Class 8 Ganita Prakash Part 2 Chapter 5 Tales by Dots and Lines: Complete NCERT Solutions, Mean Median Dot Plots, Line Graphs & Spreadsheets

Introduction

“Tales by Dots and Lines” is Chapter 5 of Ganita Prakash Part-II (Class 8). It explores mean and median through dot plots (balancing act, LHS = RHS distances), how mean/median change when values are added/removed/multiplied, frequency tables, spreadsheets for quick calculations, and advanced visualisation using line graphs, infographics and activity strips. Real-life examples (wrestler weights, coconut harvest, rainfall, space launches, daily routines) build deep understanding of data interpretation.

Basic Knowledge Required

• Mean = total sum ÷ number of values
• Median = middle value (or average of two middle) in sorted list
• Dot plots and number lines
• Frequency tables
• Basic spreadsheet formulas (=SUM, =AVERAGE)
• Line graphs for trends over time

Key Definitions

• Mean (Arithmetic Average): Balance point where sum of distances on left = sum on right.
• Median: Middle value when data is ordered; equal number of values below and above.
• Frequency: How many times a value repeats.
• Line Graph: Data points joined by lines to show change over time.
• Infographic: Visual summary using maps, colours and icons.

Important Formulas Used

1. Mean: $\frac$​

2. . Median Position

   For n observations:

   • If n is odd

   $$\text{Median position} = \frac{n+1}{2}^{th} \text{ observation}$$

3. If n is even: Median = [(n/2)th term + (n/2 + 1)th term] / 2

4. . Mean using Frequency

   $$\text{Mean} = \frac{\sum (f \times x)}{\sum f}$$

   Where

   • $f$ = frequency

   • $x$ = value (observation)

5.1 The Balancing Act

Two numbers Mean is exactly midway (e.g., 3 and 7 → mean 5).

Three numbers Figure it Out (Page 104) 1–2. Mark mean on each dot plot:

• First: Mean = 7 (balance)
• Second: Mean = 5
• Third: Mean = 12
• Fourth: Mean = 7.5

Mean is centre because total distance left = total distance right (LHS = RHS).

Not always midpoint of extremes LHS distances sum = RHS distances sum (examples with 1+1=2 vs 3+1+4=8 balanced at mean 7).

Verify for earlier collections → All satisfy LHS = RHS.

Only one centre If mean shifts left/right, distances become unequal → only one balance point.

Adding/removing values

• Add value > mean → mean increases (new balance).
• Add value < mean → mean decreases.
• Remove value > mean → mean decreases.
• Remove value < mean → mean increases.
• Add/remove value = mean → mean unchanged.

Unchanging Mean Figure it Out (Page 105–106) Possible to add/remove 2 values keeping mean same (e.g., +1 and –1 or pairs balancing). 3 values also possible (shown example: two below + one above).

Relatively Unchanged Add fixed number k to every value → mean + k. Multiply every value by m → mean × m. (Algebra proof given in book; fair-share view also works.)

Tinkering with Median Add value > median → median increases (or becomes average of two middles). Add value < median → median decreases.

Finding the Unknown Figure it Out (Page 109)

1. Wrestler weights: Sum = 392, known sum = 349 → w = 43 kg.
2. Coconuts: Total = 25.6 × 15 = 384. Actual = 384 – 3 = 381. Correct average = 381/15 = 25.4.

Mean and Median with Frequencies

Family size example: Mean = (3×3 + 4×11 + 5×9 + 6×7 + 7×3 + 8×1 + 9×1 + 10×1)/36 = 188/36 = 5.22 Median: 18th & 19th values = 5 (cumulative frequencies reach 23 at 5).

Spreadsheets

Farooq Maths = cell E5. Column B7 = Gowri Odia. Ashwin >30: Telugu, English, Maths. Science average: =AVERAGE(G2:G23) Odia vs Telugu: Compare =AVERAGE(B2:B23) and =AVERAGE(C2:C23) Total per student: =SUM(B2:G2) dragged down.

Figure it Out (Page 113–115)

1. (i) First 50 naturals mean = 25.5 (ii) First 50 odds mean = 50 (iii) First 50 multiples of 4 mean = 102
2. Missing dot at 9 (LHS = RHS = 9 units balance).
3. (i) Subtract 1 cm from mean → 149.2 cm (d) (ii) Correct = 149.2 cm
4. Album C (dots sum to 5.57 average).
5. Median = 41 (i) Any value between 26–34 (e.g., 30) (ii) One ≤26 and one ≥34 (iii) Any value between 26–34 (except extremes if affects).
6. (i) Sometimes (if odd count) (ii) Always (iii) Sometimes (iv) Never (median stays or shifts right).
7. y = 8 (sum = 83, mean 10.375 × 8 = 83).
8. Sum = 134 × 15 = 2010.
9. p can be 25, 29, 30 (median stays 29).
10. (i) Mean = 5.125 (ii) Median = 5 (iii) (b) & (c) valid; (a) false (some 0), (d) false (4 students twice). (e) New mean +1, median +1.
11. Min=1, Max=10, Mean=7.2, Median=8.

5.2 Visualising and Interpreting Data

Line Graphs Temperature Kerala/Punjab: Punjab varies more (19°C–38°C), Kerala stable ~29–33°C. Space launches: Sharp rise post-2018, USA dominant. Rainfall: West coast peak June–Aug, East Oct–Dec.

Figure it Out (Page 122)

1. Line graph: Visiting peaks Sunday, Purchasing follows.
2. (i) Weather stations average over years. (ii) Plot rounded values (Mangaluru high Jul–Aug). (iii) New Delhi: low all year (0–3 days). (iv) Mangaluru most (~100 days/year), New Delhi least (~30). (v) New Delhi: Jul–Sep; Rameswaram: Oct–Dec.

Births India: Peaks Jan & Jul–Aug; dips May–Jun.

Infographics Wheat vs Rice map: Clear west rice, north wheat divide.

What can a Strip Say? Colours: Blue=sleep, Green=school, Yellow=eat, etc. Strips: Friday (school long), Saturday (friends movie), Sunday (relax). Lunch break ~12:30 pm.

15 Most Asked FAQs – Tales by Dots and Lines (Click to Expand)

1. What happens to mean when we add a value greater than current mean?

Answer: Mean increases to restore LHS = RHS balance (example: 11 → 12.2).

2. Formula for mean with frequencies?

Answer: \( \frac{\sum (\text{value} \times \text{frequency})}{\sum \text{frequency}} \) (family size = 5.22).

3. Missing wrestler weight if mean = 39.2?

Answer: 43 kg (349 + w = 392).

4. Correct coconut average after +3 error?

Answer: 25.4 (384 – 3 = 381, 381/15 = 25.4).

5. Spreadsheet formula for total marks of Nagesh?

Answer: =SUM(B3:G3).

6. Mean of first 50 natural numbers?

Answer: 25.5 ((1+50)×50/2).

7. Median of 8,10,19…92 list?

Answer: 41 (average of 8th & 9th in sorted list).

8. Punjab vs Kerala temperature trend?

Answer: Punjab varies 19–38°C; Kerala stable ~29–33°C.

9. Peak rainfall months for Mumbai?

Answer: June–August (west coast monsoon).

10. How many space objects in 2023?

Answer: ~2900 (highest year).

11. Manoj activity strip – lunch break time?

Answer: ~12:30 pm (green school block ends).

12. Mean when every value × 5?

Answer: New mean = 5 × old mean (algebra proof).

13. States preferring rice most (infographic)?

Answer: Kerala, Tamil Nadu, West Bengal, Odisha, Andhra (+100 rice).

14. Adding value without changing median – possible range?

Answer: Between 26–34 for the given list (median stays 41).

15. Cycle rides mean & median?

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Answer: Mean 5.125, Median 5 (dot plot count).