Class 8 · Mathematics · NCERT
Chapter 5 — Tales by Dots and Lines
A complete, exam-ready guide to Mean, Median, Dot Plots, Line Graphs and Data Interpretation
📊 Mean & Median
📈 Line Graphs
🔢 Frequency Tables
💻 Spreadsheets
🗺️ Infographics
📈 Line Graphs
🔢 Frequency Tables
💻 Spreadsheets
🗺️ Infographics
📚 Contents
- Introduction & Quick Overview
- 5.1 Mean as a Centre (The Balancing Act)
- Effect of Adding/Removing Values on Mean
- Algebra of Mean — Adding/Multiplying a Fixed Number
- Tinkering with the Median
- Finding the Unknown Using Mean
- Mean & Median with Frequency Tables
- Spreadsheets — Using Technology
- 5.2 Line Graphs — Visualising Data
- Infographics & Other Data Displays
- Chapter Summary
- Important Exam Questions
Introduction — What is this Chapter About?
In Class 7, you learnt about mean and median as measures of central tendency. In this chapter, we take a deeper look at how mean and median behave when data changes, and we explore new ways to visualise data using dot plots, line graphs, and infographics.
🔢 Mean (Average)
Sum of all values ÷ Number of values. It represents the “fair share” or balance point of data.
Sum of all values ÷ Number of values. It represents the “fair share” or balance point of data.
📍 Median
The middle value when data is arranged in order. Equal number of values on both sides.
The middle value when data is arranged in order. Equal number of values on both sides.
Quick Formula Recall
Mean = (Sum of all values) ÷ (Number of values) | Median = middle value of sorted data.
Mean = (Sum of all values) ÷ (Number of values) | Median = middle value of sorted data.
5.1 Mean as a Centre — The Balancing Act
The mean is not just a formula — it is the balance point of the data. Imagine placing dots on a number line: the mean is the point where the total distance of all dots to its left equals the total distance of all dots to its right.
🔵 Key Idea: Total Distance is Equal on Both Sides
For any data set, if you calculate the sum of distances of values less than the mean from the mean, it equals the sum of distances of values greater than the mean from the mean.
Example — Collection: 6, 7, 8
Mean = (6+7+8)/3 = 7. Distance from 6 to 7 = 1 (LHS). Distance from 8 to 7 = 1 (RHS). LHS = RHS = 1 ✓
Mean = (6+7+8)/3 = 7. Distance from 6 to 7 = 1 (LHS). Distance from 8 to 7 = 1 (RHS). LHS = RHS = 1 ✓
Example — Collection: 10, 10, 11, 17
Mean = 12. LHS distances: (12−10)+(12−10)+(12−11) = 2+2+1 = 5. RHS distances: (17−12) = 5. LHS = RHS = 5 ✓
Mean = 12. LHS distances: (12−10)+(12−10)+(12−11) = 2+2+1 = 5. RHS distances: (17−12) = 5. LHS = RHS = 5 ✓
⚠️ Is Mean always the midpoint of smallest and largest value?
No! The mean is NOT always the midpoint between the minimum and maximum values. It is the point where the sum of distances on both sides is equal — this depends on how many values and where they are placed.
Why is Mean Unique?
There is only ONE such “centre.” If you move the centre to any higher or lower value, the distances on one side increase and the other decrease — they will no longer be equal.
There is only ONE such “centre.” If you move the centre to any higher or lower value, the distances on one side increase and the other decrease — they will no longer be equal.
Effect of Adding / Removing Values on Mean
➕ Including a New Value
- If new value > mean → mean increases
- If new value < mean → mean decreases
- If new value = mean → mean stays the same
➖ Removing an Existing Value
- If removed value > mean → mean decreases
- If removed value < mean → mean increases
- If removed value = mean → mean stays the same
🔄 Unchanging Mean — Adding/Removing Multiple Values
It is possible to include or remove 2 or more values without changing the mean, as long as the values added/removed are balanced around the mean (i.e., total extra distance on left = total extra distance on right).
Trick for “unchanged mean”
If you include 2 values less than mean and 1 value greater than mean, check: Sum of distances below = distance above. e.g., Mean = 9. Add 7 (dist=2), 7 (dist=2), 13 (dist=4). LHS = 2+2 = 4 = RHS ✓. Mean stays 9!
If you include 2 values less than mean and 1 value greater than mean, check: Sum of distances below = distance above. e.g., Mean = 9. Add 7 (dist=2), 7 (dist=2), 13 (dist=4). LHS = 2+2 = 4 = RHS ✓. Mean stays 9!
Common Mistake
Do NOT just count how many values are added/removed. What matters is the total distance (sum of deviations) from the mean on each side.
Do NOT just count how many values are added/removed. What matters is the total distance (sum of deviations) from the mean on each side.
Algebra of Mean — Relatively Unchanged!
📌 What if every value increases by the same fixed number?
If you add a fixed number k to every value in the dataset, the mean also increases by k.
If mean of x₁, x₂, …, xₙ = a, then mean of (x₁+k), (x₂+k), …, (xₙ+k) = a + k
Algebraic Proof (Adding k to each value)
New mean = [(x₁+k) + (x₂+k) + … + (xₙ+k)] / n = [x₁+x₂+…+xₙ + nk] / n = a + k
New mean = [(x₁+k) + (x₂+k) + … + (xₙ+k)] / n = [x₁+x₂+…+xₙ + nk] / n = a + k
📌 What if every value is subtracted by k?
New mean = a − k
📌 What if every value is multiplied by a fixed number?
If mean = a, and every value is multiplied by m, then new mean = a × m
Algebraic Proof (Multiplying each value by m)
New mean = [mx₁ + mx₂ + … + mxₙ] / n = m × (x₁+x₂+…+xₙ)/n = m × a
New mean = [mx₁ + mx₂ + … + mxₙ] / n = m × (x₁+x₂+…+xₙ)/n = m × a
Remember This!
Add/Subtract k → Mean shifts by k. Multiply/Divide by m → Mean scales by m. The shape of the dot plot stays the same; it just shifts or stretches.
Add/Subtract k → Mean shifts by k. Multiply/Divide by m → Mean scales by m. The shape of the dot plot stays the same; it just shifts or stretches.
| Operation on All Values | Effect on Mean |
|---|---|
| Add k to every value | Mean increases by k → New mean = a + k |
| Subtract k from every value | Mean decreases by k → New mean = a − k |
| Multiply every value by m | Mean multiplied by m → New mean = a × m |
| Divide every value by m | Mean divided by m → New mean = a / m |
🌟
Real Life Connection!
If everyone in a class gets 5 bonus marks in an exam, the class average also increases by exactly 5 marks. You don’t need to recalculate!
If everyone in a class gets 5 bonus marks in an exam, the class average also increases by exactly 5 marks. You don’t need to recalculate!
Tinkering with the Median
🔎 What is Median?
The median is the middle value of sorted data. There are an equal number of values less than and greater than the median.
- Odd number of values → median is the exact middle value
- Even number of values → median is the average of the two middle values
For n values (sorted): Median = value at position (n+1)/2 [if n is odd]
Median = average of values at positions n/2 and n/2 + 1 [if n is even]
Median = average of values at positions n/2 and n/2 + 1 [if n is even]
➕ Adding a New Value to Median
- New value > median → median increases or stays same
- New value < median → median decreases or stays same
- New value = median → median stays same
Example — Adding a value greater than median
Data: 3, 5, 7, 7, 11, 13 → Median = 7 (average of 4th and 5th = (7+7)/2 or pick 7 as middle).
Add value 11: New sorted data: 3, 5, 7, 7, 11, 11, 13 → New Median = 7 (middle of 7 values). Then for even count, insert 11 after 7 and compute carefully — median shifts up to 9.5 in some cases.
Data: 3, 5, 7, 7, 11, 13 → Median = 7 (average of 4th and 5th = (7+7)/2 or pick 7 as middle).
Add value 11: New sorted data: 3, 5, 7, 7, 11, 11, 13 → New Median = 7 (middle of 7 values). Then for even count, insert 11 after 7 and compute carefully — median shifts up to 9.5 in some cases.
Median vs Mean — Key Difference
Median is NOT affected by extreme values (very large or very small), but mean IS. For example, in a dataset with one very large value, the mean will be pulled up, but the median remains close to the “typical” value.
Median is NOT affected by extreme values (very large or very small), but mean IS. For example, in a dataset with one very large value, the mean will be pulled up, but the median remains close to the “typical” value.
Effect on Median — Summary
Adding a value greater than median → median may increase. Adding a value less than median → median may decrease. Adding a value equal to median → median stays the same.
Adding a value greater than median → median may increase. Adding a value less than median → median may decrease. Adding a value equal to median → median stays the same.
Finding the Unknown Using Mean
📝 How to Find a Missing Value When Mean is Given
Use the formula: Sum = Mean × Number of values. If one value is missing, let it be a variable and solve.
Missing Value (w) = (Mean × n) − (Sum of known values)
Solved Example 1 — Coach Balwan’s Wrestlers
Weights: 42, 40, 39, 33, 48, 38, 42, 35, 32, w | Mean = 39.2, n = 10
Sum of known values = 42+40+39+33+48+38+42+35+32 = 349
Total sum = 39.2 × 10 = 392
w = 392 − 349 = 43 kg
Weights: 42, 40, 39, 33, 48, 38, 42, 35, 32, w | Mean = 39.2, n = 10
Sum of known values = 42+40+39+33+48+38+42+35+32 = 349
Total sum = 39.2 × 10 = 392
w = 392 − 349 = 43 kg
Solved Example 2 — Venkayya’s Coconut Farm
Average harvest per tree = 25.6, Number of trees = 15
Total harvest = 25.6 × 15 = 384
One tree’s count was 3 more than actual → Correct total = 384 − 3 = 381
Correct mean = 381 / 15 = 25.4
Average harvest per tree = 25.6, Number of trees = 15
Total harvest = 25.6 × 15 = 384
One tree’s count was 3 more than actual → Correct total = 384 − 3 = 381
Correct mean = 381 / 15 = 25.4
Step-by-Step Method
Step 1: Find total sum using Mean × n. Step 2: Add known values. Step 3: Missing value = Total − Sum of known values.
Step 1: Find total sum using Mean × n. Step 2: Add known values. Step 3: Missing value = Total − Sum of known values.
Mean & Median with Frequency Tables
📊 Why Use Frequency Tables?
When the same values appear multiple times, we use a frequency table instead of listing every value. We multiply each value by its frequency when calculating the mean.
Mean = Σ(Value × Frequency) / Σ(Frequencies)
Solved Example — Average Family Size
| Family Size | Frequency (No. of Students) | Size × Frequency |
|---|---|---|
| 3 | 3 | 9 |
| 4 | 11 | 44 |
| 5 | 9 | 45 |
| 6 | 7 | 42 |
| 7 | 3 | 21 |
| 8 | 1 | 8 |
| 9 | 1 | 9 |
| 10 | 1 | 10 |
| Total | 36 | 188 |
Mean = 188 / 36 = 5.22
📍 Finding Median from Frequency Table
Add frequencies cumulatively from the smallest value until you reach the middle position.
- Total values (n) = 36. Middle positions = 18th and 19th.
- Cumulative frequency of 3 = 3 (positions 1–3)
- Cumulative frequency of 3 + 4 = 3 + 11 = 14 (positions 1–14, all value 4)
- Cumulative frequency of 3 + 4 + 5 = 14 + 9 = 23 (positions 15–23, all value 5)
- Both 18th and 19th values fall in positions 15–23, so both are 5.
- Median = 5
Shortcut for Median from Frequency Table
Add frequencies row by row. The first row where the cumulative frequency reaches or passes the middle position gives you the median value.
Add frequencies row by row. The first row where the cumulative frequency reaches or passes the middle position gives you the median value.
Spreadsheets — Using Technology for Data
🗂️ What is a Spreadsheet?
A spreadsheet is a digital tool with rows and columns of cells. You can enter numbers, text, or formulas. Popular software: Microsoft Excel, Google Sheets, LibreOffice Calc.
📌 Reading Cell Names
Each cell is identified by its column letter + row number. For example, cell E5 = column E, row 5.
Sudhakar’s Grade 8 Class — Example
If names are in column A, and subjects in columns B–G, then Farooq’s Maths score is in cell E5. Gowri’s data is in row 7.
If names are in column A, and subjects in columns B–G, then Farooq’s Maths score is in cell E5. Gowri’s data is in row 7.
🔢 Key Spreadsheet Formulas
| Formula | What it does | Example |
|---|---|---|
=SUM(B3:G3) | Adds all values from B3 to G3 | Total marks of Nagesh across subjects |
=AVERAGE(B3:G3) | Finds mean of B3 to G3 | Average marks of Nagesh |
=AVERAGE(B2:B23) | Mean of a full column | Class average in Odia |
📐 Cell Range Notation
- B3:G3 = All cells from B3 to G3 (a row range — Nagesh’s marks)
- B7:D7 = Cells B7, C7, D7 (Gowri’s Odia, Telugu, English)
- D2:D6 = Cells D2 to D6 (English marks of first 5 students)
Tip for Exam
Always write
Always write
= before a formula in a spreadsheet. SUM gives total, AVERAGE gives mean. The colon : means “from…to”.5.2 Visualising & Interpreting Data — Line Graphs
📌 What is a Line Graph?
A line graph connects data points with line segments. It is best used to show change over time (trends).
✅ When to Use Line Graphs
When data is collected at regular time intervals (months, years). To compare trends of 2 or more things over time.
When data is collected at regular time intervals (months, years). To compare trends of 2 or more things over time.
❌ When NOT to Use
When data is not time-based. Use bar graphs or dot plots for frequency data instead.
When data is not time-based. Use bar graphs or dot plots for frequency data instead.
🔍 Two-Step Method to Interpret a Graph
- Step 1 — Identify what is given: Look at the axes, scale, labels, units, and the number of data lines/series. Note the range of values and any obvious patterns.
- Step 2 — Infer and Interpret: Write observations in full sentences. Mention the trend (increasing/decreasing/flat), peaks, troughs, and comparisons between lines.
Example — Kerala vs Punjab Monthly Temperature
Kerala: Temperature stays mostly flat (~29°C to ~33°C) throughout the year. Low variation.
Punjab: Temperature rises sharply from Jan (~19°C) to a peak of ~38°C in June, then falls to ~23°C in December. High variation.
Conclusion: Punjab has more extreme temperatures than Kerala.
Kerala: Temperature stays mostly flat (~29°C to ~33°C) throughout the year. Low variation.
Punjab: Temperature rises sharply from Jan (~19°C) to a peak of ~38°C in June, then falls to ~23°C in December. High variation.
Conclusion: Punjab has more extreme temperatures than Kerala.
🌧️ Rainfall Graphs — Key Observations
- West Coast cities (Kovalam, Udupi, Mumbai): Peak rainfall in June–August (Southwest Monsoon)
- East Coast cities (Rameswaram, Chennai): Peak rainfall in October–December (Northeast Monsoon)
- Puri: Peak July–September despite being on the east coast
- January–March: Dry months for all cities
🚀 Space Object Launches — Reading the Graph
A steeper line segment between two points means a greater increase. A flat segment means little change.
Key Tip — Steepness of Line
Steeper slope = bigger change. Flatter slope = smaller change. This applies to any line graph where one axis is time.
Steeper slope = bigger change. Flatter slope = smaller change. This applies to any line graph where one axis is time.
Careful! Line Graphs vs Bar Graphs
Line graphs are better for trends over time. Bar graphs are better for comparing discrete categories. A line graph with 13 clusters of 4 bars (e.g., space launches by country over 13 years) would need 52 bars — very cluttered! A line graph handles this cleanly.
Line graphs are better for trends over time. Bar graphs are better for comparing discrete categories. A line graph with 13 clusters of 4 bars (e.g., space launches by country over 13 years) would need 52 bars — very cluttered! A line graph handles this cleanly.
😴 Sleep Duration Across Ages (India)
- 6-year-olds sleep ~9.5 hours/day
- Sleep decreases through teenage years into adulthood (~8 hours at age 30–50)
- After 50, sleep increases slightly again (~8.5 hours)
- The graph looks like a smooth U-curve because it has 70+ closely-spaced data points
Infographics, Activity Strips & Other Data Displays
🗺️ Infographics
An infographic presents data visually using maps, icons, colours and text together. It communicates complex information quickly and memorably.
Rice vs Wheat Infographic (India)
The infographic shows each state’s preference using a colour scale from −100 (Mostly Wheat) to +100 (Mostly Rice). A value of +100 does NOT mean no wheat is consumed — it means rice is strongly preferred. The national preference score was +13.48, meaning rice is slightly preferred overall.
The infographic shows each state’s preference using a colour scale from −100 (Mostly Wheat) to +100 (Mostly Rice). A value of +100 does NOT mean no wheat is consumed — it means rice is strongly preferred. The national preference score was +13.48, meaning rice is slightly preferred overall.
🎨 Activity Strips
An activity strip is a coloured chart with 48 boxes representing 30-minute intervals in a day (midnight to midnight). Each activity is represented by a different colour. This allows quick visual comparison of how time is spent on different days.
Reading a Strip
Look at which colour occupies the most boxes = most time spent. Look at when sleeping starts and ends = sleep schedule. Compare a school day strip vs weekend strip.
Look at which colour occupies the most boxes = most time spent. Look at when sleeping starts and ends = sleep schedule. Compare a school day strip vs weekend strip.
🌙
Fun Fact — Moon Rise and Set!
Unlike the sun which rises and sets at roughly regular times each day, the moonrise and moonset times shift by about 50 minutes later each day. This is why the moon rises late at night near full moon (purnima) and rises near sunrise around new moon (amavasya).
Unlike the sun which rises and sets at roughly regular times each day, the moonrise and moonset times shift by about 50 minutes later each day. This is why the moon rises late at night near full moon (purnima) and rises near sunrise around new moon (amavasya).
Chapter Summary — All Key Points
⚖️ Mean as Balance
Sum of distances of values to the left of mean = sum of distances to the right. Mean is the unique balance point.
Sum of distances of values to the left of mean = sum of distances to the right. Mean is the unique balance point.
➕ Adding Values
New value > mean → mean increases. New value < mean → mean decreases. New value = mean → mean unchanged.
New value > mean → mean increases. New value < mean → mean decreases. New value = mean → mean unchanged.
🔢 Shifting All Values
Add k to all → mean + k. Subtract k → mean − k. Multiply all by m → mean × m. Divide all by m → mean / m.
Add k to all → mean + k. Subtract k → mean − k. Multiply all by m → mean × m. Divide all by m → mean / m.
📍 Median Rules
Adding value > median → median may increase. Adding value < median → median may decrease. Median is unaffected by extreme outliers.
Adding value > median → median may increase. Adding value < median → median may decrease. Median is unaffected by extreme outliers.
📋 Frequency Mean
Mean = Σ(value × freq) / Σ(freq). Median: use cumulative frequency to find the middle position.
Mean = Σ(value × freq) / Σ(freq). Median: use cumulative frequency to find the middle position.
📈 Line Graphs
Best for showing change over time. Steeper slope = greater change. Use 2-step method: Identify → Infer.
Best for showing change over time. Steeper slope = greater change. Use 2-step method: Identify → Infer.
Important Exam Questions with Solutions
Q1. The mean of 8, 13, 10, 4, 5, 20, y, 10 is 10.375. Find y.
Sum = 10.375 × 8 = 83. Sum of known values = 8+13+10+4+5+20+10 = 70. y = 83 − 70 = 13
Q2. The mean of 15 values is 134. Find the sum of data.
Sum = Mean × n = 134 × 15 = 2010
Q3. Heights of 24 students averaged 150.2 cm when measured with shoes (shoes add 1 cm). What is the correct average height?
Since every value has 1 extra cm due to shoes, correct average = 150.2 − 1 = 149.2 cm. No need to re-measure everyone — just subtract 1!
Q4. Find the median of: 8, 10, 19, 23, 26, 34, 40, 41, 41, 48, 51, 55, 70, 84, 91, 92
n = 16 (even). Middle positions = 8th and 9th. Sorted: 8th value = 41, 9th value = 41. Median = (41+41)/2 = 41
Q5. The average weight of a group was 65.3 kg. One person lost 2 kg, two gained 1 kg each. What is the change in mean?
Net change in total = −2 + 1 + 1 = 0. Since total sum is unchanged and n is unchanged, mean stays the same at 65.3 kg.
Q6. True/False: “Including a value less than the mean will always decrease the mean.”
Always True. Any value less than the current mean pulls the mean downward. This is because it adds to the “left side” of the balance — to restore balance, the mean must shift left (decrease).
Q7. True/False: “Including any 4 values will not affect the median.”
Sometimes True. If 2 values are added above and 2 below the median, the median may stay the same. But adding 4 values all greater than the median will increase it. It depends on what values are added.
Q8. What does the formula =AVERAGE(B2:B23) calculate in a spreadsheet?
It calculates the arithmetic mean of all values in cells B2, B3, B4, … through B23 (i.e., 22 values). This would give the class average for the subject entered in column B.
🧮 Figure it Out — Quick Answers
Mean of first 50 natural numbers
= (1+2+…+50)/50 = (50×51/2)/50 = 51/2 = 25.5
= (1+2+…+50)/50 = (50×51/2)/50 = 51/2 = 25.5
Mean of first 50 odd numbers
Odd numbers: 1, 3, 5, …, 99. Sum = 50² = 2500. Mean = 2500/50 = 50
Odd numbers: 1, 3, 5, …, 99. Sum = 50² = 2500. Mean = 2500/50 = 50
Mean of first 50 multiples of 4
Multiples: 4, 8, 12, …, 200. Sum = 4 × (1+2+…+50) = 4 × 1275 = 5100. Mean = 5100/50 = 102
Multiples: 4, 8, 12, …, 200. Sum = 4 × (1+2+…+50) = 4 × 1275 = 5100. Mean = 5100/50 = 102
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