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Connecting the Dots

NCERT Class 7 · Mathematics Based on NCERT Class 7 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces students to the world of statistics, focusing on how to collect, organise, analyse, and interpret data to answer questions. The big idea is that a single number, like the average or median, can represent a whole set of data, but we must choose the right measure depending on the data's nature, especially when outliers are present. Students will learn to make statistical statements, compare data using mean and median, and visualise data through dot plots and bar graphs to uncover patterns and tell stories.
Key Concepts (13)
Statistical statement
A claim or summary about some phenomenon, expressed in terms of numerical values, proportions, probabilities, or predictions.
e.g. “Jemimah’s batting has been very consistent over the past year. We can expect a century from her in tomorrow’s match.”
Statistical question
A question that can be answered by collecting data.
e.g. “How tall are Grade 7 students in our school?”
Statistics
The study of collecting, organising, analysing, interpreting, and presenting data.
e.g. Finding the average height of students in a class is a part of statistics.
Average (Arithmetic Mean)
A representative number found by adding up all the values in the data and dividing by the number of values.
e.g. Shubman scored 23, 7, 10, 52, 18 in 5 matches. His average = (23+7+10+52+18) ÷ 5 = 110 ÷ 5 = 22 runs per match.
Mean
Same as the Arithmetic Mean; calculated as Sum of all values ÷ Number of values.
e.g. Mean number of flowers blooming per day = (2+7+9+4+3) ÷ 5 = 5 flowers.
Fair-share (Average as equal share)
The average can be understood as the amount each person gets if the total is shared equally among all.
e.g. Shreyas’s group has 30 guavas for 5 members. Each gets 30 ÷ 5 = 6 guavas.
Dot plot
A visual representation where data points are shown as dots on a line, helping to see variability and patterns in data.
e.g. A dot plot of onion prices in Yahapur shows dots at ₹25, ₹24, ₹26, etc., on a horizontal line from ₹10 to ₹60.
Outlier
A value which significantly deviates from the rest of the values in the data.
e.g. In Poovizhi’s family, the height of 118 cm is an outlier because all other heights are between 165 cm and 175 cm.
Median
The middle value in a sorted list of data. If there is an even number of values, it is the average of the two middle numbers.
e.g. Heights of Poovizhi’s family sorted: 118, 165, 170, 173, 175. Median = 170 cm.
Measures of central tendency
Values that represent the ‘centre’ of the data, such as the mean and median.
e.g. The mean and median both tell us about the typical height in a class.
Data visualisation
Presenting data as a picture (like a graph) to understand it more clearly.
e.g. A double bar graph showing monthly onion prices in two towns.
Clustered column graph (Double column graph)
A bar graph where two or more bars are placed side by side in each cluster to compare different categories.
e.g. A graph showing rocket launches by SpaceX, CASC, etc., for 2021, 2022, and 2023 with three adjacent bars per organisation.
Samamiti
An ancient Indian term for Arithmetic Mean, meaning ‘mean measure’ or ‘equal measure’.
e.g. Bhāskarācārya used the term samamiti for the arithmetic mean.
Formulas & Equations (2)
Mean = Sum of all the values in the data ÷ Number of values in the data
Reads as: Mean equals sum of all values divided by number of values
Variables: Sum of all values = total of the data points; Number of values = count of data points (no unit)
Use when: To find the average or representative value of a set of data, e.g., average runs, average height, average price.
⚠ Students forget to divide by the correct number of values, especially when there are missing entries (e.g., a player did not play a match, so that match is not counted).
Median = Middle value in sorted data (if odd count) OR Average of two middle values (if even count)
Reads as: Median is the middle value after sorting, or the average of the two middle values if there are an even number of values
Variables: Sorted data = data arranged in ascending order; Middle value = the value at position (n+1)/2 for odd n; Two middle values = values at positions n/2 and n/2+1 for even n
Use when: To find a central value that is not affected by outliers, e.g., finding the typical height when a very short child is in the family.
⚠ Students forget to sort the data first before finding the median. They also incorrectly average the two middle numbers without sorting.
How It Works
Calculating the Arithmetic Mean (Average)
  1. Step 1: Add up all the values in the data set.
  2. Step 2: Count the number of values in the data set.
  3. Step 3: Divide the total sum by the number of values.
💡 If a value is missing (e.g., a player did not play a match), do not include that missing entry in the count. Only count the values that are actually present.
Finding the Median
  1. Step 1: Arrange all the data values in ascending order (from smallest to largest).
  2. Step 2: If the number of values is odd, the median is the middle value.
  3. Step 3: If the number of values is even, the median is the average of the two middle values.
💡 You must sort the data first. The median is not affected by outliers, unlike the mean.
Creating and Interpreting a Dot Plot
  1. Step 1: Draw a horizontal line and mark a scale that covers the range of the data.
  2. Step 2: For each data value, place a dot above the corresponding number on the scale.
  3. Step 3: If a value occurs more than once, stack the dots vertically.
💡 Dot plots lose the original sequence (e.g., month-wise order) but make it easy to see clusters, spread, and outliers.
Reading a Clustered Column Graph (Double Bar Graph)
  1. Step 1: Identify what is given — look at the labels on the horizontal and vertical axes, the scale, and the legend.
  2. Step 2: Infer from what is given — compare the heights of bars within each cluster and across clusters to draw conclusions.
💡 Always check the scale carefully. A small change in scale can make differences look bigger or smaller than they actually are.
Exam Traps — Don't Make These Mistakes
⚠ Confusing 'average' with 'total'
✗ “Shubman performed better because his total runs are 110, which is more than Yashasvi’s 96.”
→ The total does not account for the number of matches played. Shubman played 5 matches, Yashasvi played 4. The average gives a fair comparison.
✓ “Yashasvi performed better because his average runs per match (96 ÷ 4 = 24) is higher than Shubman’s average (110 ÷ 5 = 22).”
⚠ Forgetting to sort data before finding median
✗ “For the data 3, 8, 10, 5, 4, the median is 10 because it is in the middle.”
→ The data must be sorted first. The sorted order is 3, 4, 5, 8, 10, and the middle value is 5, not 10.
✓ “First sort the data: 3, 4, 5, 8, 10. The median is the middle value, which is 5.”
⚠ Including missing values in the count for mean
✗ “A player scored 57, 13, 0, 84, (did not play), 51, 27. Average = (57+13+0+84+0+51+27) ÷ 7 = 232 ÷ 7.”
→ The player did not play in match 5, so that match should not be counted. The total runs are divided by 6 (the number of matches played), not 7.
✓ “Average = (57+13+0+84+51+27) ÷ 6 = 232 ÷ 6.”
⚠ Assuming mean is always the best representative
✗ “The average height of Poovizhi’s family is 160.2 cm, so the family is of average height.”
→ The mean is pulled down by the outlier (118 cm). Most family members are taller than 160.2 cm. The median (170 cm) is a better representative here.
✓ “Because of the outlier (118 cm), the mean (160.2 cm) is not a good representative. The median (170 cm) better represents the family’s typical height.”
⚠ Misreading the scale on a graph
✗ “In the rocket launch graph, SpaceX launched about 100 rockets in 2023 because the bar reaches the 100 mark.”
→ The scale is 1 unit = 20 rockets. The bar for SpaceX in 2023 reaches between 80 and 100, so it is approximately 90-95 rockets, not 100.
✓ “Using the scale, the bar for SpaceX in 2023 reaches about 4.5 units, so approximately 4.5 × 20 = 90 rockets.”
Compare & Contrast
Effect of outliers
Mean
Median
The mean is affected by outliers (a very high or low value pulls the mean towards it), while the median is not affected by outliers.
Requirement of data collection
Statistical question
Non-statistical question
A statistical question can be answered by collecting data that varies (e.g., “How tall are Grade 7 students?”), while a non-statistical question has a single, fixed answer (e.g., “What is the height of the tallest student in Grade 7?”).
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA (3 marks) on mean/median calculation, 1 LA (5 marks) on data interpretation with graph or table
Most asked: Calculating mean and median from given data · Identifying outliers and their effect on mean vs median · Interpreting dot plots and double bar graphs · Distinguishing between statistical and non-statistical questions
Don't skip: Word problems involving calculation of mean and median with real-life data (e.g., runs, heights, prices); Questions asking to compare mean and median and explain which is more appropriate when an outlier is present
One-Line Revision (read aloud before the exam)
  1. A statistical question is answered by collecting data that varies.
  2. The mean is calculated by dividing the sum of all values by the number of values.
  3. The median is the middle value after sorting the data.
  4. If the number of values is even, the median is the average of the two middle numbers.
  5. An outlier is a value that is very different from the rest of the data.
  6. The mean is affected by outliers, but the median is not.
  7. A dot plot shows each data point as a dot on a number line.
  8. A double bar graph (clustered column graph) compares two or more sets of data side by side.
  9. The average can also be thought of as the fair-share or equal-share value.
  10. When comparing two groups, the average is more fair than the total if group sizes are different.
  11. Data visualisation helps us see patterns and variability in data quickly.
  12. In a dot plot, the original order of data (like month-wise) is lost, but clustering and spread are easy to see.
  13. The median of Poovizhi’s family heights (118, 165, 170, 173, 175) is 170 cm.
  14. The mean of the numbers 2, 7, 9, 4, 3 is 5.
  15. A zero score in a match is counted as a value; a missing match is not counted.

MCQs — 10 Questions with Answers

Q1. Which of the following is a statistical question?

  • A. What is the height of your best friend?
  • B. How much money do students in your class typically spend on lunch each week? ✓
  • C. What is the capital of India?
  • D. Who is the tallest person in your family?

Answer: B — A statistical question requires data collection and analysis because answers vary; option B is about typical spending across many students, while A, C, and D have single or fixed answers.

Q2. The average price of mangoes at a market over 5 days is ₹40 per kg. What is the total cost?

  • A. ₹40
  • B. ₹45
  • C. ₹200
  • D. Cannot be determined ✓

Answer: D — Average tells us the mean price per day, but without knowing the quantity bought each day, we cannot calculate total cost.

Q3. Shreyas scored 10, 20, 30, 40, and 50 marks in five tests. What is his average score?

  • A. 30 ✓
  • B. 35
  • C. 40
  • D. 50

Answer: A — Sum = 10 + 20 + 30 + 40 + 50 = 150; Average = 150 ÷ 5 = 30.

Q4. A farmer collected guavas: Day 1 = 5, Day 2 = 8, Day 3 = 7. If he wants to sell equal bundles each day, how many guavas per bundle?

  • A. 5 guavas
  • B. 6 guavas ✓
  • C. 7 guavas
  • D. 8 guavas

Answer: B — Total guavas = 5 + 8 + 7 = 20; Days = 3; Average = 20 ÷ 3 ≈ 6.67, which rounds to 6 or 7 per bundle (context dependent), but exact average is 6 whole + fraction.

Q5. Two batsmen scored runs as follows — Batter A: 15, 25, 35 in three matches; Batter B: 20, 20, 20 in three matches. Who has a more consistent batting record?

  • A. Batter A (higher total)
  • B. Batter B (consistent scores with smaller spread) ✓
  • C. Both are equally consistent
  • D. Cannot compare without maximum scores

Answer: B — Batter B's scores vary less (range = 0), showing consistency, while Batter A's scores vary from 15 to 35 (range = 20).

Q6. Onion prices over 4 months: ₹20, ₹25, ₹30, ₹45. The average price is ₹30. If the store says 'typical monthly price is ₹30', what does this mean?

  • A. The price was exactly ₹30 every month
  • B. If prices were equally redistributed, each month would have ₹30 ✓
  • C. The highest price is ₹30
  • D. The price was ₹30 in two months

Answer: B — The fair-share interpretation of average shows that if total (₹120) were divided equally over 4 months, each would get ₹30.

Q7. A dot plot shows onion prices in two towns. Wahapur's dots are spread across 17–60 rupees, while Yahapur's are between 24–59 rupees. What can we infer?

  • A. Yahapur prices are less varied than Wahapur's ✓
  • B. Wahapur is always more expensive
  • C. Both towns have identical price patterns
  • D. Yahapur has more months with stable pricing

Answer: A — Yahapur's range (24–59 = 35) is smaller than Wahapur's range (17–60 = 43), meaning Yahapur has less price variation.

Q8. Nikhil ran 100 m in: 17, 18, 17, 16, 19, 17, 18 seconds. What is his average running time?

  • A. 17 seconds
  • B. 17.3 seconds ✓
  • C. 18 seconds
  • D. 19 seconds

Answer: B — Sum = 17 + 18 + 17 + 16 + 19 + 17 + 18 = 122; Count = 7; Average = 122 ÷ 7 ≈ 17.43 ≈ 17.3 seconds.

Q9. Two groups of students collected money for charity. Group 1 collected ₹150 from 5 students; Group 2 collected ₹180 from 6 students. Which group's average contribution per student is higher?

  • A. Group 1 (higher total)
  • B. Group 2 (more students)
  • C. Group 1 (₹30 per student vs ₹30 per student — equal) ✓
  • D. Group 2 (₹30 per student vs ₹25 per student)

Answer: C — Group 1: 150 ÷ 5 = ₹30 per student; Group 2: 180 ÷ 6 = ₹30 per student — both averages are equal.

Q10. Why is a dot plot more useful than a raw table for spotting which price range onions fall into most frequently?

  • A. Dot plots show exact month-wise prices
  • B. Dot plots group data and visualise clusters and spread in one glance ✓
  • C. Dot plots are always smaller than tables
  • D. Dot plots include the average directly on the plot

Answer: B — Dot plots arrange data on a number line, making it easy to see where prices cluster and how they spread, unlike a table which lists values sequentially.

Flashcards

What is a statistical question?

A question that requires collecting and analysing data because answers vary, like 'How tall are Grade 7 students in our school?'

Define the arithmetic mean (average).

The sum of all values in the data divided by the number of values.

Why is average better than total when comparing two groups of different sizes?

Because total depends on group size, but average shows performance per unit, giving a fair comparison.

What does a dot plot show?

Data points as dots on a horizontal line, revealing clusters, spread, and frequency of values.

What is the fair-share interpretation of average?

If we redistribute all values equally among all people, each person gets the average amount.

Calculate the average: 6, 2, 9, 5, 4, 6, 3, 5.

Sum = 40, Count = 8, Average = 40 ÷ 8 = 5.

What information does a dot plot lose compared to a table?

A dot plot loses the original order or sequence of data (like month-wise prices).

Name three ways to describe and compare data.

Using minimum value, maximum value, average value, sum total, or range (difference between max and min).

What is the range of data, and how is it calculated?

Range is the difference between the highest and lowest values: Range = Maximum − Minimum.

Why did ancient Indian mathematicians use the word 'sama' in arithmetic mean?

Because 'sama' means equal, showing that the mean represents the common or equalising value of a collection.

Important Board Questions

What is a statistical question? Give one example. [1 mark]

A statistical question requires data collection because answers vary; example should show variability (e.g., 'How many hours do Class 7 students study daily?').

Vaishnavi tracked hibiscus flowers blooming daily: 2, 7, 9, 4, 3. Calculate the average number of flowers blooming per day. What does this average tell us? [2 marks]

Step 1: Add all values (2 + 7 + 9 + 4 + 3 = 25). Step 2: Divide by number of days (25 ÷ 5 = 5). Interpretation: If equal flowers bloomed each day, it would be 5 flowers (fair-share idea).

Two runners trained for a 100 m race. Nikhil's times (in seconds): 17, 18, 17, 16, 19, 17, 18. Sunil's times: 20, 18, 18, 17, 16, 16, 17. Calculate the average time for each and state who ran quicker on average. Show all steps. [3 marks]

Step 1: Find Nikhil's sum (122) and divide by 7 to get average ≈ 17.4 seconds. Step 2: Find Sunil's sum (122) and divide by 7 to get average ≈ 17.4 seconds. Step 3: Compare and conclude (both are approximately equal, or state which is lower if calculations differ).

The monthly onion prices (in ₹/kg) at two towns are given: Yahapur: 25, 24, 26, 28, 30, 35, 39, 43, 49, 56, 59, 44. Wahapur: 19, 17, 23, 30, 38, 35, 42, 39, 53, 60, 52, 42. (a) Calculate the average price at each town. (b) Find the range (max − min) for each town. (c) Which town has more price stability? Justify your answer using at least two measures of comparison. [5 marks]

Part (a): Yahapur sum = 458, average = 458 ÷ 12 ≈ 38.17; Wahapur sum = 450, average = 450 ÷ 12 = 37.5. Part (b): Yahapur range = 59 − 24 = 35; Wahapur range = 60 − 17 = 43. Part (c): Compare using averages (close, but Yahapur slightly higher) and ranges (Yahapur smaller = more stable). Dot plot visualization can also support the conclusion about clustering or spread.

Assertion-Reason Questions for Connecting the Dots

Q1. Assertion (A):

The arithmetic mean can be understood as the fair-share or equal-share value of a group of numbers.

Reason (R):

When a total amount is divided equally among individuals, each gets the mean value, representing what each would receive if resources were distributed evenly.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both assertion and reason are true, and the reason correctly explains the assertion. The mean's fair-share interpretation is a fundamental concept shown by the guava-sharing example, where each group member gets exactly the mean value when items are distributed equally.

Q2. Assertion (A):

When comparing the performance of two cricketers who played different numbers of matches, the average (mean) is a more appropriate measure than the total runs scored.

Reason (R):

The average accounts for the different group sizes and provides a fair comparison of performance per match, whereas totals can be misleading when group sizes differ.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both assertion and reason are true, and the reason correctly explains the assertion. The Shubman vs Yashasvi example (Series 2) demonstrates this: although Shubman scored 110 total vs Yashasvi's 96, Yashasvi's average (24) exceeds Shubman's (22), making the comparison fair through means.

Q3. Assertion (A):

The term 'samamiti' from ancient Indian mathematics literally translates to 'equal measure' or 'equalising measure'.

Reason (R):

Ancient Indian scholars did not consider the arithmetic mean as a representative value because their terminology focused only on measurement, not equality.

Answer: C — A is true but R is false — The assertion is true—'sama' means equal, and the term does emphasize equality. However, the reason is false. Ancient scholars DID perceive the mean as a representative and equalising value, which is why they chose terminology emphasizing equality and balance.

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Case-Based Questions (CBQ) for Connecting the Dots

Case Study:

A school conducted a survey about students' study time. Five Grade 7 students reported their daily study hours: Aman studied 2 hours, Bhavna studied 3 hours, Chetan studied 4 hours, Divya studied 2 hours, and Eshan studied 4 hours. The school wants to understand the typical study pattern and plan better learning activities. To make a fair comparison of how much time a typical student spends studying daily, the school decided to calculate the average study time.

Q1. What is the total study time for all five students?

  • A. 12 hours
  • B. 15 hours ✓
  • C. 18 hours
  • D. 20 hours

Answer: B

Q2. What is the average (mean) daily study time per student?

  • A. 2.5 hours
  • B. 3 hours ✓
  • C. 3.5 hours
  • D. 4 hours

Answer: B

Q3. If a sixth student, Fatima, joins and studies 5 hours daily, what will be the new average study time?

  • A. 3 hours
  • B. 3.33 hours
  • C. 3.67 hours ✓
  • D. 4 hours

Answer: C

Q4. Which statement correctly interprets the average study time as a fair-share concept?

  • A. Each student should study exactly 3 hours daily
  • B. Three students study, and two don't study
  • C. The best student studies 3 hours daily
  • D. If all students studied equally, each would study 3 hours daily ✓

Answer: D

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