NCERT Solutions for Class 7 Mathematics Ganita Prakash II Chapter 5

NCERT solutions for Class 7 Mathematics Chapter 5: Connecting the Dots – complete answers and explanations

Class 7 Mathematics from Ganita Prakash II brings students face to face with one of the most exciting branches of mathematics – statistics. Chapter 5, Connecting the Dots, is all about how we collect information, represent it visually, and use it to draw meaningful conclusions. From understanding what makes a question statistical, to calculating averages, finding medians, spotting outliers, and interpreting double bar graphs, this chapter is packed with real-life applications that make maths feel alive. Students learn how a single number like the mean or median can represent an entire group of values – a concept that is foundational for higher mathematics and everyday reasoning. This chapter also builds critical thinking skills as students learn to question data, compare it, and even spot when a representative value might be misleading. Download the worksheet and practice alongside solutions for better clarity. Whether your child is struggling with calculating averages or needs help interpreting dot plots and clustered bar graphs, these NCERT-aligned answers will guide them step by step. Book a free trial now to get expert guidance and help your child build real confidence in mathematics.

What this NCERT chapter covers?

1. The chapter introduces the concept of statistical statements and statistical questions – questions that require data collection and analysis to answer, such as "How tall are Grade 7 students?" or "Where are onions costlier – Yahapur or Wahapur?"

2. Students learn the difference between a statistical question (which has varying answers and needs data) and a non-statistical question (which has a fixed, single answer).

3. The concept of the Arithmetic Mean (Average) is taught in depth. Students learn the formula: Mean = Sum of all values divided by the number of values, and apply it across multiple real-life contexts.

4. The idea of "Average as Fair Share" is explored through hands-on examples such as sharing guavas equally among group members, making the abstract concept of mean feel concrete and intuitive.

5. The chapter introduces Outliers – values that are significantly different from the rest of the data – and explains how outliers affect the mean more than the median.

6. Students learn the Median as an alternative representative value, including how to find it for both odd and even number of data values, and when it is a better choice than the mean.

7. Dot plots are introduced as a tool for data visualisation, helping students see how data is clustered, spread, or skewed, and how to identify minimum, maximum, and central values visually.

8. The chapter covers Double Column Graphs (also called Clustered Column Graphs or Double Bar Graphs), teaching students how to read, interpret, and construct them to compare two datasets side by side.

9. Students practise using a two-step process for reading graphs: Step 1 – Identify what is given (scale, organisation, patterns), and Step 2 – Infer from what is given (conclusions, comparisons).

10. The chapter closes with the Data Detective section, where students examine real height data from India across different years, drawing meaningful statistical conclusions and justifying or rejecting statements using data.

11. The chapter also includes Individual Projects and Small-Group Projects, encouraging students to collect real-world data and apply statistical tools to it.

12. Key terms and their meanings are reinforced throughout – range, variability, central tendency, representative value – all in the context of real and relatable situations.

How to use these NCERT solutions?

1. Before reading the answers, always attempt every question on your own first. Give yourself honest effort, even if you are not sure of the method – the process of trying activates your thinking.

2. Once you have attempted a question, compare your working step by step with the solution provided here. Do not just check the final answer – check every line of reasoning.

3. If your answer matches, move on. If it does not, read the solution carefully to understand exactly where your thinking differed and why the given approach is correct.

4. For activity-based questions (like the bat-and-ball activity or the flowering plant activity), the solutions provide explanations of what you should do. Use them as a guide to carry out the task correctly.

5. For open-ended or student-generated questions, the answer listed here is "Answers may vary." This means your personal response is valid as long as you have applied the correct method.

6. Parents and teachers can use these solutions to verify student work and to guide discussions on why certain approaches – such as using median instead of mean when there is an outlier – give more useful results.

7. All answers in this blog follow the exact order of the worksheet, section by section, so it is easy to locate any specific question quickly.

8. Use this resource alongside your worksheet for maximum benefit – refer to it for checking, not for copying, so that your understanding stays strong.

Important tips and tricks for students

1. Always remember: a statistical question expects varied answers that need data collection. A question like "What is the capital of India?" is not statistical. A question like "How many hours do students in your class sleep?" is statistical.

2. When calculating the mean, be careful with "Did not play" or absent entries. Only count the matches or days that actually happened – a dash (—) is not a zero. Zero means the value was zero; a dash means it does not exist.

3. To find the median, always sort the data in ascending order first. For an odd number of values, the median is the exact middle value. For an even number of values, the median is the average of the two middle values.

4. If a dataset has an outlier (a value that is very far from the rest), the mean gets pulled towards that outlier. The median stays stable. So if you see an extreme value in the data, the median is usually the better representative.

5. When asked to compare two groups, always compare using the same measure – mean vs. mean, or median vs. median. Comparing the total of one group to the mean of another is not valid.

6. In dot plots, each dot represents one data value. If two values are the same, two dots appear stacked above the same point. Use the dot plot to quickly spot the minimum, maximum, and where most data is clustered.

7. In a double bar graph, always check the scale used on the vertical axis and read it carefully. A common mistake is misreading the scale and estimating values incorrectly.

8. For statements based on data, always justify your answer with specific numbers from the data. Simply writing "True" or "False" without justification will not earn full marks in exams.

9. When the mean and median are close to each other, the data is balanced with no extreme outliers. When mean is less than median, there is a low outlier. When mean is greater than median, there is a high outlier.

10. For project questions, follow the given steps carefully – collect data, record it, make a dot plot, and calculate mean and median. Partial steps without the final calculations will lose marks.

NCERT solutions – complete answer key

Which of the following are statistical questions?

Explanation:
A statistical question is one where the answers vary and data must be collected to understand the pattern.

Answer:
Statistical questions:
(b) How old are the dogs that live on this street?
(c) What fraction of the students in your class like walking up a hill?
(g) What was the rainfall pattern in Barmer last year?

Representative values

What do you think of Vaishnavi's statement (that Shubman performed better since his total is 110)?

Vaishnavi's statement is not appropriate because Yashasvi played only 4 matches while Shubman played 5. The total alone is not a fair comparison when the number of matches is different. We should use the average (mean) to compare.

Can a single number represent a group of numbers?

Yes. A single number like the average (arithmetic mean) can act as a representative of a group of numbers. For example, Shubman's batting in the second series can be represented by his average of 21 runs per match, and Yashasvi's by 24 runs per match.

Figure it Out

1.
Shreyas's bounces: 6, 2, 9, 5, 4, 6, 3, 5
Average = (40 ÷ 8) = 5

2.
Explanation: Students should perform the bat-and-ball activity themselves for 7 or more attempts, record the number of bounces each time, find the total, and divide by the number of attempts to find their own average.

3.
Try This – Flowering plant activity
Explanation: Students should identify a flowering plant in their neighbourhood, track the number of flowers blooming each day over a week, find the total, and divide by 7 (number of days) to get the average number of flowers per day.

4.
Nikhil: (17+18+17+16+19+17+18) ÷ 7 = 17.43
Sunil: (20+18+18+17+16+16+17) ÷ 7 = 17.43
Answer: Both Nikhil and Sunil have the same average time of approximately 17.43 seconds. Neither ran quicker on average; their average speeds are equal.

5.
Mean enrolment during six consecutive years:
Data: 1555, 1670, 1750, 2013, 2040, 2126
Total = 1555 + 1670 + 1750 + 2013 + 2040 + 2126 = 11154
Number of years = 6
Mean = 11154 ÷ 6 = 1859
Answer: The mean enrolment in the school during this period is 1859 students.

Math Talk – Know Your Onions!

Find the average price of onions at Yahapur and Wahapur

Yahapur prices: 25, 24, 26, 28, 30, 35, 39, 43, 49, 56, 59, 44
Total = 25 + 24 + 26 + 28 + 30 + 35 + 39 + 43 + 49 + 56 + 59 + 44 = 458
Number of months = 12
Mean (Yahapur) = 458 ÷ 12 = 38.17 (approx.) rupees per kg

Wahapur prices: 19, 17, 23, 30, 38, 35, 42, 39, 53, 60, 52, 42
Total = 19 + 17 + 23 + 30 + 38 + 35 + 42 + 39 + 53 + 60 + 52 + 42 = 450
Number of months = 12
Mean (Wahapur) = 450 ÷ 12 = 37.5 rupees per kg

Answer: The average price of onions in Yahapur is approximately Rs. 38.17 per kg and in Wahapur is Rs. 37.5 per kg. Yahapur has a slightly higher average price.

Math Talk – Dot Plot Discussion

Does this visualisation capture all the data presented in the tables?

The dot plot captures all the data values but loses the month-wise (sequential) order of the data. We can see how many times each price appears, but we cannot identify which price belongs to which month.

Can we tell the price of onions in Yahapur in January from the dot plot?

No. The dot plot shows only the values and their frequency, not which month each value belongs to. We cannot tell that 25 corresponds to January.

Can you think of any other ways to compare the data?

Yes. Other ways include:
- Comparing the range (difference between maximum and minimum)
- Comparing month-by-month prices
- Comparing how many months each place had higher prices
- Using the median

What else do you wonder about?

Students may discuss things like: Why do onion prices rise in certain months? Are prices affected by rainfall or harvest seasons? How do price changes affect farmers and consumers? Students should discuss with peers, teachers, or family.

Math Talk – Height of a Family

Find the average height of each family. Can we say Yaangba's family is taller?

Yaangba's family: 169, 173, 155, 165, 160, 164
Total = 169 + 173 + 155 + 165 + 160 + 164 = 986
Number of members = 6
Mean = 986 ÷ 6 = 164.33 cm (approx.)

Poovizhi's family: 170, 173, 165, 118, 175
Total = 170 + 173 + 165 + 118 + 175 = 801
Number of members = 5
Mean = 801 ÷ 5 = 160.2 cm

Answer: Yaangba's family has a higher average height (164.33 cm) compared to Poovizhi's family (160.2 cm). However, we cannot confidently say Yaangba's family is taller because in Poovizhi's family, 4 out of 5 members are taller than 164.33 cm. The low average for Poovizhi's family is due to one young child (118 cm), who is an outlier. The median would be a better representative here.

Can you think of a number that can represent the data better?

Yes. The Median is a better representative when there are outliers. The median of Poovizhi's family is 170 cm, which better reflects the heights of most family members.

Find mean and median of Poovizhi's data without the outlier 118

Data without outlier: 165, 170, 173, 175
Total = 165 + 170 + 173 + 175 = 683
Mean = 683 ÷ 4 = 170.75 cm
Median: Sorted data = 165, 170, 173, 175 – two middle values = 170 and 173
Median = (170 + 173) ÷ 2 = 171.5 cm

Answer: Without the outlier, the mean rises from 160.2 cm to 170.75 cm and the median rises from 170 cm to 171.5 cm. Both values now more accurately represent the family's heights. The change in mean is more significant than the change in median, showing that the mean is more affected by outliers.

Are you a bookworm?

Data from the pieces of paper (as shown in the picture):
6, 8, 5, 15, 3, 0, 2, 7, 12, 40, 0, 8, 5, 1, 10

Sorted data: 0, 0, 1, 2, 3, 5, 5, 6, 7, 8, 8, 10, 12, 15, 40
Number of values = 15
Total = 0 + 0 + 1 + 2 + 3 + 5 + 5 + 6 + 7 + 8 + 8 + 10 + 12 + 15 + 40 = 122
Mean = 122 ÷ 15 = 8.13 (approx.)
Median = 8th value in sorted data = 6

Answer:
Mean = approximately 8.13 stories
Median = 6 stories

Before calculating: Since 40 is a very high outlier at the upper end, the mean will be greater than the median.
Answer confirmed: Mean (8.13) > Median (6)

The median value of 6 means that half of the class members have read 6 or more stories.

Which value is an outlier?

40 is the outlier (it is far higher than all other values in the data).

Find the mean and median without the outlier 40:

Data without 40: 0, 0, 1, 2, 3, 5, 5, 6, 7, 8, 8, 10, 12, 15
Total = 82
Number of values = 14
Mean = 82 ÷ 14 = 5.86 (approx.)
Median = average of 7th and 8th values = (5 + 6) ÷ 2 = 5.5

Answer: Without the outlier, the mean drops from 8.13 to approximately 5.86. The median changes from 6 to 5.5. The mean is more affected by removing the outlier than the median, confirming that the median is a more stable measure in the presence of outliers.

Are We on the Same Page?

Data (pages Mon–Sun): 16, 18, 20, 22, 26, 16, 10
Total = 16 + 18 + 20 + 22 + 26 + 16 + 10 = 128
Number of values = 7
Mean = 128 ÷ 7 = 18.29 (approx.)

Sorted data: 10, 16, 16, 18, 20, 22, 26
Median = 4th value = 18

Answer:
Mean = approximately 18.29 pages
Median = 18 pages

Note: The outlier 10 (Sunday, fewer pages) is at the lower end, pulling the mean slightly below the median. Mean and median are very close in this case, as the data is relatively balanced.

Math Talk – Observe variability in the three examples

(a) Mean and median close to each other: Yaangba's family heights – no extreme outlier, data is balanced.
(b) Mean < Median: Poovizhi's family heights – outlier at the lower end (118 cm) pulls the mean downward.
(c) Mean > Median: Short stories read – outlier at the higher end (40 stories) pulls the mean upward.

Math Talk – Discuss the effect on mean and median when outliers are present on both sides

When outliers are present on both sides (one very high, one very low), their effects on the mean may cancel each other out, keeping the mean close to the median. However, if one outlier is more extreme than the other, the mean will shift in that direction. The median remains largely unaffected because it depends only on the middle value(s), not the extreme values.

How Tall is Your Class?

Math Talk – How many students are taller than the class' average height (Mean = 144.4)?

From the data:
Boys: 147, 135, 130, 154, 128, 135, 134, 158, 155, 146, 146, 142, 140, 141, 144, 145, 150
Girls: 143, 136, 150, 144, 154, 140, 145, 148, 156, 150, 150

Combined class = 28 students
Students taller than 144.4 cm:
Boys above 144.4: 147, 154, 158, 155, 146, 146, 145, 150 – 8 boys
Girls above 144.4: 150, 154, 145, 148, 156, 150, 150 – 7 girls
Total = 15 students

Answer: 15 students are taller than the class average height of 144.4 cm.

How many boys are taller than the class' average height (144.4 cm)?

8 boys are taller than 144.4 cm.

Math Talk – How Long is a Minute? (Group A and Group B dot plot discussion)

Both groups performed well at estimating 1 minute.

Group A: Mean = 58.21 seconds, Median = 60 seconds.
Since Mean < Median, the data has some lower outliers pulling the mean down. A few students opened their eyes quite early (around 40–45 seconds), making the mean lower than the median.

Group B: Mean = 59.28 seconds, Median = 59.5 seconds.
Mean and median are very close to each other, suggesting Group B's data is more balanced with fewer extreme outliers. Group B's estimates are clustered more tightly around 60 seconds.

Overall, Group B performed slightly better as their estimates are closer to 60 seconds and more consistent.

Zero vs No Value

A score of 0 means the player played the match but scored zero runs. A dash (–) or "Did not play" means the player was absent and that match should not be counted in the total number of matches when calculating the average. Only matches actually played are counted in the denominator.

Figure it Out

1.
Yahapur prices sorted: 24, 25, 26, 28, 30, 35, 39, 43, 44, 49, 56, 59
Number of values = 12 (even)
Two middle values = 6th and 7th = 35 and 39
Median (Yahapur) = (35 + 39) ÷ 2 = 37 rupees per kg

Wahapur prices sorted: 17, 19, 23, 30, 35, 38, 39, 42, 42, 52, 53, 60
Number of values = 12 (even)
Two middle values = 6th and 7th = 38 and 39
Median (Wahapur) = (38 + 39) ÷ 2 = 38.5 rupees per kg

Answer: Median price in Yahapur = Rs. 37 per kg; Median price in Wahapur = Rs. 38.5 per kg.

2.
Sanskruti's class – domestic animals and pets data:
Data: 0, 1, 0, 4, 8, 0, 0, 2, 1, 1, 5, 3, 4, 0, 0, –, 10, 25, 2, –, 2, 4
(Two students were absent, so – values are excluded)
Valid data: 0, 1, 0, 4, 8, 0, 0, 2, 1, 1, 5, 3, 4, 0, 0, 10, 25, 2, 2, 4
Number of values = 20
Total = 0+1+0+4+8+0+0+2+1+1+5+3+4+0+0+10+25+2+2+4 = 72
Mean = 72 ÷ 20 = 3.6

Sorted data: 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 8, 10, 25
Number of values = 20 (even)
Two middle values = 10th and 11th = 2 and 2
Median = (2 + 2) ÷ 2 = 2

Answer:
Mean = 3.6 animals
Median = 2 animals
Description: The data has an outlier at the higher end (25 animals), which pulls the mean (3.6) above the median (2). Most students have 0 to 4 animals. The median of 2 is a better representative of the typical number of animals per household. Since mean > median, the data is skewed to the right due to the high outlier.

3.
For the median, arrange the heights in ascending order:
43, 44, 45, 46, 49, 50, 51, 52, 52, 54, 54, 55, 55, 56, 56, 57, 58, 59, 60, 60, 60, 60, 61, 61, 62, 63, 65, 66, 67
Total number of trees = 29
For odd number 29, the median is the middle value, which is the 15th term.
So, median = 56

Sum of heights = 43 + 44 + 45 + 46 + 49 + 50 + 51 + 52 + 52 + 54 + 54 + 55 + 55 + 56 + 56 + 57 + 58 + 59 + 60 + 60 + 60 + 60 + 61 + 61 + 62 + 63 + 65 + 66 + 67 = 1621
Mean = 1621 ÷ 29 = 55.89

The height of the date palm trees ranges from 43 to 67 feet, with most trees clustered around 55–60 feet. The median height is 56 feet, indicating that half of the trees are shorter than 56 feet and half are taller. The mean height is approximately 55.89 feet. 13 trees are shorter than the average height.

4.
Water usage:
(a) No, mean/median cannot be 25–30.
(b) Mean/median always lies between min and max.

5.
The weight of boys is between 2.6 kg and 4.1 kg.
The weight of girls lies between 2.5 kg and 4 kg.
The heaviest baby is a boy, and the lightest is a girl.

6.
Grade 5 (other section):
Whole class mean = 141.21, median = 142.5
Boys mean = 142.05, median = 143
Girls mean = 140.14, median = 140
Answer: Boys taller on average in this section.

7.
Sumo vs Ballet:
Average sumo = approximately 240 kg, ballet = approximately 42 kg
Ratio = approximately 6 times heavier.

Section 5.3 – Visualising data

What is the scale used in the double column graph of onion prices?

The scale used is 1 unit = 10 rupees (each grid line represents 10 rupees on the vertical axis, going from 0 to 60).

Is it now easier to compare month-wise prices in both places?

Yes. The double column (clustered) graph makes it easy to directly compare Yahapur and Wahapur prices for each month side by side. The relative heights of the bars immediately show which place had higher prices in any given month.

All it Takes is a Minute

1.
No, we cannot clearly tell which team batted first. We also cannot confidently determine who won the match from this graph alone.

2.
The blue team scored 10 runs in over 12.

3.
The red team scored the least runs in over 8.

4.
No, it is not easy to tell the target directly from this graph because it shows runs per over but not the total score clearly.

Figure it Out

1.
a. The scale used is 10 km/h per division.
b. Sample Answer:
It is interesting to see how differently animals move in air, land, and water. For example, birds like the peregrine falcon are extremely fast in the air, while animals like the cheetah are the fastest on land.
c. Sample Answer:
The cheetah (approximately 100 km/h) is about twice as fast as a lion (approximately 50 km/h).
d. Yes, a sailfish is roughly four times faster than a humpback whale. However, we cannot say that the sailfish is the fastest aquatic animal in the world, because the infographic only shows a few animals and does not include all aquatic animals.

2.
Answer:
Grade 5 – Aquatic: 6, Aerial: 13, Spaceborne: 2, None: 4
Grade 9 – Aquatic: 5, Aerial: 8, Spaceborne: 9, None: 2

Scale: 1 unit = 2 students (appropriate scale)

Explanation for drawing: Draw a double-bar graph with 4 groups on the horizontal axis (Aquatic, Aerial, Spaceborne, None). For each group, draw two bars side by side – one for Grade 5 (e.g., blue) and one for Grade 9 (e.g., red) – using the scale 1 unit = 2 students. Label axes and add a legend.

Observations:
- Aerial is the most popular choice in Grade 5 (13 students).
- Spaceborne is the most popular in Grade 9 (9 students).
- Fewer Grade 9 students chose None compared to Grade 5.
- Grade 5 students prefer Aerial over Spaceborne, while Grade 9 students show the opposite trend.

3.
(a) Marking bars for Gujarat and Delhi:
Gujarat: 2022 = 69,000; 2023 = 89,000; 2024 = 78,000
Delhi: 2022 = 62,000; 2023 = 74,000; 2024 = 81,000
Explanation: Place the top of each bar between the appropriate vertical guidelines (25,000; 50,000; 75,000; 100,000). Gujarat 2022 bar top should fall between 50,000 and 75,000 (near 69,000), Gujarat 2023 near 89,000 (between 75,000 and 100,000), Gujarat 2024 near 78,000. Delhi 2022 near 62,000, Delhi 2023 near 74,000, Delhi 2024 near 81,000.

b. Explanation:
The graph compares vehicle registrations across different states and years.

Answer:
The X-axis shows states.
The Y-axis shows number of registrations (in thousands).
The scale increases at equal intervals.

c. Most states show a steady increase in registrations from 2022 to 2024. Some states show a large rise, while others show only a small increase.

d. Assam received approximately 10,000 more registrations in 2023 compared to 2022 (approximate value from the graph).

e. The registrations increased to about 1.5 times the 2022 value (approximate comparison from the graph).

f. No, the statement is not correct.

4.
Guess of months:
Answer: Day 1 shows maximum temperature of 34°C and minimum of 16°C – this is likely a winter month (December or January in Jodhpur). Day 2 shows maximum of 43°C and minimum of 30°C – this is likely a summer month (May or June in Jodhpur).

Section 5.4 – Data Detective

Math Talk – Dot plots of School A and School B (Grades 6, 7, 8):

School A – Means:
Grade 6 Boys: 134.8, Grade 6 Girls: 137.78
Grade 7 Boys: 141.8, Grade 7 Girls: 141.83
Grade 8 Boys: 149.35, Grade 8 Girls: 147.81

School B – Means:
Grade 6 Boys: 149.84, Grade 6 Girls: 150.2
Grade 7 Boys: 156.14, Grade 7 Girls: 155.41
Grade 8 Boys: 156.14, Grade 8 Girls: 156.83

Observations:
- Students in School B are considerably taller than students in the same grades in School A.
- In School A, girls are slightly taller than boys in Grades 6 and 7, while in Grade 8 boys are slightly taller.
- In School B, the heights of boys and girls in the same grade are very similar.
- Heights increase with grade in both schools.
- The large difference between schools could be due to geographic location, nutrition, or socioeconomic factors.

Which of the following statements can be justified using the data?

1. The average heights of both boys and girls at every age increased from 1989 to 2019.
Answer: TRUE. Looking at the table, for every age from 5 to 19, both boys' and girls' heights in 2019 are higher than in 1989.

2. The average height of 13-year-old girls in 1989 is more than the average height of 14-year-old girls in 2009.
Answer: 13-year-old girls in 1989 = 143.2 cm; 14-year-old girls in 2009 = 148 cm.
Answer: FALSE. 143.2 cm < 148 cm. The statement cannot be justified.

3. The average height of 15-year-old boys in 2019 is more than the average height of 16-year-old boys in 1989.
Answer: 15-year-old boys in 2019 = 159 cm; 16-year-old boys in 1989 = 158.9 cm.
Answer: TRUE. 159 cm > 158.9 cm. The statement can be justified (by a very small margin).

4. All girls aged 13 are taller than all girls aged 11.
Answer: FALSE. The table shows averages, not individual heights. There will be individual girls aged 13 who are shorter than some girls aged 11. We cannot make this claim about all individuals from average data.

5. Throughout the age period 5 to 19, the average boy's height is more than the average girl's height.
Answer: Looking at the 2019 data (as an example): At ages 5–9, boys are slightly taller. At ages 10–12, girls are slightly taller (e.g., age 11: boys 137, girls 138.6; age 12: boys 142.2, girls 143.8).

6. Boys keep growing even beyond age 19.
Answer: CANNOT BE FULLY JUSTIFIED from this data. The data only extends to age 19 and shows near-plateau growth. The trend suggests growth slows significantly. We cannot fully justify this from the given data alone, though it is biologically possible.

Math Talk – In 2019, between which two successive ages did boys grow the most? Girls?

Boys' heights in 2019 (ages 5 to 19):
107.1, 113.1, 118.6, 123.5, 128.1, 132.6, 137.0, 142.2, 148.4, 154.4, 159.0, 162.3, 164.6, 166.0, 166.5

Differences between successive ages:
5–6: 6.0; 6–7: 5.5; 7–8: 4.9; 8–9: 4.6; 9–10: 4.5; 10–11: 4.4; 11–12: 5.2; 12–13: 6.2; 13–14: 6.0; 14–15: 4.6; 15–16: 3.3; 16–17: 2.3; 17–18: 1.4; 18–19: 0.5

Largest growth for boys: Between ages 12 and 13 (difference = 6.2 cm)

Girls' heights in 2019:
107.2, 112.9, 118.0, 122.7, 127.6, 132.8, 138.6, 143.8, 147.7, 150.4, 152.4, 153.8, 154.7, 155.2, 155.2

Differences:
5–6: 5.7; 6–7: 5.1; 7–8: 4.7; 8–9: 4.9; 9–10: 5.2; 10–11: 5.8; 11–12: 5.2; 12–13: 3.9; 13–14: 2.7; 14–15: 2.0; 15–16: 1.4; 16–17: 0.9; 17–18: 0.5; 18–19: 0.0

Largest growth for girls: Between ages 10 and 11 (difference = 5.8 cm)

Answer: In 2019, boys grew the most between ages 12 and 13. Girls grew the most between ages 10 and 11.

Math Talk – Estimate height of newborn (50 cm) and ages 1–4

Answer: If the average height of a newborn is 50 cm, we can estimate:
- Age 1: approximately 72–75 cm
- Age 2: approximately 83–86 cm
- Age 3: approximately 92–95 cm
- Age 4: approximately 98–102 cm
(These are estimates based on known growth patterns and the data showing age 5 heights around 107 cm.)

Math Talk – Estimate heights for 2029 based on trend

The data shows a consistent increase in average heights from 1989 to 2019 (roughly 0.3–0.5 cm increase per age per decade). Based on this trend, heights in 2029 would be approximately 1–2 cm taller than 2019 values at each age for both boys and girls. Students should add approximately 1 cm to the 2019 values as an estimate for 2029.

How is the graph organised? What information is presented?

The horizontal axis lists various countries from left to right (roughly from shorter to taller average heights). The vertical axis shows height in cm, starting from 145 cm. For each country, four data points are shown: B-1989 (boys' average height in 1989), B-2019 (boys' average height in 2019), G-1989 (girls' average height in 1989), and G-2019 (girls' average height in 2019). This allows comparison of how heights have changed over 30 years across different countries.

What do you find interesting?

Answers may vary.

Figure it Out

1.
a. True
Explanation: The boys' dot plot spreads across a wider range of pocket numbers.

b. True
Explanation: The middle value for boys lies further to the right compared to girls.

c. False
Explanation: Most girls have fewer pockets, so the mean is smaller.

d. True
Explanation: The highest dot for boys is farther to the right.

2.
a. Mean = (Sum of points) / (Number of games)
Suppose A scored: 10, 8, 6, 4
Total = 28
Mean = 28 ÷ 4 = 7

b. Divide by 3.
Explanation: C played only 3 games (one game marked "Did not play"). So mean must be calculated using only the games played.

c. The player with the highest average score.
From comparison, Player A performs best.

3.
Group 1: Number of students in first group = 10
Sum of marks of first group = 85 + 76 + 90 + 85 + 39 + 48 + 56 + 95 + 81 + 75 = 730
Mean = 730 ÷ 10 = 73

Arranging marks in ascending order:
39, 48, 56, 75, 76, 81, 85, 85, 90, 95
Since there are 10 scores (an even number), the median is the average of the 5th and 6th scores.
Median = (76 + 81) ÷ 2 = 78.5

Group 2: Number of students in the second group = 9
Sum of marks of second group = 68 + 59 + 73 + 86 + 47 + 79 + 90 + 93 + 86 = 681
Mean = 681 ÷ 9 = 75.67

Arranging the scores in ascending order: 47, 59, 68, 73, 79, 86, 86, 90, 93.
Since there are 9 scores (an odd number), the median is the 5th score.
Median = 79

4.
Observations:
1. Cricket is the most watched sport (1240) but participation is relatively low (320).
2. Basketball and Swimming have balanced watching and participation numbers.
3. Athletics has the least participation (105) compared to watching (510).
4. Hockey shows moderate watching (320) and participation (250).
5. Overall, watching is far more popular than participating across all sports.

5.
17 is an odd number that cannot be divided into two perfectly equal groups.
Arranging the heights of students in increasing order:
101, 102, 106, 109, 110, 110, 112, 115, 115, 115, 115, 115, 117, 120, 120, 123, 125
Median = 9th term = 115 cm

According to the conditions given in question, the height used to divide the students into two groups is 112 < height < 115.
If we take 114 cm as the required height, then
Group 1 (Less than 114 cm) – 101, 102, 106, 109, 110, 112 – Total = 7 Students
Group 2 (More than 114 cm) – 115, 115, 115, 115, 115, 117, 120, 120, 123, 125 – Total = 10 students.

6.
To analyse the heights of students in a class, we can use measures of central tendency such as the mean and median.

7.
(d) The mean height of students in the other section cannot be determined.
Explanation: Each section has the same number of students (30 students: 15 boys + 15 girls). However, knowing the mean height of students in only one section does not give any information about the heights of students in the other section.

8(a).
Estimated values for the number of skyscrapers in:
New York – 38
Tokyo – 160
London – 305

8(b).
(i) Only 12 cities have more skyscrapers than Mumbai (86).
Answer: Cities with more than 86 skyscrapers: Hong Kong (553), Shenzhen (367), New York (approximately 300), Dubai (251), Guangzhou (188), Shanghai (183), Tokyo (approximately 165), Kuala Lumpur (154), Chongqing (144), Jakarta (112), Bangkok (110), Singapore (95) – 12 cities.
Statement is TRUE.

(ii) Only 7 cities have fewer skyscrapers than Mumbai (86).
Answer: Cities with fewer than 86 skyscrapers: Seoul (82), Toronto (81), Melbourne (69), Miami (58), Istanbul (48), Moscow (46), London (approximately 20) – 7 cities.
Statement is TRUE.

(iii) The tallest building in the world is in Hong Kong.
Answer: FALSE. Hong Kong has the most skyscrapers (buildings taller than 150m), but this does not mean it has the tallest building in the world. Having the most skyscrapers (taller than 150m) is different from having the single tallest building. The tallest building in the world is the Burj Khalifa in Dubai (828 m). The infographic shows the number of buildings taller than 150m, not which city has the tallest individual building.

9.
Sum of differences = 0.5 + (-0.6) + 1.5 + (-2.4) + 0.5 = -0.5
Average difference = -0.5 ÷ 5 = -0.1 cm
Estimated values are 0.1 cm less than actual values on average.

10(a).
The dot plot showing Week 1 and Week 2 data for Aditi's Sudoku solving times is constructed from the data provided.

10(b).
Mean = Sum of values / Total number of values
410 + 400 + 370 + 340 + 360 + 400 + 320 + 330 + 310 + 320 + 290 + 380 + 280 + 270 + 230 + 220 + 240
5470 ÷ 17 = 321.76 approximately 321 seconds.

Arranging in ascending order:
220, 230, 240, 270, 280, 290, 310, 320, 320, 330, 340, 360, 370, 380, 400, 400, 410
With 17 values, the median is the 9th value.
So median = 320 seconds.

Observations:
Improvement in speed: Week 2 times are consistently lower, showing Aditi's practice is paying off.
Median shift: The median dropped from 360 seconds (Week 1) to 240 seconds (Week 2), highlighting faster average performance.

Individual Project – (open-ended activities)

(a) How Long is a Sentence?
Explanation: Students should pick two textbooks from different subjects, choose one page with a lot of text from each, count the number of words in every sentence on each page, record the data, make a dot plot for each page, and calculate the mean and median number of words per sentence for each page. Compare the two pages and describe differences in sentence length variability and central tendency.

(b) What is in a Name?
Explanation: Students should list all classmates' names, count the number of letters in each name (name length), then:
(i) Calculate mean and median name length.
(ii) Draw a dot plot of name lengths. Describe variability (minimum, maximum, range) and central tendency (mean and median).
(iii) Identify which starting letters are most and least common.
(iv) Find the median starting letter – this indicates roughly equal numbers of names start with letters in the first half (A–M) and the second half (N–Z) of the alphabet.
(v) Draw a double-bar graph categorising boys' and girls' names by vowel/consonant start and end patterns.

12. Individual Project (Long Term) – In and Out

Explanation: Over one month, students should track how many times they step out of the house each day and record this daily. After 30 days:
(i) Make a dot plot and calculate mean and median. Describe variability (are some days much more active than others?).
(ii) Look for interesting patterns (e.g., more outings on weekends, fewer on school days, seasonal variation).
(iii) Optionally, collect data from family members and compare patterns.

13. Small-Group Project

(a) Our heights vs. family's heights:
(i) Explanation: Each student collects heights of all family members, makes a dot plot, calculates mean and median, and describes variability and central tendency.
(ii) Explanation: Draw a double-bar graph showing each student's own height next to their family's mean height.
(iii) Share observations with the group – e.g., which students are taller than their family mean, which are shorter.

(b) Estimating time:
(i) Explanation: Each student (and family members) closes their eyes and opens after what they think is 1 minute or 3 minutes (no counting). The actual time elapsed is recorded. Make two dot plots – one for 1-minute estimates, one for 3-minute estimates.
(ii) Mark the estimates on the dot plot. Calculate mean and median. Describe variability.
(iii) Draw a double-bar graph showing each family's mean 1-minute and mean 3-minute estimates.
(iv) Compare estimates across students – who is better at estimating 1 minute vs 3 minutes? Are the estimates for 3 minutes roughly 3 times the 1-minute estimates?

Puzzle Time – Connect the Dots

The final 3-digit code is 415.
Here's the reasoning in plain words:
Clue 3 told us digits 0, 3, and 6 are not in the code.
Clue 1 showed that 5 is the last digit.
Clue 4 proved that 4 is in the code but not at the end.
Clue 2 showed that 1 is in the code but not in the last spot.
Clue 5 confirmed that 4 and 5 are correct but in different positions.
So the only arrangement that fits all clues is 415.

Why NCERT solutions help students?

Clear and reliable NCERT solutions like the ones in this blog are built around the exact questions and content of the textbook and worksheet, helping students understand not just the answer but the reasoning behind it. For Chapter 5 – Connecting the Dots, where the focus is on data analysis, averages, medians, outliers, and graphs, having step-by-step working helps students identify where they went wrong and what the correct approach looks like. Practising with NCERT-aligned answers also builds exam readiness, because school assessments and competitive tests alike draw heavily from this material. When students understand the logic – not just the formula – their confidence grows, their marks improve, and their curiosity about mathematics naturally deepens.

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