Class 8 › Mathematics › Chapter
Tales by Dots and Lines
The mean as a balancing point, how mean and median react to changing data, finding unknown values, frequency tables, spreadsheets, and reading line graphs, infographics and activity strips.
1. The Mean as a Balancing Point
Mean
$$\text{Mean} = \frac{\text{sum of all values}}{\text{number of values}}$$
Distances to the left of the mean add up to the distances to the right.
- Two numbers: the mean is exactly halfway between them (e.g. $3$ and $7$ give $5$).
- More numbers: the mean is not always the midpoint of the extremes. Instead, the sum of distances of the values below it equals the sum of distances of the values above it.
- Example: $10, 10, 11, 17$ has mean $12$: left $= 2 + 2 + 1 = 5$, right $= 5$.
- Only one centre: shifting the point right increases left distances and decreases right distances (and the other way round), so balance holds only at the mean.
2. How the Mean Changes
Add a value
Above the mean
Mean increases (e.g. adding $17$ to data with mean $11$ gives $12.2$).
Add a value
Below the mean
Mean decreases. A value equal to the mean leaves it unchanged.
Remove a value
Opposite effect
Removing a value above the mean lowers it; below the mean raises it.
- Unchanged mean: add values so that the total distance above the mean equals the total below (e.g. two values $2$ below and one value $4$ above a mean of $9$).
Shifting and scaling
Add $k$ to every value: new mean $= \dfrac{(x_1+k)+\dots+(x_n+k)}{n} = \dfrac{x_1+\dots+x_n}{n} + \dfrac{nk}{n} = a + k$.
Subtract $k$: new mean $= a - k$.
Multiply every value by $k$: new mean $= \dfrac{k(x_1+\dots+x_n)}{n} = ka$.
Example: data with mean $7.18$: add $10$ $\to$ $17.18$; double $\to$ $14.36$. The mean keeps the same relative position.
3. Tinkering with the Median
Median: the middle value of the sorted data. For an even number of values, it is the average of the two middle values.
- Include a value greater than the median: the median increases (or stays if values tie). Example: $3, 7, 8, 12, 14$ has median $8$; adding $11$ makes it $\tfrac{8+11}{2} = 9.5$.
- Include a value less than the median: the median decreases.
- Adding or removing a value equal to the middle one can keep the median unchanged; adding one value above and one below also keeps it.
- The median depends on position, the mean on every value.
4. Finding the Unknown
Key relation
$$\text{Sum of values} = \text{mean} \times \text{number of values}$$
- Missing value: weights of $10$ wrestlers with mean $39.2$ and known sum $349$: $349 + w = 392 \Rightarrow w = 43$ kg.
- Correcting an error: mean $25.6$ over $15$ trees gives total $384$. One count was $3$ too high, so correct total $= 381$ and correct mean $= \tfrac{381}{15} = 25.4$.
- Systematic shift: all $24$ heights measured $1$ cm too high (shoes), mean $150.2$ cm: just subtract $1$ to get $149.2$ cm; no re-measuring needed.
- Examples: mean of $8, 13, 10, 4, 5, 20, y, 10$ is $10.375$: sum $= 83$, known $= 70$, so $y = 13$. Mean $134$ over $15$ values gives sum $2010$.
- Mean of an arithmetic list: first $50$ natural numbers: $25.5$; first $50$ odd numbers: $50$; first $50$ multiples of $4$: $102$ (the mean is the average of first and last).
5. Mean and Median with Frequencies
Mean from a frequency table
$$\text{Mean} = \frac{\sum (\text{value} \times \text{frequency})}{\sum \text{frequency}}$$
- Common error: averaging the listed values only (ignoring frequencies). For family sizes $3$ to $10$ with frequencies $3, 11, 9, 7, 3, 1, 1, 1$: mean $= \tfrac{188}{36} \approx 5.22$, not $6.5$.
- Median: with $n = 36$ values, take the average of the $18$th and $19$th values.
- Quick method: add frequencies cumulatively from the smallest value ($3$, then $3 + 11 = 14$, then $14 + 9 = 23$). Positions $15$ to $23$ are all $5$, so the median is $5$.
6. Spreadsheets
Spreadsheet: a digital grid of rows and columns of boxes called cells that can hold text, numbers or formulae (e.g. Excel, Google Sheets, LibreOffice Calc).
| Idea | Example |
|---|---|
| Cell name = column letter + row number | Farooq's Maths marks are in E5 |
Range Start:End | B3:G3 is Nagesh's row; D2:D6 is a column of $5$ English marks |
| Total of a range | =SUM(B3:G3) |
| Average of a range | =AVERAGE(B7:D7) |
- A formula starts with
=. A formula can be copied to the other rows or columns to save time. - Class average for a subject:
=AVERAGEover that subject's column.
7. Line Graphs
Line graph: points joined by line segments; used mainly to show change over time (trends).
Two-step reading
Step 1: Identify what is given. Title, axes, units and scale, legend (colours and marker shapes), peaks, lows, rises and falls.
Step 2: Infer and interpret. Summarise trends in words, compare lines, and note what the graph does not show.
- Steepness: the steeper the segment, the greater the change between those two points.
- Why not a clustered column graph? With many years or categories (e.g. $13 \times 4 = 52$ bars, or $70$ ages) a bar graph is crowded; a line graph stays clear.
- Smooth curves: appear when there are very many close data points (e.g. sleep versus age).
- Different marker shapes help when printed in black and white or for colour-blind readers.
- Method of collection: knowing how data was produced shows its limits (bias, missing groups). Monthly maximum temperature is the highest recorded value across stations; monthly average rainfall averages that month over several years.
- Examples: Punjab's maximum temperature swings from about $19^\circ$C (Jan) to $38^\circ$C (Jun) while Kerala stays between about $29^\circ$C and $33^\circ$C. West-coast cities (Udupi, Mumbai, Kovalam) peak in Jun-Aug; Rameswaram in Oct-Dec; Chennai peaks in Nov.
- Valid inference check: do not claim what the graph does not show (e.g. "Nepal launched nothing" when only three countries are plotted).
8. Infographics and Activity Strips
Infographic
Wheat vs Rice
States are coloured on a scale from $-100$ (mostly wheat) to $+100$ (mostly rice), showing the difference in per capita consumption. $+100$ does not mean no wheat is eaten. National preference: $+13.48$ (rice).
Activity strip
Time use
A strip of 48 boxes, each a 30-minute slot from midnight to midnight, coloured by activity (sleep, eating, school, travel, leisure). Used to compare days and people.
- Always read the legend and scale first; ask who collected the data and how.
- Data often raises new questions (why does Punjab vary more than Kerala?).
9. Key Results and Quick Revision
| Situation | Effect on mean | Effect on median |
|---|---|---|
| Add value above | Increases | Increases or stays |
| Add value below | Decreases | Decreases or stays |
| Add value equal to mean | No change | Depends on position |
| Add $k$ to all values | $a + k$ | Also $+k$ |
| Multiply all by $k$ | $ka$ | Also $\times k$ |
Common mistakes:
• Taking the mean as the midpoint of the smallest and largest values.
• Ignoring frequencies when finding mean or median from a table.
• Using the wrong positions for the median (average the two middle values when $n$ is even).
• Reading a graph without checking the axis scale and units.
• Concluding things the graph does not show.
• Forgetting
• Taking the mean as the midpoint of the smallest and largest values.
• Ignoring frequencies when finding mean or median from a table.
• Using the wrong positions for the median (average the two middle values when $n$ is even).
• Reading a graph without checking the axis scale and units.
• Concluding things the graph does not show.
• Forgetting
= at the start of a spreadsheet formula.