Tales by Dots and Lines Class 8: Mean, Median, Frequency Tables, Spreadsheets and Line Graphs

What Tales by Dots and Lines is about

Key concepts

The mean as a balance point

NCERT section 5.1 The Balancing Act

The mean (average) of $n$ values is $\frac{x_1+x_2+\cdots +x_n}{n}$. Two ways to picture it:

• Fair share: if everyone's values were pooled and shared equally, each would get the mean.
• Balance point: on a dot plot, the total distance of the values below the mean equals the total distance of the values above it, just like a see-saw balancing.

For two numbers the mean is exactly halfway between them, but for more numbers it need not be the midpoint of the smallest and largest. For 4, 7, 9, 14, 16 the mean is 10: distances below are $6+3+1=10$ and above are $4+6=10$.

The mean is the only such balance point: move the point up and the distances below grow while those above shrink, so it no longer balances.

Worked example

Find the mean of 4, 7, 9, 14, 16 and show that it is the balance point. Is it the midpoint of 4 and 16?

Try it first, then show the solutionSolution

1. Sum $=50$, so mean $=\frac{50}{5}=10$.
2. Distances below 10: $6+3+1=10$; above 10: $4+6=10$. They balance.
3. The midpoint of 4 and 16 is also 10 here, but that is a coincidence; for 10, 10, 11, 17 the mean is 12 while the midpoint is 13.5.

Answer: Mean 10, the balance point (here it happens to equal the midpoint).

How the mean changes

NCERT section 5.1 The Balancing Act

Including or removing a value:

• including a value greater than the mean increases the mean; smaller decreases it; equal leaves it unchanged;
• removing a value greater than the mean decreases the mean, and so on.

To keep the mean the same when including several values, their distances above and below the mean must balance. For data with mean 10, including 7, 8 and 15 keeps the mean 10 ($-3-2+5=0$).

Changing every value:

• add (or subtract) the same number $c$ to every value: the mean goes up (or down) by $c$;
• multiply every value by $k$: the mean is multiplied by $k$.

*Algebra:* $\frac{(x_1+3)+\cdots +(x_n+3)}{n}=\frac{x_1+\cdots +x_n+3n}{n}=a+3$, and $\frac{5x_1+\cdots +5x_n}{n}=5a$ by the distributive property.

Worked example

Five values have a mean of 12. Find the new mean if (a) 18 is included, (b) 12 is included, (c) a value of 20 is removed instead.

Try it first, then show the solutionSolution

1. Sum $=5\times 12=60$.
2. (a) $\frac{60+18}{6}=13$ (up, since 18 > 12).
3. (b) $\frac{60+12}{6}=12$ (unchanged).
4. (c) $\frac{60-20}{4}=10$ (down, since 20 > 12 was removed).

Answer: (a) 13 (b) 12 (c) 10

How the median changes

NCERT section 5.1 The Balancing Act

The median is the middle value of the sorted data (the mean of the two middle values when there is an even number of values).

• Including a value greater than the median moves the median up or keeps it the same; including a smaller value moves it down or keeps it.
• The median does not care how far away the new value is: including 20 or 2000 to 3, 5, 8, 11, 14 gives the same new median, $\frac{8+11}{2}=9.5$. The mean, by contrast, would change a lot. So the median is less affected by extreme values.
• Including one value above and one below the median keeps the median the same.

The median can stay unchanged even when a bigger value is included if the middle values are equal: 2, 5, 5, 9 has median 5, and 2, 5, 5, 9, 10 also has median 5.

Worked example

The data 3, 5, 8, 11, 14 has median 8. Find the median after including (a) 20, (b) 1.

Try it first, then show the solutionSolution

1. (a) 3, 5, 8, 11, 14, 20: middle values 8 and 11, median $=9.5$.
2. (b) 1, 3, 5, 8, 11, 14: middle values 5 and 8, median $=6.5$.
3. A larger value raised the median; a smaller value lowered it.

Answer: (a) 9.5 (b) 6.5

Finding missing and corrected values

NCERT section 5.1 The Balancing Act

Use sum = mean × number of values.

Missing value: if 6 values have mean 15, their sum is 90; subtract the known values to find the missing one.

Corrections: if one value was recorded wrongly, the true total = recorded total − wrong value + correct value; then divide by the number of values again. You do not need the full data list, only the mean, the count and the change.

Example: the mean of 20 marks is 42.5 (total 850). A mark of 86 was entered as 68, so the true total is $850-68+86=868$, and the true mean is 43.4.

Worked example

The mean of 12, 18, 9, 20, 14 and $w$ is 15. Find $w$.

Try it first, then show the solutionSolution

1. Sum of all six $=6\times 15=90$.
2. Known values add to $12+18+9+20+14=73$.
3. $w=90-73=17$.

Answer: $w=17$

Mean and median from a frequency table

NCERT section 5.1 The Balancing Act

When values repeat, a frequency table lists each value and how often it occurs.

Mean: multiply each value by its frequency, add these products, and divide by the total frequency (not by the number of different values):
$\text{mean}=\frac{\sum (\text{value}\times \text{frequency})}{\sum \text{frequency}}$.

Median: find the middle position(s), then add frequencies one by one (cumulative totals) until you pass that position.

Example: pages read per day by 25 students: 1 (2 students), 2 (5), 3 (8), 4 (6), 5 (3), 6 (1).

• Mean $=\frac{2+10+24+24+15+6}{25}=\frac{81}{25}=3.24$.
• Median is the 13th value. Cumulative: 2, 7, 15, ..., so positions 8 to 15 are all 3: median 3.

Worked example

Shoe sizes of 25 students: size 4 (3), 5 (7), 6 (9), 7 (5), 8 (1). Find the mean and the median.

Try it first, then show the solutionSolution

1. Sum of value × frequency $=12+35+54+35+8=144$.
2. Mean $=\frac{144}{25}=5.76$.
3. Median is the 13th value; cumulative frequencies 3, 10, 19: positions 11 to 19 are size 6, so the median is 6.

Answer: Mean 5.76, median 6.

Spreadsheets

NCERT section 5.1 The Balancing Act

A spreadsheet is a digital notebook of rows and columns of cells. Columns are labelled A, B, C, ... and rows 1, 2, 3, ..., so each cell has a name such as E5 (column E, row 5).

• A block of cells along a row or column is written Start:End: B3:G3 is row 3 from column B to G; D2:D6 is column D from row 2 to row 6.
• A cell can hold a formula beginning with '='. =SUM(B3:G3) adds those cells; =AVERAGE(B7:D7) finds their mean.
• If a number in the sheet changes, every formula using it updates automatically.

Spreadsheets save time and avoid arithmetic slips when there are many numbers, such as the marks of a whole class in six subjects.

Worked example

In a marks sheet, student names are in column A and Maths marks are in column E for rows 2 to 23. Write formulas for (a) the class average in Maths, (b) the total marks of the student in row 4 if subject marks are in columns B to G.

Try it first, then show the solutionSolution

1. (a) The Maths marks are E2:E23, so type =AVERAGE(E2:E23).
2. (b) Row 4 marks are B4:G4, so type =SUM(B4:G4).

Answer: =AVERAGE(E2:E23) and =SUM(B4:G4)

Line graphs

NCERT section 5.2 Visualising and Interpreting Data

A line graph plots values at points and joins them with straight segments. It is best for showing how something changes over time (temperature by month, launches by year).

Reading a line graph in two steps:
1. Identify what is given: title, what each axis shows, the scale, the units, what each line or marker stands for, and the source of the data.
2. Infer and interpret: describe trends (rising, falling, flat), peaks and lows, compare lines, and state conclusions carefully.

Steepness: the steeper a segment, the faster the change; a rising segment means increase, a falling one decrease, a flat one no change.

Why lines instead of columns? With many time points or several series (for example 13 years × 4 groups), a column graph would need dozens of bars and look crowded; a line graph shows the trends clearly. Different marker shapes help if the graph is printed in black and white or for readers who find colours hard to tell apart.

Careful inferences: a graph showing only some countries cannot tell you about others; missing data is not the same as zero.

Worked example

A town's monthly maximum temperatures (°C) from January to June are 22, 24, 29, 35, 39, 36. Between which two months is the line steepest, and what is the mean of these six values?

Try it first, then show the solutionSolution

1. Month-to-month changes: +2, +5, +6, +4, −3.
2. The largest change is +6, from March to April, so that segment is steepest (upwards).
3. Mean $=\frac{22+24+29+35+39+36}{6}=\frac{185}{6}\approx 30.8$ °C.

Answer: March to April; mean about 30.8 °C.

Infographics and activity strips

NCERT section 5.2 Visualising and Interpreting Data

An infographic presents data visually (maps with colour scales, icons, strips) so a message can be understood quickly. To read one:

• find the key or colour scale and what each shade or symbol means;
• note the units and the range (for example, a scale from −100 to +100 comparing two quantities);
• look for overall patterns (such as a clear divide between regions) and exceptions.

Activity strips: a day can be shown as a strip of 48 boxes, one for each half hour, coloured by activity. Counting boxes gives time spent: 16 boxes of sleep is 8 hours. Comparing strips for different days shows patterns, such as weekends having more leisure time.

Good data reading also asks how the data was collected: this shows its limits (bias, missing values) and how confidently conclusions can be drawn.

Worked example

On a 48-box day strip (half-hour boxes), a student colours 17 boxes for sleep, 12 for school and study, 3 for meals and 5 for travel. How many hours go to each, and how much time is left for other activities?

Try it first, then show the solutionSolution

1. Each box is half an hour: sleep $8.5$ h, school and study 6 h, meals 1.5 h, travel 2.5 h.
2. Total used $=17+12+3+5=37$ boxes $=18.5$ h.
3. Left: $48-37=11$ boxes $=5.5$ hours.

Answer: 8.5 h, 6 h, 1.5 h, 2.5 h; 5.5 hours left.

Maths in real life: Tales by Dots and Lines

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Mean, Median and How They Change with the Data
• Spreadsheets and Line Graphs

Tales by Dots and Lines formula sheet

Mean

$\bar{x}=\frac{x_1+x_2+\cdots +x_n}{n}$

Sum from mean

$\text{sum}=\text{mean}\times n$

Add c to each value

$\text{new mean}=\bar{x}+c$

Multiply each value by k

$\text{new mean}=k\bar{x}$

Correcting a value

$\text{new mean}=\frac{\text{old sum}-\text{wrong}+\text{right}}{n}$

Mean from frequencies

$\bar{x}=\frac{\sum fx}{\sum f}$

Median position

$\text{odd }n:\frac{n+1}{2}\text{th value};\text{even }n:\text{mean of }\frac{n}{2}\text{th and }\left(\frac{n}{2}+1\right)\text{th}$

Spreadsheet formulas

$=\text{SUM}(B3:G3),=\text{AVERAGE}(E2:E23)$

How the school exam asks Tales by Dots and Lines

The question patterns to expect, and the method that works for each.

Tricks to remember Tales by Dots and Lines

Short memory aids. Say each one aloud once, then use it on a question the same day.

• See-saw mean

  The mean sits where the see-saw of data balances: distances left = distances right.

  Use it when: Understanding and checking the mean.

• Mean follows the crowd's change

  Add 5 to everyone, the mean adds 5; double everyone, the mean doubles.

  Use it when: Transformed data.

• Totals first

  For missing or wrong values, turn the mean into a total, fix the total, then divide again.

  Use it when: Correction questions.

• Count with frequency

  Divide by the total frequency, never by the number of rows in the table.

  Use it when: Frequency-table means.

• Steeper means faster

  On a line graph, the steepest segment shows the biggest change.

  Use it when: Interpreting trends.

Tales by Dots and Lines: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section E style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 1 + 2 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Tales by Dots and Lines

The slips examiners see most often, and what to write instead.

• ✗Mistake: Ignoring frequencies when finding the mean

  Only the distinct values are averaged.

  Wrong working

  Values 1 to 6 with frequencies: mean $=\frac{1+2+3+4+5+6}{6}=3.5$.

  Right working

  Mean $=\frac{\sum fx}{\sum f}=\frac{81}{25}=3.24$.

  ✓Do this: Multiply each value by its frequency and divide by the total frequency.

  Usually costs: Whole answer.

• ✗Mistake: Finding the median without sorting

  The middle of the list as written is taken.

  Wrong working

  Median of 23, 8, 15, 42, 31, 19 is $\frac{15+42}{2}=28.5$.

  Right working

  Sorted: 8, 15, 19, 23, 31, 42; median $=\frac{19+23}{2}=21$.

  ✓Do this: Sort first, every time.

  Usually costs: Answer mark.

• ✗Mistake: Recomputing a corrected mean from the wrong count

  The number of values is changed by mistake.

  Wrong working

  One of 20 marks corrected: new mean $=\frac{850-68+86}{21}$.

  Right working

  Still 20 values: $\frac{868}{20}=43.4$.

  ✓Do this: A correction replaces a value; the count stays the same.

  Usually costs: 1 mark.

• ✗Mistake: Assuming the mean is the midpoint of the extremes

  It is true for two numbers, so it is assumed for more.

  Wrong working

  Mean of 10, 10, 11, 17 is $\frac{10+17}{2}=13.5$.

  Right working

  Mean $=\frac{48}{4}=12$.

  ✓Do this: Always add all values and divide by the count.

  Usually costs: Answer mark.

• ✗Mistake: Over-reading a graph

  Conclusions go beyond the data shown.

  Wrong working

  The graph shows three countries, so no other country launched anything.

  Right working

  Countries not shown are simply not in the graph; their data is unknown, not zero.

  ✓Do this: State only what the graph shows, and say what it does not show.

  Usually costs: 1 mark in interpretation items.

NCERT exercise guide: Tales by Dots and Lines

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

Practice questions on Tales by Dots and Lines

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   If every value in a data set is doubled, the mean
   (a) stays the same
   (b) increases by 2
   (c) doubles
   (d) halves

   Hint

   Distributive property.

   Answer

   (c) doubles

2. Q2 · 1 mark · MCQExam level

   Which graph is most suitable for showing a city's monthly temperature over a year?
   (a) pie chart
   (b) line graph
   (c) pictograph
   (d) frequency table

   Hint

   Change over time.

   Answer

   (b) line graph

3. Q3 · 1 mark · Very short answerExam level

   The mean of 5 numbers is 18. What is their sum?

   Hint

   Mean × count.

   Answer

   90

4. Q4 · 1 mark · Very short answerExam level

   Write the spreadsheet formula to find the average of the values in cells C2 to C21.

   Hint

   Start:End.

   Answer

   =AVERAGE(C2:C21)

5. Q5 · 2 marks · Very short answerHarder exam level

   Find the median of 7, 9, 13, 16, 20, 24. What will it be if 30 is included?

   Hint

   Even then odd count.

   Answer

   14.5; with 30 it becomes 16.

6. Q6 · 2 marks · Very short answerHarder exam level

   The mean of 7 values is 20. If the value 26 is removed, find the new mean.

   Hint

   Total 140.

   Answer

   $\frac{140-26}{6}=19$

7. Q7 · 2 marks · Very short answerHarder exam level

   The average age of 4 friends is 13 years. When a fifth friend joins, the average becomes 13.4 years. How old is the fifth friend?

   Hint

   Compare totals.

   Answer

   $5\times 13.4-4\times 13=67-52=15$ years

8. Q8 · 3 marks · Short answerHarder exam level

   True or false? Give a reason or example. (i) Including a value greater than the mean always increases the mean. (ii) Including a value greater than the median always increases the median.

   Hint

   Try 2, 5, 5, 9.

   Answer

   (i) True: it adds more distance above the balance point, so the mean moves up. (ii) False: 2, 5, 5, 9 has median 5, and after including 10 the median is still 5.

9. Q9 · 2 marks · Very short answerHarder exam level

   Pages read per day: 1 (2 students), 2 (5), 3 (8), 4 (6), 5 (3), 6 (1). Find the mean.

   Hint

   $\sum fx÷\sum f$.

   Answer

   $\frac{81}{25}=3.24$ pages

10. Q10 · 5 marks · Long answerHarder exam level

    A city's monthly maximum temperatures (°C) from January to June were 18, 21, 27, 31, 30, 33. (a) Describe how to draw the line graph. (b) Between which months is the line steepest? (c) In which month does it fall? (d) Find the mean temperature.

    Hint

    Find month-to-month changes.

    Answer

    (a) Months on the horizontal axis, °C on the vertical axis (say 0 to 40), plot and join the six points. (b) Changes are +3, +6, +4, −1, +3, so February to March is steepest. (c) It falls from April to May. (d) $\frac{160}{6}\approx 26.7$ °C.

11. Q11 · 2 marks · Very short answerHOTS / competency

    Give five whole numbers whose mean is 12 and median is 9.

    Hint

    Sum 60, middle 9.

    Answer

    For example 3, 5, 9, 10, 33 (sum 60, middle 9; many answers).

12. Q12 · 3 marks · Short answerHOTS / competency

    The data 6, 8, 10, 12, 14 has mean 10. Can you include two values, both greater than 10, and keep the mean at 10? Explain.

    Hint

    Balance of distances.

    Answer

    No. Both new values add distance above the mean and nothing below, so the balance point must move up; the mean would increase.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.Describe the mean as a balance point.Show answer

   The total distance of values below the mean equals the total distance of values above it.

2. 2.A new value exactly equal to the mean joins the data. Does the mean change?Show answer

   No, it stays the same.

3. 3.If 4 is added to every value, what happens to the mean?Show answer

   It increases by 4.

4. 4.If every value is tripled, what happens to the mean?Show answer

   It is tripled.

5. 5.Which is less affected by one very large value, mean or median?Show answer

   The median.

6. 6.How do you find a missing value from the mean?Show answer

   Total = mean × count; subtract the known values.

7. 7.How is the mean of a frequency table found?Show answer

   $\frac{\sum fx}{\sum f}$.

8. 8.What does the cell name E5 mean?Show answer

   Column E, row 5.

9. 9.What does =AVERAGE(B7:D7) do?Show answer

   Finds the mean of cells B7, C7 and D7.

10. 10.What kind of data is a line graph best for?Show answer

    Data that changes over time.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you show that the mean balances the distances on both sides?
• ☐Can you predict the new mean after including or removing a value?
• ☐Can you include three values without changing the mean?
• ☐Can you find the new mean when all values are increased or doubled?
• ☐Can you find how the median changes when a value is included?
• ☐Can you find a missing value from the mean?
• ☐Can you correct a mean after a recording error?
• ☐Can you find the mean and median from a frequency table?
• ☐Can you write SUM and AVERAGE formulas for a spreadsheet range?
• ☐Can you find the steepest segment of a line graph and explain it?

Tales by Dots and Lines: frequently asked questions