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Tales by Dots and Lines

NCERT Class 8 · Mathematics Based on NCERT Class 8 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter revisits the concepts of mean and median, exploring them through dot plots and the idea of balance — the sum of distances on either side of the mean is equal. It then extends to how the mean changes when values are added, removed, or scaled, and introduces line graphs as a tool to visualise data over time. The one big idea a student must walk away with is that the mean is the unique 'centre' of data where total distances balance, and line graphs effectively show trends and patterns across time.
Key Concepts (15)
Mean (Arithmetic Mean)
The sum of all values in a data set divided by the number of values.
e.g. If the weights of 5 students are 40 kg, 42 kg, 38 kg, 45 kg, and 35 kg, the mean weight is (40+42+38+45+35) ÷ 5 = 200 ÷ 5 = 40 kg.
Median
The middle value when data is arranged in ascending or descending order. If there are an even number of values, the median is the average of the two middle values.
e.g. For the data 3, 5, 7, 9, 11, the median is 7. For 4, 6, 8, 10, the median is (6+8)÷2 = 7.
Centre of Data (Mean as Balance Point)
The mean is the unique point such that the sum of distances of all values less than the mean equals the sum of distances of all values greater than the mean.
e.g. For data 10, 10, 11, 17 (mean = 12), distances left: (12-10)+(12-10)+(12-11)=2+2+1=5; distances right: (17-12)=5. They are equal.
Effect of Including a Value on Mean
Including a value greater than the mean increases the mean; including a value less than the mean decreases the mean; including a value equal to the mean does not change the mean.
e.g. If class average height is 150 cm and a new student of height 160 cm joins, the new average will be greater than 150 cm.
Effect of Removing a Value on Mean
Removing a value greater than the mean decreases the mean; removing a value less than the mean increases the mean; removing a value equal to the mean does not change the mean.
e.g. If a team's average score is 50 and a player who scored 60 leaves, the new average will be less than 50.
Effect of Adding a Constant to All Values
If a fixed number is added to every value in a data set, the mean increases by that same fixed number.
e.g. If the mean of marks is 65 and every student gets 5 grace marks, the new mean becomes 70.
Effect of Multiplying All Values by a Constant
If every value in a data set is multiplied by a fixed number, the mean is also multiplied by that same fixed number.
e.g. If the mean of lengths is 10 cm and every length is doubled, the new mean becomes 20 cm.
Median with Frequencies
When data is given with frequencies, the median is found by cumulatively adding frequencies until the middle position(s) are reached.
e.g. For family sizes: 3 (freq 3), 4 (freq 11), 5 (freq 9), total 36 values. The 18th and 19th values fall in the '5' category, so median = 5.
Line Graph
A graph made by connecting data points with line segments, used to visualise data over time.
e.g. A line graph showing monthly maximum temperatures in Kerala and Punjab across months of the year.
Infographic
A visual representation of information or data, often using colours, maps, and icons to communicate insights quickly.
e.g. A map of India showing rice vs wheat preference using a colour scale from -100 to +100.
Activity Strip
A strip of paper divided into 48 boxes (each representing 30 minutes) coloured to show different activities across a 24-hour day.
e.g. Manoj's strip for a weekday showing sleeping, eating, classes, travel, and hobbies in different colours.
Unchanging Mean (Including/Removing Multiple Values)
It is possible to include or remove multiple values such that the mean remains unchanged, e.g., including two values less than the mean and one greater than the mean in a balanced way.
e.g. If mean is 10, including 8 and 9 (both less) and 13 (greater) can keep the mean unchanged if the total sum balances.
Finding the Unknown Value Using Mean
If the mean and all but one value are known, the missing value can be found using the formula: Mean = (Sum of known values + unknown) ÷ Total number of values.
e.g. Mean of 10 weights is 39.2 kg, sum of 9 known weights is 349 kg, so unknown weight = (39.2 × 10) - 349 = 392 - 349 = 43 kg.
Correcting an Error Using Mean
If one value is incorrectly recorded, the correct mean can be found by adjusting the total sum and recalculating.
e.g. Average harvest per tree was 25.6 from 15 trees, but one tree's count was 3 more than actual. Correct total = (25.6×15) - 3 = 384 - 3 = 381, correct average = 381÷15 = 25.4.
Spreadsheet
A digital tool with rows and columns of cells used to organise and calculate data using formulae like =SUM() and =AVERAGE().
e.g. Sudhakar uses a spreadsheet to calculate total marks of each student using =SUM(B3:G3) and class average in Science using =AVERAGE(H2:H23).
Formulas & Equations (6)
Mean = (Sum of all values) ÷ (Number of values)
Reads as: Mean equals sum of all values divided by number of values
Variables: Sum of all values: total of all data points (no unit); Number of values: count of data points (dimensionless)
Use when: When asked to calculate the average of a data set in any exam question
⚠ Students forget to divide by the correct number of values, especially when data is given with frequencies — they may divide by the number of distinct values instead of the total count.
Median = Middle value (for odd n) or average of two middle values (for even n) after sorting
Reads as: Median is the middle value when data is sorted; if even number of values, median is the average of the two middle values
Variables: n: total number of values (dimensionless)
Use when: When asked to find the median of a data set, especially with frequencies or in word problems
⚠ Students forget to sort the data first before finding the middle value, or they incorrectly average the two middle numbers without checking if n is even.
New mean after adding constant k to all values = Old mean + k
Reads as: New mean equals old mean plus k
Variables: k: the constant added to every value (same unit as data)
Use when: When every value in the data set is increased or decreased by the same fixed number
⚠ Students think the mean changes by k only for some values, not all — they may try to recalculate the entire sum instead of using the shortcut.
New mean after multiplying all values by constant m = Old mean × m
Reads as: New mean equals old mean multiplied by m
Variables: m: the constant multiplier (dimensionless)
Use when: When every value in the data set is multiplied or divided by the same fixed number
⚠ Students forget to multiply the mean by the same factor, or they add the factor instead of multiplying.
Sum of values = Mean × Number of values
Reads as: Sum of values equals mean multiplied by number of values
Variables: Sum of values: total of all data points; Mean: arithmetic mean; Number of values: count of data points
Use when: When the mean and number of values are given, and you need to find the total sum (e.g., to find a missing value or correct an error)
⚠ Students may divide instead of multiply, or use the wrong number of values when frequencies are involved.
Mean with frequencies = (Σ f × x) ÷ (Σ f)
Reads as: Mean equals sum of (frequency times value) divided by sum of frequencies
Variables: f: frequency of each value; x: the value; Σ f: total number of observations
Use when: When data is presented in a frequency table and you need to calculate the mean
⚠ Students forget to multiply each value by its frequency and simply add the distinct values, then divide by the number of distinct values.
How It Works
Finding the Mean as the Balance Point
  1. Step 1: Arrange the data values on a number line or dot plot.
  2. Step 2: Calculate the mean using the formula (sum ÷ count).
  3. Step 3: For each value less than the mean, calculate the distance (mean - value).
  4. Step 4: For each value greater than the mean, calculate the distance (value - mean).
  5. Step 5: Verify that the sum of distances on the left equals the sum of distances on the right.
💡 The mean is the only point where this balance holds — any other point will have unequal sums of distances.
Effect of Including a New Value on the Mean
  1. Step 1: Compare the new value to the current mean.
  2. Step 2: If new value > mean, the new mean will be greater than the old mean.
  3. Step 3: If new value < mean, the new mean will be less than the old mean.
  4. Step 4: If new value = mean, the mean remains unchanged.
💡 The mean always moves toward the new value — it increases if the new value is larger, decreases if smaller.
Effect of Removing an Existing Value on the Mean
  1. Step 1: Compare the removed value to the current mean.
  2. Step 2: If removed value > mean, the new mean will be less than the old mean.
  3. Step 3: If removed value < mean, the new mean will be greater than the old mean.
  4. Step 4: If removed value = mean, the mean remains unchanged.
💡 Removing a value has the opposite effect on the mean compared to including it.
Effect of Adding a Constant to All Values
  1. Step 1: Identify the constant k being added to every value.
  2. Step 2: The new mean = old mean + k.
  3. Step 3: Algebraically, (x1+k + x2+k + ... + xn+k)/n = (x1+...+xn)/n + (nk)/n = old mean + k.
💡 The relative positions of data points do not change — the entire dot plot shifts by k units.
Effect of Multiplying All Values by a Constant
  1. Step 1: Identify the constant m being multiplied to every value.
  2. Step 2: The new mean = old mean × m.
  3. Step 3: Algebraically, (m×x1 + m×x2 + ... + m×xn)/n = m × (x1+...+xn)/n = m × old mean.
💡 The spread of data changes — the dot plot stretches or shrinks by factor m.
Finding the Median from a Frequency Table
  1. Step 1: Find the total number of observations (sum of all frequencies).
  2. Step 2: Determine the position(s) of the median: if n is odd, position = (n+1)/2; if n is even, positions = n/2 and (n/2)+1.
  3. Step 3: Cumulatively add frequencies from the smallest value upward until you reach the median position(s).
  4. Step 4: The value corresponding to that cumulative frequency is the median (or average of two values for even n).
💡 Do not write out all values individually — use cumulative frequency to find the median quickly.
Finding the Missing Value Using Mean
  1. Step 1: Write the formula: Mean = (Sum of known values + unknown) ÷ Total number of values.
  2. Step 2: Multiply both sides by total number of values to get: Sum of known values + unknown = Mean × Total number.
  3. Step 3: Subtract the sum of known values from both sides to isolate the unknown.
  4. Step 4: Calculate the result.
💡 Always check that you have the correct total number of values including the unknown.
Correcting an Error in Data Using Mean
  1. Step 1: Calculate the incorrect total sum using the given (incorrect) mean and number of values.
  2. Step 2: Adjust the total sum by subtracting the overcount or adding the undercount.
  3. Step 3: Divide the corrected total by the same number of values to get the correct mean.
💡 Only the total sum changes — the number of values remains the same unless a value is added or removed.
Reading and Interpreting a Line Graph
  1. Step 1: Identify what is on the horizontal axis (usually time) and vertical axis (the measured quantity).
  2. Step 2: Note the scale and units used on both axes.
  3. Step 3: Observe the trend — is the line going up (increase), down (decrease), or flat (constant)?
  4. Step 4: Compare different lines if multiple data sets are shown (e.g., different states or countries).
  5. Step 5: Draw conclusions and ask further questions about the data.
💡 Line graphs are best for showing change over time — steeper lines mean faster change.
Exam Traps — Don't Make These Mistakes
⚠ Mean is always the midpoint of extremes
✗ The mean is always exactly halfway between the smallest and largest value.
→ This is only true for two numbers. For three or more numbers, the mean is the balance point where total distances on both sides are equal, not necessarily the midpoint of extremes.
✓ The mean is the unique centre such that the sum of distances of values less than the mean equals the sum of distances of values greater than the mean.
⚠ Median changes with every new value
✗ If we include a new value, the median will always change.
→ If the new value is equal to the current median, or if the data set is large and the new value falls within the middle range, the median may remain the same.
✓ Including a value equal to the median does not change the median. Including a value greater than the median may increase it; including a value less than the median may decrease it.
⚠ Mean with frequencies — dividing by number of distinct values
✗ Average family size = (3+4+5+6+7+8+9+10) ÷ 8 = 6.5.
→ The student ignored the frequencies. Each value occurs multiple times, so the sum must be (value × frequency) for each, and divided by the total number of observations (sum of frequencies), not the number of distinct values.
✓ Average = (3×3 + 4×11 + 5×9 + ...) ÷ (3+11+9+...) = 188 ÷ 36 = 5.22.
⚠ Adding a constant to some values, not all
✗ If every student gets 5 extra marks, the new mean is old mean + 5.
→ This statement is actually correct if every student gets the extra marks. The trap is when students apply this rule when only some values change — the rule only works when the same constant is added to every single value.
✓ If a fixed number is added to every value in the collection, the new mean = old mean + that fixed number. If only some values change, recalculate the total sum and divide.
⚠ Median from frequency table — forgetting to sort
✗ For data 10 (freq 2), 5 (freq 3), 8 (freq 4), the median is 8 because it has the highest frequency.
→ The data must be sorted in ascending order before finding the median. The median is not the mode (most frequent value).
✓ Sort the values: 5,5,5,8,8,8,8,10,10. Total n=9, median position = 5th value, which is 8. The median is 8 because it is the middle value after sorting.
⚠ Removing a value less than median decreases median
✗ Removing a value less than the median will always decrease the median.
→ Removing a value less than the median can actually increase the median because the middle position shifts upward. For example, data: 1, 2, 3, 4, 5 (median=3). Remove 1 (less than median): new data 2,3,4,5 (median=3.5, which increased).
✓ Removing a value less than the median may increase the median because the data set becomes smaller and the middle shifts toward higher values.
Compare & Contrast
Effect of including a new value
Mean
Median
Including a value greater than the mean always increases the mean; including a value greater than the median may or may not increase the median depending on where it falls in the sorted order.
Effect of adding a constant to all values
Mean
Median
Adding a constant k to every value increases the mean by exactly k, but the median also increases by exactly k because all values shift equally.
Best use case
Line Graph
Clustered Column Graph
Line graphs are best for showing change over time (trends), while clustered column graphs are better for comparing categories at a single point in time. Line graphs become cluttered with too many categories; column graphs become cluttered with too many time points.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA-I (2 marks), 1 SA-II (3 marks), 1 LA (5 marks) — total approximately 12 marks
Most asked: Calculating mean and median from raw data and frequency tables · Effect on mean when a value is added, removed, or changed · Finding the missing value using mean · Effect of adding a constant or multiplying by a constant on the mean · Reading and interpreting line graphs
Don't skip: Word problems involving finding a missing value or correcting an error using mean (5 marks); Effect on mean and median when data is modified (3 marks); Interpreting a line graph and drawing conclusions (3 marks)
One-Line Revision (read aloud before the exam)
  1. Mean = sum of all values ÷ number of values.
  2. Median is the middle value when data is sorted; for even n, average the two middle values.
  3. The mean is the unique point where sum of distances on left equals sum of distances on right.
  4. Including a value greater than the mean increases the mean; including a value less than the mean decreases it.
  5. Removing a value greater than the mean decreases the mean; removing a value less than the mean increases it.
  6. Adding a constant k to every value increases the mean by k.
  7. Multiplying every value by a constant m multiplies the mean by m.
  8. To find a missing value, use: unknown = (mean × total number) – sum of known values.
  9. To correct an error, adjust the total sum and recalculate the mean.
  10. Line graphs show change over time; steeper lines mean faster change.
  11. Median from a frequency table is found using cumulative frequencies.
  12. The mean can remain unchanged even when multiple values are added or removed if the total sum balances.
  13. Spreadsheets use =SUM() and =AVERAGE() to quickly calculate totals and averages.
  14. An infographic uses colours and maps to communicate data insights visually.
  15. An activity strip divides a 24-hour day into 48 half-hour blocks to show daily routines.

MCQs — 10 Questions with Answers

Q1. What is the mean of the numbers 3 and 7?

  • A. 5 ✓
  • B. 4
  • C. 10
  • D. 3.5

Answer: A — Mean = (3 + 7) ÷ 2 = 10 ÷ 2 = 5, which is exactly halfway between the two numbers.

Q2. If the mean of 5 numbers is 20, what is their sum?

  • A. 4
  • B. 25
  • C. 100 ✓
  • D. 20

Answer: C — Sum = Mean × Count = 20 × 5 = 100.

Q3. The original dataset has a mean of 15. If every value is increased by 4, what is the new mean?

  • A. 11
  • B. 15
  • C. 19 ✓
  • D. 60

Answer: C — When you add a fixed number to all values, the mean increases by that same number: 15 + 4 = 19.

Q4. Which of the following represents the median of an ordered dataset with 8 values?

  • A. The 4th value
  • B. The average of the 4th and 5th values ✓
  • C. The sum of all values divided by 8
  • D. The middle value

Answer: B — With 8 (even) values, the median is the average of the two middle values at positions 4 and 5.

Q5. Mira has marks in 5 subjects with mean 72. One of her marks was incorrectly recorded as 80 instead of 75. What should be her correct mean?

  • A. 70
  • B. 71 ✓
  • C. 72
  • D. 75

Answer: B — Original sum = 72 × 5 = 360; Corrected sum = 360 − 80 + 75 = 355; Correct mean = 355 ÷ 5 = 71.

Q6. A teacher collected family size data from 10 students. The sizes are: 4, 5, 4, 6, 5, 5, 7, 4, 5, 6. Using a frequency table, which value appears most frequently?

  • A. 4 (appears 3 times)
  • B. 5 (appears 4 times) ✓
  • C. 6 (appears 2 times)
  • D. 7 (appears 1 time)

Answer: B — Counting occurrences: 4 appears 3 times, 5 appears 4 times, 6 appears 2 times, 7 appears 1 time.

Q7. If all values in a dataset are multiplied by 2, how does the mean change?

  • A. It increases by 2
  • B. It is multiplied by 2 ✓
  • C. It stays the same
  • D. It is divided by 2

Answer: B — Using algebra: new mean = (2x₁ + 2x₂ + ... + 2xₙ)/n = 2(x₁ + x₂ + ... + xₙ)/n = 2 × original mean.

Q8. A farmer recorded coconut harvest from 15 trees with average 25.6 coconuts per tree. Later he found one tree's count was 3 more than actual. What is the correct average?

  • A. 25.0
  • B. 25.2
  • C. 25.4 ✓
  • D. 25.8

Answer: C — Incorrect total = 25.6 × 15 = 384; Correct total = 384 − 3 = 381; Correct mean = 381 ÷ 15 = 25.4.

Q9. In a spreadsheet with student marks, cell E5 contains Farooq's score in which subject if columns are: A(Name), B(Odia), C(Telugu), D(English), E(Maths)?

  • A. Odia
  • B. Telugu
  • C. English
  • D. Maths ✓

Answer: D — Column E is the 5th column which corresponds to Maths, so E5 is the Maths score in row 5.

Q10. The mean of a dataset is 10. If you remove a value of 10 from the data, what happens to the mean?

  • A. It stays at 10 ✓
  • B. It increases
  • C. It decreases
  • D. It becomes zero

Answer: A — Removing a value equal to the mean does not change the balance, so the mean remains unchanged.

Flashcards

What does it mean for the mean to be a 'balance point'?

The sum of distances from the mean to values below it equals the sum of distances to values above it.

If the mean of 10 numbers is 39.2 and 9 numbers sum to 349, what is the missing number?

The missing number is 43 kg, calculated as (39.2 × 10) − 349 = 392 − 349.

What happens to the mean if you add 5 to every value in the dataset?

The mean increases by exactly 5.

What happens to the mean if you multiply every value in the dataset by 3?

The mean is multiplied by 3.

How do you find the median when you have 36 data values with frequencies listed?

Add frequencies successively from smallest value until you reach position 18 and 19, then average those two middle values.

In a frequency table, how do you calculate the mean?

Multiply each value by its frequency, sum all products, then divide by the total count of all frequencies.

If the average family size is 5.22 and there are 36 students, what is the total of all family sizes?

The total is 5.22 × 36 = 187.92, or approximately 188 family members.

Can the mean remain unchanged when you remove one value and add a different value?

Yes, if the value you remove and the value you add are equal to each other (both equal to the original mean, or differ equally on opposite sides).

In a spreadsheet, what does cell D7 represent?

It is the cell in column D and row 7 where the intersection contains one data value.

If one tree's coconut count was incorrectly recorded as 3 MORE than actual, how do you correct the mean?

Subtract 3 from the total harvest count, then recalculate the mean with the corrected total.

Important Board Questions

Define the mean of a dataset. Is it always a value that exists in the data? [1 mark]

Mean is the sum of all values divided by the count. Give one example where mean is not in the original data (like mean of 3 and 4 is 3.5).

The mean weight of 8 wrestlers is 42 kg. If one wrestler's weight is missing and the sum of the other 7 weights is 285 kg, find the missing weight. [2 marks]

Use the formula: Mean = Sum ÷ Count. Calculate total sum first (42 × 8), then subtract the known sum (285) to find the missing value.

A dataset has values: 10, 15, 20, 25, 30. (a) Find the mean and verify it is the balance point. (b) What will be the new mean if every value is increased by 5? Show your working. [3 marks]

(a) Mean = (10+15+20+25+30)/5 = 20. Check: distances below 20 are 10+5=15, distances above 20 are 5+10=15 (equal). (b) Add 5 to the original mean: 20 + 5 = 25.

A class has the following family size distribution: Size 3 (frequency 3), Size 4 (frequency 11), Size 5 (frequency 9), Size 6 (frequency 7), Size 7 (frequency 3), Size 8 (frequency 1), Size 9 (frequency 1), Size 10 (frequency 1). Calculate the mean family size and identify the median family size using the frequency table. [5 marks]

Mean = [Σ(value × frequency)] / (total frequency) = [(3×3)+(4×11)+(5×9)+(6×7)+(7×3)+(8×1)+(9×1)+(10×1)] / 36. For median with 36 values, find positions 18 and 19 by adding cumulative frequencies: 3+11=14, then 3+11+9=23, so positions 15–23 are all 5. Median is 5.

Assertion-Reason Questions for Tales by Dots and Lines

Q1. Assertion (A):

The mean is the only value where the sum of distances to all values less than it equals the sum of distances to all values greater than it.

Reason (R):

If we move to any value larger or smaller than the mean, the distances on one side increase while those on the other side decrease, breaking this balance.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains why the mean is the unique centre. The reason provides the logical justification for why no other value can serve as the centre.

Q2. Assertion (A):

When a constant number k is added to all values in a dataset, the new mean increases by k.

Reason (R):

The mean represents the average value, so adding k to each value shifts all data points equally upward by k.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both the assertion and reason are true, but the reason is not the correct explanation. The reason uses intuitive reasoning rather than the algebraic proof. The correct explanation involves factoring: [(x₁+k) + (x₂+k) + ... + (xₙ+k)]/n = (x₁+x₂+...+xₙ+nk)/n = old mean + k.

Q3. Assertion (A):

If the mean of a dataset is m, and we add a value equal to m, the mean will remain m.

Reason (R):

Adding any value to a dataset always increases the mean.

Answer: C — A is true but R is false — The assertion is true—adding a value equal to the mean preserves the mean by the balancing property. However, the reason is false—adding a value equal to the mean does not increase the mean; it keeps it the same. Adding a value greater than the mean increases it, not a value equal to the mean.

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Case-Based Questions (CBQ) for Tales by Dots and Lines

Case Study:

A school conducted a height survey of 10 students in Class 8. The heights (in cm) recorded were: 145, 152, 148, 155, 150, 148, 153, 149, 151, 147. The physical education teacher wants to understand the 'centre' of the height data and analyze how it changes when one student's height is updated. The current mean height is 149.8 cm. After a recount, one student's height recorded as 155 cm is found to be 160 cm.

Q1. What is the new mean height after correcting the measurement from 155 cm to 160 cm?

  • A. 150.3 cm ✓
  • B. 150.5 cm
  • C. 150.8 cm
  • D. 151.2 cm

Answer: A

Q2. By how much did the mean height increase due to this correction?

  • A. 0.3 cm ✓
  • B. 0.5 cm
  • C. 1 cm
  • D. 5 cm

Answer: A

Q3. If all students' heights are increased by 5 cm (e.g., due to better posture measurement), what would be the new mean?

  • A. 154.3 cm
  • B. 155 cm
  • C. 155.3 cm ✓
  • D. 156 cm

Answer: C

Q4. Which of the following would NOT change the mean height?

  • A. Adding a student with height exactly 150.3 cm
  • B. Removing a student with height 148 cm
  • C. Adding two students: one with 148 cm and one with 152.6 cm ✓
  • D. Multiplying all heights by 2

Answer: C

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