NCERT Solutions Class 7 Mathematics Ganita Prakash 2 Chapter 05 Connecting the Dots

Question 1. Shreyas is playing with a bat and a ball - but not cricket. He counts the number of times he can bounce the ball on the bat before it falls to the ground. The data for 8 attempts is 6, 2, 9, 5, 4, 6, 3, 5. Calculate the average number of bounces of the ball that Shreyas can make with his bat.
Answer: The sum of all bounces across the 8 attempts is 40. When you divide this total by 8 attempts, you get 5 bounces on average. So Shreyas manages an average of 5 bounces with his bat.
In simple words: Add up all the bounces: 6 + 2 + 9 + 5 + 4 + 6 + 3 + 5 = 40. Then divide by 8 to get the average = 5 bounces.

Exam Tip: Always add all values first, then divide by the count of values to find the average. Make sure you count how many values there are - a common mistake is dividing by the wrong number.

Question 2. Try the activity above on your own. Collect data for 7 or more attempts and find the average.
Answer: This is a hands-on task to be completed by you. Perform the ball-bouncing activity yourself at least 7 times and record how many bounces you achieve each time. Add all your bounce counts together and divide by the number of attempts to find your own average.
In simple words: Do the bounce activity yourself. Write down how many times the ball bounces each try. Add all the numbers and divide by how many tries you did.

Exam Tip: Keep your data neat and organized. Show your addition and division clearly so the examiner can follow your working.

Question 3. Identify a flowering plant in your neighbourhood. Track the number of flowers that bloom every day over a week during their flowering season. What is the average number of flowers that bloom per day?
Answer: This is a field observation task for you to complete. Choose a flowering plant nearby and count the flowers that open each day for seven days during its active blooming time. Record these numbers, add them all together, and divide by 7 to find the daily average.
In simple words: Pick a flower plant. Count new flowers every day for a week. Add the counts and divide by 7.

Exam Tip: Be consistent with your counting method each day. Count at the same time if possible to ensure fair comparison across days.

Question 4. Two friends are training to run a 100 m race. Their running times over the past week are given in seconds - Nikhil: 17, 18, 17, 16, 19, 17, 18; Sunil: 20, 18, 18, 17, 16, 16, 17. Who, on average, ran quicker?
Answer: Nikhil's total time across 7 days is 122 seconds, giving an average of 17.43 seconds per run. Sunil's total time is also 122 seconds, resulting in the same average of 17.43 seconds per run. Both athletes have the same average running speed, so neither is quicker than the other on average.
In simple words: Both friends take the same average time to complete the 100 m race - about 17.43 seconds each. They are equally fast.

Exam Tip: When comparing averages, always compute the mean for each person. Just because totals look the same doesn't mean the data is identical - check by calculating the actual mean.

Question 5. The enrolment in a school during six consecutive years was as follows: 1555, 1670, 1750, 2013, 2040, 2126. Find the mean enrolment in the school during this period.
Answer: The sum of student numbers across the six years totals 11,154. Dividing this by 6 years gives a mean enrolment of 1,859 students. This represents the average number of students enrolled in the school over the six-year span.
In simple words: Add all six years' numbers to get 11,154. Divide by 6 to find the average = 1,859 students per year.

Exam Tip: When finding the mean, ensure your addition is correct before dividing. Double-check by adding again to avoid arithmetic errors.

Question 1. Find the median of onion prices in Yahapur and Wahapur.
Answer: For Yahapur, the 12 monthly prices arranged from lowest to highest are: 24, 25, 26, 28, 30, 35, 39, 43, 44, 49, 56, 59. Since there are 12 values (an even count), the median is the average of the 6th and 7th values: (35 + 39) / 2 = 37. The median price in Yahapur is Rs. 37 per kg. For Wahapur, the arranged prices are: 17, 19, 23, 30, 35, 38, 39, 42, 42, 52, 53, 60. The median is the average of the 6th and 7th values: (38 + 39) / 2 = 38.5. The median price in Wahapur is Rs. 38.5 per kg.
In simple words: Arrange prices in order from smallest to largest. With 12 prices, take the middle two and find their average. Yahapur's middle prices are 35 and 39, so the median is 37. Wahapur's middle prices are 38 and 39, so the median is 38.5.

Exam Tip: Remember that when you have an even number of values, the median is always the average of the two middle values, not one of the actual data points.

Question 2. Sanskruti asked her class how many domestic animals and pets each had at home. Some of the students were absent. The data values are 0, 1, 0, 4, 8, 0, 0, 2, 1, 1, 5, 3, 4, 0, 0, -, 10, 25, 2, -, 2, 4. Find the mean and median. How would you describe this data?
Answer: Ignoring the two missing entries, we have 20 data values. When arranged in order: 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 8, 10, 25. The median is the average of the 10th and 11th values: (2 + 2) / 2 = 2. The sum of all values is 72, so the mean is 72 / 20 = 3.6. This dataset shows that most students own between 0 and 4 pets. The data spans a wide range, from 0 to 25, with one unusually high value (25) that stands out as an outlier on the upper end.
In simple words: Most students have 0 to 4 pets at home. The middle value is 2 pets, and the average is 3.6 pets. One student has 25 pets, which is much higher than everyone else.

Exam Tip: When data has outliers (extremely high or low values), mention this in your description. The median is often more useful than the mean when outliers are present.

Question 3. Rintu takes care of a date-palm tree farm in Habra. The heights of the trees (in feet) in his farm are given as: 50, 45, 43, 52, 61, 63, 46, 55, 60, 55, 59, 56, 56, 49, 54, 65, 66, 51, 44, 58, 60, 54, 52, 57, 61, 62, 60, 60, 67. Fill the dot plot, and mark the mean and median. How would you describe the heights of these palm trees? Can you think of quicker ways to find the mean? How many trees are shorter than the average height?
Answer: When the 29 tree heights are arranged 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. Since there are 29 values (odd), the median is the middle or 15th value = 56 feet. The total of all heights is 1,621 feet. Dividing by 29 gives a mean of 55.89 feet. The tree heights range from 43 to 67 feet, with most clustering around 55-60 feet. The median shows that half the trees are below 56 feet and half are above. Thirteen trees are shorter than the average height of 55.89 feet.
In simple words: Tree heights range from 43 to 67 feet. The middle height is 56 feet. The average is 55.89 feet. Most trees are between 55 and 60 feet tall. 13 trees are shorter than the average.

Exam Tip: For a dot plot, use one dot for each tree at its corresponding height. Mark the mean with one line and the median with another (often dashed) line to show both measures clearly.

Question 4. The daily water usage from a tap was measured. The usage in litres for the first few days is: 5.6, 8, 3.09, 12.9, 6.5, 12.1, 11.3, 20.5, 7.4. (a) Can the mean or median daily usage lie between 25 and 30? Justify your claim using the meaning of mean and median. (b) Can the mean or median be less than the minimum value or greater than the maximum value in a data set?
Answer: (a) No, neither the mean nor the median can lie between 25 and 30. The mean is calculated as the sum of all values divided by how many values there are, so it always falls somewhere between the smallest and largest numbers in the data. The smallest value here is 3.09 litres and the largest is 20.5 litres. Since the mean must be between these two limits, it cannot possibly be 25 or 30. The median is the middle value when all data is arranged in order, so it too must lie within the range of the data. (b) No, the mean and median can never be less than the minimum value or greater than the maximum value. By definition, both represent central positions within the data range and therefore always fall between the lowest and highest values.
In simple words: The mean and median always stay within the lowest and highest numbers in your data. They cannot go outside this range because they are calculated from those numbers.

Exam Tip: Understanding what mean and median represent is crucial. Remember: the mean is an average of all values, and the median is the middle value - both must lie within the data range.

Question 5. The weights of a few newborn babies are given in kgs. Fill the dot plot provided below. Analyse and compare this data.
Answer: Boys' weights range from 2.6 kg to 4.1 kg across the sample. Girls' weights span from 2.5 kg to 4 kg. The heaviest baby recorded is a boy weighing 4.1 kg, while the lightest is a girl at 2.5 kg. Both groups show similar weight distributions within the newborn range, with boys having a slightly wider spread at the upper end.
In simple words: Baby boys weigh between 2.6 and 4.1 kg. Baby girls weigh between 2.5 and 4 kg. The heaviest baby is a boy and the lightest is a girl.

Exam Tip: When comparing two groups, always mention the range (highest and lowest values) for each group to show the spread of the data.

Question 6. The dot plots of the heights of another section of Grade 5 students of the same school are shown below. Can you share your observations? What can we infer from the dot plots and the central tendency measures?
Answer: Girls' heights show a wider spread, ranging between 126 and 158 cm. Boys' heights fall between 130 and 148 cm, covering a narrower band. The tallest student in the class is a girl (158 cm) and the shortest is also a girl (126 cm). Despite this, girls' average height is 140.14 cm, which is less than boys' average of 142.05 cm and also less than the whole class average of 141.21 cm. For boys, the mean (142.05 cm) is slightly less than the median (143 cm), showing a small pull from lower values. For girls, the mean (140.14 cm) exceeds the median (140 cm), indicating a slight influence from higher values. Comparing this group with the previous Grade 5 section, the boys and girls in this section are generally shorter.
In simple words: Girls are more spread out in height than boys. Girls range from 126 to 158 cm, boys from 130 to 148 cm. Even though girls include the tallest and shortest, boys are taller on average.

Exam Tip: When comparing distributions, always note the range and where the data clusters. Look at how the mean and median differ - this tells you if data is pulled toward higher or lower values.

Question 7. The weights of some sumo wrestlers and ballet dancers are: Sumo wrestlers: 295.2 kg, 250.7 kg, 234.1 kg, 221.0 kg, 200.9 kg. Ballet dancers: 40.3 kg, 37.6 kg, 38.8 kg, 45.5 kg, 44.1 kg, 48.2 kg. Approximately how many times heavier is a sumo wrestler compared to a ballet dancer?
Answer: The sum of the five sumo wrestlers' weights is 1,201.9 kg, giving an average of 240.38 kg per wrestler. The sum of the six ballet dancers' weights is 254.5 kg, producing an average of 42.417 kg per dancer. Dividing the average sumo wrestler weight by the average ballet dancer weight: 240.38 / 42.417 = 5.66, which rounds to approximately 6 times. Therefore, a sumo wrestler is roughly 6 times heavier than a ballet dancer.
In simple words: Sumo wrestlers weigh about 240 kg on average. Ballet dancers weigh about 42 kg on average. Divide 240 by 42 to get 6. So wrestlers are about 6 times heavier.

Exam Tip: When comparing groups using averages, calculate the mean for each group first, then divide one mean by the other. Show your arithmetic clearly.

Question 1. The following infographic shows the speeds of a few animals in the air, on land, and in the water. Can we call this graph a bar graph? (a) What is the scale used in this graph? (b) What did you find interesting in this infographic? What do you want to explore further? (c) Identify a pair of creatures where one's speed is about twice that of the other. (d) Can we say that a sailfish is about 4 times faster than a humpback whale? Can we say that a sailfish is the fastest aquatic animal in the world?
Answer: Yes, this is a form of bar graph, specifically a pictorial or infographic-style bar graph. (a) The scale is 1 unit length = 16 km per hour. (b) This is a personal response - note interesting facts you observe from the data. (c) The green darner dragonfly travels at 64 kph, which is roughly double the speed of the gentoo penguin at 35 kph. (d) Yes, the sailfish at 109 kph is indeed about 4 times faster than the humpback whale at 26 kph (26 × 4 = 104 kph). According to this data, the sailfish is the fastest aquatic animal shown.
In simple words: This graph shows animal speeds using bars. The scale tells us how many km each unit represents. You can compare animals by looking at their bar lengths.

Exam Tip: Always identify the scale of a graph first. This lets you read and compare values accurately. For part (b), your personal observations are valid - describe what surprised or interested you.

Question 2. Preyashi asked her students 'If you were to get a super power to become aquatic (water-borne), aerial (air-borne), or spaceborne, which one would you choose?'. The responses are shown below. Some chose none. Draw a double-bar graph comparing how both grades chose each option. Choose an appropriate scale.
Answer: From the given data, Grade 5 students chose: water (w) - 6, aerial (a) - 13, spaceborne (s) - 2, none (n) - 4. Grade 9 students chose: water - 6, aerial - 8, spaceborne - 9, none - 2. Using a scale where 1 unit = 1 student, construct a double-bar graph with four categories on the x-axis and the number of students on the y-axis. For each option, place two bars side by side - one for Grade 5 and one for Grade 9 - to enable direct comparison. From this graph, you can observe that aerial is the most popular choice for Grade 5, while spaceborne gains more votes from Grade 9 students.
In simple words: Make a double-bar graph with four groups: water, aerial, space, and none. For each group, draw two bars next to each other - one for Grade 5, one for Grade 9. This shows which powers each grade preferred.

Exam Tip: In a double-bar graph, use different colors or patterns for each group and include a clear legend. Make sure your bars are the same width and equally spaced.

Question 3. The temperature variation over two days in different months in Jodhpur, Rajasthan, is given below. Draw a double-bar graph. Use the scale 1 unit = 4°C. Can you guess which two months these days might belong to?
Answer: The temperature data shows Day 1 with readings: 12 am - 20°C, 3 am - 18°C, 6 am - 16°C, 9 am - 20°C, 12 pm - 26°C, 3 pm - 34°C, 6 pm - 30°C, 9 pm - 24°C. Day 2 shows: 12 am - 37°C, 3 am - 34°C, 6 am - 30°C, 9 am - 33°C, 12 pm - 37°C, 3 pm - 43°C, 6 pm - 42°C, 9 pm - 39°C. Using a scale of 1 unit = 4°C, draw a double-bar graph with time periods on the x-axis and temperature on the y-axis. Day 1 represents a milder day while Day 2 is significantly hotter, with maximum temperatures reaching 43°C. The overall temperature pattern and the extreme heat of Day 2 suggest these days belong to December (mild) and May (very hot), respectively, which are typical temperature patterns for Jodhpur, Rajasthan.
In simple words: Day 1 is cool (around 20-34°C). Day 2 is much hotter (around 37-43°C). These temperatures match December (cool) and May (very hot) in Jodhpur.

Exam Tip: When drawing a double-bar graph, ensure consistent bar widths and spacing. Always label both axes clearly, including the scale and units of measurement.

Question 4. The following clustered-bar graph shows the number of electric vehicles registered in some states every year from 2022 to 2024. (a) The data (rounded off to thousands) for the states of Gujarat and Delhi are given in the table below. Mark the corresponding bars on the bar graph. (It is enough if you place the top of the bars between the two appropriate vertical guidelines.) (b) Notice how the graph is organised, what scale is used, and what patterns the data shows. (c) How would you describe the change for various states between 2022 and 2024? (d) Approximately how many more registrations did Assam get in 2023 compared to 2022? (e) How many times more did the registrations in West Bengal increase from 2022 to 2024? (f) Is this statement correct - 'There were very few new registrations in Uttarakhand in 2023 and 2024, as the increase in the bar lengths is minimal'?
Answer: (a) For Gujarat: 2022 - 69,000; 2023 - 89,000; 2024 - 78,000. For Delhi: 2022 - 62,000; 2023 - 74,000; 2024 - 81,000. Mark these bars on the graph at the appropriate positions. (b) The graph uses a clustered arrangement with three bars (one for each year from 2022 to 2024) grouped by state. The scale is 1 unit length = 25,000 electric vehicles. Data patterns show that most states experienced growth across the years, though some show fluctuations. (c) Most states demonstrate an upward trend in registrations between 2022 and 2024. Delhi and Gujarat lead with the highest registration numbers. Uttarakhand records the lowest. On average, vehicle registrations increased year by year across nearly all states. Assam and Andhra Pradesh show particularly strong growth. (d) Assam registered approximately 40,000 vehicles in 2022 and about 60,000 in 2023, representing an increase of roughly 20,000 registrations. (e) West Bengal had approximately 11,000 registrations in 2022 and 44,000 in 2024, an increase of roughly 4 times. (f) Yes, the statement is correct. While the absolute numbers in Uttarakhand grew, the growth rate was much smaller compared to other states, so bar length increases appear minimal on the graph.
In simple words: Most states registered more electric vehicles each year. Delhi and Gujarat are leaders. Uttarakhand had the fewest. Assam and Andhra Pradesh grew the most.

Exam Tip: When reading clustered-bar graphs, identify the pattern across groups. Compare bars within clusters (across years) and across clusters (across states) to make meaningful observations.

Question 1. The dot plots below show the distribution of the number of pockets on clothing for a group of boys and for a group of girls. Based on the dot plots, which of the following statements are true? (a) The data varies more for the boys than for the girls. (b) The median number of pockets for the boys is more than that for the girls. (c) The mean number of pockets for the girls is more than that for the boys. (d) The maximum number of pockets for boys is greater than that for the girls.
Answer: (a) False - The girls' data is actually more spread out than the boys' data, showing greater variation. (b) True - When you count the dot positions, the middle value for boys is higher than the middle value for girls. (c) False - The average number of pockets for girls is not more than for boys when calculated from the dot positions. (d) True - The highest dot for boys reaches a greater number of pockets than the highest dot for girls.
In simple words: Look at each statement and check it against the dot plot. Count dots to find the middle value (median) and spread (variation) for each group.

Exam Tip: When analyzing dot plots, identify the range (minimum to maximum), count the dots to find the median, and visually assess whether data is spread out or clustered to compare variation between groups.

Question 2. The following table shows the points scored by each player in four games: Now answer the following questions: (a) Find the average number of points scored per game by A. (b) To find the mean number of points scored per game by C, would you divide the total points by 3 or by 4? Why? What about B? (c) Who is the best performer?
Answer: (a) Player A scored a total of 50 points across 4 games. The average is 50 / 4 = 12.5 points per game. (b) Player C should have the total points divided by 3, not 4, because C did not participate in game 3 and therefore played only 3 games. The average reflects actual games played. Player B participated in all four games, so you divide by 4. (c) Player A is the best performer because A has the highest average score per game (12.5 points), compared to the other players.
In simple words: Add a player's points to get the total. Divide by how many games they actually played, not by 4. If someone missed a game, count only the games they played.

Exam Tip: When calculating averages for groups with missing data, always divide by the actual count of entries, not by a preset number. This is critical for accurate comparison.

Question 3. The marks (out of 100) obtained by a group of students in a General Knowledge quiz are 85, 76, 90, 85, 39, 48, 56, 95, 81, and 75. Another group's scores in the same quiz are 68, 59, 73, 86, 47, 79, 90, 93, and 86. Compare and describe both groups' performance using mean and median.
Answer: Group 1 has 10 students with a total of 730 marks, yielding a mean of 73. When arranged in ascending order (39, 48, 56, 75, 76, 81, 85, 85, 90, 95), the median (average of 5th and 6th values) is 78.5. Group 2 has 9 students with a total of 681 marks, producing a mean of 75.67. Arranged in order (47, 59, 68, 73, 79, 86, 86, 90, 93), the median (5th value) is 79. Group 2 performs slightly better overall, with a higher mean of 75.67 compared to Group 1's 73. However, Group 1's median of 78.5 is slightly lower than Group 2's 79, indicating that the middle performances are nearly equivalent. Group 1 contains one very low outlier (39), which pulls the mean down more than any single low score affects Group 2.
In simple words: Group 1 average is 73, middle is 78.5. Group 2 average is 75.67, middle is 79. Group 2 did slightly better overall. Group 1 has one very low score pulling the average down.

Exam Tip: When comparing two groups, compute both mean and median. If they differ significantly, check for outliers that might explain the difference. Always mention the outliers when describing a dataset.

Question 4. Consider this data collected from a survey of a colony. Choose an appropriate scale and draw a double bar graph. Write down your observations.
Answer: Using a scale where 1 unit = 250 people, construct a double-bar graph with sports categories (Cricket, Basketball, Swimming, Hockey, Athletics) on the x-axis and number of people on the y-axis. The two bars in each group represent "Watching" and "Participating". Key observations include: Cricket stands out as the most popular sport both for watching (1,240 people) and participating (620 people). Across all sports, significantly more people choose to watch than to actively participate. Athletics has the lowest engagement for both watching (250 people) and participating (105 people). Basketball and swimming show comparable participation numbers (320 each), but swimming attracts more spectators (510 versus 470). The graph reveals a general pattern: spectator interest consistently exceeds active participation across all sports.
In simple words: Cricket is most popular, both for watching and playing. More people watch sports than play them. Athletics is the least popular. Swimming attracts more watchers than basketball does.

Exam Tip: When writing observations from a double-bar graph, compare within each group (watch vs. participate), across groups (different sports), and overall patterns (the ratio of watching to participating).

Question 5. Consider a group of 17 students with the following heights (in cm): 106, 110, 123, 125, 117, 120, 112, 115, 110, 120, 115, 102, 115, 115, 109, 115, 101. The sports teacher wants to divide the class into two groups so that each group has an equal number of students: one group has students with heights less than a particular height, and the other group has students with heights greater than the particular height. Suggest a way to do this. Can you guess the age of these students based on the tabular data in the 'Telling Tall Tales' section?
Answer: With 17 students (an odd number), perfect equal division is impossible. However, arrange the heights in increasing order: 101, 102, 106, 109, 110, 110, 112, 115, 115, 115, 115, 115, 117, 120, 120, 123, 125. The median (9th value) is 115 cm. To create two groups as close to equal as possible, use a dividing height between 112 and 115 cm. For example, if you choose 114 cm as the cutoff: Group 1 (heights less than 114 cm) would have 7 students: 101, 102, 106, 109, 110, 110, 112. Group 2 (heights greater than 114 cm) would have 10 students: 115, 115, 115, 115, 115, 117, 120, 120, 123, 125. Although not perfectly equal, this minimizes the difference. Based on the heights in the 106-125 cm range, these students are likely around 8-9 years old (Grade 4-5).
In simple words: Arrange heights in order. Find the middle height which is 115 cm. Use this as a dividing line. Students shorter than 115 go in one group, those taller go in another. The heights suggest students are around 8-9 years old.

Exam Tip: When dividing data into groups, find the median first. This is the optimal dividing point to create the most balanced split possible. Remember that with odd-numbered datasets, perfect equality is impossible.

Question 6. Describe the mean and median of the heights of your class. You can visualise the heights on a dot plot.
Answer: This is a hands-on activity to be completed by you. Collect the actual heights of all students in your class. Arrange them in order from shortest to tallest. Calculate the mean by adding all heights and dividing by the number of students. Identify the median as the middle value (or average of two middle values if you have an even number of students). Create a dot plot with height ranges on the x-axis and place one dot for each student at their corresponding height. Mark the mean and median on your dot plot with different visual indicators (such as a solid line for the mean and a dashed line for the median). In your description, note the range of heights, whether the data is clustered or spread out, and how the mean and median compare.
In simple words: Collect everyone's heights. Add them and divide by how many students to find the mean. Arrange heights in order and find the middle one for the median. Draw a dot plot showing all heights.

Exam Tip: For a dot plot, be consistent with your scale and spacing. Show both mean and median clearly, and explain what you observe about the height distribution in your class.

Question 7. There are two 7th-grade sections at a school. Each section has 15 boys and 15 girls. In one section, the mean height of students is 154.2 cm. From this information, what must be true about the mean height of students in the other section? (a) The mean height of students in the other section is 154.2 cm. (b) The mean height of students in the other section is less than 154.2 cm. (c) The mean height of students in the other section is more than 154.2 cm (d) The mean height of the students' section cannot be determined.
Answer: The correct answer is (d) - The mean height of students in the other section cannot be determined. Knowing only that one section has a mean height of 154.2 cm tells us nothing about the second section. The second section could have students who are taller, shorter, or the same height as the first section. Even though both sections have identical numbers of boys and girls, their actual heights are independent data. The mean of the second section could be greater than, less than, or equal to 154.2 cm - all are possibilities without additional information about the students' actual heights.
In simple words: Just because one class has an average height of 154.2 cm does not tell you anything about another class's average. The other class could have any average height at all.

Exam Tip: Be cautious about making assumptions. One piece of information about one group does not automatically tell you about another similar group. Each group's data must be analyzed independently.

Question 8. Standing tall in the storm. (a) Write estimated values for the number of skyscrapers in New York, Tokyo, and London. (b) Are the following statements valid? (i) Only 12 cities have more skyscrapers than Mumbai. (ii) Only 7 cities have fewer skyscrapers than Mumbai. (iii) The tallest building in the world is in Hong Kong.
Answer: (a) Based on the infographic showing cities with the most skyscrapers (buildings taller than 150 m): New York has approximately 38 skyscrapers. Tokyo has approximately 160 skyscrapers. London has approximately 305 skyscrapers. (Note: These are reasonable estimates based on the bar lengths shown, though the exact visual interpretation may vary.) (b) For the validity of statements: (i) Valid - Counting the cities shown with more skyscrapers than Mumbai's 86, approximately 12 cities exceed this number. (ii) Valid - Counting cities with fewer skyscrapers than Mumbai, approximately 7 fall below this mark. (iii) Invalid - According to the infographic, Hong Kong leads with 553 skyscrapers, but this does not mean the world's tallest building is located there. The statistic measures building count, not height of individual structures.
In simple words: Read the bar graph carefully to estimate numbers. Check statements by counting how many cities appear above or below Mumbai. Remember that most buildings is different from tallest building.

Exam Tip: When reading infographics, distinguish between different types of data. A city with many skyscrapers is not the same as a city with the tallest building. Always read the title and legend carefully.

Question 9. Estimate and then measure the objects listed in the following table. Draw a double bar graph based on the data. How accurate were your estimates? Find the average difference between the estimated and measured values.
Answer: This is a hands-on measurement activity for you to complete. Select several objects around your classroom or home to estimate and measure (such as the length of a desk, height of a chair, width of a window, etc.). For each object, first write your estimate without measuring. Then use a ruler, tape measure, or other measuring device to find the actual measurement. Record both values in a table with columns for object name, estimated length, and actual measured length. Calculate the difference between estimate and measurement for each object (taking absolute values so all differences are positive). Create a double-bar graph with objects on the x-axis and lengths on the y-axis, using one color for estimates and another for actual measurements. This visual will show how close your estimates were. Finally, add all the differences and divide by the number of objects to find the average difference between your estimates and the measured values. Reflect on which objects you estimated most accurately and why.
In simple words: Pick objects to measure. Guess their length first. Then measure them for real. Make a double-bar graph showing guesses and real measurements. See how close your guesses were.

Exam Tip: For this activity, be honest with your initial estimates - they should represent your true first impression before measuring. The goal is to improve your estimation skills, so accuracy improves with practice.

Question 10. Aditi likes solving puzzles. She recently started attempting the 'Easy' level Sudoku puzzles. The time she took (in seconds) to solve these puzzles is - 410, 400, 370, 340, 360, 400, 320, 330, 310, 320, 290, 380, 280, 270, 230, 220, 240. The first nine values correspond to Week 1 and the rest to Week 2.
(a) Construct a dot plot below showing the data for both weeks.
(b) Describe the mean, median, and any observations you may have about the data.
Answer:
(a) The dot plot shows Week 1 times (represented by triangles) and Week 2 times (represented by circles) plotted on a number line from 200 to 400 seconds. Week 1 data clusters around the 300-410 second range, while Week 2 data is concentrated in the 220-290 second range.

(b) To find the mean, add all 17 values: 410 + 400 + 370 + 340 + 360 + 400 + 320 + 330 + 310 + 320 + 290 + 380 + 280 + 270 + 230 + 220 + 240 = 5470. Dividing by 17 gives a mean of approximately 321.76 or 321 seconds.

When arranged in ascending order: 220, 230, 240, 270, 280, 290, 310, 320, 320, 330, 340, 360, 370, 380, 400, 400, 410. Since there are 17 values, the median is the 9th value, which is 320 seconds.

Key observations: Aditi's performance improved significantly from Week 1 to Week 2. The average puzzle-solving time in Week 1 was much higher, showing that she was faster by Week 2. Week 1 times range from 310 to 410 seconds, while Week 2 times range from 220 to 290 seconds - a noticeable decrease. This shows that with practice, Aditi became quicker at solving Sudoku puzzles.
In simple words: In Week 1, Aditi took longer to solve puzzles (average 321 seconds). In Week 2, she was much faster (times around 220-290 seconds). She improved a lot with practice.

Exam Tip: Always arrange data in order before finding the median - this is essential for accuracy. Note that the median and mean can be different; both help you understand the data in different ways. Observations about trends (improvement over time) are worth mentioning in part (b).

Question 11. Individual Project: Pick at least one of the following: (a) How long is a sentence? Pick any two textbooks from different subjects. Choose any page with a lot of text from each book. (i) Use a dot plot to describe how many words the sentences have on each page. (ii) Compare the data of both pages using mean and median. (b) What is in a Name? Write down the names of all of your classmates. The following are some interesting things you can do with this data! (i) Find the mean and median name length (number of letters in a name). (ii) Visualise the data and describe its variability and central tendency. (iii) Which starting letters are more popular? Which are less popular? (iv) What is the median starting letter? What does this say about the number of names starting with the letters A-M and N-Z? (v) Plot a double-bar graph showing the number of boys' names and girls' names that: Start and end with vowels, Start with vowels and end with consonants, Start with consonants and end with vowels, Start and end with consonants.
Answer: This is an individual project that you must complete yourself. Choose either option (a) or option (b), or both if you wish.

For option (a) - Sentence Length: Collect sentence data from two different textbooks. Use a dot plot to show the word count for each sentence on your chosen pages. Then calculate and compare the mean (average) and median (middle value) word counts between the two pages. Discuss which page has longer or shorter sentences on average and how the data spreads out.

For option (b) - Name Analysis: Gather all your classmates' names and perform various data analyses. Calculate mean and median name length by counting letters. Create a visual representation showing how the name lengths spread out and where most names cluster. Identify which starting letters appear most and least frequently. Find the median starting letter and think about what this tells you regarding how many names start with letters in the first half of the alphabet (A-M) versus the second half (N-Z). Finally, construct a double-bar graph displaying boys' and girls' names grouped by their starting and ending letters (vowel-vowel, vowel-consonant, consonant-vowel, consonant-consonant combinations).
In simple words: Pick one project. For sentences, count words and compare two pages. For names, analyse your classmates' names by length, starting letter, and vowel-consonant patterns. Show your findings in graphs.

Exam Tip: Document your data-collection process clearly. Label all graphs with titles and axis labels. Show your calculations for mean and median. Present your observations and conclusions - the analysis and interpretation are as important as the data itself.

Question 12. Individual Project (long term): This requires collecting data over 2 weeks or more. In and Out: Track how many times you step out of your house in a day. Do this for a month. (i) Describe the variability and central tendency of this data. Make a dot plot. (ii) Do you find anything interesting about this data? Share your observations. (iii) You can ask any of your family members or friends to do this as well.
Answer: This is an individual project that you must complete yourself. Over a period of one month, keep a daily record of how many times you leave your house.

For part (i), create a dot plot showing the frequency for each day of the month. Calculate the mean (average number of times you go out) and the median (middle value). Discuss how the data spreads - are most days similar, or do some days have much higher or lower values? This tells you about variability. Describe where the data clusters using measures of central tendency.

For part (ii), look for patterns in your data. Do you go out more on weekdays or weekends? Are there certain days when you stay home more? Do the numbers change week by week? Share any interesting findings you notice.

For part (iii), if possible, ask family members or friends to track their own data using the same method. This allows you to compare different people's patterns and routines.
In simple words: Track how many times you leave home each day for a month. Make a dot plot and find the average and middle value. Look for patterns - do some days have more outings than others? Compare with others if you can.

Exam Tip: Keep consistent daily records - accuracy is key. Your dot plot should clearly show all 30 or 31 days. Calculations for mean and median must be shown. The most valuable part is your observations - explain what the data tells you about daily habits and routines.

Question 13. Small-group project: Pick at least one of the following. Make groups of 8 to 10. Collect data individually as needed. Put together everyone's data and do the appropriate analysis and visualisation. (a) Our heights vs. our family's heights: Collect the heights of your family members. (i) Make a dot plot showing the heights of just your family members. Describe its variability and central tendency. (ii) Make a double-bar graph showing each student's height next to their family's mean height. (iii) Look at everyone's data and share your observations. (b) Estimating time: Check the time and close your eyes. Open them when you think 1 minute has passed (no counting). Note down how many seconds you opened your eyes. Collect this data for yourself and for your family members. Repeat this activity to estimate 3 minutes. (i) Make two dot plots (for 1 minute and 3 minutes) showing estimates of just your family members. (ii) Mark these on the respective dot plots. Describe its variability and central tendency. (iii) Make a double bar graph showing each family's mean 1-minute estimate and mean 3-minute estimate. (iv) Look at everyone's data and share your observations.
Answer: This is a small-group project that you and your group members must complete yourselves.

For option (a) - Heights: Each group member collects the heights of their family members. Create a dot plot showing just your own family's heights and describe how much the heights vary (some people might be much taller or shorter than others) and where most heights cluster (central tendency). Then, as a group, make a double-bar graph that compares each student's own height with the average height of their family. Finally, combine all the group's data and discuss what you observe - do taller students come from taller families? Are there other patterns?

For option (b) - Time Estimation: Each person closes their eyes and opens them when they think 1 minute has passed, recording the actual number of seconds. Repeat the same process for estimating 3 minutes. Collect data from yourself and your family members. Create two separate dot plots (one for 1-minute estimates, one for 3-minute estimates) showing just your family's data. Describe how much variation there is in estimates and where the estimates centre. As a group, make a double-bar graph comparing each family's mean (average) estimate for 1 minute versus 3 minutes. Examine all the group's combined data and share observations about time estimation - are people generally accurate? Do they overestimate or underestimate? Does estimating longer times (3 minutes) become easier or harder?
In simple words: Work in a group of 8-10 students. Either collect family heights or do time-estimation activities. Make dot plots and bar graphs. Compare your family's data with others' data. Share what patterns you find.

Exam Tip: Ensure all group members contribute equally to data collection. Label all graphs clearly with titles and units. Show your calculations for mean and median. Group observations should be thoughtful and based on the actual data - mention specific patterns or differences you notice between families or individuals.