Question 1

Find the mean of the following data and share your observations:
(i) The first 50 natural numbers.
(ii) The first 50-odd numbers.
(iii) The first 50 multiples of 4.

Accepted Answer

(i) The mean of the first 50 natural numbers.
The first 50 natural numbers.
1, 2, 3,………, 50
The sum of n natural numbers is $\frac{n(n+1)}{2}$
So for n = 50
the sum = $\frac{50(50+1)}{2}$
= $\frac{50 \times 51}{2}$
= 1275
Mean = $\frac{\text { Sum of the first } 50 \text { natural numbers }}{\text { Total number of natural numbers }}$
= $\frac {1275}{50}$
= 25.5
Observations:
The mean of the first n natural numbers is always $\left(\frac{n+1}{2}\right)$
For n = 50, $\frac {51}{2}$ = 25.5
(ii) Mean of the first 50 odd numbers.
The first 50-odd numbers
1, 3, 5, 7, 9,…………, 99
Sum of the first n odd numbers = n
2
So for n (= 50) the sum = (50)
2
= 2500
Mean = $\frac {2500}{50}$ = 50
Observations:
The mean of the first n odd numbers is always n.
For 50-odd numbers, the mean is 50.
(iii) Mean of the first 50 multiples of 4.
The first 50 multiples of 4 = 4, 8, 12, 16,………, 200
Sum of multiples of 4 = 4 + 8 + 12 + ……. + 200
= 4(1 + 2 + 3 + ….. + 50)
= 4 × (1275)
= 5100
Mean = $\frac {5100}{50}$ = 102
Observations:
The mean of the first n multiples of 4 is 4 times the mean of the first n natural numbers.
Mean = 4 × 25.5 = 102

Question 2

The dot plot below shows a collection of data and its average, but one dot is missing. Mark the missing value so that the mean is 9 (as shown below).

Accepted Answer

From the dot plot, the data values
4, 7, 8, 8, 9, 9, 9, 9, 9, 11, x
Number of observations = 11
Mean = 9
Sum of data values = 4 + 7 + 8 + 8 + 9 + 9 + 9 + 9 + 9 + 11 = 83
Total sum = 9 × 11 = 99
Missing number = 99 – 83 = 16

Question 3

Sudhakar, the class teacher, asks Shreyas to measure the heights of all 24 students in his class and calculate the average height. Shreyas informs the teacher that the average height is 150.2 cm. Sudhakar discovers that the students were wearing uniform shoes when the measurements were taken and the shoes added 1 cm to the height.
(i) Should the teacher get all the heights measured again, without the shoes, to find the correct average height? Or is there a simpler way?
(ii) What is the correct a

Accepted Answer

(i) The teacher does not need to remeasure all the heights. Since the shoes add 1 cm to every student’s height, the average height with shoes is 1 cm more than the actual average height. So the correct average height is 1 cm less than the measured average height.
(ii) Correct average height = 150.2 cm – 1 cm = 149.2 cm
∴ Option (d) is correct.

Question 4

The three dot plots below show the lengths, in minutes, of songs of different albums. Which of these has a mean of 5.57 minutes? Explain how you arrived at the answer.

Accepted Answer

We can determine it by examining the dot plots. In dot plot A, most song lengths are between 5 and 6.5 minutes, so the mean is likely around 5.57 minutes. In dot plots B and C, all song lengths are below 5.57 minutes, so their means cannot exceed 5.57. Therefore, dot plot A is the one with a mean of 5.57 minutes.
Check:
For Album A
Data values = 5, 5, 5.25, 5.5, 5.75, 6, 6.5
Mean = $\frac{5+5+5.25+5.5+5.75+6+6.5}{7}$
= $\frac {39}{7}$
= 5.57 minutes
For Album B
Data values = 0.5, 0.75, 1.5, 1.5, 2, 3.75, 4.25, 5
Mean = $\frac{0.5+0.75+1.5+1.5+2+3.75+4.25+5}{8}$
= $\frac {19.25}{8}$
= 2.41 minutes
For Album C
Data values = 3.5, 3.5, 3.5, 4, 4, 4, 4.25, 4.5
Mean = $\frac{3.5+3.5+3.5+4+4+4+4.25+4.5}{8}$
= $\frac {30.75}{8}$
= 3.9 minutes
Album (A) has a mean of 5.57 minutes.

Question 5

Find the median of 8, 10, 19, 23, 26, 34, 40, 41, 41, 48, 51, 55, 70, 84, 91, 92.
(i) If we include one value to the data (in the given list) without affecting the median, what could that value be?
(ii) If we include two values to the data without affecting the median what could the two values be?
(iii) If we remove one value from the data without affecting the median what could the value be?

Accepted Answer

The given data set is 8, 10, 19, 23, 26, 34, 40, 41, 41, 48, 51, 55, 70, 84, 91, 92
The number of observations (n) = 16 (even)
Median = average of $\left(\frac{n}{2}\right)^{t h}$ and $\left(\frac{n}{2}+1\right)^{t h}$ terms
= average of 8th and 9th terms
= $\frac{41+41}{2}$
= $\frac {82}{2}$
= 41
(i) To keep the median 41 with an odd number of values (n = 17), the new value must be placed at the median itself in the order list for making the middle value the 9th value. So the value could be 41.
(ii) With n = 18 (even) the median is the average of the 9th and 10th values.
To keep the median 41, we need to add two numbers whose sum is 82.
For this, one value should be less than 41 and the other greater than 41.
For example, the two values could be 40 and 42.
(iii) With n = 15 (odd) the median is the 8th value that is 41.
So we can remove another 41 from the ordered list.

Question 6

Examine the statements below and justify whether the statement is always true, sometimes true, or never true.
(i) Removing a value less than the median will decrease the median.
(ii) Including a value less than the mean will decrease the mean.
(iii) Including any 4 values will not affect the median.
(iv) Including 4 values less than the median will increase the median.

Accepted Answer

(i) Sometimes true: Removing a value less than the median can decrease the median. But if the data set has an even number of elements and the removed value is below the lower of the two middle values, the median may stay the same.
(ii) Always true: The mean is the sum of all values divided by the number of values. Adding a value less than the mean decreases the total sum less than proportionally to the increase in the number of values, thus decreasing the mean.
(iii) Sometimes true: Including any 4 values could shift the median depending on whether those values are above or below the median and on the original number of observations.
(iv) Never true: Including values less than the median will either keep the median the same or decrease it, but it will never increase it.

Question 7

The mean of the numbers 8, 13, 10, 4, 5, 20, y, 10 is 10.375. Find the value of y.

Accepted Answer

Numbers = 8, 13, 10, 4, 5, 20, y, 10
Total numbers = 8
Mean = 10.375
Total Sum = 10.375 × 8 = 83
Sum of numbers = 8 + 13 + 10 + 4 + 5 + 20 + y + 10 = 70 + y
So 83 = 70 + y
y = 83 – 70 = 13

Source: This question is from Tales by Dots and Lines, Mathematics &mdash; Class 8, CBSE Board.
Key Concepts CoveredThis question tests your understanding of the following concepts from the chapter Tales by Dots and Lines: Mean, Numbers, Value, Tales by Dots and Lines, Mathematics. These are fundamental topics in Mathematics that students are expected to master as part of the CBSE Class 8 curriculum.A thorough understanding of these concepts will help you answer similar questions confidently in your CBSE examinations. These topics are frequently tested in both objective and subjective sections of Mathematics papers. We recommend revising the relevant section of your textbook alongside practising these solved examples to build a strong foundation.
How to Approach This QuestionRead the question carefully and identify what is being asked. Break down complex questions into smaller parts. Use the terminology and concepts discussed in this chapter. Structure your answer logically — begin with a definition or key statement, then provide supporting details. Review your answer to ensure it addresses all parts of the question completely.
Key Points to RememberAlways show your working steps clearly.
Verify your answer by substituting values back into the equation.
Practice similar problems from the textbook exercises.
Memorise important formulae and their conditions of applicability.
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Question 8

The mean of a set of data with 15 values is 134. Find the sum of the data.

Accepted Answer

Mean = 134
Total values = 15
Sum of data = 134 × 15 = 2010

Source: This question is from Tales by Dots and Lines, Mathematics &mdash; Class 8, CBSE Board.
Key Concepts CoveredThis question tests your understanding of the following concepts from the chapter Tales by Dots and Lines: Mean, Set, Data, Values, Sum, Tales by Dots and Lines. These are fundamental topics in Mathematics that students are expected to master as part of the CBSE Class 8 curriculum.A thorough understanding of these concepts will help you answer similar questions confidently in your CBSE examinations. These topics are frequently tested in both objective and subjective sections of Mathematics papers. We recommend revising the relevant section of your textbook alongside practising these solved examples to build a strong foundation.
How to Approach This QuestionRead the question carefully and identify what is being asked. Break down complex questions into smaller parts. Use the terminology and concepts discussed in this chapter. Structure your answer logically — begin with a definition or key statement, then provide supporting details. Review your answer to ensure it addresses all parts of the question completely.
Key Points to RememberAlways show your working steps clearly.
Verify your answer by substituting values back into the equation.
Practice similar problems from the textbook exercises.
Memorise important formulae and their conditions of applicability.
Related Questions from Tales by Dots and LinesQ1: Find the mean of the following data and share your observations: (i) The first 50 natural numbers....
Q10: The number of times students rode their cycles in a week is shown in the dot plot below. Four...
Q11: A dart-throwing competition was organised inaschool. The number of throws participants took to hit...
Q13: Small group project: Make a group of 3-4 members. Do at least one of the following: (i) Track daily...
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Question 9

Consider the data: 12, 47, 8, 73, 18, 35, 39, 8, 29, 25, p. Which of the following numbers could be p if the median of this data is 29?
(i) 10
(ii) 25
(iii) 40
(iv) 100
(v) 29
(vi) 47
(vii) 30

Accepted Answer

Arranging the data (excluding p)
8, 8, 12, 18, 25, 29, 35, 39, 47, 73
Total observations = 11 (odd)
Median = $\left(\frac{n+1}{2}\right)^{t h}$ term
= 6th term
= 29
So, p must be ≥ 29 to keep the value 29 in the middle.
So possible values of p are (iii) 40, (iv) 100, (v) 29, (vi) 47, and (vii) 30.

Question 10

The number of times students rode their cycles in a week is shown in the dot plot below. Four students rode their cycles twice in that week.
(i) Find the average number of times students rode their cycles.
(ii) Find the median number of times students rode their cycles.
(iii) Which of the following statements are valid? Why?
(a) Everyone used their cycle at least once.
(b) Almost everyone used their cycle a few times.
(c) Some students cycled more than once on some days.
(d) Exactly 5 students h

Accepted Answer

From the dot plot Data
(i) Sum = (0 × 3) + (1 × 1) + (2 × 4) + (3 × 7) + (4 × 7) + (5 × 5) + (6 × 4) + (7 × 6) + (8 × 3) + (10 × 2)
= 0 + 1 + 8 + 21 + 28 + 25 + 24 + 42 + 24 + 20
= 193
Total number of students = 42
Average = $\frac {193}{42}$ = 4.59 times
(ii) Total number of students = 42 (even)
So Median = average of 21st and 22nd terms
The 21st and 22nd terms lie in the cumulative frequency corresponding to 4.
So median = $\frac{4+4}{2}$ = 4
Median number of times students rode their cycles = 4
(iii) (a) Invalid
(b) Valid
(c) Valid
(d) Invalid
(e) Next week’s mean and median will be Mean = previous mean + 1
= 4.59 + 1
= 5.59
Median: Now the 21st and 22nd terms lie in the cumulative frequency corresponding to 5.
So Median = $\frac{5+5}{2}$ = 5

Question 11

A dart-throwing competition was organised inaschool. The number of throws participants took to hit the bull’s eye (the centre circle) is given in the table below. Describe the data using its minimum, maximum, mean, and median.

Accepted Answer

The minimum number of trials is 1.
The maximum number of trials is 10.
Mean = $\begin{gathered}
(1 \times 1)+(2 \times 0)+(3 \times 0)+(4 \times 1) \
+(5 \times 4)+(6 \times 9)+(7 \times 12)+ \
(8 \times 15)+(9 \times 10)+(10 \times 10) \
\hline 62
\end{gathered}$
= $\begin{gathered}
1+0+0+4+20+54+84 \
+120+90+100 \
\hline 62
\end{gathered}$
= $\frac {473}{62}$
= 7.6 trials
Median:
Median = average of $\left(\frac{62}{2}\right)^{t h}$ term and $\left(\frac{62}{2}+1\right)^{t h}$ term
= average of 31th and 32th terms
The 31st and 32nd terms lie in the cumulative frequency corresponding to 8.
5.2 Visualising and Interpreting Data
Figure It Out (Pages 122-123)

Question 12

Individual Project: Create your own activity strip for each day of the week.
(i) Do you eat and sleep at regular times every day? Typically, how long do you spend outdoors?
(ii) Calculate the average time spent per activity. Represent this average day using a strip.
(iii) Similarly, track the activities of any adult at home. Compare your data with theirs.

Accepted Answer

Collect and analyse your own data to answer the question.

Source: This question is from Tales by Dots and Lines, Mathematics &mdash; Class 8, CBSE Board.
Key Concepts CoveredThis question tests your understanding of the following concepts from the chapter Tales by Dots and Lines: Individual, Project, Create, Own, Activity, Strip. These are fundamental topics in Mathematics that students are expected to master as part of the CBSE Class 8 curriculum.A thorough understanding of these concepts will help you answer similar questions confidently in your CBSE examinations. These topics are frequently tested in both objective and subjective sections of Mathematics papers. We recommend revising the relevant section of your textbook alongside practising these solved examples to build a strong foundation.
How to Approach This QuestionRead the question carefully and identify what is being asked. Break down complex questions into smaller parts. Use the terminology and concepts discussed in this chapter. Structure your answer logically — begin with a definition or key statement, then provide supporting details. Review your answer to ensure it addresses all parts of the question completely.
Key Points to RememberAlways show your working steps clearly.
Verify your answer by substituting values back into the equation.
Practice similar problems from the textbook exercises.
Memorise important formulae and their conditions of applicability.
Related Questions from Tales by Dots and LinesQ1: Find the mean of the following data and share your observations: (i) The first 50 natural numbers....
Q10: The number of times students rode their cycles in a week is shown in the dot plot below. Four...
Q11: A dart-throwing competition was organised inaschool. The number of throws participants took to hit...
Q13: Small group project: Make a group of 3-4 members. Do at least one of the following: (i) Track daily...
Q14: The following graphs show the sunrise and sunset times across the year at 4 locations in India....
Practice more questions from Tales by Dots and Lines &mdash; Mathematics, Class 8 CBSE

Question 13

Small group project: Make a group of 3-4 members. Do at least one of the following:
(i) Track daily sleep time of all your family members for a week. Daily sleep time includes nighttime sleep, naps, and any sleep during the day.
(a) Represent this on strips.
(b) Put together the data of all your group members. Calculate the average and median sleep time of children, adults, elderly.
(c) Share your findings and observations.
(ii) When do schools start and end? On a weekday, Manoj’s school starts

Accepted Answer

Collect and analyse your own data to answer the question.

Source: This question is from Tales by Dots and Lines, Mathematics &mdash; Class 8, CBSE Board.
Key Concepts CoveredThis question tests your understanding of the following concepts from the chapter Tales by Dots and Lines: Small, Group, Project, Make, Members, Least. These are fundamental topics in Mathematics that students are expected to master as part of the CBSE Class 8 curriculum.A thorough understanding of these concepts will help you answer similar questions confidently in your CBSE examinations. These topics are frequently tested in both objective and subjective sections of Mathematics papers. We recommend revising the relevant section of your textbook alongside practising these solved examples to build a strong foundation.
How to Approach This QuestionRead the question carefully and identify what is being asked. Break down complex questions into smaller parts. Use the terminology and concepts discussed in this chapter. Structure your answer logically — begin with a definition or key statement, then provide supporting details. Review your answer to ensure it addresses all parts of the question completely.
Key Points to RememberAlways show your working steps clearly.
Verify your answer by substituting values back into the equation.
Practice similar problems from the textbook exercises.
Memorise important formulae and their conditions of applicability.
Related Questions from Tales by Dots and LinesQ1: Find the mean of the following data and share your observations: (i) The first 50 natural numbers....
Q10: The number of times students rode their cycles in a week is shown in the dot plot below. Four...
Q11: A dart-throwing competition was organised inaschool. The number of throws participants took to hit...
Q14: The following graphs show the sunrise and sunset times across the year at 4 locations in India....
Q15: We all know the typical sunrise and sunset timings. Do you know when the moon rises and sets? Does...
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Question 14

The following graphs show the sunrise and sunset times across the year at 4 locations in India. Observe how the graphs are organised. Are you able to identify which lines indicate the sunrise and which indicate the sunset?
Answer the following questions based on the graphs:
(i) At which place does the sun rise the earliest in January? What is the approximate day length at this place in January?
(ii) Which place has the longest day length over the year?
(iii) Share your observations — what do you

Accepted Answer

Yes, we can identify the lines.
(i) In January, the sunrise was earliest in Kibithu. (approx 6:00)
The day length is (16:00-6:00) = 10 hours
(ii) The place with the longest day length over the year is Ghuar Moti.
(iii) Observations are
Longest day length – Ghuar Moti
Shortest day length – Kanyakumari
Earliest sunrise in Jan – Kibithu
Latest sunset in Jan – Ghuar Moti
I am curious to find out the seasonal variation of sunrise and sunset, or day length, for any particular location.

Question 15

We all know the typical sunrise and sunset timings. Do you know when the moon rises and sets? Does it follow a regular pattern like the sun? Let’s find out. The following graph shows the moonrise and moonset times over a month:
(i) Find out on what dates amavasya (new moon) and purnima (full moon) were in this month.
(ii) What do you notice? What do you wonder?

Accepted Answer

(i) Amavasya – Day 21
Purnima – Day 7-8
(ii) Observation: The difference between moonrise and moonset changes throughout the month.
Around the middle of the month, the gap between moonrise and moonset is the maximum.
I wonder
Why do the moonrise and moonset times change every day?
Why is the time gap between moonrise and moonset maximum on some days and minimum on others?