Class 7 Mathematics Ch 12 Mastery Guide: Data Handling, Frequency Distributions, Central Tendency & Graphs (FBISE)
🗺️ Teacher & Parent Roadmap: Unit 12 (Data Handling)
In our modern data-driven world, Data Handling is one of the most practical and empowering branches of mathematics. Grade 7 students learn how raw facts and figures are systematically transformed into actionable knowledge. From conducting surveys and organizing raw data into grouped frequency distributions with tally marks, to calculating the three key measures of central tendency (Mean, Median, Mode) and constructing impactful visual diagrams (Pie Charts, Bar Graphs, and Histograms), this unit builds vital quantitative literacy and critical reasoning skills.
- Differentiate between Primary vs. Secondary Data and Ungrouped (Raw) vs. Grouped Data.
- Construct tabular Frequency Distributions using tally marks, determining Class Limits ($l_1, l_2$), Class Width ($h$), and Class Marks / Midpoints ($x = \frac{l_1 + l_2}{2}$).
- Compute the Arithmetic Mean ($\bar{X}$) for ungrouped data ($\bar{X} = \frac{\sum x}{n}$) and grouped data ($\bar{X} = \frac{\sum fx}{\sum f}$).
- Determine the Median (the middle-most value of an ordered dataset for odd and even $n$) and Mode (the most frequent value).
- Construct and interpret Pie Graphs (central angle $\theta = \frac{\text{Value}}{\text{Total}} \times 360^\circ$), Bar Graphs (discrete bars with equal spacing), and Histograms (continuous adjacent rectangles without gaps).
- Calculate the Range ($\text{Maximum} - \text{Minimum}$) as a measure of data spread.
- Unordered Median Trap: Picking the middle number from an unsorted list without first arranging the data in ascending or descending order!
- Even Count Median: Forgetting that when $n$ is even, the median is the arithmetic mean of the two middle numbers ($\frac{\text{middle}_1 + \text{middle}_2}{2}$).
- Bar Graph vs. Histogram Confusion: Leaving gaps between bars in a histogram (wrong! histograms represent continuous classes and bars must touch) or joining bars in a bar graph of discrete categories.
- Pie Chart Total Angle: Calculating sector angles out of $100^\circ$ instead of the full circle's $360^\circ$.
📊 Core Concepts & Visual Foundations
2.1 The Data Life Cycle & Classification of Data
Data refers to distinct pieces of numerical information, observations, or facts gathered for analysis.
- Primary Data: Data collected first-hand directly by the investigator (e.g., measuring student heights with a tape measure, classroom questionnaires).
- Secondary Data: Data obtained from existing secondary sources (e.g., published census reports, weather department records, internet archives).
- Ungrouped Data (Raw Data): Data in its original unorganized list form, exactly as collected.
- Grouped Data: Data condensed into orderly classes or intervals with corresponding frequencies.
2.2 Frequency Distribution & Anatomy of Class Intervals
A frequency distribution is a tabular summary displaying the frequencies (count of occurrences) of different data classes.
2.3 Measures of Central Tendency (Mean, Median, Mode)
Central tendency describes the central or typical value of a probability distribution or dataset.
$$\text{Ungrouped: } \bar{X} = \frac{\sum x}{n} = \frac{x_1 + x_2 + \dots + x_n}{n}$$ $$\text{Grouped: } \bar{X} = \frac{\sum fx}{\sum f}$$ Represents the balance point (fulcrum) of all values.
$$\text{If } n \text{ is odd: Median} = \text{Value at position } \left(\frac{n+1}{2}\right)$$ $$\text{If } n \text{ is even: Median} = \frac{\text{Term }\frac{n}{2} + \text{Term }\left(\frac{n}{2}+1\right)}{2}$$ Represents the positional exact middle after sorting.
$$\text{Mode} = \text{Observation occurring with the highest frequency}$$ Unimodal (1 peak), Bimodal (2 peaks), Multimodal (3+), or No Mode.
2.4 Statistical Visualizations: Pie Graphs, Bar Graphs & Histograms
📝 100% Solved Exercises (Step-by-Step for Class 7)
Exercise 12.1: Frequency Distribution Tables & Class Intervals 6 Questions + Check Point
Q1. Make a frequency distribution table using two as the size of class (i.e. 0–1, 2–3, ...) for the following 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
Minimum value = 0, Maximum value = 10. Class size = 2.
• 0 – 1: Values: 0, 1, 1 $\implies$ Tally:
/// $\implies$ Frequency = 3• 2 – 3: Values: 2, 2, 3, 3, 3 $\implies$ Tally:
//// $\implies$ Frequency = 5• 4 – 5: Values: 4, 4, 4, 5, 5, 5 $\implies$ Tally:
//// / $\implies$ Frequency = 6• 6 – 7: Values: 6, 6, 6, 7, 7, 7, 7 $\implies$ Tally:
//// // $\implies$ Frequency = 7• 8 – 9: Values: 8, 8, 9 $\implies$ Tally:
/// $\implies$ Frequency = 3• 10 – 11: Values: 10 $\implies$ Tally:
/ $\implies$ Frequency = 1Total Frequency ($\sum f$) = 3 + 5 + 6 + 7 + 3 + 1 = 25.
Q2. Factory Workers Age Table:
15–24 (10), 25–34 (15), 35–44 (20), 45–54 (25), 55–64 (30), 65–74 (9), 75–84 (1).
(ii) Frequency of 7th class (75–84): $\mathbf{1\text{ worker}}$.
(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 (Class size = 3):
• 1–3: 9 mistakes | • 4–6: 7 mistakes | • 7–9: 7 mistakes | • 10–12: 5 mistakes | • 13–15: 2 mistakes. Total = 30.
Q4. Hospital disease deaths (Class size = 5):
• 1–5: 3 | • 6–10: 4 | • 11–15: 5 | • 16–20: 4 | • 21–25: 3 | • 26–30: 1. Total = 20.
Exercise 12.2: Mean, Median, Mode, Pie Graphs, Bar Graphs & Histograms 11 Questions
Q1. Calculate Arithmetic Mean:
(i) $5, 3, 12, 8, 22 \implies \text{Mean} = \frac{50}{5} = \mathbf{10}$.
(ii) $103, 105, 108, 112, 122 \implies \text{Mean} = \frac{550}{5} = \mathbf{110}$.
(iii) 10 marks: $35, 67, 30, 52, 57, 49, 76, 66, 36, 22 \implies \text{Mean} = \frac{490}{10} = \mathbf{49}$.
Q2. Heights of 10 boys:
(i) Tallest boy = $\mathbf{151\text{ cm}}$.
(ii) Shortest boy = $\mathbf{128\text{ cm}}$.
(iii) Mean height = $\frac{1421}{10} = \mathbf{142.1\text{ cm}}$.
(iv) Boys with height $< 142.1\text{ cm}$: $135, 128, 139, 132 \implies \mathbf{4\text{ boys}}$.
Q3. Frequency Table Mean:
(i) Values $x \in \{0,1,2,3,4,5\}$, frequencies $f \in \{12,18,15,9,3,3\}$:
$$\sum f = 60, \quad \sum fx = 0 + 18 + 30 + 27 + 12 + 15 = 102 \implies \text{Mean} = \frac{102}{60} = \mathbf{1.7}$$
(ii) Values $x \in \{15,16,17,18,19,20\}$, frequencies $f \in \{12,18,15,9,3,3\}$:
$$\sum f = 60, \quad \sum fx = 180 + 288 + 255 + 162 + 57 + 60 = 1002 \implies \text{Mean} = \frac{1002}{60} = \mathbf{16.7}$$
Q4. Math Test Marks Mean:
(i) 10 marks sum to $100 \implies \text{Mean} = \frac{100}{10} = \mathbf{10}$.
(ii) When 21 is added: $\text{New Sum} = 121, n=11 \implies \text{New Mean} = \frac{121}{11} = \mathbf{11}$.
Q5. Find the Median:
(i) $84, 95, 72, 88, 72, 67, 80, 91$ (8 values) $\to$ Sorted: $67, 72, 72, 80, 84, 88, 91, 95 \implies \text{Median} = \frac{80+84}{2} = \mathbf{82}$.
(ii) $70, 73, 69, 68, 75, 81, 82$ (7 values) $\to$ Sorted: $68, 69, 70, 73, 75, 81, 82 \implies \text{Median} = \mathbf{73}$.
(iii) 9 values $\to$ Sorted: $40, 40, 41, 42, 44, 45, 47, 47, 48 \implies \text{Median} = \mathbf{44}$.
(iv) 8 values $\to$ Sorted: $10, 12, 15, 18, 20, 30, 49, 50 \implies \text{Median} = \frac{18+20}{2} = \mathbf{19}$.
(v) 8 values $\to$ Sorted: $43, 46, 47, 47, 49, 50, 51, 55 \implies \text{Median} = \frac{47+49}{2} = \mathbf{48}$.
Q6. Find the Mode:
(i) $3, 4, 4, 6, 7, 5, 4, 5 \implies \mathbf{4}$ (appears 3 times).
(ii) $5, 8, 7, 6, 9, 7, 5, 7, 5, 6, 9, 7 \implies \mathbf{7}$ (appears 4 times).
(iii) $3, 7, 5, 4, 9, 5, 0, 2, 5, 1, 6, 3, 4, 5 \implies \mathbf{5}$ (appears 4 times).
Q8. Sports Week Pie Graph Sector Angles (Total = 710):
• Cricket: $\frac{200}{710} \times 360^\circ = \mathbf{101.4^\circ}$
• Hockey: $\frac{140}{710} \times 360^\circ = \mathbf{71.0^\circ}$
• Football: $\frac{80}{710} \times 360^\circ = \mathbf{40.6^\circ}$
• Badminton: $\frac{150}{710} \times 360^\circ = \mathbf{76.1^\circ}$
• Volleyball: $\frac{80}{710} \times 360^\circ = \mathbf{40.6^\circ}$
• Table Tennis: $\frac{60}{710} \times 360^\circ = \mathbf{30.4^\circ}$
Q10. Shahzad's Marks Bar Graph:
(i) Shows marks of Shahzad in 5 subjects.
(ii) Best subject = Social Science (70 marks).
(iii) Poorest subject = Mathematics (50 marks).
(iv) Average marks = $\frac{60+50+70+55+65}{5} = \mathbf{60\text{ marks}}$.
(v) Average of Math & Science = $\frac{50+55}{2} = \mathbf{52.5\text{ marks}}$.
(vi) Median marks = $\mathbf{60\text{ marks}}$.
Review Exercise 12: Objective MCQs & Conceptual Solutions 10 MCQs + Review Proofs
- The data collected from the first information and written in ordinary form is: (a) ungrouped data.
- The data written in systematic form or any particular order is: (b) grouped data.
- The most repeated value in the data is: (b) mode.
- In data $2, 5, 4, 5, 6, 5, 3, 4, 2$, mode is: (d) 5.
- Arithmetic mean of $3000, 5000, 4000, 2000, 6000$ is: (b) 4000.
- Lower limits of $(41–45)$ and $(46–50)$ are: (c) 41, 46.
- Upper limits of $(31–40)$ and $(41–50)$ are: (a) 40, 50.
- Median of $25, 15, 10, 5, 20$ (sorted: $5, 10, 15, 20, 25$) is: (c) 15.
- Mean, Median and Mode describe: (a) Measure of central tendency.
- Difference between largest and smallest value is: (d) range.
🌟 Data Handling Master Formula Cheatsheet (Grade 7 FBISE)
- Ungrouped Mean: $\bar{X} = \frac{\sum x}{n}$
- Grouped Mean: $\bar{X} = \frac{\sum fx}{\sum f}$
- Median (Odd $n$): Term at $\frac{n+1}{2}$
- Median (Even $n$): $\frac{\text{Term }\frac{n}{2} + \text{Term }\left(\frac{n}{2}+1\right)}{2}$
- Mode: Most frequent observation
- Class Mark (Midpoint): $x = \frac{l_1 + l_2}{2}$
- Class Size ($h$): $l_2 - l_1 + 1$
- Range: $\text{Maximum} - \text{Minimum}$
- Total Observations: $n = \sum f$
- Sector Angle ($\theta$): $\frac{\text{Category Value}}{\text{Total}} \times 360^\circ$
- Bar Graph: Discrete bars with equal gaps
- Histogram: Continuous touching rectangles
- Total Pie Chart Angle: $360^\circ$
More Chapter Notes for Class 7 (FBISE)
MathematicsTest Your Knowledge on Chapter 12: Class 7 Mathematics Ch 12 Mastery Guide: Data Handling, Frequency Distributions, Central Tendency & Graphs (FBISE)
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