24. Chapter 24. Data and Their Representations Angles NIOS Class 10 Mathematics Textbook Solutions
Chapter 24. Data and Their Representations NIOS Class 10 Mathematics Textbook Solutions for Exam Preparation
Chapter 24. DATA AND THEIR REPRESENTATIONS
CHECK YOUR PROGRESS 24.1
1. Fill in the blanks with suitable word(s) so that the following sentences give the proper meaning:
(a) Statistics, in singular sense, means the subject which deals with _______, _____, analysis of data as well as drawing of meaningful _______ from the data.
(b) Statistics is used, in a plural sense, meaning _______________.
(c) The data are said to be __________ if the investigator himself is responsible for its collection.
(d) Data taken from governmental or private agencies in the form of published reports are called __________ data.
(e) Statistics is the science which deals with collection, organisation, analysis and interpretation of the ____________.
Answer: (a) Statistics, in singular sense, means the subject which deals with classification, organisation , analysis of data as well as drawing of meaningful inferences from the data.
(b) Statistics is used, in a plural sense, meaning numerical data .
(c) The data are said to be primary if the investigator himself is responsible for its collection.
(d) Data taken from governmental or private agencies in the form of published reports are called secondary data.
(e) Statistics is the science which deals with collection, organisation, analysis and interpretation of the numerical data .
2. Javed wanted to know the size of shoes worn by the maximum number of persons in a locality. So, he goes to each and every house and notes down the information on a sheet. The data so collected is an example of ___________ data.
Answer: Javed wanted to know the size of shoes worn by the maximum number of persons in a locality. So, he goes to each and every house and notes down the information on a sheet. The data so collected is an example of primary data.
3. To find the number of absentees in each day of each class from I to XII, you collect the information from the school records. The data so collected is an example of ___________ data.
Answer: To find the number of absentees in each day of each class from I to XII, you collect the information from the school records. The data so collected is an example of secondary data.
CHECK YOUR PROGRESS 24.2
1. Give an example of a raw data and an arrayed data.
Answer: Raw data: Raw data is the data collected in its original form, without any arrangement.
For example: Marks obtained by 10 students in a test: 18, 12, 15, 20, 16, 14, 19, 13, 17, 11
Arrayed data: When the same data is arranged in ascending or descending order, it is called arrayed data.
For example (Ascending order): 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 .
2. Heights (in cm) of 30 girls in Class IX are given below:
140 140 160 139 153 146 151 150 150 154 148 158 151 160 150 149 148 140
148 153 140 139 150 152 149 142 152 140 146 148
Determine the range of the data.
Solution: We arrange the numbers in ascending order:
139, 139, 140, 140, 140, 140, 140, 142, 146, 146, 148, 148, 148, 149, 149, 150, 150, 150, 150, 150, 151, 151, 152, 152, 153, 153, 154, 158, 160, 160 .
Therefore, the range of the data = 160 – 139 = 21 (in cm).
3. Differentiate between a primary data and secondary data.
Answer: The difference between a primary data and secondary data :
|
Primary Data |
Secondary Data |
|
Primary data is the data collected first-hand by the investigator for a specific purpose. |
Secondary data is the data that has already been collected by someone else and is used by the investigator. |
|
It is original data. |
It is not original to the investigator. |
|
For example: A teacher collects marks of students in a class for a survey. |
For example: Population data taken from the Census report or a government publication. |
4. 30 students of Class IX appeared for mathematics olympiad. The marks obtained by them are given as follows:
46 31 74 68 42 54 14 93 72 53 59 38 16 88 27 44 63 43 81 64
77 62 53 40 71 60 8 68 50 58
Construct a grouped frequency distribution of the data using the classes 0 – 9 , 10 – 19 etc. Also, find the number of students who secured marks more than 49.
Solution: We construct the frequency distribution table :
|
Marks |
Number of students |
|
0 – 9 10 – 19 20 – 29 30 – 39 40 – 49 50 – 59 60 – 69 70 – 79 80 – 89 90 – 99 |
1 2 1 2 5 6 6 4 2 1 |
|
Total |
30 |
Therefore, the number of students who secured marks more than 49 is 19 .
5. Construct a frequency table with class intervals of equal sizes using 250 – 270 (270 not included) as one of the class interval for the following data:
268 230 368 248 242 310 272 342 310 300 300 320 315 304 402 316 406 292 355 248 210 240 330 316 406 215 262 238
Solution: We construct a frequency distribution table :
|
Class interval |
Frequency |
|
210 – 230 230 – 250 250 – 270 270 – 290 290 – 310 310 – 330 330 – 350 350 – 370 370 – 390 390 – 410 |
2 5 2 2 4 6 2 2 0 3 |
|
Total |
25 |
6. Following is the frequency distribution of ages (in years) of 40 teachers in a school:
|
Age (in years) |
Number of teachers |
|
25 – 31 31 – 37 37 – 43 43 – 49 49 – 55 |
12 15 7 5 1 |
|
Total |
40 |
(i) What is the class size?
(ii) What is the upper class limit of class 37 – 43 ?
(iii) What is the lower class limit of class 49 – 55 ?
Solution: (i) The class size is 6 .
(ii) The upper class limit of class 37 – 43 is 43 .
(iii) The lower class limit of class 49 – 55 is 49 .
CHECK YOUR PROGRESS 24.3
1. Construct a cumulative frequency distribution for each of the following distributions:
(i)
|
Classes |
1 – 5 |
6 – 10 |
11 – 15 |
16 – 20 |
21 – 25 |
26 – 30 |
|
Frequency |
4 |
6 |
10 |
13 |
6 |
2 |
(ii)
|
Classes |
0 – 10 |
10 – 20 |
20 – 30 |
30 – 40 |
40 – 50 |
50 – 60 |
|
Frequency |
3 |
10 |
24 |
32 |
9 |
7 |
Solution: We construct the cumulative frequency distribution table
|
Classes |
Frequency |
Cumulative frequency |
|
1 – 5 6 – 10 11 – 15 16 – 20 20 – 25 26 – 30 |
4 6 10 13 6 2 |
4 10 20 33 39 41 |
|
Total |
41 |
|
(ii) We construct the cumulative frequency distribution table
|
Classes |
Frequency |
Cumulative frequency |
|
0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 |
3 10 24 32 9 7 |
3 13 37 69 78 85 |
|
Total |
85 |
|
2. Construct a cumulative frequency distribution from the following data:
|
Heights (in cm) |
110 – 120 |
120 – 130 |
130 – 140 |
140 – 150 |
150 – 160 |
Total |
|
Number of students |
14 |
30 |
60 |
42 |
14 |
160 |
How many students have their heights less than 150 cm ?
Solution: We construct the cumulative frequency distribution table:
|
Heights (in cm) |
Number of students |
Cumulative frequency |
|
110 – 120 120 – 130 130 – 140 140 – 150 150 – 160 |
14 30 60 42 14 |
14 44 104 146 160 |
|
Total |
160 |
|
There are 14 students whose heights are less than 150 cm.
CHECK YOUR PROGRESS 24.4
1. Fill in the blanks:
(i) A bar graph is a graphical representation of numerical data using _______ of equal width.
(ii) In a bar graph, bars are drawn with _________ spaces in between them.
(iii) In a bar graph, heights of rectangles are _________ to their respective frequencies.
Answer: (i) A bar graph is a graphical representation of numerical data using bars of equal width.
(ii) In a bar graph, bars are drawn with equal spaces in between them.
(iii) In a bar graph, heights of rectangles are proportional to their respective frequencies.
2. The following bar graph shows how the members of the staff of a school come to school.
Photo
Study the bar graph and answer the following questions:
(i) How many members of staff come to school on bicycle?
(ii) How many member of staff come to school by bus?
(iii) What is the most common mode of transfport of the members of staff ?
Solution: (i) There are 2 members of staff who come to school by bicycle.
(ii) There are 6 members of staff who come to school by bus.
(iii) The most common mode of transport for the members of staff is by bus.
3. The bar graph given below shows the number of players in each team of 4 given games:
Photo
Read the bar graph and answer the following questions:
(i) How many players play in the volley ball team?
(ii) Which game is played by the maximum number of players?
(iii) Which game is played by only 3 players?
Solution: (i) There are 6 players in the volleyball team.
(ii) Football is played by the maximum number of players.
(iii) Table tennis is played by only 3 players.
4. The following bar graph shows the number of trees planted by an agency in different years:
Photo
Study the above bar graph and answer the following questions:
(i) What is the total number of trees planted by the agency from 2003 to 2008?
(ii) In which year is the number of trees planted the maximum?
(iii) In which year is the number of trees planted the minimum?
(iv) In which year, the number of trees planted is less than the number of trees planted in the year preceding it?
Solution: (i) The total number of trees planted by the agency from 2003 to 2008 = 200 + 400 + 600 + 800 + 1000 + 1200 + 1400 + 1600 = 5900 .
(ii) The number of trees planted the maximum is 2007 (with 1600 trees).
(iii) The number of trees planted the minimum is 2003 (with 400 trees) .
(iv) The number of trees planted in 2007 is less than the number of trees planted in 2008.
5. The expenditure of a company under different heads (in lakh of rupees) for a year is given below:
|
Head |
Expenditure (in lakhs of rupees) |
|
Salary of employees Travelling allowances Electricity and water Rent Others |
200 100 50 125 150 |
Construct a bar chart to represent this data.
Solution: Photo in mobile
CHECK YOUR PROGRESS 24.5
1. Fill in the blanks:
(i) In a histogram, the class intervals are generally taken along ________.
(ii) In a histogram, the class frequencies are generally taken along _______.
(iii) In a histogram, the areas of rectangles are proportional to the _______ of the respective classes.
(iv) A histogram is a graphical representation of a __________.
Answer: (i) In a histogram, the class intervals are generally taken along horizontal axis .
(ii) In a histogram, the class frequencies are generally taken along vertical axis .
(iii) In a histogram, the areas of rectangles are proportional to the frequency of the respective classes.
(iv) A histogram is a graphical representation of a continuous grouped frequency distribution.
2. The daily earnings of 26 workers are given below:
|
Daily earnings (in Rs) |
150 – 200 |
200 – 250 |
250 – 300 |
300 – 350 |
350 – 400 |
|
Number of workers |
4 |
8 |
5 |
6 |
3 |
Draw a histogram to represent the data.
Solution: Photo in mobile
3. Draw a frequency polygon for the data in Question 2 above by
(i) drawing a histogram (ii) without drawing a histogram
Solution: (i) We construct the frequency polygon table :
|
Daily earnings (in Rs) |
Class marks |
Number of workers |
|
150 – 200 200 – 250 250 – 300 300 – 350 350 – 400 |
175 225 275 325 375 |
4 8 5 6 3 |
(ii) We construct the frequency polygon table :
|
Daily earnings (in Rs) |
Class marks |
Number of workers |
|
150 – 200 200 – 250 250 – 300 300 – 350 350 – 400 |
175 225 275 325 375 |
4 8 5 6 3 |
4. Observe the histogram given below and answer the following questions:
(i) What information is given by the histogram?
(ii) In which class (group) is the number of students maximum?
(iii) How many students have the height of 145 cm and above?
(iv) How many students have the height less than 140 cm?
(v) How many students have the height more than or equal to 140 but less than 155?
Photo
Solution: (i) The histogram gives information about the heights (in cm) of the students.
(ii) The maximum number of students are in the height group 145–150 cm.
(iii) There are 15 students whose heights are 145 cm and above.
(iv) There are 4 students whose heights are less than 140 cm.
(v) There are 13 students whose heights are more than or equal to 140 cm but less than 155 cm.
TERMINAL EXERCISE
1. Fill in the blanks by appropriate words/phrases to make each of the following statements true:
(i) When the data are condensed in classes of equal size with frequencies, they are called ________ data and the table is called _______ table.
(ii) When the class limits are adjusted to make them continuous, the class limits are renamed as ________.
(iii) The number of observations falling in a particular class is called its _______.
(iv) The difference between the upper limit and lower limit of a class is called _________.
(v) The sum of frequencies of a class and all classes prior to that class is called ________ frequency of that class.
(vi) Class size = Difference between ________ and _____ of the class.
(vii) The raw data arranged in ascending or descending order is called an _______ data.
(viii) The difference between the maximum and minimum observations occuring in the data is called the _________ of the raw data.
Answer: (i) When the data are condensed in classes of equal size with frequencies, they are called group data and the table is called frequency table.
(ii) When the class limits are adjusted to make them continuous, the class limits are renamed as true limits .
(iii) The number of observations falling in a particular class is called its frequency .
(iv) The difference between the upper limit and lower limit of a class is called class size .
(v) The sum of frequencies of a class and all classes prior to that class is called cumulative frequency frequency of that class.
(vi) Class size = Difference between upper limit and lower limit of the class.
(vii) The raw data arranged in ascending or descending order is called an arrayed data.
(viii) The difference between the maximum and minimum observations occuring in the data is called the range of the raw data.
2. The number of TV sets in each of 30 households are given below:
1, 2, 2, 4, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 3 , 1 , 2 , 2 , 1 , 2 , 0 , 3 , 3 , 1 , 2 , 1 , , 1 , 0 , 1 , 1
Construct a frequency table for the data.
Solution: We have,
|
Number of TV sets |
Number of hours |
|
0 1 2 3 4 |
2 15 8 4 1 |
|
Total |
30 |
3. The number of vehicles owned by each of 50 families are listed below:
2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 0, 1, 1, 2, 3, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 1, 0, 1, 2, 1, 2, 1, 1, 4, 1 , 3, 1, 1, 1, 2, 2, 2, 2, 1, 1, 3, 2, 1, 2
Construct a frequency distribution table for the data.
Solution: We have,
|
Number of vehicles |
Number of families |
|
0 1 2 3 4 |
2 27 16 4 1 |
|
Total |
50 |
4. The weight (in grams) of 40 New Year’s cards were found as:
10.4 6.3 8.7 7.3 8.8 9.1 6.7 11.1 14.0 12.2 11.3 9.4 8.6 7.1 8.4 10.0 9.1 8.8 10.3 10.2 7.3 8.6 9.7 10.9 13.6 9.8 8.9 9.2 10.8 9.4 6.2 8.8 9.4 9.9 10.1 11.4 11.8 11.2 10.1 8.3
Prepare a grouped frequency distribution using the class 5.5 - 7.5 , 7.5 - 9.5 etc.
Solution: We have,
|
Weight (in grams) |
Number of cards |
|
5.5 – 7.5 7.5 – 9.5 9.5 – 11.5 11.5 – 13.5 13.5 – 15.5 |
6 15 15 2 2 |
|
Total |
40 |
5. The lengths, in centimetres, to the nearest centimeter of 30 carrots are given below:
15 21 20 10 18 18 16 18 20 20 18 16 13 15 15 16 13 14 14 16 12 15 17 12 14 15 13 11 14 17
Construct a frequency table for the data using equal class sizes and taking one class as 10 – 12 (12 excluded).
Solution: We have,
|
Length (in cm) |
Number of carrots |
|
10 – 12 12 -14 14 – 16 16 – 18 18 – 20 20 – 22 |
2 5 9 6 4 4 |
|
Total |
30 |
6. The following is the distribution of weights (in kg) of 40 persons:
|
Weight |
40 – 45 |
45 – 50 |
50 – 55 |
55 – 60 |
60 – 65 |
65 – 70 |
Total |
|
Number of persons |
4 |
5 |
10 |
7 |
6 |
8 |
40 |
(i) Determine the class marks of the classes 40 – 45 , 45 – 50 etc.
(ii) Construct a cumulative frequency table.
Solution: (i) We have,
|
Weight |
Class marks |
Number of person |
|
40 – 45 45 – 50 50 – 55 55 – 60 60 – 65 65 – 70 |
42.5 47.5 52.5 57.5 62.5 67.5 |
4 5 10 7 6 8 |
|
Total |
|
40 |
(ii) We Construct a cumulative frequency table:
|
Weight |
Number of person |
Cumulative frequency |
|
40 – 45 45 – 50 50 – 55 55 – 60 60 – 65 65 – 70 |
4 5 10 7 6 8 |
4 9 19 26 32 40 |
|
Total |
40 |
|
7. The class marks of a distribution and the corresponding frequencies are given below:
|
Class marks |
5 |
15 |
25 |
35 |
45 |
55 |
65 |
75 |
|
Frequency |
2 |
6 |
10 |
15 |
12 |
8 |
5 |
2 |
Determine the frequency table and construct the cumulative frequency table.
Solution: We have,
|
Weight |
Number of person |
Cumulative frequency |
|
5 15 25 35 45 55 65 75 |
2 6 10 15 12 8 5 2 |
2 8 18 33 45 63 68 60 |
|
Total |
60 |
|
8. For the following frequency table
|
Classes |
15 – 20 |
20 – 25 |
25 – 30 |
30 – 35 |
35 – 40 |
40- 45 |
45 – 50 |
Total |
|
Frequency |
2 |
6 |
10 |
15 |
12 |
8 |
5 |
25 |
(i) Write the lower limit of the class 15 – 20.
(ii) Write the class limits of the class 25 – 30.
(iii) Find the class mark of the class 35 – 40.
(iv) Determine the class size.
(v) Form a cumulative frequency table.
Solution: (i) The lower limit of the class 15 – 20 is 15 .
(ii) The lower class limit is 25 and upper class limit is 30.
(iii) The lower class limit is 35 and upper class limit is 40.
The class mark 
(iv) We have,
|
Classes |
Class marks |
frequency |
|
15 – 20 20 – 25 25 – 30 30 – 35 35 – 40 40 – 45 45 – 50 |
17.5 23.5 28.5 33.5 38.5 42.5 47.5 |
2 3 5 7 4 3 1 |
|
Total |
|
25 |
(v) We have,
|
Classes |
Frequency |
Cumulative frequency |
|
15 – 20 20 – 25 25 – 30 30 – 35 35 – 40 40 – 45 45 – 50 |
2 3 5 7 4 3 1 |
2 5 10 17 21 24 25 |
|
Total |
25 |
|
9. Given below is a cumulative frequency distribution table showing marks obtained by 50 students of a class.
|
Marks |
Below 20 |
Below 40 |
Below 60 |
Below 80 |
Below 100 |
|
Number of students |
15 |
24 |
29 |
34 |
50 |
Form a frequency table from the above data.
Solution: We have,
|
Marks |
Number of students (C.F.) |
Frequency |
|
0 – 20 20 – 40 40 – 60 60 – 80 60 – 100 |
15 24 29 34 50 |
15 9 5 5 16 |
|
Total |
|
50 |
10. Draw a bar graph to represent the following data of sales of a shopkeeper:
|
Day |
Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
Saturday |
|
Sales (in Rs) |
16000 |
18000 |
17500 |
9000 |
85000 |
16500 |
Solution: We have,
|
Day |
Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
Saturday |
|
Sales (in Rs) |
16000 |
18000 |
17500 |
9000 |
85000 |
16500 |
Given photo
11. Study the following bar graph and answer the following questions:
Fig. 24.19
(i) What is the information given by the bar graph?
(ii) On which day is number of students born the maximum?
(iii) How many more students were born on Thursday than that on Tuesday.
(iv) What is the total number of students in the class?
Solution:
12. The times (in minutes) taken to complete a crossword at a competition were noted for 50 competitors are recorded in the following table:
|
Time (in minutes ) |
Number of competitors |
|
20 – 25 25 – 30 30 – 35 35 – 40 40 – 45 45 – 50 |
8 10 9 12 6 5 |
(i) Construct a histogram for the data.
(ii) Construct a frequency polygon.
Solution: photo in laptop
13. Construct a frequency polygon for the data in question 12 without drawing a histogram.
Solution: Photo
Plot the midpoint against the frequency: (22.5,8), (27.5,10), (32.5,9), (37.5,12), (42.5,6), (47.5,5)
14. The following histogram shows the number of literate females in the age group 10 to 40 (in years) in a town:
Photo Fig. 24.20
Study the above histogram and answer the following questions:
(i) What was the total number of literate females in the town in the age group 10 to 40?
(ii) In which age group, the number of literate females was the highest?
(iii) In which two age groups was the number of literate females the same?
(iv) State true or false: The number of literate females in the age group 25-30 is the sum of the numbers of literate females in the age groups 20-25 and 35-40.
Solution:
Write the correct option:
15. The sum of the class marks of the classes 90-120 and 120-150 is
(A) 210 (B) 220 (C) 240 (D) 270
Answer: (C) 240
[ Class mark of 90 – 120
Class mark of 120 – 150
The sum of the class mark = 105 + 135 = 240 ]
16. The range of the data : 28, 17, 20, 16, 19, 12, 30, 32, 10 is
(A) 22 (B) 28 (C) 30 (D) 32
Answer: (A) 22 .
[ We arrange the data in ascending order : 10 , 12 , 16 , 17 , 19 , 20 , 28 , 30 , 32 .
The range = 32 – 10 = 22 ]
17. In a frequency distribution, the mid-value of a class is 12 and its width is 6. The lower limit of the class is:
(A) 6 (B) 9 (C) 12 (D) 18
Answer: (B) 9
[ Let, x and y be the lower class limit and upper class limit respectively .
A/Q, 
And 
The lower class limit is 9 and the upper class limit is 15 . ]
18. The width of each of five continuous classes in a frequency distribution is 5 and the lower limit of the lowest (first) class is 10. The upper limit of the highest (last) class is
(A) 15 (B) 20 (C) 30 (D) 35
Answer: (D) 35
[ Here, Number of classes = 5 and Class width = 5
Lower limit of the first class = 10
First class: lower limit = 10, upper limit = 10 + 5 = 15
Class = 10 – 15
Second class: 15 – 20 , Third class: 20 – 25 , Fourth class: 25 – 30 , Fifth (highest) class: 30 – 35
The upper limit of the last class is 35.
19. The class marks (in order) of a frequency distribution are 10, 15, 20, .... The class corresponding to the class mark 15 is
(A) 11.5-18.5 (B) 17.5-22.5 (C) 12.5-17.5 (D) 13.5-16.5
Answer: (C) 12.5–17.5
[ The class marks are: 10, 15, 20, ...
The difference between consecutive class marks = 15 − 10 = 5 .
So, the class width = 5.
Thus, the class is 12.5 – 17.5. ]
20. For drawing a frequency polygon of a continuous frequency distribution, we plot the points whose ordinates are the frequencies of the respective classes and abcissae are respectively:
(A) class marks of the classes (B) lower limits of the classes
(C) upper limits of the classes (D) upper limits of preceding classes
Answer: (A) class marks of the classes
[ For a frequency polygon of a continuous frequency distribution, we plot points where the x-coordinates (abscissae) are the class marks (midpoints) of the classes, and the y-coordinates (ordinates) are the corresponding frequencies.]
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