Uses of Data — Questions and Answers
Welcome to HSLC Guru. This page gives complete, step-by-step solutions to Chapter 5 Uses of Data of ASSEB (Assam State School Education Board) Class 8 General Mathematics. Pictographs, bar graphs, histograms, pie charts and probability are all explained with worked figures and calculations.
Summary
The data we collect may be organised or unorganised. To reach a meaningful conclusion the data must be arranged systematically. The number of times a particular entry occurs is called its frequency, and a frequency distribution table can be built using tally marks.
Data can be displayed graphically using a pictograph, bar graph, double bar graph, histogram (for continuous class intervals, with no gaps between bars) or a pie chart. In a pie chart the central angle of each part is $\frac{\text{value of the part}}{\text{total value}} \times 360^\circ$.
An experiment whose outcome cannot be predicted exactly in advance is a random experiment. The probability of an event $= \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$. Tossing a coin and rolling a die are random experiments.
Summary: This page covers ASSEB Class 8 General Mathematics Chapter 5 — Uses of Data. It explains organising data with tally marks and frequency distribution tables, displaying it with pictographs, bar graphs, double bar graphs, histograms and pie charts (central angle = value/total × 360°), and introduces random experiments, equally likely outcomes, events and probability = favourable/total outcomes, with full worked answers to Exercises 5.1, 5.2 and 5.3.
Textbook Questions and Answers
In-text Examples (Pictograph, Bar graph, Double bar graph)
Pictograph: In a bookshop, the books sold on the first four days of a week are shown in a pictograph where one book symbol (📖) = 5 books. Monday has 4 symbols, Tuesday 5, Wednesday 3 and Thursday 2.
(i) On which day were the most books sold?
Answer: Tuesday has the most symbols (5), so $5 \times 5 = 25$ books were sold — the maximum.
(ii) What is the total number of books sold on Tuesday and Wednesday together?
Answer: Tuesday $= 5 \times 5 = 25$ and Wednesday $= 3 \times 5 = 15$. Total $= 25 + 15 = 40$ books.
Bar graph: Number of students of a school for the last five years — 2014: 250, 2015: 330, 2016: 300, 2017: 280, 2018: 450 (Figure 5.1).
(i) In which year is the number of students maximum and how many?
Answer: In 2018, with 450 students.
(ii) What is the difference between the maximum and minimum number of students?
Answer: Maximum $= 450$ (2018), minimum $= 250$ (2014). Difference $= 450 – 250 = 200$.
(iii) In which years is the number of students 300 or above?
Answer: 2015 (330), 2016 (300) and 2018 (450).
Double bar graph: Boys and girls in the same school — 2014: 100, 150; 2015: 170, 160; 2016: 150, 150; 2017: 150, 130; 2018: 200, 250 (Figure 5.2).
(i) In which year is the number of boys maximum?
Answer: In 2018 (200 boys).
(ii) In which year is the number of boys minimum?
Answer: In 2014 (100 boys).
(iii) In which year are the numbers of boys and girls the same?
Answer: In 2016 (150 each).
Examples 1–7
Example 1: Grouped frequency table (Table-4) for ages of 60 people — $0$–$10$: 7, $10$–$20$: 9, $20$–$30$: 11, $30$–$40$: 10, $40$–$50$: 9, $50$–$60$: 8, $60$–$70$: 6 (Total 60). Answer the questions.
Answer: (i) Class width $= 10$ (since $10-0=10,\ 20-10=10,\ \dots$). (ii) Highest frequency class is $20$–$30$ (frequency $11$). (iii) Lowest frequency class is $60$–$70$ (frequency $6$). (iv) Upper limit of $40$–$50$ is $50$. (v) Classes with the same frequency are $10$–$20$ and $40$–$50$ (both $9$).
Example 2: Marks obtained by 78 students in a Mathematics Olympiad (Table-5) — $0$–$10$: 2, $10$–$20$: 14, $20$–$30$: 18, $30$–$40$: 10, $40$–$50$: 13, $50$–$60$: 8, $60$–$70$: 6, $70$–$80$: 7 (Total 78). Read the histogram.
Answer: (i) Students scoring more than 30 but less than 60 $= 10 + 13 + 8 = 31$. (ii) Students scoring 60 or more $= 6 + 7 = 13$. (iii) The class interval with the maximum number of students is $20$–$30$ (18 students).
Example 3: Runs scored by 4 students in an inter-school cricket match — Ariphul 70, Paramjit 65, Raju 30, Joseph 15 (Total 180). Draw a pie chart.
Answer: Central angle $= \frac{\text{runs}}{180} \times 360^\circ$ —
$$\text{Ariphul}=\frac{70}{180}\times 360^\circ=140^\circ,\quad \text{Paramjit}=\frac{65}{180}\times 360^\circ=130^\circ$$
$$\text{Raju}=\frac{30}{180}\times 360^\circ=60^\circ,\quad \text{Joseph}=\frac{15}{180}\times 360^\circ=30^\circ$$
Example 4: Savings and expenditure (in %) from a monthly salary — Savings $25\%$, Children’s education $25\%$, Food $30\%$, Others $20\%$. Express in a pie chart.
Answer: Central angle $=$ percent $\times 360^\circ \div 100$ —
$$\text{Savings}=\frac{25}{100}\times 360^\circ=90^\circ,\ \ \text{Education}=\frac{25}{100}\times 360^\circ=90^\circ$$
$$\text{Food}=\frac{30}{100}\times 360^\circ=108^\circ,\ \ \text{Others}=\frac{20}{100}\times 360^\circ=72^\circ$$
Example 5: A die has two faces with one dot each, three faces with two dots each, and the remaining face with three dots. When rolled, find the probability of getting (i) one dot (ii) two dots (iii) three dots.
Answer: The die has 6 faces, so total outcomes $= 6$. (i) Faces with one dot $= 2$, so $P(\text{one dot})=\frac{2}{6}=\frac{1}{3}$. (ii) Faces with two dots $= 3$, so $P(\text{two dots})=\frac{3}{6}=\frac{1}{2}$. (iii) Faces with three dots $= 1$, so $P(\text{three dots})=\frac{1}{6}$. There is no face with four dots, so $P(\text{four dots})=\frac{0}{6}=0$.
Example 6: A coin has a head on both sides. When tossed, what is the probability of getting a head, and of getting a tail?
Answer: Total outcomes $= 2$. As both faces are heads, favourable outcomes for head $= 2$. So $P(\text{head})=\frac{2}{2}=1$ (a sure event). There is no tail, so $P(\text{tail})=\frac{0}{2}=0$.
Example 7: An unbiased die is tossed. Find the probability of (i) getting a number 4 or greater than 4 (ii) getting an even or an odd number.
Answer: Outcomes are $1,2,3,4,5,6$. (i) Numbers 4 or greater $\{4,5,6\}$ — 3 outcomes, so $P=\frac{3}{6}=\frac{1}{2}$. (ii) Even $\{2,4,6\}$ and odd $\{1,3,5\}$ together give all 6 outcomes, so $P=\frac{6}{6}=1$; the event is sure to happen.
Think and Say
(a) Is throwing a die in a Ludo game a random experiment? What are its outcomes?
Answer: Yes, throwing a die is a random experiment because it can be repeated the same way again and again and its outcome cannot be predicted beforehand. The outcomes are $1, 2, 3, 4, 5, 6$.
(b) Instead of using the staircase of a two-storeyed building, if a person jumps down, what may be the result?
Answer: Jumping down carries a very high chance of injury or danger, whereas coming down by the staircase has much lower uncertainty. For safety, one should always use the staircase.
(c) Two coins are tossed together. (i) Are the chances of getting two heads and of getting only one head the same? (ii) Are the chances of getting two heads and of getting two tails the same?
Answer: The outcomes are HH, HT, TH, TT (4 in all). (i) Two heads (HH) has chance $\frac{1}{4}$, while exactly one head (HT, TH) has chance $\frac{2}{4}=\frac{1}{2}$ — not the same. (ii) Two heads (HH) has chance $\frac{1}{4}$ and two tails (TT) has chance $\frac{1}{4}$ — these two are equal.
Exercise 5.1
1. The choice of 46 students of Class VIII about four colours — white (W), red (R), black (B) and yellow (Y) — is given below (each student chooses only one colour). Prepare a frequency distribution table using tally marks and draw a bar graph to explain it.
W, R, R, Y, B, B, B, Y, R, W, W, R,
Y, B, B, Y, B, R, R, W, B, B, R, Y,
Y, B, W, Y, Y, R, W, W, R, R, B, B,
R, Y, B, W, W, B, Y, B, W, W
Answer: Counting each colour gives the table below (Total $= 46$).
| Colour | Tally marks | Frequency |
|---|---|---|
| White (W) | ||||| ||||| | | 11 |
| Red (R) | ||||| ||||| | | 11 |
| Black (B) | ||||| ||||| |||| | 14 |
| Yellow (Y) | ||||| ||||| | 10 |
| Total | 46 |
2. The monthly savings (in rupees) of 35 members of ‘Jonaki Self Help Group’ are given below. Taking $110$–$120,\ 120$–$130,\ 130$–$140\ \dots$ as class intervals, prepare a frequency table using tally marks.
110, 125, 110, 140, 150, 150, 150, 110, 180, 180, 110, 140, 140, 120, 120, 120, 140, 140, 170, 175, 175, 145, 145, 140, 175, 120, 125, 130, 135, 135, 155, 145, 145, 175, 185
Answer: Placing each value in its class (an upper-limit value goes into the next class) gives —
| Class interval (savings) | Tally marks | Frequency |
|---|---|---|
| 110 – 120 | |||| | 4 |
| 120 – 130 | ||||| | | 6 |
| 130 – 140 | ||| | 3 |
| 140 – 150 | ||||| ||||| | 10 |
| 150 – 160 | |||| | 4 |
| 160 – 170 | 0 | |
| 170 – 180 | ||||| | 5 |
| 180 – 190 | ||| | 3 |
| Total | 35 |
3. Draw a histogram from the frequency table of Q.2 and answer: (i) Which class interval has the maximum number of members? (ii) How many members save 150 or more? (iii) In which class intervals are there an equal number of members?
Answer: (i) The class interval with the maximum members is $140$–$150$ (10 members). (ii) Members saving 150 or more $= 4 + 0 + 5 + 3 = 12$. (iii) $110$–$120$ and $150$–$160$ have an equal number (4 each); also $130$–$140$ and $180$–$190$ have an equal number (3 each).
4. The daily study time of the Class VIII students of a school is shown in the histogram (Figure 5.5). (i) For how many hours do most students study daily? (ii) How many students study for more than 5 hours daily? (iii) How many students study for less than 4 hours daily?
Answer: The bar values are $1$–$2$: 4, $2$–$3$: 10, $3$–$4$: 8, $4$–$5$: 14, $5$–$6$: 12, $6$–$7$: 8, $7$–$8$: 4 (Total 60). (i) The tallest bar (14) is over $4$–$5$, so most students study 4 to 5 hours daily. (ii) Students studying more than 5 hours $= 12 + 8 + 4 = 24$. (iii) Students studying less than 4 hours $= 4 + 10 + 8 = 22$.
5. The heights (in cm) of 30 students are given below. Taking appropriate class intervals, prepare a frequency distribution table using tally marks.
136, 138, 140, 140, 154, 160, 158, 147, 139, 153, 162, 162, 173, 137, 142, 156, 162, 164, 185, 143, 145, 182, 152, 163, 174, 138, 142, 152, 144, 146
Answer: Taking class intervals $130$–$140,\ 140$–$150,\ \dots$ (width $10$), counting gives (Total $= 30$) —
| Height (cm) | Tally marks | Frequency |
|---|---|---|
| 130 – 140 | ||||| | 5 |
| 140 – 150 | ||||| |||| | 9 |
| 150 – 160 | ||||| | | 6 |
| 160 – 170 | ||||| | | 6 |
| 170 – 180 | || | 2 |
| 180 – 190 | || | 2 |
| Total | 30 |
Exercise 5.2
1. The favourite sports of 60 persons are given below. Draw a pie chart. Cricket 20, Football 18, Kabaddi 12, Badminton 10.
Answer: Central angle of each sport $= \frac{\text{number of persons}}{60} \times 360^\circ$ (i.e. 1 person $= 6^\circ$) —
$$\text{Cricket}=\frac{20}{60}\times 360^\circ=120^\circ,\quad \text{Football}=\frac{18}{60}\times 360^\circ=108^\circ$$
$$\text{Kabaddi}=\frac{12}{60}\times 360^\circ=72^\circ,\quad \text{Badminton}=\frac{10}{60}\times 360^\circ=60^\circ$$
2. 600 persons were present to watch a football match. The pie chart shows how many came by different vehicles or on foot (on foot $150^\circ$, cycle $120^\circ$, scooter/motor cycle $60^\circ$, four wheeler $30^\circ$). (i) How many came on foot? (ii) What is the difference between those who came by scooter/motor cycle and those by four wheeler? (iii) What type of vehicle was used by 200 persons?
Answer: Total persons $= 600$, so $1^\circ$ represents $\frac{600}{360} = \frac{5}{3}$ persons. (i) On foot $= \frac{150}{360} \times 600 = 250$ persons. (ii) Scooter/motor cycle $= \frac{60}{360} \times 600 = 100$; four wheeler $= \frac{30}{360} \times 600 = 50$; difference $= 100 – 50 = 50$ persons. (iii) 200 persons correspond to $\frac{200}{600} \times 360^\circ = 120^\circ$, which is the cycle sector. So 200 persons used a cycle.
3. In a school of 720 students, the numbers in classes Six, Seven, Eight, Nine and Ten are — Six 120, Seven 140, Eight 200, Nine 80, Ten 180 (Total 720). Express the data in a pie chart.
Answer: Central angle $= \frac{\text{number of students}}{720} \times 360^\circ$ (i.e. 1 student $= 0.5^\circ$) —
$$\text{Six}=\frac{120}{720}\times 360^\circ=60^\circ,\ \text{Seven}=70^\circ,\ \text{Eight}=\frac{200}{720}\times 360^\circ=100^\circ$$
$$\text{Nine}=\frac{80}{720}\times 360^\circ=40^\circ,\quad \text{Ten}=\frac{180}{720}\times 360^\circ=90^\circ$$
4. A pie chart is drawn for 180 students who like stories, novels, tales, poems and autobiographies (Stories $90^\circ$, Novels $60^\circ$, Tales $58^\circ$, Poems $72^\circ$, Autobiographies $80^\circ$). (i) How many students like to read novels? (ii) What do most students like to read and how many? (iii) What is the total number of students who like poems and autobiographies?
Answer: Here $1^\circ = \frac{180}{360} = 0.5$ student. (i) Novels $= \frac{60}{360} \times 180 = 30$ students. (ii) The largest angle is Stories ($90^\circ$), so most students like stories; number $= \frac{90}{360} \times 180 = 45$. (iii) Poems $= \frac{72}{360} \times 180 = 36$ and Autobiographies $= \frac{80}{360} \times 180 = 40$; total $= 36 + 40 = 76$ students.
5. The numbers of trees in a fruit garden are — Mango 30, Jackfruit 50, Guava 20 (Total 100). Determine the central angle of each sector and draw a pie chart.
Answer: Central angle $= \frac{\text{number of trees}}{100} \times 360^\circ$ —
$$\text{Mango}=\frac{30}{100}\times 360^\circ=108^\circ,\ \ \text{Jackfruit}=\frac{50}{100}\times 360^\circ=180^\circ,\ \ \text{Guava}=\frac{20}{100}\times 360^\circ=72^\circ$$
Exercise 5.3
1. A judge writes some topics on pieces of paper for an extempore speech competition and keeps them hidden. If the topics are marked A, B, C and D, what are the possible outcomes of a competitor’s choice if (i) the competitor may select only one piece; (ii) the competitor may select any two pieces?
Answer: (i) Selecting one piece gives 4 outcomes — $A,\ B,\ C,\ D$. (ii) Selecting two pieces gives 6 outcomes — $\{A,B\},\ \{A,C\},\ \{A,D\},\ \{B,C\},\ \{B,D\},\ \{C,D\}$.
2. Find all possible outcomes when two unbiased coins are tossed together.
Answer: Each coin gives H (head) or T (tail). For two coins the outcomes are 4 — HH, HT, TH, TT.
3. In a coloured pencil box there are 4 violet, 3 blue and 5 red pencils. If one pencil is chosen at random, what are the chances of it being (i) violet (ii) blue?
Answer: Total pencils $= 4 + 3 + 5 = 12$. (i) $P(\text{violet})=\frac{4}{12}=\frac{1}{3}$. (ii) $P(\text{blue})=\frac{3}{12}=\frac{1}{4}$.
4. Some events for the experiment of tossing a die are given below. Express each event using its outcomes — (i) getting a square number (ii) getting an odd number greater than 1 (iii) getting an even number greater than 6 (iv) getting a prime number (v) getting an odd prime number (vi) getting an even prime number.
Answer: The outcomes are $1,2,3,4,5,6$. (i) Square numbers — $\{1, 4\}$ (since $1=1^2,\ 4=2^2$). (ii) Odd numbers greater than 1 — $\{3, 5\}$. (iii) Even numbers greater than 6 — none, i.e. no outcome (an impossible event). (iv) Prime numbers — $\{2, 3, 5\}$. (v) Odd prime numbers — $\{3, 5\}$. (vi) Even prime numbers — $\{2\}$.
5. A bag has 15 red, 10 blue and 5 yellow marbles. If a marble is drawn at random, what is the probability that it is (i) red (ii) blue (iii) yellow (iv) blue or yellow?
Answer: Total marbles $= 15 + 10 + 5 = 30$. (i) $P(\text{red})=\frac{15}{30}=\frac{1}{2}$. (ii) $P(\text{blue})=\frac{10}{30}=\frac{1}{3}$. (iii) $P(\text{yellow})=\frac{5}{30}=\frac{1}{6}$. (iv) $P(\text{blue or yellow})=\frac{10+5}{30}=\frac{15}{30}=\frac{1}{2}$.
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. The number of times a particular entry occurs is called its — (a) class width (b) frequency (c) central angle (d) probability
Answer: (b) frequency.
2. The angle at the centre of a complete circle is — (a) $90^\circ$ (b) $180^\circ$ (c) $270^\circ$ (d) $360^\circ$
Answer: (d) $360^\circ$.
3. In a histogram, the bars — (a) have big gaps (b) have no gaps (c) have small gaps (d) none of these
Answer: (b) have no gaps.
4. The probability of getting a head when an unbiased coin is tossed is — (a) $0$ (b) $\frac{1}{4}$ (c) $\frac{1}{2}$ (d) $1$
Answer: (c) $\frac{1}{2}$.
5. The class width of the interval $10$–$20$ is — (a) $5$ (b) $10$ (c) $15$ (d) $20$
Answer: (b) $10$.
6. The probability of getting an even number when a die is rolled is — (a) $\frac{1}{6}$ (b) $\frac{1}{3}$ (c) $\frac{1}{2}$ (d) $\frac{2}{3}$
Answer: (c) $\frac{1}{2}$ (even outcomes $\{2,4,6\}$, so $\frac{3}{6}=\frac{1}{2}$).
7. The probability of a sure event is — (a) $0$ (b) $\frac{1}{2}$ (c) $1$ (d) $2$
Answer: (c) $1$.
8. The probability of an impossible event is — (a) $0$ (b) $\frac{1}{2}$ (c) $1$ (d) $360$
Answer: (a) $0$.
9. A graph that represents data using symbols is called a — (a) bar graph (b) pictograph (c) pie chart (d) histogram
Answer: (b) pictograph.
10. In a pie chart of 180 people, a sector of $90^\circ$ represents how many people? (a) $30$ (b) $45$ (c) $60$ (d) $90$
Answer: (b) $45$ (since $\frac{90}{360}\times 180 = 45$).
Fill in the Blanks
- Data collected directly from the main source are called __________ data. (primary)
- The difference between the upper limit and lower limit of a class interval is called the __________. (class width)
- An experiment whose outcome cannot be predicted exactly in advance is called a __________ experiment. (random)
- The central angle of a pie sector $= \frac{\text{value of the part}}{\text{total value}} \times$ __________. ($360^\circ$)
- When a coin is tossed, the total number of outcomes is __________. (2)
True or False
- In a histogram there are gaps between the bars. — False
- The probability of any event always lies between $0$ and $1$. — True
- Tossing two coins together gives 3 possible outcomes. — False (4: HH, HT, TH, TT)
- The probability of a sure event is $1$. — True
- The sum of all central angles in a pie chart is $360^\circ$. — True
Short Answer Questions
1. What is meant by frequency?
Answer: The frequency of an entry is the number of times that particular entry occurs in the data.
2. What is a random experiment? Give an example.
Answer: A random experiment is one that can be repeated under the same conditions but whose outcome cannot be predicted exactly in advance. Example — tossing a coin.
3. What is the probability of getting a prime number when an unbiased die is rolled?
Answer: The prime numbers on a die are $\{2, 3, 5\}$ — 3 outcomes. So $P = \frac{3}{6} = \frac{1}{2}$.
4. What central angle represents $25\%$ in a pie chart?
Answer: $\frac{25}{100} \times 360^\circ = 90^\circ$.
Key Terms
| Term | Meaning |
|---|---|
| Data | A collection of numbers or facts gathered for study |
| Frequency | The number of times an entry occurs |
| Tally mark | A small stroke used for counting |
| Pictograph | A graph that shows data using symbols |
| Bar graph | A graph using bars of uniform width |
| Histogram | A bar graph of class intervals with no gaps |
| Pie chart | Data shown in the sectors of a circle |
| Central angle | The angle a sector makes at the centre |
| Random experiment | An experiment whose outcome is unpredictable |
| Probability | Favourable outcomes ÷ total outcomes |
| Event | A collection of outcomes of an experiment |