Ch 5: Tales by Dots and Lines Quick revision notes Class 8th Mathematics (Ganita Prakash-II)

Quick revision notes

Ch 5: Tales by Dots and Lines Quick revision notes Class 8th Mathematics (Ganita Prakash-II)

Class 8 · Mathematics (Ganita Prakash) · Chapter 5 : Tales by Dots and Lines · All Board · ENGLISH · 8 views

Ye material inke liye bhi hai: All Board BIHAR BOARD CBSE CHHATTISGARH BOARD JHARKHAND BOARD MP BOARD NIOS RAJASTHAN BOARD UP BOARD

⚖️

Section 5.1

The Balancing Act — Mean as Centre

📌 What is Mean?

\[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} \]

📌 What is Median?

The middle value when data is sorted in order. Equal number of values on both sides.

Dot plot showing mean = 5 for values 3 and 7

3 and 7 on a number line — their mean (5) is exactly halfway between them

🎯 Mean as the Balance Point

The mean is not always the midpoint of the two extreme values. Instead, it is the point where total distances on the left = total distances on the right.

Four dot plot collections — mark the mean

Four collections: find where the mean sits as the balance point

LHS RHS distance dot plots showing mean as balance point

Mean = 7 (LHS = 1 = RHS) · Mean = 5 (LHS = 4 = RHS) · Mean = 12 (LHS 5 = RHS 5) · Mean = 7.5 (LHS 7 = RHS 7)

The Balance Rule

\[ \sum (\text{distances on LHS of mean}) = \sum (\text{distances on RHS of mean}) \]

There is only one such centre — the mean is unique.

Two diagrams showing LHS ≠ RHS when centre is not the mean

Shifting the centre away from the mean breaks the balance — LHS ≠ RHS

🔄

Section 5.1 continued

How the Mean Shifts — Rules & Algebra

Three dot plots showing mean shifting from 11 to 12.2 when new value added

Adding a new value (17) greater than the mean (11) → new mean increases to 12.2

➕ Add value > mean

Mean increases

➕ Add value < mean

Mean decreases

➖ Remove value < mean

Mean increases

➖ Remove value > mean

Mean decreases

💡

Adding or removing a value equal to the mean keeps the mean unchanged.

🔢 Rule 1 — Adding a Constant to Every Value

If every value increases by a fixed number \(k\), the mean also increases by \(k\).

\[ \text{Original mean} = a = \frac{x_1 + x_2 + \cdots + x_n}{n} \] \[ \text{New mean} = \frac{(x_1+k)+(x_2+k)+\cdots+(x_n+k)}{n} = \frac{\sum x_i + nk}{n} = a + k \]

✖️ Rule 2 — Multiplying Every Value by a Constant

If every value is multiplied by a fixed number \(k\), the mean also gets multiplied by \(k\).

\[ \text{New mean} = \frac{kx_1 + kx_2 + \cdots + kx_n}{n} = k \cdot \frac{\sum x_i}{n} = ka \]

📐 Example: Add 10 to all

Data: 8, 3, 10, 13, 4, 6, 7, 7, 8, 8, 5

\[ \text{Mean} = 7.18 \]

After adding 10 to every value:

\[ \text{New mean} = 7.18 + 10 = 17.18 \]

📐 Example: Double all values

Same data, every value ×2:

\[ \text{New mean} = 2 \times 7.18 = 14.36 \]

The mean doubles — the dot plot stretches out on the number line.

Summary of Algebra Rules

\[ \text{Each value} + k \Rightarrow \text{Mean} + k \qquad \text{Each value} \times k \Rightarrow \text{Mean} \times k \]

🎯

Section 5.1 continued

Tinkering with Median

📌 What is Median?

The middle value in sorted data — equal number of values on both sides. For even number of values, it is the average of the two middle values.

\[ \text{Odd } n: \text{ median} = \text{middle value} \qquad \text{Even } n: \text{ median} = \frac{\text{middle}_1 + \text{middle}_2}{2} \]

Three dot plots showing median = 8, new value inserted, median becomes 9.5

Original median = 8 · Adding value 11 (> median) → new median = 9.5

➕ Add value > median

Median may increase

➕ Add value < median

Median may decrease

⚠️

Key difference: Unlike the mean, the median is not affected by how far away an extreme value is — only by its position.

🔍

Section 5.1 continued

Finding the Unknown Value

🔑 The Technique

When one value is missing but the mean is known, use the mean formula to find it:

\[ \text{Mean} = \frac{\text{Sum of known values} + w}{n} \implies w = (\text{Mean} \times n) - \text{Sum of known values} \]

🤼 Example 1 — Kushti Players

10 players, weights: 42, 40, 39, 33, 48, 38, 42, 35, 32, w

Mean = 39.2

\[ \frac{349 + w}{10} = 39.2 \] \[ 349 + w = 392 \] \[ w = 43 \text{ kg} \]

Handwritten list of wrestler weights with average 39.2

Coach Balwan's record — one weight smudged!

🥥 Example 2 — Coconut Harvest

15 trees, average harvest = 25.6 per tree.

One tree's count recorded 3 more than actual.

\[ z = 25.6 \times 15 = 384 \text{ (total, incorrect)} \] \[ \text{Actual total} = 384 - 3 = 381 \] \[ \text{Correct average} = \frac{381}{15} = 25.4 \]

Venkayya's coconut farm — one count was off by 3!

📊

Section 5.1 continued

Mean and Median with Frequencies

📌 Mean with a Frequency Table

Don't just add the distinct values! Multiply each value by its frequency, then divide by total frequency.

\[ \text{Mean} = \frac{\sum (\text{value} \times \text{frequency})}{\sum \text{frequency}} \]

Family Size	Frequency	Value × Freq
3	3	9
4	11	44
5	9	45
6	7	42
7	3	21
8,9,10	1 each	27
Total	36	188

\[ \text{Mean} = \frac{188}{36} = 5.22 \]

Vertical dot chart showing family size frequency distribution

Frequency dot chart — family sizes in a class

📌 Median with a Frequency Table

For 36 values → median = average of 18th and 19th value when sorted. Use cumulative frequencies:

1
Frequency of 3 = 3 → positions 1–3 are value 3
2
Freq of 3 + freq of 4 = 14 → positions 4–14 are value 4
3
Freq of 3+4+5 = 23 → positions 15–23 are value 5
✓
18th and 19th values are both 5 → Median = 5

💡

Common mistake: Adding just the distinct values (3+4+5…+10)/8 = 6.5 is wrong — it ignores frequency. Always multiply value × frequency!

📈

Section 5.2

Visualising Data — Line Graphs

📌 What is a Line Graph?

Data points connected by line segments. Best used to show change over time — trends, rises, falls, patterns.

✅ Use Line Graph When

Showing change over time, comparing trends, many data points that would make a bar graph crowded

❌ Bar Graph Problem

13 years × 4 countries = 52 bars — too crowded! Line graph shows the same trends clearly.

Clustered bar graph of monthly max temperature Kerala and Punjab

Bar graph: Monthly Maximum Temperature in Kerala & Punjab (2023) — harder to see trends

Line graph of monthly max temperature Kerala and Punjab

Line graph of the same data — trends now clearly visible!

Annotated line graph showing peaks and lows for Kerala and Punjab

Annotated line graph: Punjab peaks in June ~38°C; Kerala stays flat at ~29–33°C year-round

🔍 Two-Step Process to Read a Graph

1

Identify what is given: What is on each axis? What scale? What are the units? What patterns do you notice?
2

Infer and interpret: What conclusions can you draw? What trends exist? What questions does the data raise?

🌡️ Punjab Temperature Pattern

Rises Jan → Jun (peak ~38°C), drops Jul–Dec (low ~19°C Jan). Wide variation between summer and winter.

🌴 Kerala Temperature Pattern

Stays mostly flat: 29–33°C all year. Peak ~33°C in April, low ~29°C in July. Very stable climate.

📌

Key Insight: Understanding how data is collected helps interpret it correctly. Temperature data from weather stations gives monthly maximum — not average or minimum. Data quality and collection method matter!

📊 Spreadsheets — Quick Calculations

A spreadsheet is a digital notebook with rows and columns of cells. Cells are named by column letter + row number (e.g., E5).

Formula	What it does
`=SUM(B3:G3)`	Total marks for student in row 3 across columns B to G
`=AVERAGE(B2:B24)`	Average of all values in column B, rows 2 to 24

⭐ Quick Summary

Mean is the balance point: sum of distances on LHS = sum on RHS. Only one such centre exists.

Adding a value greater than the mean → mean increases. Adding a value less → mean decreases. Adding a value equal → mean unchanged.

If every value increases by \(k\): new mean \(= a + k\). If every value is multiplied by \(k\): new mean \(= ka\).

Median: adding a value greater than median may increase it; less may decrease it. Median is position-based, not distance-based.

Missing value from mean: \(w = (\text{Mean} \times n) - \text{Sum of known values}\).

Mean with frequency: \(\text{Mean} = \dfrac{\sum (\text{value} \times \text{freq})}{\sum \text{freq}}\). Never just average the distinct values!

Line graphs show change over time — connect data points with lines. Best when data has many points or trends matter more than exact values.

Reading any graph: Step 1 — identify axes, scale, units, patterns. Step 2 — infer, interpret, conclude.

Notes by @edugrown · Ganita Prakash Grade 8 · Chapter 5 · Tales by Dots and Lines