) 
9.5 – 14.5 
5 
5 
14.5 – 19.5 
10 
15 
19.5 – 24.5 
15 
30 
24.5 – 29.5 
20 
50 
29.5 – 34.5 
10 
60 
34.5 – 39.5 
5 
65 


(a) Highest income of lowest 50% workers
32.5^{th} item lies in the 50^{th} cumulative frequency and the corresponding class interval is 24.5 – 29.5.
(b) Minimum income earned by top 25% workers
In order to calculate the minimum income earned by top 25% workers, we need to ascertain Q_{3}.
48.75^{th} item lies in 50^{th} item and the corresponding class interval is 24.5 – 29.5.
(c) Maximum income earned by lowest 25% workers
In order to calculate the maximum income earned by lowest 25% workers, we need to ascertain Q_{1}.
16.25^{th} item lies in the 30^{th} cumulative frequency and the corresponding class interval is 19.5 – 24.5
Page No 73:
Question 9:
The following table gives production yield in kg. per hectare of wheat of 150 farms in a village. Calculate the mean, median and mode values.
Production yield (kg. per hectare) 
50 – 53 
53 – 56 
56 – 59 
59 – 62 
62 – 65 
65 – 68 
68 – 71 
71 – 74 
74 – 77 
Number of farms 
3 
8 
14 
30 
36 
28 
16 
10 
5 
Answer:
(i) Mean
Production Yield 
No. of farms 
Mid value 
A = 63.5 

50 – 53 
3 
51.5 
–12 
–4 
–12 
53 – 56 
8 
54.5 
–9 
–3 
–24 
56 – 59 
14 
57.5 
–6 
–2 
–28 
59 – 62 
30 
60.5 
–3 
–1 
–30 
62 – 65 
36 
63.5 
0 
0 
0 
65 – 68 
28 
66.5 
+3 
+1 
28 
68 – 71 
16 
69.5 
+6 
+2 
32 
71 – 74 
10 
72.5 
+9 
+3 
30 
74 – 77 
5 
75.5 
+12 
+4 
20 
Total 



= 63.5 + 0.32
= 63.82 kg per hectare
(ii) Median
Class Interval 
Frequency (f) 
CF 
50 – 53 
3 
3 
53 – 56 
8 
11 
56 – 59 
14 
25 
59 – 62 
30 
55 
62 – 65 
36 
91 
65 – 68 
28 
119 
68 – 71 
16 
135 
71 – 74 
10 
145 
74 – 77 
5 
150 
Total 

75^{th} item lies in the 91^{st} cumulative frequency and the corresponding class interval is 62 – 65.
(iii) Mode
Grouping Table




Class Interval 
I 
II 
III 
IV 
V 
VI 











50 – 53 
3 






11 
22 
25 
52 

53 – 56 
8 







56 – 59 
14 






44 



59 – 62 
30 







62 – 65 






44 

54 

65 – 68 
28 







68 – 71 
16 






26 
15 
31 



71 – 74 
10 







74 – 77 
5 








Analysis Table
Column 
50 – 53 
53 – 56 
56 – 59 
59 – 62 
62 – 65 
65 – 68 
68 – 71 
71 – 74 
74 – 77 
I 




√ 




II 




√ 
√ 



III 



√ 
√ 




IV 



√ 
√ 
√ 



V 




√ 
√ 
√ 


VI 


√ 
√ 
√ 




Total 
– 
– 
1 
3 
6 
3 
1 
– 
– 
Modal class = 62 – 65
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