Class 7 Notes

Class 7 Mathematics Notes Chapter 3 (Data Handling) – Mathematics Book

Philoid

Philoid

— 6 min read

Mathematics
Alright class, let's get focused. Today we're diving deep into Chapter 3, 'Data Handling'. This is a crucial chapter, not just for your school exams, but the concepts form the basis for quantitative sections in many government exams. Pay close attention!

Chapter 3: Data Handling - Detailed Notes for Exam Preparation

1. Introduction to Data

  • What is Data? Data is a collection of facts, figures, or information gathered through observation or measurement. Examples: Marks obtained by students, temperature recorded daily, runs scored by a cricketer.
  • Raw Data: Data collected in its original form, without any organization or processing. It can be difficult to interpret directly.
  • Need for Organization: Raw data needs to be organized systematically to:
    • Understand it easily.
    • Extract meaningful information.
    • Compare different values.
    • Make decisions based on the data.

2. Organizing Data

  • Frequency Distribution Table: A common way to organize data. It shows how often each value (or group of values) occurs in a dataset.

    • Frequency: The number of times a particular observation (value) occurs in the data.
    • Tally Marks: A system used for counting frequencies quickly. We use vertical bars (|) for each occurrence. Every fifth mark is drawn diagonally across the previous four (||||), forming groups of 5 for easy counting.

  • Example: Marks obtained by 15 students in a test (out of 10): 7, 8, 5, 6, 7, 9, 8, 7, 4, 5, 6, 7, 8, 9, 7

    Marks Tally Marks Frequency
    4
    5
    6
    7
    8
    9
    Total=15

3. Representative Values (Measures of Central Tendency)

These are single values that attempt to describe the center or typical value of a dataset.

  • a) Arithmetic Mean (or Average):

    • Definition: The most common representative value. It is calculated by dividing the sum of all observations by the total number of observations.
    • Formula:
      Mean = (Sum of all observations) / (Number of observations)
    • Use: Useful when data points are relatively close together. It is affected by extreme values (outliers).
    • Example: For the marks data above (7, 8, 5, 6, 7, 9, 8, 7, 4, 5, 6, 7, 8, 9, 7):
      Sum = 7+8+5+6+7+9+8+7+4+5+6+7+8+9+7 = 103
      Number of observations = 15
      Mean = 103 / 15 ≈ 6.87

  • b) Mode:

    • Definition: The observation that occurs most frequently in the dataset.
    • How to find: Arrange data (optional but helpful) or use a frequency table. Identify the value with the highest frequency.
    • Use: Useful for categorical data (e.g., most popular shirt size) or when identifying the most common item. A dataset can have more than one mode (multimodal) or no mode if all values occur with the same frequency.
    • Example: For the marks data (7, 8, 5, 6, 7, 9, 8, 7, 4, 5, 6, 7, 8, 9, 7), the mark '7' appears 5 times, which is more than any other mark.
      Mode = 7

  • c) Median:

    • Definition: The middle value of a dataset when it is arranged in ascending (smallest to largest) or descending (largest to smallest) order.
    • How to find:
      1. Arrange the data in ascending or descending order.
      2. Count the number of observations (n).
      3. If 'n' is odd: The median is the single middle value. The position is (n+1)/2-th term.
      4. If 'n' is even: The median is the average of the two middle values. The positions are n/2-th and (n/2 + 1)-th terms. Median = (Value at n/2 + Value at n/2 + 1) / 2.
    • Use: Useful when the data has extreme values (outliers) because it is not affected by them as much as the mean.
    • Example: For the marks data, first arrange in ascending order: 4, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9
      Number of observations (n) = 15 (odd)
      Position of median = (15 + 1) / 2 = 16 / 2 = 8th term.
      The 8th value in the ordered list is 7.
      Median = 7

  • d) Range:

    • Definition: The difference between the highest observation and the lowest observation in a dataset. It gives an idea of the spread or variability of the data.
    • Formula: Range = Highest Observation - Lowest Observation
    • Example: For the marks data (4, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9):
      Highest Observation = 9
      Lowest Observation = 4
      Range = 9 - 4 = 5

4. Use of Bar Graphs

  • Purpose: To represent data visually using bars (rectangles) of uniform width. The height (or length) of the bars is proportional to the value they represent. They are excellent for comparing quantities across different categories.
  • Key Features:
    • Title: Explains what the graph represents.
    • Axes: Horizontal axis (X-axis) usually shows categories, Vertical axis (Y-axis) usually shows the numerical values. Axes must be labelled.
    • Scale: A consistent scale must be chosen for the axis representing numerical values (usually Y-axis). The scale determines the height of the bars.
    • Bars: Should be of equal width and have equal spacing between them.
  • Double Bar Graph:
    • Purpose: Used to compare two sets of data simultaneously for the same categories. Example: Comparing marks of two different terms for the same set of students.
    • Key: A legend or key is needed to distinguish between the two sets of bars (e.g., using different colours or patterns).

5. Chance and Probability

  • Chance: Refers to the likelihood or possibility of an event happening. Many situations in real life involve uncertainty (e.g., raining tomorrow, winning a lottery).
  • Experiment: An action or trial whose result is uncertain (e.g., tossing a coin, rolling a die).
  • Outcome: A possible result of an experiment (e.g., getting 'Heads' when tossing a coin; getting a '4' when rolling a die).
  • Event: A specific outcome or a collection of outcomes from an experiment (e.g., getting an even number when rolling a die - outcomes are 2, 4, 6).
  • Probability: The measure of the likelihood that an event will occur. It is calculated as the ratio of the number of favourable outcomes to the total number of possible outcomes.
    • Formula:
      Probability of an event (P(E)) = (Number of outcomes favourable to the event) / (Total number of possible outcomes)
    • Range of Probability: Probability values always lie between 0 and 1 (inclusive).
      • P(E) = 0 means the event is impossible.
      • P(E) = 1 means the event is certain.
      • Values between 0 and 1 indicate varying degrees of likelihood.
    • Example: Probability of getting a 'Head' when tossing a fair coin.
      Total possible outcomes = {Head, Tail} = 2
      Favourable outcome = {Head} = 1
      P(Head) = 1 / 2
    • Example: Probability of getting a '5' when rolling a standard six-sided die.
      Total possible outcomes = {1, 2, 3, 4, 5, 6} = 6
      Favourable outcome = {5} = 1
      P(Getting a 5) = 1 / 6
    • Example: Probability of getting an odd number when rolling a standard six-sided die.
      Total possible outcomes = {1, 2, 3, 4, 5, 6} = 6
      Favourable outcomes = {1, 3, 5} = 3
      P(Odd Number) = 3 / 6 = 1 / 2

Key Takeaways for Exams:

  • Know the definitions and formulas for Mean, Median, Mode, and Range.
  • Understand how to calculate each measure for a given dataset.
  • Know when Median is preferred over Mean (presence of outliers).
  • Be able to construct and interpret Frequency Distribution Tables (using Tally Marks).
  • Understand the purpose and components of Bar Graphs and Double Bar Graphs. Be able to read information from them.
  • Understand the basic concepts of probability, including the formula and the range (0 to 1). Be able to calculate simple probabilities.

Multiple Choice Questions (MCQs)

  1. The arithmetic mean of the first 5 natural numbers (1, 2, 3, 4, 5) is:
    a) 2
    b) 3
    c) 4
    d) 15

  2. The mode of the data: 2, 3, 5, 6, 3, 4, 3, 5, 2, 3 is:
    a) 2
    b) 5
    c) 3
    d) 6

  3. What is the median of the following numbers: 15, 12, 18, 9, 10, 16?
    a) 12
    b) 13.5
    c) 15
    d) 14

  4. The range of the data: 35, 42, 21, 55, 28, 31, 21 is:
    a) 21
    b) 55
    c) 34
    d) 30

  5. A standard six-sided die is rolled once. What is the probability of getting a prime number? (Prime numbers on a die are 2, 3, 5)
    a) 1/6
    b) 1/3
    c) 1/2
    d) 2/3

  6. Which measure of central tendency is most affected by extreme values (outliers) in the data?
    a) Mean
    b) Median
    c) Mode
    d) Range

  7. A coin is tossed twice. What is the probability of getting at least one Head? (Possible outcomes: HH, HT, TH, TT)
    a) 1/4
    b) 1/2
    c) 3/4
    d) 1

  8. A double bar graph is used to:
    a) Represent data using pictures.
    b) Compare two sets of data simultaneously across categories.
    c) Show the frequency of data values.
    d) Find the average of the data.

  9. The tally mark representation for the number 7 is:
    a) |||| ||
    b) ||||| ||
    c) |||| ||
    d) ||| ||||

  10. If the probability of an event happening is 0.7, what is the probability of the event not happening?
    a) 0
    b) 1
    c) 0.3
    d) 0.7

Answer Key for MCQs:

  1. b) 3 (Sum=15, Count=5, Mean=15/5=3)
  2. c) 3 (Occurs 4 times, more than any other number)
  3. b) 13.5 (Ordered: 9, 10, 12, 15, 16, 18. n=6 (even). Middle values are 12 and 15. Median = (12+15)/2 = 27/2 = 13.5)
  4. c) 34 (Highest=55, Lowest=21. Range = 55-21 = 34)
  5. c) 1/2 (Favourable outcomes {2, 3, 5} = 3. Total outcomes = 6. P = 3/6 = 1/2)
  6. a) Mean
  7. c) 3/4 (Favourable outcomes {HH, HT, TH} = 3. Total outcomes = 4. P = 3/4)
  8. b) Compare two sets of data simultaneously across categories.
  9. c) |||| ||
  10. c) 0.3 (P(not E) = 1 - P(E) = 1 - 0.7 = 0.3)

Study these notes carefully. Practice solving problems from your textbook and other resources. Good luck with your preparation!

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