Class 9 Mathematics Chapter 14: Data Handling and Probability Previous Year Questions (2020–2025)

Reading Chapter 14 three times won't improve your exam score as much as solving five well-chosen previous year questions. Here's why: (1) **Pattern Recognition** — CBSE examiners repeat question types. Mean calculations almost always appear as 1-mark fill-in-the-blank or 3-mark numerical problems. Pie chart interpretation questions follow predictable setups. (2) **Speed Training** — In the exam hall, you have ≈90 seconds per mark. PYQs force you to work at exam pace, exposing gaps in your calculation fluency. (3) **Confidence** — Seeing that the 5-mark question on probability uses the same sample-space logic you've already solved twice removes exam anxiety. (4) **Marking Scheme Familiarity** — By studying solutions, you learn exactly where CBSE gives partial credit: writing the formula (1 mark), substituting values (1 mark), final answer (1 mark). This structure repeats across years. Rather than passive reading, active question-solving is the shortcut to 90+ in this chapter.

These five question types appear almost every year. Each is worth 1 mark and typically requires a single numerical answer or short definition. **Q1: Mean of a Simple Dataset** The marks obtained by 5 students in a test are 12, 15, 18, 21, and 24. Find the mean. Solution: Mean = (12 + 15 + 18 + 21 + 24) ÷ 5 = 90 ÷ 5 = **18 marks** **Q2: Mode Identification** In the dataset 5, 8, 8, 8, 12, 15, 15, 20, what is the mode? Solution: The value 8 appears 3 times (most frequent). Mode = **8** **Q3: Median of Odd Number of Values** Find the median of 7, 11, 5, 19, 3 (arrange first: 3, 5, 7, 11, 19). Solution: Middle value of 5 numbers = 3rd value = **7** **Q4: Basic Probability Formula** A fair die is rolled once. What is the probability of getting an even number? Solution: Even numbers on a die: 2, 4, 6 (3 outcomes). Total outcomes = 6. P(even) = 3/6 = **1/2 or 0.5** **Q5: Sample Space Counting** A coin is tossed twice. How many outcomes are in the sample space? Solution: Sample space = {HH, HT, TH, TT}. Total outcomes = **4** These questions test foundational understanding without multi-step reasoning, making them high-confidence marks if you practise them regularly.

Three-mark questions require formula application, intermediate calculations, and proper presentation. These five patterns dominate past papers. **Q1: Mean with Frequency Distribution** The following table shows the number of students and their heights: | Height (cm) | 140 | 145 | 150 | 155 | 160 | | Frequency | 4 | 6 | 8 | 5 | 2 | Find the mean height. Solution: Sum of (height × frequency) = (140×4) + (145×6) + (150×8) + (155×5) + (160×2) = 560 + 870 + 1200 + 775 + 320 = 3725 Total frequency = 4 + 6 + 8 + 5 + 2 = 25 Mean = 3725 ÷ 25 = **149 cm** **Q2: Median from Grouped Data** Find the median class from a cumulative frequency table where N = 50 and the median class has cumulative frequency just ≥ N/2 = 25. Solution: Identify the class interval where CF ≥ 25; state it as **median class** with reasoning (1 mark for identifying, 1 for CF, 1 for class name). **Q3: Pie Chart Angle Calculation** In a survey, 360 students chose sports: 120 chose cricket, 100 chose football, 80 chose tennis, 60 chose badminton. Draw a pie chart. Solution: Cricket angle = (120/360) × 360° = 120° Football angle = (100/360) × 360° ≈ 100° Tennis angle = (80/360) × 360° ≈ 80° Badminton angle = (60/360) × 360° = 60° (1 mark for formula, 1 for calculations, 1 for angles summing to 360°) **Q4: Probability with Conditions** A bag has 5 red, 3 blue, and 2 green balls. A ball is drawn at random. What is the probability that it is not red? Solution: Total = 5 + 3 + 2 = 10. Not red = 3 + 2 = 5. P(not red) = 5/10 = **1/2** (1 mark for identifying total, 1 for counting favourable outcomes, 1 for final answer) **Q5: Bar Graph Interpretation** A bar graph shows rainfall (in mm) over 5 months. If July shows 150 mm, August shows 180 mm, how much more rain fell in August than July? Solution: Difference = 180 − 150 = **30 mm** (Read values from graph: 1 mark each, subtract: 1 mark) These questions test calculation accuracy and data interpretation skills that appear consistently across CBSE question papers.

Five-mark questions combine multiple concepts, require detailed working, and test deeper understanding. These three represent the highest-frequency patterns. **Q1: Mean, Median, Mode from Grouped Frequency Distribution** | Class Interval | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 | | Frequency | 5 | 10 | 15 | 12 | 8 | Find (i) mean, (ii) median class, (iii) mode class. **Full Solution:** (i) **Mean Calculation** (2 marks) Midpoints: 15, 25, 35, 45, 55 Sum of (midpoint × frequency) = (15×5) + (25×10) + (35×15) + (45×12) + (55×8) = 75 + 250 + 525 + 540 + 440 = 1830 Total frequency = 5 + 10 + 15 + 12 + 8 = 50 Mean = 1830 ÷ 50 = **36.6** (ii) **Median Class** (1.5 marks) Cumulative frequency: 5, 15, 30, 42, 50 N/2 = 50/2 = 25 Median class is 30–40 (CF = 30 ≥ 25, and previous CF = 15 < 25) (iii) **Mode Class** (1.5 marks) Highest frequency = 15 (class 30–40) Mode class = **30–40** **Q2: Probability with Two-Stage Events (Tree Diagram or Listing)** A box has 4 red and 6 blue marbles. Two marbles are drawn without replacement. Find the probability that: (i) both are red, (ii) one is red and one is blue, (iii) at least one is blue. **Full Solution:** Total marbles = 10. Drawing without replacement. (i) **Both Red** (1.5 marks) P(both red) = P(1st red) × P(2nd red | 1st red) = (4/10) × (3/9) = 12/90 = **2/15** (ii) **One Red, One Blue** (1.5 marks) P(R then B) + P(B then R) = (4/10 × 6/9) + (6/10 × 4/9) = 24/90 + 24/90 = 48/90 = **8/15** (iii) **At Least One Blue** (2 marks) P(at least 1 blue) = 1 − P(both red) = 1 − 2/15 = **13/15** **Q3: Pie Chart Construction from Real Data + Interpretation** A school surveyed 720 students on their favourite subject: - Science: 240 students - Mathematics: 180 students - English: 150 students - Social Studies: 150 students Construct a pie chart. If 25% of science students are girls, how many science-loving girls are there? **Full Solution:** (i) **Pie Chart Angles** (2 marks) Science: (240/720) × 360° = 120° Mathematics: (180/720) × 360° = 90° English: (150/720) × 360° = 75° Social Studies: (150/720) × 360° = 75° (Sketch pie chart with these angles; 1 mark for correct formula, 1 mark for accurate drawing) (ii) **Calculation from Pie Chart Data** (3 marks) Science students = 240 25% are girls = 0.25 × 240 = **60 girls** like science (1 mark for identifying 240, 1 mark for applying percentage, 1 mark for final answer) These 5-mark questions test your ability to synthesize multiple skills: data organization, formula application, calculation accuracy, and interpretation.

The 2024–25 CBSE rationalized syllabus has refined Chapter 14 focus areas. Key shifts to watch: **1. Emphasis on Grouped Data Over Raw Data** Older papers (pre-2023) often had mean/median questions on ungrouped datasets. The 2024–25 onwards pattern favours **frequency distributions and class intervals**, requiring cumulative frequency calculations and median class identification. Practise grouped frequency tables extensively. **2. Real-World Data Interpretation** Expect more multi-part questions where you **read a pie chart or bar graph, then answer probability or statistical questions** on that data. For example: "A pie chart shows 5 sectors. Calculate missing frequency, then find probability that a randomly selected item belongs to sector X." **3. Sample Space and Probability Shift to Compound Events** Simple single-event probability (rolling a die: P(even) = 1/2) is now often combined with **two-stage, three-stage, or conditional probability**. Without-replacement scenarios appear frequently. Tree diagrams are now expected in solutions. **4. Reduced Calculation, Increased Reasoning** Examiners are moving away from "calculate mean of 50 data points" and toward **conceptual questions**: "Explain why mode is more appropriate than mean for this dataset." This means understanding *when* to use which measure, not just mechanical calculation. **5. Integration with Other Topics** Data Handling now bleeds into geometry (finding angles for pie charts) and algebra (working with expressions in frequency distributions). Expect questions that span multiple chapters. **6. Focus on Median Over Mean** In recent years, median-related questions (finding median class, interpreting position) outnumber mean calculations by roughly 2:1. Invest extra time in cumulative frequency curves and median class identification.

**Step 1: Read the Entire Question (30 seconds)** Don't assume it's asking for mean just because it lists numbers. Check: Is this grouped or ungrouped? Probability or statistics? Multi-part or single? Reading carefully saves wrong answers. **Step 2: Identify Question Type & Mark Value (20 seconds)** 1-mark questions: **Short answer**. Write formula + one-line answer. No lengthy working needed. 3-mark questions: **Show formula, substitution, and answer**. Examiners award marks for method, even if final answer has a small error. 5-mark questions: **Label each sub-part clearly (i), (ii), (iii)**. Allocate ≈1 minute per mark. If stuck on (ii), move to (iii) and return later. **Step 3: For Statistical Questions (Mean, Median, Mode)** - **Ungrouped data:** Arrange in order first (for median/mode). Then calculate. - **Grouped data:** Build a frequency table with midpoints, cumulative frequencies, angles (if pie chart). Double-check that frequencies/angles sum correctly (Σf = N, Σangles = 360°). - **Common trap:** Forgetting to divide by total frequency when calculating mean. Always write mean = (Σfx) ÷ (Σf). **Step 4: For Probability Questions** - **Step 1:** List the sample space clearly (even if large, list at least 3–4 outcomes to show understanding). - **Step 2:** Count total outcomes. Count favourable outcomes. - **Step 3:** Write P(event) = favourable/total. Simplify the fraction. - **Two-stage events:** Use multiplication rule or tree diagram. P(A and B) = P(A) × P(B | A). **Step 5: For Data Interpretation (Bar/Pie Charts)** - **Reading:** Use a ruler for bar graphs. Estimate pie chart sectors by angles (90° = 1/4, 120° = 1/3). - **Construction:** If asked to draw a pie chart, measure angles with a protractor. Label sectors clearly. - **Calculation from chart:** Always show the intermediate step (e.g., 240/720 = 1/3, then angle = 120°). **Step 6: Time Management** - **Allocate time by marks:** 90 seconds for 1-mark, 3 minutes for 3-mark, 5 minutes for 5-mark. Set a timer on your phone during practice. - **Prioritise:** Attempt 1-mark and 3-mark questions first (high confidence, quick marks). Return to 5-mark if time permits. - **If stuck:** Move on. Partial marks on attempted questions beat zero marks on unattempted ones. **Step 7: Final 5-Minute Check** - Frequencies sum to N? ✓ - Pie angles sum to 360°? ✓ - Probabilities between 0 and 1? ✓ - Units included (cm, mm, etc.)? ✓ - Fractions simplified? ✓ This systematic approach converts anxiety into structured action, helping you avoid careless errors and maximize marks in Chapter 14.

Solving 13 questions is only half the work. Here's how to extract full value: **Day 1–2: Solve Without Answers** Set a timer. Solve all 13 questions in exam conditions (no notes, calculator if permitted in your school, pen only). Mark your own rough attempts. Don't look at solutions yet. This reveals your actual weak spots. **Day 3–4: Study Solutions Deeply** For every incorrect answer, read the solution twice: 1. First read: Understand the logic and method. 2. Second read: Identify *exactly where* your approach diverged (Did you forget a step? Misread the question? Arithmetic error?). Jot down the mistake type: **Conceptual** (you didn't know the formula), **Procedural** (you knew the formula but applied it wrong), or **Careless** (arithmetic or reading). **Day 5–7: Solve Again** Re-solve only the questions you got wrong (or weren't fully confident on). This time, you'll work faster and more accurately. Repeat until you achieve 100% success. **Ongoing: Expand Beyond 13** These 13 questions are a curated core. Search your CBSE textbook's exercise section and your school's past papers for additional questions. Aim to solve 50+ total questions from Chapter 14 before your final exam. At cbsetutor.ai, each question type has 3–5 video explanations, helping you understand not just the answer but the examiner's mindset. **Week Before Exam: Speed Rounds** Solve full Chapter 14 mock tests (mix of 1-, 3-, and 5-mark questions) under timed conditions. Aim to complete them 10 minutes ahead of the allotted time. This builds exam-day confidence and reduces panic.