Statistics- Exercise 14.1 – Class 10

  1. A survey was conducted by a group of students as a part of their environment awareness program, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.
number of plants 0-1 2-4 4-6 6-10 8-10 10-12 12-14
number of houses 1 2 1 5 6 2 3

Which method did you use for finding the mean, and why?

Solution:

To find the class mark (xi) for each interval, the following relation is used.

Class mark 𝒙𝒊 = 𝑼𝒑𝒑𝒆𝒓 𝒍𝒊𝒎𝒊𝒕+𝑳𝒐𝒘𝒆𝒓 𝒍𝒊𝒎𝒊𝒕/𝟐

xi and fixi can be calculated as follows.

Number of plants number of houses xi fixi
0 – 2 1 1 1
2 – 4 2 3 2 x 3 = 6
4 – 6 1 5 1 x 5 = 5
6 – 8 5 7 5 x 7 = 35
8 – 10 6 9 6  x 9 = 54
10 – 12 2 11 2 x 11 = 22
12 – 14 3 13 3 x13 = 39
Total 20   162

From the table, it can be observed that

∑fi = 20

∑fixi = 162

Statistics- Exercise 14.1 – Class 10

= 162/20 = 8.1

Therefore, mean number of plants per house is 8.1.

Here, direct method has been used as the values of class marks (xi) and fi are small.


  1. Consider the following distribution of daily wages of 50 worker of a factory.
Daily wages(in Rs) 100-120 120-140 140-160 160-180 180-200
number of workers 12 14 8 6 10

Find the mean daily wages of the workers of the factory by using an appropriate method.

Solution:

To find the class mark for each interval, the following relation is used.

𝒙𝒊 = 𝑼𝒑𝒑𝒆𝒓 𝒍𝒊𝒎𝒊𝒕 + 𝑳𝒐𝒘𝒆𝒓 𝒍𝒊𝒎𝒊𝒕/𝟐

Class size (h) of this data = 20

Taking 150 as assured mean (a), di, ui, and fiui can be calculated as follows.

Daily wages(in  Rs) Number of workers(fi) xi di =  xi  – 150 ui=di/20 fiui
100 – 120 12 110 -40 -2 -24
120 -140 14 130 -20 -1 -14
140 – 160 8 150 0 0 0
160  – 180 6 170 20 1 6
180 – 200 10 190 40 2 20
total 50 -12

From the table, it can be observed that

∑fi = 50

∑fixi = -12

Statistics- Exercise 14.1 – Class 10

 

=150+(-12/50)20

 = 150 – 24/5

= 150 – 4.8 = 145.2

Therefore, mean number of plants per house is 145.20


  1. The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs.18. Find the missing frequency f.
Daily pocket allowance(in Rs) 11 – 13 13 – 15 15 – 17 17 – 19  19 – 21 21 – 23 23 – 25
Number of workers 7 6 9 13 f 5 4

Solution:

To find the class mark (xi) for each interval, the following relation is used.

𝒙𝒊 = 𝑼𝒑𝒑𝒆𝒓 𝒍𝒊𝒎𝒊𝒕 + 𝑳𝒐𝒘𝒆𝒓 𝒍𝒊𝒎𝒊𝒕/𝟐

Given that, mean pocket allowance,

Taking 18 as assured mean (a), di and fidi are calculated as follows.

Daily pocket allowance(in  Rs) Number of children (fi) class mark xi di =  xi  – 18 fidi fiui
11 – 13 7 12 -6 – 42 -24
13 -15 6 14 -4 -24 -14
15 – 17 9 16 -2 -18 0
17  – 19 13 18 0 0 6
19 – 21 f 20 2 2f 20
21 – 23 5 22 4 20 -12
23 – 25 4 24 6 24 24
total fi = 44 + f

 

2f – 40

From above,

fi = 44 + f

fiui = 2f – 40

 

Statistics- Exercise 14.1 – Class 10

18 = 18 +(2f-40)/(44+f)

2f – 40 = 0

2f = 40

f = 20

Hence, the missing frequency, f, is 20.


4: Thirty women were examined in a hospital by a doctor and the number of heart beats per minute were recorded and summarized as follows. Fine the mean heart beats per minute for these women, choosing a suitable method.

Number of heart beats per minute 65 – 68 68 – 71 71 – 74 74 – 77 77 – 80 80 – 83 83 – 86
Number of women 2 4 3 8 7 4 2

Solution:

To find the class mark of each interval (xi), the following relation is used.

𝒙𝒊 = 𝑼𝒑𝒑𝒆𝒓 𝒍𝒊𝒎𝒊𝒕 + 𝑳𝒐𝒘𝒆𝒓 𝒍𝒊𝒎𝒊𝒕/𝟐

Class size, h, of this data = 3

Taking 75.5 as assumed mean (a), di, ui, fiui are calculated as follows.

Number of heart beats per minute Number of women fi xi di = xi − 75 𝒖𝒊 = 𝒅𝒊/𝟑 fiui
65 – 68 2 66.5 -9 -3 -6
68 – 71 4 69.5 -6 -2 -8
71 – 74 3 72.5 -3 -1 -3
74 – 77 8 75.5 0 0 0
77 – 80 7 78.5 3 1 7
80 – 83 4 81.5 6 2 8
83 – 86 2 84.5 9 3 6
total 30 4

From the table

fi = 30

fiui = 4

Statistics- Exercise 14.1 – Class 10

= 75.5+(4/30)x3

= 75.5 + 4/30 x 3

= 75.5 – 0.4

= 75.9

Therefore, mean hear beats per minute for these women are 75.9 beats per minute.


  1. In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes.
Number of mangoes 50 – 52 53 – 55 56 – 58 59 – 61 62 – 64
Number of boxes 15 110 135 115 25

Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?

Solution:

Number of mangoes Number of boxes fi
50 – 52 15
53 – 55 110
56 – 58 135
59 – 61 115
62 – 64 25

It can be observed that class intervals are not continuous. There is a gap of 1 between two class intervals. Therefore, 1/2 has to be added to the upper class limit and 1/2 has to be subtracted from the lower class limit of each interval.

Class mark (xi) can be obtained by using the following relation.

𝒙𝒊 = 𝑼𝒑𝒑𝒆𝒓 𝒍𝒊𝒎𝒊𝒕 + 𝑳𝒐𝒘𝒆𝒓 𝒍𝒊𝒎𝒊𝒕/𝟐

Class size (h) of this data = 3

Here we are using Step deviation method

Taking 57 as assumed mean (a), di, ui, fiui are calculated as follows.

class interval fi xi di = xi-57 ui = di/3 fiui
49.5 – 52.5 15 51 -6 -2 -30
52.5 – 55.5 110 54 -3 -1 -110
55.5 – 58.5 135 57 0 0 0
58.5 – 61.5 115 60 3 1 115
61.5 – 64.5 25 63 6 2 50
total 400 25

fi = a + (∑fiui/∑fi)xh

= 57 + (25/400)x3

= 57 + 3/16 = 57 + 0.1875

= 57.1875 = 57.19

Mean number of mangoes kept in a packing box is 57.19.


6: The table below shows the daily expenditure on food of 25 households in a locality

Daily expenditure(in Rs) 100 – 150 150 – 200 200 – 250 250 – 300 300 – 350
Number of households 4 5 12 2 2

Find the mean daily expenditure on food by a suitable method.

Solution:

To find the class mark (xi) for each interval, the following relation is used.

𝒙𝒊 = 𝑼𝒑𝒑𝒆𝒓 𝒍𝒊𝒎𝒊𝒕 + 𝑳𝒐𝒘𝒆𝒓 𝒍𝒊𝒎𝒊𝒕/𝟐

Class size = 50

Taking 225 as assumed mean (a), di, ui, fiui are calculated as follows.

Daily expenditure(in Rs) fi xi di = xi – 225 ui == di/50 fiu­i
100 – 150 4 125 -100 -2 -8
150 – 200 5 175 -50 -1 -5
200 – 250 12 225 0 0 0
250 – 300 2 275 50 1 2
300 – 350 2 325 100 2 4
total 25 -7

fi = 25

fiui = a + (fiui/ ∑fi)xh

= 225 + (-7/25)x50

= 225 – 14

= 211

Therefore, mean daily expenditure on food is Rs 211.


  1. To find out the concentration of SO2 in the air (in parts per million, i.e., ppm), the data was collected for 30 localities in a certain city and is presented below:
concentration of SO2 Frequency
00.0 – 0.04 4
0.04 – 0.08 9
0.08 – 0.12 9
0.12 – 0.16 2
0.16 – 0.20 4
0.20 – 0.24 2

Find the mean concentration of SO2 in the air.

Solution:

To find the class marks for each interval, the following relation is used.

𝒙𝒊 = 𝑼𝒑𝒑𝒆𝒓 𝒍𝒊𝒎𝒊𝒕 + 𝑳𝒐𝒘𝒆𝒓 𝒍𝒊𝒎𝒊𝒕/𝟐

Class size of this data = 0.04

Taking 0.14 as assumed mean (a), di, ui, fiui are calculated as follows.

Concentration of SO2 frequency fi class mark xi di = xi – 0.14 ui = di/0.04 fiui
0.00-0.04 4 0.02 -0.12 -3 -12
0.04-0.08 9 0.06 -0.08 -2 -18
0.08-0.12 9 0.10 -0.04 -1 -9
0.12-0.16 2 0.14 0 0 0
0.16-0.20 4 0.18 0.04 1 4
0.20-0.24 2 0.22 0.08 2 4
total 30       -31

∑fi = 30

∑fiui = -31

Statistics- Exercise 14.1 – Class 10

 

=0.14 +(-31/30)(0.04)

=0.14 – 0.04133

=0.09867

=0.099ppm

Therefore, mean concentration of SO2 in the air is 0.099 ppm.


  1. A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent.
Number of days 0-6 6-10 10-14 14-20 20-28 28-38 38-49
number of students 11 10 7 4 4 3 1

Solution:

To find the class mark of each interval, the following relation is used.

𝒙𝒊 = 𝑼𝒑𝒑𝒆𝒓 𝒍𝒊𝒎𝒊𝒕 + 𝑳𝒐𝒘𝒆𝒓 𝒍𝒊𝒎𝒊𝒕/𝟐

Taking 17 as assumed mean (a), di and fidi are calculated as follows.

Number of days number of students fi xi di = xi – 17 fidi
0-6 11 3 -14 -154
6-10 10 8 -9 -90
10-14 7 12 -5 -35
14-20 4 17 0 0
20-28 4 24 7 28
28-38 3 33 16 48
38-40 1 39 22 22
total 40     -181

∑fi = 40

∑fidi = -181

Statistics- Exercise 14.1 – Class 10

= 17 + (-181/40)

= 17 – 4.525

= 12.475

= 12.48

Therefore, the mean number of days is 12.48 days for which a student was absent.


9: The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.

Literacy rate(in%) 45-55 55-65 65-75 75-85 85-95
number of cities 3 10 11 8 3

Solution:

To find the class marks, the following relation is used.

𝒙𝒊 = 𝑼𝒑𝒑𝒆𝒓 𝒍𝒊𝒎𝒊𝒕 + 𝑳𝒐𝒘𝒆𝒓 𝒍𝒊𝒎𝒊𝒕/𝟐

Class size (h) for this data = 10 Taking 70 as assumed mean (a), di, ui, and fiui are calculated as follows.

Literacy rate(in%) number of cities fi xi di = xi – 17 ui = di/10 fidi
45-55 3 50 -20 -2 -6
55-65 10 60 -10 -1 -10
65-75 11 70 0 0 0
75-85 8 80 10 1 8
85-95 3 90 20 2 6
total 35       -2

∑fi = 35

∑fidi = -2

Statistics- Exercise 14.1 – Class 10

= 70 + (-2/35)x10

= 17 – 20/35 = 17 – 4/7

= 70 – 0.57

= 69.43

Therefore, mean literacy rate is 69.43%


 

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