Chapter Overview
4.1
Collecting & Organising Data
4.2
Pie Charts / Circle Graphs
4.3
Chance & Probability
2024
Reprint / Exercises
What You'll Learn
Data & Information
Difference between raw data and organized information; methods of collecting and representing data.
Graphs & Charts
Pictographs, bar graphs, double bar graphs, pie charts — how to draw and read them; calculating central angles.
Probability Basics
Random experiments, equally likely outcomes, event, probability formula and simple examples (coin, die, spinner, cards).
Applications
Using pie charts for percentages, using probability to model chance in real life (weather prediction, exit polls).
Why this matters
Data handling is the first step in statistical thinking — collect, organize, represent and interpret. Probability gives a mathematical language to describe uncertainty and chance, which helps in experiments, predictions, and decision making.
Detailed Chapter Summary
4.1 Looking for information — Data & Representation
Data are numbers or observations collected for a purpose (examples: runs scored, marks obtained, number of books read). Raw data needs organization — tables, tally marks, frequency distributions. Graphical representation gives quick visual insights: pictographs, bar graphs, double bar graphs.
4.1.1 Pictograph
Pictorial representation using a symbol to represent a fixed quantity. Example in PDF: symbol represents 100 cars. Read off values by counting symbols (fractions allowed for partial symbols).
4.2 Circle Graphs / Pie Charts
A circle graph divides a circle into sectors, each proportional to the frequency or percentage of the category. Central angle for a category = fraction of whole × 360°. E.g., chocolate 50% ⇒ central angle \(= \tfrac{50}{100}\times360^\circ = 180^\circ\).
Steps to draw a pie chart
- Compute total and convert each category to fraction of total or percentage.
- Compute central angle \(= \text{fraction} \times 360^\circ\).
- Using protractor, mark each sector and shade/label it.
4.3 Chance and Probability
Random experiment: outcome cannot be predicted exactly (coin toss, die throw). Equally likely outcomes: each outcome has same chance. Probability (when outcomes equally likely):
Definition: \(\displaystyle P(E) = \dfrac{\text{Number of outcomes favourable to }E}{\text{Total number of equally likely outcomes}}\)
Examples: coin toss ⇒ \(P(\text{Head}) = \dfrac{1}{2}\). Throw a die ⇒ \(P(2) = \dfrac{1}{6}\).
4.3.1 Outcomes and Events
Events may be single outcomes or collections of outcomes (e.g., even number when a die is thrown: {2,4,6}). For equally likely outcomes, probability of an event is the count of favourable outcomes divided by total count.
Applications of probability
Exit polls, weather forecasting, assessing risks — probabilistic thinking helps approximate uncertain outcomes using data.
Questions and Answers from Chapter 4
Short Questions (1 mark each)
Q S1. What is data?
Answer: Data are raw observations or numbers collected for study, e.g., marks, runs, counts.
Q S2. What is a pictograph?
Answer: A pictograph uses symbols where each symbol represents a fixed number to show data pictorially.
Q S3. How many degrees are in a full circle?
Answer: \(360^\circ\).
Q S4. If chocolate represents 50% in a pie chart, what is its central angle?
Answer: \(0.5\times360^\circ = 180^\circ\).
Q S5. What is a double bar graph?
Answer: A bar graph that shows two sets of values side by side for comparison.
Q S6. Define a random experiment.
Answer: An experiment with outcomes that cannot be predicted exactly in advance (e.g., coin toss).
Q S7. What are equally likely outcomes?
Answer: Outcomes that have the same chance of occurring (e.g., faces of a fair die).
Q S8. Write the probability of getting a tail on a fair coin.
Answer: \(P(\text{Tail})=\dfrac{1}{2}.\)
Q S9. Write the probability of getting a 7 on a single die throw.
Answer: \(0\) (impossible), because die faces are 1–6.
Q S10. If a spinner has 3 green, 1 blue, 1 red sector, probability of green = ?
Answer: \(\dfrac{3}{5}.\)
Q S11. In the pie chart example, vanilla is 25%. What's its central angle?
Answer: \(0.25\times360^\circ = 90^\circ.\)
Q S12. What does each sector of a pie chart represent?
Answer: A part of the whole, proportional to its frequency or percentage.
Q S13. What is the total angle used when drawing a pie chart?
Answer: \(360^\circ\).
Q S14. For data 320,120,160,80,40 (total 720), central angle for 320 is?
Answer: \(\dfrac{320}{720}\times360^\circ = 160^\circ.\)
Q S15. When are outcomes said to be equally likely?
Answer: When every outcome has the same chance of occurring (e.g., fair die faces).
Medium Questions (3 marks each)
Q M1. A pie chart shows chocolate 50%, vanilla 25%, others 25%. Find the angles for each sector.
Answer: Chocolate \(=180^\circ\), Vanilla \(=90^\circ\), Others \(=90^\circ\). (Work: multiply percentage by 360°.)
Q M2. From table: Watches sold (July 1000, Aug 1500, Sep 1500, Oct 2000, Nov 2500, Dec 1500). Draw a simple conclusion: which month had max production?
Answer: November with 2500 watches. (Explanation: compare numbers or draw bar heights to see max.)
Q M3. A group of 360 people choose seasons: Summer 90, Rainy 120, Winter 150. Find central angles and draw pie chart (angles only).
Answer: Total 360. Summer: \(90/360\times360=90^\circ\). Rainy: \(120/360\times360=120^\circ\). Winter: \(150/360\times360=150^\circ\).
Q M4. From baker's shop example (total 720): compute central angles for each item: ordinary bread 320, biscuits 120, cakes 160, fruit bread 80, others 40.
Answer: Ordinary bread: \(320/720\times360=160^\circ\). Biscuits: \(120/720\times360=60^\circ\). Cakes: \(160/720\times360=80^\circ\). Fruit bread: \(80/720\times360=40^\circ\). Others: \(40/720\times360=20^\circ\).
Q M5. If a coin is tossed twice, list sample space and probability of exactly one head.
Answer: Sample space \(S=\{HH,HT,TH,TT\}\). Exactly one head outcomes: {HT,TH} ⇒ \(P=\dfrac{2}{4}=\dfrac{1}{2}.\)
Q M6. From a well shuffled deck of 52, what is probability of selecting an ace?
Answer: There are 4 aces. So \(P=\dfrac{4}{52}=\dfrac{1}{13}.\)
Q M7. Numbers 1–10 on slips; one chosen. Find \(P(6)\), \(P(<6)\), \(P(>6)\).
Answer: Total 10. \(P(6)=\dfrac{1}{10}\). \(P(<6)=\dfrac{5}{10}=\dfrac{1}{2}\). \(P(>6)=\dfrac{3}{10}\) (numbers 7,8,9,10? actually 4 numbers so \(\dfrac{4}{10}=\dfrac{2}{5}\) — careful). (Correct: >6 are 7,8,9,10 ⇒ 4 outcomes ⇒ \(4/10=2/5\)).
Q M8. A spinner has 3 green,1 blue,1 red sectors. What is probability of non-blue sector?
Answer: Total 5 sectors. Non-blue = 4 ⇒ \(P=\dfrac{4}{5}\).
Q M9. If total marks 540 and a subject has 105 marks, find its central angle in pie chart of marks.
Answer: Fraction \(=105/540\). Angle \(=105/540\times360^\circ = 70^\circ\).
Q M10. From spinner with unequal sectors, explain how to compute probability of landing on a particular color.
Answer: Count number of sectors of that color and divide by total sectors (if sectors equal sized). If sectors unequal sized, use central angles (favourable angle / 360°).
Q M11. A bag has 4 red + 2 yellow identical balls. If a ball is drawn, probability red = ?
Answer: Total 6. Red favourable 4 ⇒ \(P=\dfrac{4}{6}=\dfrac{2}{3}.\)
Q M12. Given percentages: Chocolate 50%, Vanilla 25%, Other 25%. If 1000 CDs made, how many of each?
Answer: Chocolate: 50% of 1000 = 500. Vanilla: 250. Others: 250.
Q M13. When drawing a pie chart, why must sum of sector angles equal \(360^\circ\)?
Answer: Because the pie chart represents whole (100%); the central angles partition the circle so sum of their measures covers full circle ⇒ \(360^\circ\).
Q M14. A die thrown once: probability of getting prime number?
Answer: Prime numbers on die: 2,3,5 ⇒ 3 outcomes ⇒ \(P=\dfrac{3}{6}=\dfrac{1}{2}.\)
Q M15. Explain 'favourable outcomes' with an example.
Answer: Favourable outcomes are those that satisfy the event. Example: event "even on die" ⇒ favourable = {2,4,6} ⇒ 3 outcomes; total 6; probability = \(3/6\).
Long Questions (detailed answers)
Q L1. Draw a pie chart for time spent by a child: Sleep 8h, School 6h, Homework 4h, Play 4h, Others 2h. Show working.
Answer:
Total = \(8+6+4+4+2 = 24\) hours. Compute fraction and central angles:
- Sleep: fraction \(=8/24=1/3\). Angle \(=1/3\times360^\circ=120^\circ\).
- School: \(6/24=1/4\). Angle \(=90^\circ\).
- Homework: \(4/24=1/6\). Angle \(=60^\circ\).
- Play: \(4/24=1/6\). Angle \(=60^\circ\).
- Others: \(2/24=1/12\). Angle \(=30^\circ\).
Using a protractor draw central angles sequentially: 120°, then 90°, then 60°, 60°, 30° and label sectors.
Q L2. A survey of flavour preference: Chocolate 50%, Vanilla 25%, Others 25%. If 200 students surveyed, find counts and draw pie chart angles. Also check sum consistency.
Answer:
Counts: Chocolate = 50% of 200 = 100. Vanilla = 50. Others = 50. Angles: Chocolate \(=0.5\times360=180^\circ\). Vanilla \(=90^\circ\). Others \(=90^\circ\). Sum of angles \(=180+90+90=360^\circ\) consistent.
Q L3. From the baker's sales (ordinary 320, biscuits 120, cakes 160, fruit 80, others 40): convert sales to central angles and explain how sectors relate to sales.
Answer:
Total = 720. Central angles: ordinary \(=320/720\times360=160^\circ\), biscuits \(=60^\circ\), cakes \(=80^\circ\), fruit \(=40^\circ\), others \(=20^\circ\). Larger the sales, larger the central angle — visual proportion.
Q L4. A bag contains 4 red and 2 yellow balls. A ball is drawn without looking. Find \(P(\text{red}), P(\text{yellow})\) and compare.
Answer:
Total outcomes = 6. \(P(\text{red})=4/6=2/3\). \(P(\text{yellow})=2/6=1/3\). So red is twice as likely as yellow.
Q L5. Explain equally likely outcomes using die experiments and show how frequencies approach probabilities with many trials.
Answer:
Each die face 1–6 has equal chance 1/6. When throwing the die many times, relative frequency of each face tends to approach \(1/6\) (law of large numbers intuition). Table of tallies for increasing tosses shows counts getting closer for each face.
Q L6. An experiment: numbers 1–10 on slips; one slip chosen. Find probability of even number and justify steps.
Answer:
Even numbers: {2,4,6,8,10} ⇒ 5 favourable. Total 10 ⇒ \(P=\dfrac{5}{10}=\dfrac{1}{2}\). Steps: list sample space, identify event, count favourable outcomes, divide by total.
Q L7. Show how to use percentages to create a pie chart for family expenditure example (one exercise in PDF) and solve one question from it.
Answer:
If expenditure percentages are: Food 40%, Rent 30%, Education 15%, Savings 15% (example). Central angles: food \(=0.4\times360=144^\circ\), rent \(=108^\circ\), education \(=54^\circ\), savings \(=54^\circ\). If savings = ₹3000 corresponds to 15%, then 1% = ₹200 ⇒ clothes (10%) = ₹2000 (example problem technique).
Q L8. A spinner with unequal-sized sectors: if sector angles are 40°, 120°, 200°, find probability of landing on the 120° sector.
Answer:
Total \(=360^\circ\). Probability = favourable angle / 360° = \(120/360 = 1/3\).
Q L9. Explain the relation between fraction of total and central angle, and derive formula.
Answer:
Fraction of total = \( \dfrac{\text{category}}{\text{total}} \). Circle is \(360^\circ\) so central angle \(= \dfrac{\text{category}}{\text{total}}\times360^\circ\). This is direct proportionality: angle ∝ category size.
Q L10. Define event, sample space and favourable outcomes with suitable examples from chapter.
Answer:
Sample space \(S\) = set of all possible outcomes (e.g., {1,2,3,4,5,6} for die). Event \(E\) = subset (e.g., even numbers {2,4,6}). Favourable outcomes = elements of \(E\). Probability = \(n(E)/n(S)\).
Q L11. A spinner has 3 green,1 blue,1 red sectors. If pointer spun twice, find probability both times land on green (assume replacement).
Answer:
Single spin \(P(G)=3/5\). Two independent spins: \(P(G,G)=(3/5)\times(3/5)=9/25.\)
Q L12. A pie chart is given for marks; if student scored 105 and total 540, verify angle and discuss how pie chart shows relative performance.
Answer:
Angle \(=105/540\times360 = 70^\circ\). Pie chart sector of 70° visually smaller than subjects with larger marks; helps compare proportions quickly.
Q L13. A group of 360 people: Blue 18, Green 9, Red 6, Yellow 3. Draw pie chart fractions and angles, show method.
Answer:
Total 36. Fractions: Blue \(=18/36=1/2\) ⇒ angle \(=180^\circ\). Green \(=9/36=1/4\) ⇒ \(90^\circ\). Red \(=6/36=1/6\) ⇒ \(60^\circ\). Yellow \(=3/36=1/12\) ⇒ \(30^\circ\).
Q L14. Suppose we toss a fair coin 100 times and get 48 heads. Discuss how this compares to expected probability.
Answer:
Expected heads = \(100\times 1/2 =50\). Observed 48 is close — illustrates relative frequency approaching theoretical probability as trials increase.
Q L15. Create a small real-life example where probability can be used to make decisions (explain steps).
Answer:
Example: weather-based picnic decision. Use historical data: if it rained on 4 of last 10 same-day events ⇒ \(P(\text{rain})=0.4\). If P(rain) > threshold (say 0.3), plan indoor alternative. Steps: collect data, compute frequency, estimate probability, decide threshold and act.
