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DATA HANDLING (Statistics) - Q&A


EXERCISE 24

1. Multiple Choice Type:

Choose the correct answer from the options given below.

(i) Lower class limit of 15-18 is:
(a) 15
(b) 18
(c) (18-15)/2
(d) (15+18)/2
Answer: (a) 15
In the class interval 15-18, the smaller number 15 is the lower class limit.

(ii) Upper class limit of 5-12.5 is:
(a) 5
(b) 12.5
(c) (12.5-5)/2
(d) (12.5+5)/2
Answer: (b) 12.5
In the class interval 5-12.5, the larger number 12.5 is the upper class limit.

(iii) If the upper and the lower limits of a class interval are 16 and 10, the class-mark is:
(a) 6
(b) 3
(c) 13
(d) none of these
Answer: (c) 13
Class mark = (Upper limit + Lower limit) / 2 = (16 + 10) / 2 = 26 / 2 = 13.

(iv) If the lower and the upper limits of a class interval are 7.5 and 12.5, the class-mark is:
(a) 10
(b) 2.5
(c) 7.5
(d) 12.5
Answer: (a) 10
Class mark = (7.5 + 12.5) / 2 = 20 / 2 = 10.

(v) In a pie-chart, an angle of 30° represents 80 articles. The number of articles represented by 105° are:
(a) 280
(b) 22 6/7
(c) 200
(d) none of these
Answer: (a) 280
30° represents 80 articles.
1° represents 80/30 articles.
105° represents (80/30) × 105 = 8 × 35 = 280 articles.

(vi) In a pie-chart, 76 articles are represented by 19°, how many articles will be represented by 76°?
(a) 19
(b) 76
(c) 304
(d) none of these
Answer: (c) 304
19° represents 76 articles.
1° represents 76/19 = 4 articles.
76° represents 76 × 4 = 304 articles.

2. Hundred students from a certain locality use different modes of travelling to school as given below. Draw a bar graph.

BusCarRickshawBicycleWalk
321624208

Answer:
To draw the bar graph:
1. Draw the x-axis (horizontal) and mark the modes of transport: Bus, Car, Rickshaw, Bicycle, Walk.
2. Draw the y-axis (vertical) and mark the scale (e.g., 1 unit = 4 students). Mark values 0, 4, 8, 12, ... up to 36.
3. Draw rectangular bars of equal width for each mode with heights corresponding to the data:
- Bus: Height 32
- Car: Height 16
- Rickshaw: Height 24
- Bicycle: Height 20
- Walk: Height 8

3. Mr. Mirza's monthly income is ₹ 7,200. He spends ₹ 1,800 on rent, ₹ 2,700 on food, ₹ 900 on education of his children, ₹ 1,200 on other things and saves the rest. Draw a pie-chart to represent it.

Answer:
Total Income = ₹ 7,200.
First, find the savings: 7200 - (1800 + 2700 + 900 + 1200) = 7200 - 6600 = ₹ 600.
Calculation of Central Angles:
- Rent: (1800 / 7200) × 360° = (1/4) × 360° = 90°
- Food: (2700 / 7200) × 360° = (3/8) × 360° = 135°
- Education: (900 / 7200) × 360° = (1/8) × 360° = 45°
- Others: (1200 / 7200) × 360° = (1/6) × 360° = 60°
- Savings: (600 / 7200) × 360° = (1/12) × 360° = 30°
(Check: 90+135+45+60+30 = 360°)
Draw a circle and divide it into sectors with these angles.

4. The percentage of marks obtained in different subjects by Ashok Sharma (in an examination) are given below. Draw a bar graph to represent it.

EnglishHindiMathsScienceSocial Studies
8560355070

Answer:
To draw the bar graph:
1. X-axis: Subjects (English, Hindi, Maths, Science, Social Studies).
2. Y-axis: Marks (Scale: 1 cm = 10 marks). Range 0 to 90.
3. Draw bars with heights:
- English: 85
- Hindi: 60
- Maths: 35
- Science: 50
- Social Studies: 70

5. The following table shows the market position of different brands of tea leaves. Draw a pie-chart to represent the above information.

BrandsABCDothers
% Buyers3520201510

Answer:
Total percentage = 100%. Total angle = 360°.
Calculation of Central Angles:
- Brand A: (35 / 100) × 360° = 126°
- Brand B: (20 / 100) × 360° = 72°
- Brand C: (20 / 100) × 360° = 72°
- Brand D: (15 / 100) × 360° = 54°
- Others: (10 / 100) × 360° = 36°
Draw the pie chart using these angles.

6. Students of a small school use different modes of travel to school as shown below : Draw a suitable bar graph.

ModesBusCarBicyleAutoOn foot
No. of students14298503416

Answer:
To draw the bar graph:
1. X-axis: Modes of travel.
2. Y-axis: No. of students. Scale: 1 unit = 20 students.
3. Heights of bars:
- Bus: 142
- Car: 98
- Bicycle: 50
- Auto: 34
- On foot: 16

7. For the following table, draw a bar-graph.

ABCDEF
230400350200380160

Answer:
To draw the bar graph:
1. X-axis: Categories (A, B, C, D, E, F).
2. Y-axis: Value. Scale: 1 unit = 50.
3. Heights of bars:
- A: 230
- B: 400
- C: 350
- D: 200
- E: 380
- F: 160

8. Manoj appeared for ICSE examination 2018 and secured percentage of marks as shown in the following table. Represent the above data by drawing a suitable bar graph.

SubjectsHindiEnglishMathsScienceSocial Study
Marks as percent6045424875

Answer:
To draw the bar graph:
1. X-axis: Subjects.
2. Y-axis: Marks %. Scale: 1 unit = 10%.
3. Heights of bars:
- Hindi: 60
- English: 45
- Maths: 42
- Science: 48
- Social Study: 75

9. For the data given above in question number 8, draw a suitable pie-graph.

Answer:
Sum of the values = 60 + 45 + 42 + 48 + 75 = 270.
Calculation of Central Angles:
- Hindi: (60 / 270) × 360° = 80°
- English: (45 / 270) × 360° = 60°
- Maths: (42 / 270) × 360° = 56°
- Science: (48 / 270) × 360° = 64°
- Social Study: (75 / 270) × 360° = 100°
(Check: 80+60+56+64+100 = 360°)
Draw the pie chart with these sectors.

10. Mr. Kapoor compares the prices (in ₹) of different items at two different shops A and B. Examine the following table carefully and represent the data by a double bar graph.

ItemsTea-setMixerCoffee-makerDinner set
Price (in ₹) at shop A900700600600
Price (in ₹) at Shop B950800700500

Answer:
To draw the double bar graph:
1. X-axis: Items (Tea-set, Mixer, Coffee-maker, Dinner set).
2. Y-axis: Price. Scale: 1 unit = ₹ 100.
3. For each item, draw two adjacent bars (one for Shop A, one for Shop B):
- Tea-set: A=900, B=950
- Mixer: A=700, B=800
- Coffee-maker: A=600, B=700
- Dinner set: A=600, B=500

11. The following table shows the modes of transport used by boys and girls for going to the same school. Draw a double bar graph representing the above data.

ModeBusBicycleWalkingOther sources
Number of boys80602085
Number of girls90753560

Answer:
To draw the double bar graph:
1. X-axis: Modes of transport.
2. Y-axis: Number of students. Scale: 1 unit = 10 students.
3. Draw adjacent bars for Boys and Girls for each mode:
- Bus: Boys=80, Girls=90
- Bicycle: Boys=60, Girls=75
- Walking: Boys=20, Girls=35
- Other sources: Boys=85, Girls=60



Test yourself

1. Multiple Choice Type: Choose the correct answer from the options given below.

(i) The number of times a data, in the set, occurs is called:
(a) upper-limit
(b) class-mark
(c) frequency
(d) class-limit
Answer: (c) frequency

(ii) The difference between the greatest and the smallest values of observations is called:
(a) frequency
(b) range
(c) class-mark
(d) class-limit
Answer: (b) range

(iii) The difference between the upper and lower class-limits of a class-interval is called:
(a) width of the class-interval
(b) frequency
(c) class-limit
(d) class-mark
Answer: (a) width of the class-interval

(iv) In a bar-graph, if the widths of all the bars are kept same, their heights are proportional to their :
(a) class-size
(b) class-mark
(c) class-limits
(d) frequency
Answer: (d) frequency

(v) In a pie-chart, the angle corresponding to different components is:
(a) (value of the component / total value of all the components)
(b) (total value of all the components / value of the component) × 360°
(c) (total value of all the components / 360°)
(d) (value of the component / total value of all the components) × 360°
Answer: (d) (value of the component / total value of all the components) × 360°

(vi) Consider the following class intervals of a grouped data: Class Interval 10-25, 25-40, ..., 55-70.
Statement 1: Class mark of the 3rd class interval is 46.5.
Statement 2: If the class mark of the 2nd class interval is 77.5, the class interval is 60-85.
Which of the following options is correct?
(a) Both the statements are true.
(b) Both the statements are false.
(c) Statement 1 is true, and statement 2 is false.
(d) Statement 1 is false, and statement 2 is true.
Answer: (b) Both the statements are false.
Explanation:
Intervals are 10-25, 25-40, 40-55. 3rd is 40-55. Midpoint = (40+55)/2 = 47.5. Statement 1 says 46.5 (False).
Statement 2: If interval is 60-85, Mark = (60+85)/2 = 72.5. Statement says 77.5 (False).

The following questions are Assertion-Reason based questions. Choose your answer based on the codes given below.
(1) Both A and R are correct, and R is the correct explanation for A.
(2) Both A and R are correct, and R is not the correct explanation for A.
(3) A is true, but R is false.
(4) A is false, but R is true.

(vii) Assertion (A): If in a pie chart representing the number of students opting for different streams in college admission out of a total admission of 3300, the central angle for the sector representing Mathematics is 48° then the number of students who opted for Mathematics is 440.
Reason (R): Central angle for sector (Component) = (Value of the component / Total value) × 360°
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (a) (1)
Check A: Value = (Angle / 360) × Total = (48/360) × 3300 = 440. A is correct. R is the correct formula.

(viii) Assertion (A): Class size of the following class intervals is 10. 1-10, 11-20, 21-30, etc.
Reason (R): The difference between the upper limit and lower limit is the class size.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (c) (3)
A is True (1 to 10 is 10 items). R is False for inclusive series (10-1 = 9, not 10). Class size is (Upper - Lower + 1) or difference of boundaries.

(ix) Assertion (A): The given bar graph shows the heights of six mountain peaks.
Reason (R): The space between consecutive bars may be of any suitable value, but the spaces between all the consecutive bars must the same.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (b) (2)
A is True (Graph shows P, Q, R, S, T, V). R is True (Standard rule). R does not explain why the graph shows 6 peaks.

(x) Assertion (A): The distribution of land in Pacific Housing Society is shown in the pie chart below. The total land area for the project is 144000 m². The ratio of the area kept open to the ratio of the area for apartment construction is 5:13.
Reason (R): In a pie chart, the central angle of a sector subtended by its arc is proportional to the value it represents.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (d) (4)
R is True. A is likely False because the angles given (90, 36, 234) do not correspond to a 5:13 ratio (100°:260°) for any logical combination of sectors.

2. Draw a bar-graph to represent the following data:

Articles:ABCDEFG
Price of articles:20025015015010050350

Answer:
Construct a bar graph:
X-axis: Articles A, B, C, D, E, F, G.
Y-axis: Price. Scale: 1 unit = 50.
Heights: A=200, B=250, C=150, D=150, E=100, F=50, G=350.

3. Study the given graph and then answer the following questions:
(i) Which classes have the larger number of students ?
(ii) Which class has the equal number of girls and boys ?
(iii) What is the total number of students in class VIII ?
(iv) What is the total number of students (from the class VI to class X) ?
Answer:
(Based on the graph where Boys are left bar, Girls are right bar):
(i) Class VI (Approx 110 students) and VII (Approx 110 students).
(ii) Class VIII (Both bars appear at height 30).
(iii) 30 + 30 = 60 students.
(iv) Total = VI(60+50) + VII(50+60) + VIII(30+30) + IX(40+40) + X(20+30) = 110 + 110 + 60 + 80 + 50 = 410 students.

4. The following table shows the number of students in various classes:

ClassVIVIIVIIIIXX
No. of students36030054150216

Draw a pie-graph to represent the above data.
Answer:
Total students = 360 + 300 + 54 + 150 + 216 = 1080.
Calculation of Central Angles:
- VI: (360 / 1080) × 360° = 120°
- VII: (300 / 1080) × 360° = 100°
- VIII: (54 / 1080) × 360° = 18°
- IX: (150 / 1080) × 360° = 50°
- X: (216 / 1080) × 360° = 72°
(Check: 120+100+18+50+72 = 360°)
Draw the pie graph with these angles.

5. The given pie-graph represents the number of students in different classes. If the total of all the students in the class is 1080; use the graph to find the number of students in each class.
(Angles: VI=80°, VII=90°, VIII=70°, IX=65°, X=55°)
Answer:
Total students = 1080.
- Class VI: (80 / 360) × 1080 = 80 × 3 = 240 students.
- Class VII: (90 / 360) × 1080 = 90 × 3 = 270 students.
- Class VIII: (70 / 360) × 1080 = 70 × 3 = 210 students.
- Class IX: (65 / 360) × 1080 = 65 × 3 = 195 students.
- Class X: (55 / 360) × 1080 = 55 × 3 = 165 students.

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Quick Review Flashcards - Click to flip and test your knowledge!
Question
In the singular sense, how is the word 'statistics' defined?
Answer
It refers to the whole subject as a branch of knowledge used in collecting, analyzing, presenting, and interpreting numerical data.
Question
In the plural sense, what does the term 'statistics' imply?
Answer
It implies the collection of numerical data in a systematic manner with a definite object in view.
Question
Term: Data
Answer
Definition: A set of numerical facts collected with a definite object in view.
Question
What is a 'set of data'?
Answer
A group of numbers representing numerical facts for a specific variable, such as the heights of children.
Question
Mention two common sources for obtaining data.
Answer
Individuals (through direct inquiry) and government sources (such as birth rates or price indices).
Question
Term: Tabulation
Answer
Definition: The arrangement of collected data in a systematic form, generally in the form of a table.
Question
Term: Frequency
Answer
Definition: A number that indicates how many times a particular data value appears in a given set of data.
Question
Term: Raw Data
Answer
Definition: Data recorded in its original, unorganized form as initially collected.
Question
What is 'arrayed data' or an 'array'?
Answer
Data that has been arranged in either ascending or descending order of magnitude.
Question
Term: Frequency Distribution
Answer
Definition: A tabular arrangement of data showing the frequency of each observation.
Question
In a frequency table, what are the three standard columns used for construction?
Answer
The observation value (or class interval), tally marks, and frequency.
Question
What is the standard procedure for marking the fifth stroke in a tally mark group?
Answer
A stroke is made across the first four strokes to create a bundle of five.
Question
Term: Grouped Frequency Distribution
Answer
Definition: A frequency table where data is organized into specific groups called class intervals.
Question
In the class interval $10-20$, what is the term for the number $10$?
Answer
The lower class-limit.
Question
In the class interval $10-20$, what is the term for the number $20$?
Answer
The upper class-limit.
Question
In a grouped frequency distribution, which class interval would include the value $30$ if the intervals are $20-30$ and $30-40$?
Answer
The class interval $30-40$.
Question
Term: Class Mark
Answer
Definition: The value that is midway between the lower class limit and the upper class limit of a class interval.
Question
Formula: Class mark of a class-interval
Answer
$\frac{\text{lower class limit} + \text{upper class limit}}{2}$
Question
Calculate the class mark for the interval $10-20$.
Answer
$15$
Question
Calculate the class mark for the interval $50-60$.
Answer
$55$
Question
Term: Bar Graph (Bar Chart)
Answer
Definition: A graphical representation of numerical data using rectangular bars of equal width with varying heights.
Question
In a bar graph, what does the width of the bars represent?
Answer
The width does not represent anything; it must simply be equal for all bars.
Question
In a bar graph, to what is the height of each bar directly proportional?
Answer
The number or frequency of the data it represents.
Question
What is the requirement for spacing between consecutive bars in a bar graph?
Answer
The space must be equal between all consecutive bars.
Question
In the construction of a bar graph, what is the horizontal line $OX$ called?
Answer
The x-axis.
Question
In the construction of a bar graph, what is the vertical line $OY$ called?
Answer
The y-axis.
Question
Term: Double Bar Graph
Answer
Definition: A graph where two sets of bars are drawn side by side to compare two collections of data.
Question
In Example 4, if store D sold $385$ articles and store B sold $275$ articles, what is the percentage by which D's sales exceed B's?
Answer
$40\%$
Question
In Example 5's double bar graph, what indicates the class with the maximum number of students?
Answer
The class with the highest combined height of both bars (Class II).
Question
Term: Pie Graph (Pie Chart)
Answer
Definition: A graphical representation where numerical data is shown using the sectors of a circle.
Question
Term: Sector
Answer
Definition: The region of a circle enclosed by two radii and the arc between them.
Question
Term: Central Angle
Answer
Definition: An angle whose vertex is at the center of a circle.
Question
Formula: Central angle for a component in a pie chart
Answer
$\frac{\text{value of the component}}{\text{total value of all components}} \times 360^\circ$
Question
In a pie chart, what is the sum of all central angles?
Answer
$360^\circ$
Question
Formula: Percentage of a component in a pie chart using its central angle
Answer
$\frac{\text{central angle}}{360^\circ} \times 100\%$
Question
How is the ratio of two components determined directly from a pie chart?
Answer
By finding the ratio of their respective central angles.
Question
Term: Range
Answer
Definition: The difference between the greatest and the smallest values of observations in a data set.
Question
What does the 'width of the class-interval' represent?
Answer
The difference between the upper limit and the lower limit of that specific interval.
Question
If the lower and upper limits of a class interval are $7.5$ and $12.5$, what is the class-mark?
Answer
$10$
Question
In a pie chart, if a central angle of $30^\circ$ represents $80$ articles, how many articles does an angle of $105^\circ$ represent?
Answer
$280$ articles
Question
In a pie chart, if $19^\circ$ represents $76$ articles, how many articles does $76^\circ$ represent?
Answer
$304$ articles
Question
Calculate the class mark for the interval $15-18$.
Answer
$16.5$
Question
Assertion: The heights of different bars in a bar graph are parallel to the y-axis. Is this true?
Answer
Yes, the vertical bars are parallel to the y-axis.
Question
In Example 7, if there are $480$ players and the central angle for game D is $75^\circ$, how many players play game D?
Answer
$100$ players
Question
In Example 7, if the central angle for game B is $144^\circ$, what percentage of players play game B?
Answer
$40\%$
Question
Statement: 'The class mark of the second class interval is $77.5$, the class interval is $60-85$.' Is this statement mathematically correct?
Answer
No, because the average of $60$ and $85$ is $72.5$.
Question
Cloze: In a bar graph, the space between consecutive bars must be _____.
Answer
the same (or equal)
Question
Cloze: The midpoint of a class interval's base on the x-axis corresponds to the _____ of that interval.
Answer
class mark
Question
Process: What is the first step in constructing a grouped frequency distribution table?
Answer
Determine the range and decide on the class intervals.
Question
Process: How do you determine the size of a sector in a pie chart?
Answer
Divide $360^\circ$ in proportion to the component's value relative to the total value.
Question
Concept: Why is numerical data arranged in a systematic form like tabulation?
Answer
To get a fair idea of the essential points and make the data easier to interpret.
Question
Example: Find the class mark of the interval $10-25$.
Answer
$17.5$
Question
In Example 3, if the total speed of all vehicles is $240$ km/hr and the speed of a Scooter is $40$ km/hr, what is the central angle for the Scooter?
Answer
$60^\circ$
Question
In a pie chart, what is the relationship between the central angle of a sector and the value it represents?
Answer
The central angle is directly proportional to the value it represents.
Question
If a data set is $\{5, 7, 3, 8, 7, 5, 5, 3, 5, 8, 7\}$, what is the frequency of the number $7$?
Answer
$3$
Question
What is the difference between an 'ungrouped' and a 'grouped' frequency distribution?
Answer
Ungrouped shows frequencies for individual values, while grouped shows frequencies for ranges of values (intervals).
Question
In Example 4, which store sold the largest number of articles based on the bar graph?
Answer
Store D
Question
In Example 5, what is the ratio of girls to boys in class V if there are $20$ girls and $30$ boys?
Answer
$2:3$
Question
How do you calculate the ratio of two components 'A' and 'C' from a pie chart using only their central angles?
Answer
Divide the central angle of A by the central angle of C.
Question
In a pie chart, what does a sector represent?
Answer
A specific part or component of the total numerical data.