Chapter 12: Data Handling — Comprehensive Solved Notes & Solution Manual
Mastery Guide: Data Handling — Frequency Distribution, Central Tendency & Statistical Diagrams
Comprehensive student mastery guide, formulas, and step-by-step solved textbook manual for Class 8 Mathematics.
📖 Unit Overview & Target Learning Outcomes
By mastering this unit, Class 8 students will be able to:
- Distinguish Data Formats: Differentiate clearly between raw/ungrouped data (unorganized primary observations) and systematically organized grouped data.
- Construct Frequency Distribution Tables: Form class intervals, determine class width ($h$), record tally marks, and compute class frequencies ($f$).
- Identify Class Interval Properties: Recognize Lower Class Limits ($l_1$), Upper Class Limits ($l_2$), and calculate Class Mid-points ($x = \frac{l_1 + l_2}{2}$).
- Calculate Measures of Central Tendency:
- Arithmetic Mean ($\bar{X}$): For ungrouped data ($\bar{X} = \frac{\sum X}{n}$) and grouped/frequency data ($\bar{X} = \frac{\sum fx}{\sum f}$).
- Median: Locate the middle ranked observation in ordered datasets for both odd and even sample sizes.
- Mode: Identify the most frequently occurring value(s) in unimodal, bimodal, or multimodal datasets.
- Construct & Interpret Statistical Diagrams: Accurately draw and analyze Pie Graphs (using sector angles: $\theta = \frac{\text{Value}}{\text{Total}} \times 360^\circ$), Bar Graphs (with uniform spacing), and Histograms (adjacent rectangles with continuous class intervals).
💡 Kid-Friendly Tips & Memory Hooks
• Mean: "Mean is the Generous Share" (Add all & divide equally by $n$).
• Median: "Median is the Middle of the Road" (Sort first, then pick the exact middle).
• Mode: "Mode is the Most Popular" (The number that shows up the most).
A full circle is always $360^\circ$. Every sector angle is simply its fraction of the total multiplied by $360^\circ$: $$\text{Sector Angle} = \frac{\text{Component Value}}{\text{Total Sum}} \times 360^\circ$$
• Bar Graph: Has gaps between bars (used for distinct categories or discrete items).
• Histogram: Bars touch each other with zero gaps (used for continuous class intervals).
🌍 Real-World Connections
- Climatology & Weather Forecasting: Meteorologists record daily temperatures and rainfall figures across decades, grouping them into intervals to detect long-term climate patterns and flood risks.
- Retail & Inventory Optimization (e.g., Uniform Shops): Clothing store owners tally customer age distributions to decide which uniform sizes to stock in large quantities, minimizing unsold waste.
- Sports Analytics: Cricket and football analysts compute batting averages (Mean), strike rates, and plot run-rate bar charts to evaluate team performance.
- National Census & Economic Planning: The Pakistan Bureau of Statistics groups income, age, and literacy data to allocate government resources, hospitals, and schools effectively.
🔑 Study Cues & Essential Inquiries
- Why can a single extreme value (outlier) drastically skew the Mean, while the Median remains completely unaffected?
- Can a dataset have more than one Mode, or no Mode at all? (Yes: bimodal/multimodal, or no mode if all counts are equal).
- When constructing a grouped frequency table, how does changing the class size ($h$) affect the clarity of the distribution?
- Why do the rectangles in a histogram have no space between them, unlike a standard vertical bar chart?
🌟 Section-by-Section Conceptual Breakdown
12.1 Data Classification & Frequency Distribution
Data refers to any collection of numerical facts, measurements, or observations gathered from any field of inquiry.
- Raw / Ungrouped Data: Data collected in its original, unorganized sequence directly from the source (e.g., student test scores recorded randomly).
- Grouped Data: Data organized systematically into classes, categories, or intervals alongside their corresponding frequencies.
- Frequency ($f$): The total number of times a specific observation occurs in the dataset.
- Class Limits ($l_1, l_2$): Each class interval is bounded by a lower limit ($l_1$) and an upper limit ($l_2$). For example, in the class $10 - 19$, $l_1 = 10$ and $l_2 = 19$.
- Class Size / Width ($h$): The difference between two consecutive lower class limits: $h = l_{1,\text{next}} - l_{1,\text{current}}$.
- Class Mid-point ($x$): $$x = \frac{l_1 + l_2}{2}$$
12.2 Measures of Central Tendency (Mean, Median & Mode)
A measure of central tendency is a single representative value that describes the center of a data distribution:
- Arithmetic Mean ($\bar{X}$):
• Ungrouped data: $\bar{X} = \frac{\sum X}{n}$
• Grouped data: $\bar{X} = \frac{\sum fx}{\sum f}$ - Median:
• Odd $n$: Observation at position $\frac{n+1}{2}$.
• Even $n$: Average of observations at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$. - Mode: The observation with the highest frequency.
12.3 Statistical Displays: Pie Graphs, Bar Graphs & Histograms
Visual diagrams transform numerical tables into intuitive graphic insights:
🎯 Unit Synthesis Summary
Unit 12 establishes essential statistical foundations for middle and secondary school mathematics under the Federal Board curriculum. Raw statistical data collected from experiments or surveys is condensed into grouped frequency distribution tables using structured class intervals and tally marks. Summary statistics—specifically the Mean ($\bar{X} = \frac{\sum fx}{\sum f}$), Median (the central ranked value), and Mode (the most frequent value)—provide critical numerical benchmarks for analyzing data distributions. Finally, visual representations—including Pie Graphs based on $360^\circ$ circular sectors, Bar Graphs for discrete comparison, and Histograms for continuous class intervals—enable intuitive, real-world data interpretation and decision making.
📝 Solved Textbook Exercises — Step-by-Step Manual
Exercise 12.1 • Frequency Distribution & Grouped Data
Q1. Make a frequency distribution table using two as the size of class (i.e. 0-1, 2-3, ...) for the given data:
3, 7, 6, 10, 9, 8, 7, 6, 5, 1, 2, 4, 8, 7, 5, 3, 2, 1, 0, 7, 6, 5, 4, 3, 4
- Step 1: Minimum value $= 0$, Maximum value $= 10$. Total observations $n = 25$.
- Step 2: Form class intervals of width $h = 2$: $0-1, 2-3, 4-5, 6-7, 8-9, 10-11$.
- Step 3: Calculate frequencies:
• $0–1$: $0, 1, 1 \implies \mathbf{3}$
• $2–3$: $2, 2, 3, 3, 3 \implies \mathbf{5}$
• $4–5$: $4, 4, 4, 5, 5, 5 \implies \mathbf{6}$
• $6–7$: $6, 6, 6, 7, 7, 7, 7 \implies \mathbf{7}$
• $8–9$: $8, 8, 9 \implies \mathbf{3}$
• $10–11$: $10 \implies \mathbf{1}$ - Final Answer: Total Frequency $\sum f = 25$.
Q2. Factory workers age distribution table analysis:
- (i) Total number of workers: $\sum f = 10 + 15 + 20 + 25 + 30 + 9 + 1 = \mathbf{110\text{ workers}}$.
- (ii) Frequency of 7th class ($75-84$): $\mathbf{1}$.
- (iii) Workers of ages ($45-74$): $25 + 30 + 9 = \mathbf{64\text{ workers}}$.
- (iv) Workers having ages less than 35 years: $10 + 15 = \mathbf{25\text{ workers}}$.
- (v) Size of each age group: $25 - 15 = \mathbf{10\text{ years}}$.
Q3. Mistakes per page by 30 students taking 3 as interval size:
• $0–2: 5 \quad|\quad 3–5: 9 \quad|\quad 6–8: 6 \quad|\quad 9–11: 5 \quad|\quad 12–14: 5$
Final Answer: Total Frequency $\sum f = 30$.
Q4. Disease deaths across 20 hospitals taking 5 as class size:
• $1–5: 3 \quad|\quad 6–10: 4 \quad|\quad 11–15: 5 \quad|\quad 16–20: 4 \quad|\quad 21–25: 3 \quad|\quad 26–30: 1$
Final Answer: Total Frequency $\sum f = 20$.
Q5. Marks of 80 students in computer subject of 10th class:
(a) Class size 5 ($51-55, \dots, 96-100$): Frequencies: $1, 5, 11, 7, 19, 15, 8, 7, 5, 2 \implies \text{Total} = \mathbf{80}$.
(b) Class size 7 ($51-57, \dots, 93-99$): Frequencies: $2, 12, 13, 20, 19, 7, 7 \implies \text{Total} = \mathbf{80}$.
Q6. Marks of 20 students in mathematics taking 10 as interval size:
• $80–89: 2 \quad|\quad 90–99: 4 \quad|\quad 100–109: 2 \quad|\quad 110–119: 2 \quad|\quad 120–129: 4 \quad|\quad 130–139: 3 \quad|\quad 140–149: 3$
Final Answer: Total Frequency $\sum f = 20$.
Exercise 12.2 • Central Tendency & Statistical Displays
Q1. Find the mean:
• (i) $5, 3, 12, 8, 22 \implies \bar{X} = \frac{50}{5} = \mathbf{10}$.
• (ii) $103, 105, 108, 112, 122 \implies \bar{X} = \frac{550}{5} = \mathbf{110}$.
• (iii) Marks of 10 students $\implies \bar{X} = \frac{490}{10} = \mathbf{49}$.
Q2. Heights of 10 boys:
• (i) Tallest: $\mathbf{151\text{ cm}}$.
• (ii) Shortest: $\mathbf{128\text{ cm}}$.
• (iii) Mean: $\bar{X} = \frac{1421}{10} = \mathbf{142.1\text{ cm}}$.
• (iv) Heights $< 142.1\text{ cm}$: $\mathbf{4\text{ boys}}$.
Q3. Grouped Frequency Means:
• (a) $\sum f = 60, \sum fx = 102 \implies \bar{X} = \frac{102}{60} = \mathbf{1.7}$.
• (b) $\sum f = 60, \sum fx = 1002 \implies \bar{X} = \frac{1002}{60} = \mathbf{16.7}$.
Q4. Test Marks Mean & Added Value:
• (i) Mean $= \frac{100}{10} = \mathbf{10}$.
• (ii) With 21 added: New Mean $= \frac{100 + 21}{11} = \frac{121}{11} = \mathbf{11}$.
Q5. Find the median:
• (i) $84, 95, 72, 88, 72, 67, 80, 91 \implies \mathbf{82}$.
• (ii) $70, 73, 69, 68, 75, 81, 82 \implies \mathbf{73}$.
• (iii) $45, 47, 42, 40, 48, 40, 47, 44, 41 \implies \mathbf{44}$.
• (iv) $10, 30, 12, 15, 50, 49, 18, 20 \implies \mathbf{19}$.
• (v) $43, 46, 49, 51, 47, 55, 50, 47 \implies \mathbf{48}$.
Q6. Find the mode:
• (i) $3, 4, 4, 6, 7, 5, 4, 5 \implies \mathbf{\text{Mode} = 4}$.
• (ii) $5, 8, 7, 6, 9, 7, 5, 7, 5, 6, 9, 7 \implies \mathbf{\text{Mode} = 7}$.
• (iii) $3, 7, 5, 9, 5, 0, 2, 5, 1, 6, 3, 4, 5 \implies \mathbf{\text{Mode} = 5}$.
Q7. Combined Mean, Median & Mode:
• (i) $\mathbf{\text{Mean} = 12.19}, \quad \mathbf{\text{Median} = 13}, \quad \mathbf{\text{Mode} = 15}$.
• (ii) $\mathbf{\text{Mean} = 12.67}, \quad \mathbf{\text{Median} = 12}, \quad \mathbf{\text{Mode} = 11}$.
Q8. Sports Week Pie Graph Angles ($n = 710$):
Cricket: $\mathbf{101.41^\circ}$, Hockey: $\mathbf{70.99^\circ}$, Football: $\mathbf{40.56^\circ}$, Badminton: $\mathbf{76.06^\circ}$, Volleyball: $\mathbf{40.56^\circ}$, Table Tennis: $\mathbf{30.42^\circ}$. Total $= \mathbf{360.00^\circ}$.
Q9. Family Expenditure Pie Graph Angles ($n = 185$):
Food: $\mathbf{97.30^\circ}$, Clothing: $\mathbf{58.38^\circ}$, Rent: $\mathbf{38.92^\circ}$, Light: $\mathbf{68.11^\circ}$, Education: $\mathbf{29.19^\circ}$, Fuel: $\mathbf{38.92^\circ}$, Misc: $\mathbf{29.19^\circ}$. Total $= \mathbf{360.00^\circ}$.
Q10. Shahzad Marks Bar Graph:
• (i) Display: Marks scored by Shahzad across school subjects.
• (ii) Best Subject: Mathematics (70 marks).
• (iii) Poor Subject: Urdu (40 marks).
• (iv) Overall Average: $\mathbf{55\text{ marks}}$.
• (v) Math & Science Average: $\mathbf{60\text{ marks}}$.
• (vi) Median: $\mathbf{55\text{ marks}}$.
Q11. Uniform Shop Customer Ages (52 customers):
• $5–8: f = 5\ (34.62^\circ) \quad|\quad 9–12: f = 11\ (76.15^\circ) \quad|\quad 13–16: f = 18\ (124.62^\circ) \quad|\quad 17–20: f = 8\ (55.38^\circ) \quad|\quad 21–24: f = 10\ (69.23^\circ)$
Final Answer: Total Frequency $= \mathbf{52}$, Total Sector Angles $= \mathbf{360.00^\circ}$.
Review Exercise 12 • Official Solutions
- Data in ordinary form: (a) ungrouped data
- Data in systematic form: (b) grouped data
- Most repeated value: (b) mode
- Mode of $\{2, 5, 4, 5, 6, 5, 3, 4, 2\}$: (c) 5
- Mean of $\{3000, 5000, 4000, 2000, 6000\}$: (b) 4000
- Lower limits of $(41–45)$ and $(46–50)$: (c) 41, 46
- Upper limits of $(31–40)$ and $(41–50)$: (a) 40, 50
- Median of $\{25, 15, 10, 5, 20\}$: (c) 15
- Mean, Median, Mode describe: (a) Measure of central tendency
- Largest minus smallest value: (d) range
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