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CBSE · Class 8 · Mathematics Part 2

Part 2, Chapter 5: Tales by Dots and Lines

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी · हिंदी

Mean as the balance point

From daily life

Four friends have 2, 4, 6 and 12 sweets. They pool them and share equally: 24 ÷ 4 = 6 each. That fair share is the mean.

Simple definition

Mean = sum of all values ÷ number of values. On a dot plot the mean is the balance point: the distances of values below it and above it add up to the same total.

024681012146(−4) + (−2) + 0 + 6 = 0

In the figure: Below 6: 4 + 2 = 6. Above 6: 6. They balance.

Rule 1: Find the mean — Add all values, then divide by how many there are.

Live figure

Mean as the balance point

In the app you can move this figure with your finger.

Open in the app

In the figure: Move the sliders and watch the mean change.

  • Easy: 4, 6, 8: 18 ÷ 3 = 6.
  • Medium: 2, 9, 10, 7: 28 ÷ 4 = 7.
  • Exam-type: 12, 15, 18, 20, 25: 90 ÷ 5 = 18.

Rule 2: Check the balance — Value minus mean: the negatives and positives cancel to 0.

  • Easy: 2, 4, 12 (mean 6): −4, −2, +6 add to 0.
  • Medium: 5, 5, 8, 10 (mean 7): −2, −2, +1, +3 add to 0.
  • Exam-type: 1 and 9 (mean 5): −4 and +4.

Rule 3: Mean and sum — Sum = mean × number of values. Use it to find a hidden value.

  • Easy: Mean 8 of 5 numbers: sum 40.
  • Medium: Mean 12 of 6 numbers: sum 72.
  • Exam-type: Mean 15 of 4 numbers, three are 10, 12, 20: 60 − 42 = 18 is the fourth.
Remember

The mean always lies between the smallest and the largest value, and it need not be one of the values.

Common mistake

Dividing by the wrong count, for example forgetting a value that is 0. A zero still counts as a value.

Shortcut

Add all, divide by how many. Memory clue: "Pool and share": pool everything, then share equally.

Spot the difference

PointMeanMedian
How foundsum ÷ countmiddle value after sorting
Data 2, 4, 6, 1265
Pulled by a very big valueyesvery little

Practice questions

Easy
  1. 1Find the mean of 3, 5 and 10.
  2. 2Find the mean of 12 and 20.
  3. 3The mean of 4 numbers is 9. Find their sum.
Medium
  1. 4Find the mean of 6, 8, 9, 12 and 15.
  2. 5The mean of 4 numbers is 15. Three of them are 10, 12 and 20. Find the fourth.
  3. 6The data 5, 7, 12 has mean 8. Find the total distance of the values below the mean from the mean.
Challenging
  1. 7The mean of 5 numbers is 22. The mean of 3 of them is 20. Find the mean of the other two.
  2. 8For what value of x do 4, x and 10 have mean 8?
Show answers (try them yourself first)
  1. 16
  2. 216
  3. 336
  4. 410
  5. 518
  6. 64
  7. 725
  8. 810

How the mean changes

From daily life

Five students have a mean of 50 marks. A sixth student scores 80. The new mean is 330 ÷ 6 = 55. A high score pulls the mean up.

Simple definition

Adding a value above the mean raises it, below the mean lowers it, equal to the mean leaves it unchanged. If every value gets +k, the mean gets +k. If every value is multiplied by k, the mean is multiplied by k.

Live figure

How the mean changes

In the app you can move this figure with your finger.

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In the figure: Slide the extra score and watch the mean move.

Rule 1: Add or remove one value — New mean = (old sum ± the value) ÷ new count.

  • Easy: Mean 10 of 4 numbers (sum 40), add 20: 60 ÷ 5 = 12.
  • Medium: Mean 8 of 5 numbers (sum 40), remove 12: 28 ÷ 4 = 7.
  • Exam-type: Mean 6 of 3 numbers (sum 18), add 6: 24 ÷ 4 = 6, unchanged.

Rule 2: Add the same number to every value — The mean goes up (or down) by exactly that number.

  • Easy: 4, 6, 8 (mean 6) with +5 each: mean 11.
  • Medium: 10, 20, 30 (mean 20) with −4 each: mean 16.
  • Exam-type: 2, 3, 7 (mean 4) with +10 each: mean 14.

Rule 3: Multiply every value — The mean is multiplied by the same number.

  • Easy: 2, 4, 6 (mean 4) times 3: mean 12.
  • Medium: 10, 20, 30 (mean 20) times half: mean 10.
  • Exam-type: 5, 7, 9 (mean 7) times 2: mean 14.
Remember

Adding a new value changes the count as well as the sum, so always recompute from sum and count.

Common mistake

Thinking that +5 on each of 10 values adds 50 to the mean. The sum rises by 50, but the mean rises by only 5.

Shortcut

Same change for all, same change for the mean. Memory clue: "The mean follows the crowd".

Spot the difference

PointOne value addedEvery value changed
Countgoes up by 1stays the same
Sumgoes up by that valuechanges for every value
Meanmoves toward the new valuechanges by exactly +k

Practice questions

Easy
  1. 1The mean of 5, 7 and 9 is 7. Each value gets +3. Find the new mean.
  2. 2The mean of 2, 4 and 6 is 4. Each value is multiplied by 5. Find the new mean.
  3. 3The mean of 4 numbers is 10. A fifth number 10 is added. Find the new mean.
Medium
  1. 4The mean of 5 numbers is 20. A sixth number 38 is added. Find the new mean.
  2. 5The mean of 6 numbers is 15. The number 30 is removed. Find the new mean.
  3. 6The mean of 3 numbers is 12. Each is multiplied by 4 and then 2 is subtracted from each. Find the new mean.
Challenging
  1. 7The mean of 4 numbers is 14. Two more numbers, 8 and another, are added and the mean stays 14. Find the other number.
  2. 8The mean mark of 10 students is 60. The teacher adds 4 marks to everyone. Then one student whose new mark is 64 leaves. Find the mean of the remaining 9.
Show answers (try them yourself first)
  1. 110
  2. 220
  3. 310
  4. 423
  5. 512
  6. 646
  7. 720
  8. 864

Unknown values, median and frequency tables

From daily life

A boy scored 55, 65 and 75 in three tests and wants a mean of 70 after four tests. He needs 4 × 70 − 195 = 85 in the last one.

Simple definition

Missing value = (mean × count) − sum of known values. Median: sort the values; take the middle one, or the mean of the two middle ones if the count is even. From a frequency table: mean = (sum of value × frequency) ÷ total frequency.

Live figure

Unknown values, median and frequency tables

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In the figure: Change the mean and a known score to find the missing score.

Rule 1: Find the missing value — Total needed minus total known.

  • Easy: Mean 9 of 4 numbers, known 5, 8, 12: 36 − 25 = 11.
  • Medium: Mean 7 of 5 numbers, known 3, 6, 8, 9: 35 − 26 = 9.
  • Exam-type: Mean 80 in 5 tests, scores 70, 75, 85, 90: 400 − 320 = 80.

Rule 2: Find the median — Sort first. Then take the middle (or the mean of the two middle ones).

Live figure

Unknown values, median and frequency tables

In the app you can move this figure with your finger.

Open in the app

In the figure: Change the frequencies and watch the mean.

  • Easy: 3, 9, 5, 1, 7 sorted 1, 3, 5, 7, 9: median 5.
  • Medium: 8, 2, 6, 4 sorted 2, 4, 6, 8: (4 + 6) ÷ 2 = 5.
  • Exam-type: 1, 2, 3, 100: median (2 + 3) ÷ 2 = 2.5, while the mean is 26.5.

Rule 3: Mean from a frequency table — Multiply each value by its frequency, add, divide by the total frequency.

  • Easy: Values 1, 2, 3 with frequencies 2, 3, 5: 23 ÷ 10 = 2.3.
  • Medium: Values 10, 20 with frequencies 3, 1: 50 ÷ 4 = 12.5.
  • Exam-type: Values 0, 1, 2, 3 with frequencies 1, 2, 3, 4: 20 ÷ 10 = 2.
Remember

A very large value changes the mean a lot but the median very little, so the median is a better summary for data with extreme values.

Common mistake

Taking the middle value without sorting, or finding the mean of the values in a frequency table without using the frequencies.

Shortcut

Total needed, minus total known. Memory clue: "Sort, then split in the middle" for the median.

Spot the difference

PointOdd number of valuesEven number of values
Median isthe middle valuemean of the two middle values
Example1, 3, 51, 3, 5, 9
Median34

Practice questions

Easy
  1. 1Find the median of 4, 9 and 2.
  2. 2The mean of 3 numbers is 6. Two of them are 5 and 7. Find the third.
  3. 3Find the median of 1, 2, 3 and 4.
Medium
  1. 4The mean of 5 numbers is 10. Four of them are 6, 8, 12 and 14. Find the fifth.
  2. 5The values 2, 4, 6 have frequencies 1, 2, 1. Find the mean.
  3. 6Find the median of 10, 3, 8, 6, 15 and 1.
Challenging
  1. 7The mean of 5 numbers is 18. The largest, 30, is removed. Find the mean of the other four.
  2. 8The data 2, 4, 6, 8, x has median 6 and mean 6. Find x.
Show answers (try them yourself first)
  1. 14
  2. 26
  3. 32.5
  4. 410
  5. 54
  6. 67
  7. 715
  8. 810

Line graphs

From daily life

A doctor notes your temperature every 3 hours and joins the dots. Going up, going down, staying flat: the line tells the story of the fever at a glance.

Simple definition

A line graph joins points (time, value) in order of time. Rising line = increase, falling line = decrease, flat line = no change. A steeper line means a faster change.

5092214276+12 / 2 = 6

In the figure: Between 2 and 4 the line rises 12 in 2 steps: 6 per step, the steepest part.

Rule 1: Read values and changes — Read the height at each point. Change = later value − earlier value.

  • Easy: Points (2019, 40), (2020, 55), (2021, 50): value in 2020 is 55.
  • Medium: From 2019 to 2020 the rise is 55 − 40 = 15.
  • Exam-type: From 2020 to 2021 the fall is 55 − 50 = 5.

Rule 2: Find the biggest change — Work out the change for each neighbouring pair and compare.

  • Easy: Values 10, 14, 30, 32 in 2018 to 2021: changes +4, +16, +2.
  • Medium: The biggest rise is +16, between 2019 and 2020.
  • Exam-type: The total change from 2018 to 2021 is 32 − 10 = 22.

Rule 3: Rate and in-between values — Rate = change ÷ steps. Halfway between two points, take the halfway value (an estimate).

  • Easy: From (1, 10) to (5, 30): 20 ÷ 4 = 5 per step.
  • Medium: At step 3 (halfway) the estimate is (10 + 30) ÷ 2 = 20.
  • Exam-type: A flat part from (6, 40) to (9, 40): rate 0.
Remember

The line between two readings is only an estimate: the graph does not tell us what happened exactly in between.

Common mistake

Thinking the highest point is the fastest change. The fastest change is the steepest part, not the tallest point.

Shortcut

Up is a rise, down is a fall, flat is still. Memory clue: "Steeper means speedier".

Spot the difference

PointLine graphBar graph
Best forchange over timecomparing categories
Points arejoined by linesshown as separate bars
Exampletemperature each hourmarks of five classes

Practice questions

Easy
  1. 1Weekly values are 5, 8 and 6 in weeks 1, 2 and 3. In which week is the value highest?
  2. 2In the same graph, find the rise from week 1 to week 2.
  3. 3What does a flat part of a line graph show?
Medium
  1. 4A line goes from (2, 10) to (4, 26). Find the rate of change per step.
  2. 5Values in steps 1 to 4 are 40, 34, 29, 35. Find the largest fall between neighbouring steps.
  3. 6Points (1, 12) and (3, 20) are joined by a straight line. Estimate the value at step 2.
Challenging
  1. 7A line graph has the points (0, 5), (2, 9), (4, 21), (6, 27). Find the highest rate of change per step.
  2. 8A plant is 20, 26, 32 and 38 cm tall in months 1 to 4 and keeps growing at the same rate. Find its height in month 6.
Show answers (try them yourself first)
  1. 1week 2
  2. 23
  3. 3no change
  4. 48
  5. 56
  6. 616
  7. 76
  8. 850 cm

Choosing and reading graphs

From daily life

A school office keeps a bar graph for each class strength, a line graph for the daily attendance, and a pie chart for how the fund is spent. Each picture answers a different question.

Simple definition

Choose by purpose: change over time uses a line graph; comparing categories uses a bar graph; parts of a whole use a pie chart; repeated values of one set use a dot plot. In a pie chart angle = share × 360°. Always check the title, labels and where the axis starts.

90°108°162°

In the figure: 25% of 360° = 90°; 30% = 108°; 45% = 162°.

Rule 1: Choose the graph — Ask: time, categories or parts of a whole?

Live figure

Choosing and reading graphs

In the app you can move this figure with your finger.

Open in the app

In the figure: Change the bars and compare the sizes.

  • Easy: Marks of five students: bar graph.
  • Medium: Temperature through a week: line graph.
  • Exam-type: Share of expenses: pie chart.

Rule 2: Pie chart angles — Angle = share of the whole × 360°.

  • Easy: 1/4 of the whole: 90°.
  • Medium: 30%: 30/100 × 360° = 108°.
  • Exam-type: 45 of 90 students: 1/2 × 360° = 180°.

Rule 3: Read carefully — Check where the axis starts, the scale and the labels.

  • Easy: Bars of real values 52 and 56 with the axis starting at 50 are drawn 2 and 6 tall: three times, though the values are almost equal.
  • Medium: An axis that starts at 0 shows the true size of the bars.
  • Exam-type: Equal steps on the axis make the graph fair.
Remember

A pie chart can show only parts of ONE whole; its sectors always add to 360° (100%).

Common mistake

Using a pie chart to show change over time, or not checking where the axis starts.

Shortcut

Time: line. Category: bar. Whole: pie. Memory clue: "Time Line, Category Bar, Whole Pie".

Spot the difference

PointBar graphPie chart
Showsamountsshares of a whole
Total of all partscan be anything360° (100%)
Examplebooks in five librarieshow a day is spent

Practice questions

Easy
  1. 1Which graph is best for monthly rainfall through a year?
  2. 2Find the angle for a 25% sector in a pie chart.
  3. 3Which graph is best for comparing runs scored by six players?
Medium
  1. 4In a pie chart, 40 of 200 students like football. Find the angle.
  2. 5A sector of 135° stands for a share of 480 people. How many people is that?
  3. 6A sector of 54° is what percentage of the whole?
Challenging
  1. 7In a pie chart of 360 students, science has 150° and maths has 90°. The rest is languages. How many students take languages?
  2. 8A bar graph axis starts at 40. Bars for the real values 45 and 55 are drawn 5 and 15 units tall. How many times taller does the second bar look?
Show answers (try them yourself first)
  1. 1line graph
  2. 290°
  3. 3bar graph
  4. 472°
  5. 5180
  6. 615%
  7. 7120
  8. 83 times