Tales by Dots and Lines — Class 8 Mathematics (Ganita Prakash Part 2)
Most people meet averages every day and misuse them almost as often. This chapter fixes that — not by giving you a better formula, but by giving you a better picture.
1. About the Chapter
This is Chapter 5 of Ganita Prakash Part 2 (pages 103–133), the twelfth chapter of the Class 8 course. It has two halves:
- 5.1 The Balancing Act — the mean and the median seen afresh. The mean stops being "sum ÷ count" and becomes the balance point of the data. From that one idea flows everything else: what happens when a value joins or leaves, why adding 10 to everything adds 10 to the average, how to find a missing value, how to handle frequency tables, and how a spreadsheet does all of it at once.
- 5.2 Visualising and Interpreting Data — line graphs, infographics, activity strips and data stories, together with a two-step method for reading any of them honestly.
What is not in this chapter
Older notes for "Chapter 12" often cover the coordinate plane, quadrants, y = mx + c, histograms, pie charts and scatter plots. None of that is in this chapter, and the coordinate plane is not in Class 8 Ganita Prakash at all. Pie charts belong to the previous chapter, Proportional Reasoning — 2. For completeness those topics are kept in the appendix at the end of this page, clearly labelled as later material.
Key idea
The mean is the point where the total distance to the values on its left equals the total distance to the values on its right.
Everything in the first half of the chapter is a consequence of that sentence.
2. The Mean as a Balance Point
The first surprise: the mean is not the midpoint
Take the data 10, 10, 11, 17. Its mean is (10 + 10 + 11 + 17)/4 = 48/4 = 12.
The midpoint of the smallest and largest value is (10 + 17)/2 = 13.5. That is not 12. So whatever "centre" means here, it is not "halfway between the extremes".
What the mean actually balances
Measure every value from 12:
| Side | Values | Distances | Total |
|---|---|---|---|
| Below the mean | 10, 10, 11 | 2, 2, 1 | 5 |
| Above the mean | 17 | 5 | 5 |
The two totals are equal. Picture the number line as a see-saw carrying one unit of weight on each dot: it balances exactly at 12. One value far out on the right is held up by three values close in on the left.
Why this always works
For values x₁, x₂, …, xₙ with mean a, the signed distances add to
(x₁ − a) + (x₂ − a) + … + (xₙ − a) = (x₁ + x₂ + … + xₙ) − na = na − na = 0
A signed total of zero means the negative part (left distances) exactly cancels the positive part (right distances).
And there is only one such point
Suppose someone proposes 12.5 as the centre of 10, 10, 11, 17. Every left distance grows (2.5, 2.5, 1.5 → total 6.5) and the right distance shrinks (4.5). The left side is now heavier. Move below 12 and the right side wins instead.
Every shift away from 12 breaks the balance in a definite direction, so 12 is the unique balance point.
Why this matters. The balance picture answers questions the formula makes you recompute. Is 17 the mean of a dot plot? Add up the deviations from 17: if they come to +18 rather than 0, then no — and the true mean is 17 + 18/25 = 17.72, found without adding a single one of the 25 values.
3. What Moves the Mean
Including or removing a value
| Action | Effect on the mean |
|---|---|
| Include a value greater than the mean | Mean increases |
| Include a value less than the mean | Mean decreases |
| Include a value equal to the mean | Mean unchanged |
| Remove a value greater than the mean | Mean decreases |
| Remove a value less than the mean | Mean increases |
The balance reading: extra weight to the right of the pivot tips the plank right, so the pivot must move right.
The fair-share reading: four friends with 12 sweets each have a fair share of 12. A fifth arrives with 20 — more than a fair share, so there is surplus to spread and every share rises to 13.6. Arrive with 7 instead and everyone's share falls to 11. Arrive with exactly 12 and nobody is affected.
The unchanging mean
Values can be added without changing the mean exactly when their own deviations cancel — that is, when the new values themselves average to the current mean.
Take 4, 6, 8, 10, 12 with mean 8:
- Add 5 and 11 (deviations −3, +3) → 56/7 = 8 ✓
- Add 6, 7 and 11 (deviations −2, −1, +3) → 64/8 = 8 ✓ — two below, one above
- Add 10, 12 and 2 (deviations +2, +4, −6) → 64/8 = 8 ✓ — two above, one below
Notice the cost of the arrangement: with two values on one side, the lone value on the other has to be pushed further out, because it carries the whole imbalance by itself.
Changing every value at once
Add c to every value ⟹ the mean rises by c.
((x₁ + c) + … + (xₙ + c))/n = (x₁ + … + xₙ + nc)/n = a + c
Multiply every value by c ⟹ the mean is multiplied by c.
(cx₁ + … + cxₙ)/n = c(x₁ + … + xₙ)/n = ca
For the data 8, 3, 10, 13, 4, 6, 7, 7, 8, 8, 5 (sum 79, mean 79/11 ≈ 7.18):
| Change | New mean |
|---|---|
| +10 to every value | 189/11 = 17.18 |
| −1 from every value | 6.18 |
| ×2 on every value | 158/11 = 14.36 |
In the dot-plot picture, adding 10 slides the whole plot 10 steps right without changing a single gap — so the balance point slides 10 steps too. The mean keeps the same relative position inside the data.
A whole class in 1 cm shoes. Shreyas measures 24 students and reports an average height of 150.2 cm. The shoes add 1 cm to every reading. Nobody needs re-measuring: every value is too big by the same amount, so the mean is too big by exactly 1 cm. The correct average is 149.2 cm.
4. Working Backwards from an Average
An average never tells you the individual values — but it always tells you the total:
sum = mean × count
That single reversal solves an entire family of questions.
A missing value
Coach Balwan's ten wrestlers weigh 42, 40, 39, 33, 48, 38, 42, 35, 32 and one smudged figure w, with mean 39.2 kg.
Total must be 39.2 × 10 = 392. The nine known weights sum to 349. So w = 392 − 349 = 43 kg.
A wrongly recorded value
Venkayya's 15 trees averaged 25.6 coconuts, but one tree's count was written 3 too high.
- Recorded total = 25.6 × 15 = 384 — recovered without knowing a single tree's count
- True total = 384 − 3 = 381
- Correct average = 381/15 = 25.4
Notice the size of the effect: an error of 3 spread over 15 trees moves the average by only 3/15 = 0.2.
The general rule: an error of e in one reading shifts the mean by e ÷ n. This is why one large-looking mistake barely disturbs an average taken over many values — and why averages over few values are fragile.
5. Tinkering with the Median
The median is the middle value of the sorted data. It responds to counts and positions, not to sizes — which makes its behaviour different from the mean's.
- Include a value greater than the median → there are now more values above than below, so the middle slides up.
- Include a value less than the median → the middle slides down.
- Include a value equal to the median → no change.
For 2, 5, 7, 8, 9, 10 the median is 7.5. Including 11 gives seven values, 2, 5, 7, 8, 9, 10, 11, with median 8 — up, as predicted.
Why the median is so unmoved by outliers
Take 8, 10, 19, 23, 26, 34, 40, 41, 41, 48, 51, 55, 70, 84, 91, 92 — sixteen values, median (41 + 41)/2 = 41.
Because both middle values are 41, this data is unusually forgiving:
| Change | Effect on the median |
|---|---|
| Include any one value | Median stays 41 |
| Include two values | Stays 41 only if one is ≤ 41 and one is ≥ 41 |
| Remove any one value | Median stays 41 |
There is a spare 41 waiting to take over the middle position. Replace 92 by 92,000 and the median still does not move — but the mean would leap.
Choosing between them. The mean uses every value, so it can be recovered from a total and is the right summary when there are no extreme outliers. The median depends only on position, so it is unshaken by a few very large or small values. If one house on a street sells for a hundred times the rest, the mean price describes no house on the street while the median describes a typical one.
6. Mean and Median from a Frequency Table
A class reports its family sizes:
| Family size | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|
| Number of students | 3 | 11 | 9 | 7 | 3 | 1 | 1 | 1 |
The trap
It is tempting to write (3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)/8 = 52/8 = 6.5. This is wrong. It averages the eight distinct sizes, giving family size 10 — reported by one student — the same weight as family size 4, reported by eleven.
The right method
mean = Σ(value × frequency) ÷ Σ(frequency)
(3×3) + (4×11) + (5×9) + (6×7) + (7×3) + (8×1) + (9×1) + (10×1) = 9 + 44 + 45 + 42 + 21 + 8 + 9 + 10 = 188
Students = 3 + 11 + 9 + 7 + 3 + 1 + 1 + 1 = 36
Mean = 188/36 ≈ 5.22 — a long way from 6.5.
The median without writing 36 numbers
With 36 values the median is the average of the 18th and 19th. Build running totals from the smallest value up:
| Family size | Frequency | Positions occupied |
|---|---|---|
| 3 | 3 | 1 – 3 |
| 4 | 11 | 4 – 14 |
| 5 | 9 | 15 – 23 |
| 6 | 7 | 24 – 30 |
| 7 | 3 | 31 – 33 |
| 8, 9, 10 | 1 each | 34, 35, 36 |
Positions 18 and 19 both land in the 15–23 block, so both are 5 and the median is 5.
The mean (5.22) sits slightly above the median (5) because the few large families pull the balance point right while leaving the middle position where it is.
No family has 5.22 members. An average is a fair share, not a description of any member of the group. This is the most common error in all of statistics — and it returns later in the chapter, when a graph showing rural 15-year-olds averaging 1 h 19 min on hobbies is misread as a promise about every child.
7. A Shortcut Worth Knowing: Evenly Spaced Data
If the data is evenly spaced, the dots are symmetric about the middle, so the balance point is simply
mean = (first + last) ÷ 2
| Data | Mean |
|---|---|
| First 50 natural numbers: 1, 2, …, 50 | (1 + 50)/2 = 25.5 |
| First 50 odd numbers: 1, 3, …, 99 | (1 + 99)/2 = 50 |
| First 50 multiples of 4: 4, 8, …, 200 | (4 + 200)/2 = 102 |
No addition needed at all. And notice the second rule at work: multiplying every natural number by 4 multiplies the mean by 4, so 4 × 25.5 = 102 ✓.
8. Spreadsheets
Sudhakar has 21 students × 6 subjects = 126 marks and wants every student's total and every subject's average — 27 calculations, each a chance to slip.
Reading a spreadsheet
Columns carry letters across the top, rows carry numbers down the side, and a cell is named column letter + row number.
With A = Name, B = Odia, C = Telugu, D = English, E = Maths, F = Social Science, G = Science, and row 1 holding the headings:
- Farooq is the 4th student, so he is in row 5 → his Maths mark is in cell E5
- B7 holds Gowri's Odia mark
Ranges and formulas
A run of cells is written Start:End.
| What you want | Formula |
|---|---|
| Nagesh's total across all six subjects | =SUM(B3:G3) |
| Gowri's average in Odia, Telugu, English | =AVERAGE(B7:D7) |
| Class average in Science (21 students, rows 2–22) | =AVERAGE(G2:G22) |
| Is the Odia average above the Telugu average? | =AVERAGE(B2:B22) > AVERAGE(C2:C22) |
Type the subject-average formula in the cell just below the last row and drag it sideways: every subject's average appears at once. Better still, correct one mark and every affected total and average updates by itself.
(For the record, that last comparison returns FALSE: Odia averages 32.52 and Telugu 34.05.)
9. Line Graphs
A line graph joins successive data points with segments. It is the natural choice for showing change over time.
The two-step reading process
Step 1 — Identify what is given. How is the graph organised? What scale is used? Which line is which — and how was the data produced in the first place? (The Kerala/Punjab graph shows monthly maximum temperature, which is the highest reading among a state's weather stations. That tells you it describes the hottest spot on the hottest day, not a typical day.)
Step 2 — Infer and interpret. Describe each trend, then summarise. Punjab climbs from about 19 °C in January to about 38 °C in June, dips, then falls to about 23 °C in December. Kerala stays between about 29 °C and 33 °C all year. Conclusion: Punjab's temperature varies far more than Kerala's.
Why a line and not bars
The space-launch data covers 13 years and 4 series. As a clustered column graph that is 52 bars — and to follow one country you must hop from the first bar of one cluster to the first bar of the next. A line turns each country's history into one continuous shape, so rises, falls and changes of steepness are read instantly. The steepness itself becomes meaningful: a steeper segment means a larger change that year.
The same argument decides the sleep-across-ages graph, which holds 80 closely spaced points and would need about 70 columns. It reads as a smooth curve showing sleep falling from about 9.5 hours at age 6 to about 8 hours between 30 and 50, then rising to about 8.5 hours by 75.
Drawing one well
- Label both axes and mark the scale
- Choose a scale that comfortably fits the largest value
- Use a different marker shape as well as a different colour for each series, so the graph survives greyscale printing and works for readers who find colours hard to distinguish
- Add a legend
10. Reading Data Honestly
This is the part of the chapter that stays with you longest. Four traps, each shown by a real example from the book:
Trap 1 — Treating an average as a statement about individuals
A graph of "Average Daily Time Spent on Hobbies and Games" shows rural 15-year-olds at about 1 h 19 min. The claim "all rural kids aged 15 spend at least 1 hour" is false: an average of 1 h 19 min is perfectly consistent with many children spending 20 minutes and others three hours.
Trap 2 — Reading a missing series as zero
The space-launch graph plots the world total plus the USA, China and Russia. Nepal is not on it. That does not mean Nepal launched nothing — and the proof is on the graph itself: the three country lines do not add up to the world total, so unshown countries certainly exist.
Trap 3 — Confusing two lines
In 2000, urban electricity stood at about 91% and urban kerosene at about 10%. Swap them and a true statement becomes false. Read the legend before taking any numbers off a graph.
Trap 4 — Answering a question the graph never addressed
The household-lighting graph records each home's primary energy source. It therefore says nothing whatever about power cuts — a household counted as an electricity user may still face daily outages. No amount of careful reading can make that graph settle that question.
One more: a lower bound is not an exact count
A dot plot shows how many times each of 42 students cycled in a week; 5 students exceeded 7 rides, and a week has 7 days, so at least 5 students rode twice on some day. But "exactly 5" is unsupportable — a student with 7 rides might also have doubled up on one day and skipped another. The plot records weekly totals only.
11. Infographics and Activity Strips
Infographics
The Wheat vs Rice map shades every state on a scale from −100 (mostly wheat) through 0 (both equally) to +100 (mostly rice), with a national figure of +13.48, "Rice Wins".
- Most rice: Manipur +100, Nagaland +99, Mizoram +97, Tripura +96, Meghalaya +95 — the entire top of the list is the North-East
- Most wheat: Rajasthan −93, Haryana −81, Punjab −78, Madhya Pradesh −60, Delhi −45 — all north-western
- Most balanced: Bihar +3, then Maharashtra −15, Uttarakhand −18, Himachal −19
A red line across the map marks the split, and it tracks climate: rice needs plentiful water and dominates the high-rainfall south, east and North-East; wheat suits the drier north-western plains.
Activity strips
Manoj records each day on a strip of 48 boxes, one per 30 minutes from midnight to midnight, colouring each box by activity. Three strips decode as Friday (full school day), Saturday (half day) and Sunday (no school), and turning them into a time budget gives:
| Sleep | Eat | Wash/exercise | School & study | Free time | Travel | |
|---|---|---|---|---|---|---|
| Friday | 10.5 h | 1.5 h | 1 h | 7 h | 3 h | 1 h |
| Saturday | 10.5 h | 1.5 h | 1 h | 4 h | 5.5 h | 1.5 h |
| Sunday | 10.5 h | 1.5 h | 1.5 h | 2 h | 7.5 h | 1 h |
The finding the picture alone would never give you: Manoj sleeps exactly 10.5 hours and eats exactly 1.5 hours on all three days. What swings wildly is the trade between study and free time. He holds his sleep constant by shifting bedtime and by napping — an hour after lunch on Saturday, half an hour at 17:00 on Sunday.
Each row totals exactly 24 hours, which is the check that no box has been misread.
12. Worked Examples
Example 1 — Is 17 the average?
A dot plot holds 25 values: 14 (×2), 15 (×2), 16 (×3), 17 (×5), 18 (×4), 19 (×4), 20 (×3), 21 (×1), 23 (×1).
Use the balance test rather than adding 25 numbers.
- Below 17: 3+3, 2+2, 1+1+1 → shortfall 13
- Above 17: 1×4, 2×4, 3×3, 4×1, 6×1 → surplus 31
Surplus wins by 18, so 17 is not the mean. Spreading 18 over 25 values shifts the balance point by 18/25 = 0.72, giving mean = 17.72. (Check: total 443, and 443/25 = 17.72 ✓)
Example 2 — Two new students
A class of 24 averages 150.2 cm. Two students join, 149 cm and 152 cm.
- Old total = 24 × 150.2 = 3604.8 cm — no re-measuring of the original 24 needed
- New total = 3604.8 + 149 + 152 = 3905.8 cm
- New mean = 3905.8/26 = 150.22 cm — a slight increase, because the newcomers average 150.5 cm, above 150.2
The median, however, cannot be determined. The mean says nothing about where the middle student stands.
Example 3 — Weights that cancel
A group's mean weight is 65.3 kg and median 67 kg. This month one person loses 2 kg and two gain 1 kg each.
Change in total = −2 + 1 + 1 = 0, and nobody joined or left, so the mean stays exactly 65.3 kg.
The median cannot be determined — if the middle person is the one who lost 2 kg it falls, if it is a gainer it rises, otherwise it stays. That information is not given.
Example 4 — A mean grid
Fill a 3 × 3 grid with 9 distinct numbers so every row, column and diagonal averages 10.
Three cells averaging 10 must sum to 30, so this is a magic square with constant 30. Take the standard 1-to-9 magic square (constant 15) and add 5 to every entry:
| 7 | 12 | 11 |
|---|---|---|
| 14 | 10 | 6 |
| 9 | 8 | 13 |
The centre is forced to be 10: the four lines through the centre cover it four times and every other cell once, giving 4 × 30 = (sum of all nine) + 3 × centre = 90 + 3 × centre.
Example 5 — Building data to order
- 3 numbers with mean 8 — the total must be 24: 8, 8, 8 or 1, 3, 20
- 4 numbers with median 15.5 — the 2nd and 3rd must sum to 31: 10, 15, 16, 20 or 3, 11, 20, 40; the outer two can be anything
- 6 numbers with mean > median — push one value far right: 1, 2, 3, 4, 5, 100 has median 3.5 and mean 19.17
A mean condition fixes the total, leaving the spread free. A median condition fixes only the middle, leaving the extremes free.
Example 6 — Three false statements
All three are false, and one counterexample each is enough:
| Claim | Counterexample |
|---|---|
| The average of two even numbers is even | 2 and 4 average to 3 |
| The average of two multiples of 5 is a multiple of 5 | 5 and 10 average to 7.5 |
| The average of five multiples of 5 is a multiple of 5 | 5, 5, 5, 5, 10 average to 6 |
Averaging divides, and division can strip away the very property being claimed.
13. Common Mistakes
- Averaging the distinct values instead of the data. The family-size table gives 188/36 = 5.22, not 52/8 = 6.5. Always multiply by the frequencies.
- Expecting the mean to be the midpoint of the extremes. For 10, 10, 11, 17 the midpoint is 13.5 and the mean is 12.
- Assuming more values means a bigger mean. Compare the new value with the current mean — that alone decides the direction.
- Reading an average as a fact about individuals. No family has 5.22 members.
- Re-measuring everything when only the total was needed. An average carries the total inside it.
- Treating a missing series as zero. Nepal's absence from a chart is not evidence about Nepal.
- Answering a question the graph does not address. Lighting sources say nothing about power cuts.
- Reading the wrong line. Check the legend first — 91% and 10% are easy to swap.
- Reporting a lower bound as an exact count. "At least 5" is supportable; "exactly 5" is not.
14. Real-World Applications
- Weather and climate — monthly average rainfall and rainy days are built exactly as in this chapter: many years of readings averaged month by month, then plotted so the monsoon's shape becomes visible. It is why New Delhi peaks in July–August (south-west monsoon) while Rameswaram peaks in November (north-east monsoon).
- Public health — the monthly live-births graph is real published data, and health systems use exactly this kind of seasonal line to plan maternity staffing and vaccine orders months ahead.
- Price monitoring — the iodised-salt series is a government price table. Comparing an absolute rise with a proportional one (West Bengal's ₹14.52 against Assam's 106%) is the everyday work of tracking inflation.
- Spreadsheets at work —
=SUMand=AVERAGEover a range are the first two formulas anyone learns on the job. - Time-use studies — Manoj's 48-box strip is a simplified time-use diary, the instrument national statistical offices use to measure how a population spends its day.
- Reading the news — every chart in a newspaper invites the four traps above.
15. Conclusion
The chapter's own summary puts it plainly:
- Last year the mean was a fair share. Here it became a balance point — the place where the distances to the left and to the right are equal.
- Values inserted above the mean raise it and values below lower it, and the median behaves similarly, though it responds to counts rather than sizes.
- Line graphs visualise change over time.
- Examining data leads to new questions.
That last point is not decoration. Every worked example here ends somewhere useful: why do births peak in August–October? Why is Mizoram's salt always the dearest? Why does the moon rise 50 minutes later each day, and what does that have to do with the length of a month? The techniques are worth learning because they let you ask better questions, not merely answer set ones.
This chapter is the direct foundation for Class 9–10 Statistics, where the frequency-table method here becomes mean, median and mode of grouped data.
Appendix — Related Topics from Other Chapters and Later Classes
The material below is not part of this chapter. It is kept here because older notes for "Chapter 12" often mixed it in.
Pie charts — Ganita Prakash Part 2, Chapter 1
Pie charts belong to Proportional Reasoning — 2, where they arise as a proportional-reasoning tool.
Angle for a category = (category ÷ total) × 360°, and all angles must sum to 360°.
The coordinate plane — Class 9
Not in Class 8 Ganita Prakash at all. In Class 9 Coordinate Geometry you will meet the x- and y-axes, the origin (0, 0), the four quadrants, and the fact that (3, 5) and (5, 3) are different points.
Straight-line graphs, y = mx + c — Class 9
Direct proportion y = kx passes through the origin; the general linear form y = mx + c has slope m and y-intercept c.
Histograms — Class 9
For continuous data grouped into class intervals, with bars touching. A bar graph, by contrast, shows discrete categories and its bars have gaps.
Scatter plots and correlation — later study
Plotting pairs (x, y) to reveal a relationship between two variables. Class 8 works with one variable at a time.
Mode, range and spread — Class 9–11
This chapter covers the mean and median only. The mode, the range and measures of spread such as standard deviation come later, though the range does appear informally here when the dart data is described by its minimum and maximum.
