Class 8 Mathematics Notes Chapter 5 (Data Handling) – Mathematics Book

Alright class, let's get straight into Chapter 5: Data Handling. This chapter is fundamental, not just for your exams but also for understanding how information is presented and interpreted in the real world, which is often tested in government exams. Pay close attention to the concepts and definitions.

Chapter 5: Data Handling - Detailed Notes for Exam Preparation

1. What is Data?

• Data is a collection of facts, figures, or information gathered for a specific purpose.
• Raw Data: Data collected in its original form, often unorganized. Example: Marks obtained by 10 students in a test: 25, 30, 15, 45, 50, 15, 30, 45, 25, 30.
• Observation: Each individual numerical figure in a set of data is called an observation.
• Array: Arranging raw data in ascending or descending order of magnitude. Example (from above): 15, 15, 25, 25, 30, 30, 30, 45, 45, 50.

2. Organizing Data: Why and How?

• Raw data is difficult to interpret. Organizing it helps in drawing meaningful conclusions quickly.

• Frequency: The number of times a particular observation occurs in the data.

• Frequency Distribution Table: A table that shows the different observations and their corresponding frequencies.

  • Ungrouped Frequency Distribution Table: Used when the number of distinct observations is manageable.

    • Steps:

      1. Create three columns: Observation, Tally Marks, Frequency.
      2. List all distinct observations in the first column (preferably in ascending order).
      3. Go through the raw data one by one. For each observation, put a tally mark (|) against it in the second column. A group of four tally marks is crossed by the fifth (||||).
      4. Count the tally marks for each observation and write the total (frequency) in the third column.
      5. Sum the frequencies; it should equal the total number of observations.

    • Example (using marks data):

      Marks	Tally Marks	Frequency
      15
      25
      30
      45
      50
      	Total	10

  • Grouped Frequency Distribution Table: Used when the range of data is large. Data is grouped into convenient intervals called 'Class Intervals'.

    • Class Interval: A range into which data is grouped (e.g., 0-10, 10-20, 20-30).
    • Lower Class Limit: The smallest value in a class interval (e.g., 10 in 10-20).
    • Upper Class Limit: The largest value in a class interval (e.g., 20 in 10-20).
    • Class Size (or Width): The difference between the upper class limit and the lower class limit (e.g., 20 - 10 = 10). Class sizes should generally be equal.
    • Convention: The upper limit of a class interval is usually excluded from that interval, and included in the next. E.g., 10-20 includes values from 10 up to (but not including) 20. The value 20 would fall in the 20-30 interval. This is the exclusive method.
    • Range: The difference between the highest and lowest observation in the raw data. Helps decide the number and size of class intervals.

3. Graphical Representation of Data
Visual representations make data easier to understand and compare.

• A. Pictograph: Represents data using pictures or symbols. Less common in higher analysis due to limitations in representing exact values and large numbers. Requires a key.
• B. Bar Graph (or Bar Chart):
  • Represents data using rectangular bars of uniform width.
  • The length (or height) of the bars is proportional to the value they represent.
  • Bars are drawn with equal spacing between them.
  • Used for comparing discrete categories.
  • Double Bar Graph: Used to compare two sets of data simultaneously for the same categories (e.g., marks in two different subjects for the same set of students). Requires a key to distinguish the bars.
• C. Histogram:
  • Used to represent grouped frequency distributions (continuous data).
  • Consists of rectangular bars adjacent to each other (no gaps between bars).
  • The width of each bar corresponds to the class interval.
  • The height (or area) of each bar is proportional to the frequency of that class interval.
  • The bars are drawn on the x-axis (representing class intervals) and frequencies on the y-axis.
  • Important Distinction: Bar graphs have gaps between bars (represent discrete categories), while histograms have no gaps (represent continuous intervals).
  • Jagged Line (or Kink): If the first class interval does not start from zero, a jagged line is shown on the x-axis near the origin to indicate a break in the scale.
• D. Circle Graph (or Pie Chart):
  • Represents data as parts of a whole circle.
  • The whole circle represents the total quantity (100%).
  • The size of each sector (slice) is proportional to the value or fraction it represents.
  • Calculating Central Angle: The angle of each sector at the centre of the circle is calculated as:
    Central Angle for a component = (Value of the component / Total Value) × 360°
    OR
    Central Angle for a component = (Fraction of the component) × 360°
  • Used effectively to show the proportion or percentage distribution of different categories.

4. Chance and Probability

• Chance: The likelihood or possibility of an event occurring.
• Experiment: An action or operation which results in some well-defined outcomes.
• Random Experiment: An experiment where all possible outcomes are known, but the exact outcome cannot be predicted in advance (e.g., tossing a coin, rolling a die).
• Outcome: A possible result of an experiment.
• Equally Likely Outcomes: Outcomes that have the same chance of occurring (e.g., getting Heads or Tails when tossing a fair coin).
• Event: A specific outcome or a collection of outcomes from an experiment (e.g., getting an even number when rolling a die is an event with outcomes {2, 4, 6}).
• Probability: The measure of the likelihood of an event occurring.
  • Formula:
    Probability of an Event (P(E)) = (Number of outcomes favourable to the event) / (Total number of possible outcomes in the experiment)
  • Range of Probability: The probability of any event E lies between 0 and 1, inclusive.
    0 ≤ P(E) ≤ 1
  • Impossible Event: An event that cannot occur. Its probability is 0. (e.g., getting a 7 when rolling a standard six-sided die).
  • Certain Event: An event that is sure to occur. Its probability is 1. (e.g., getting a number less than 7 when rolling a standard six-sided die).

Key Takeaways for Exams:

• Understand the difference between raw and organized data.
• Be proficient in creating and interpreting frequency distribution tables (both ungrouped and grouped). Know the terms: class interval, limits, size.
• Know the purpose and construction of Bar Graphs, Histograms, and Pie Charts. Crucially, distinguish between Bar Graphs and Histograms.
• Be able to calculate central angles for Pie Charts.
• Understand the basic concepts of probability and be able to calculate the probability of simple events using the formula. Remember the range of probability (0 to 1).

Multiple Choice Questions (MCQs)

1. The number of times a particular observation occurs in a given data is called its:
   (A) Range
   (B) Frequency
   (C) Class Interval
   (D) Tally Mark

2. A graph that displays data using adjacent rectangular bars representing grouped frequency distributions is called a:
   (A) Bar Graph
   (B) Pictograph
   (C) Histogram
   (D) Pie Chart

3. In a grouped frequency distribution, the class interval 10-20 (exclusive method) includes the value:
   (A) 10 only
   (B) 20 only
   (C) Both 10 and 20
   (D) Values from 10 up to, but not including, 20

4. The range of the data: 35, 50, 21, 42, 15, 30, 28, 45 is:
   (A) 35
   (B) 50
   (C) 15
   (D) 35 (Calculated as 50 - 15)

5. In a Pie Chart, the central angle for a component representing 25% of the total value is:
   (A) 25°
   (B) 90°
   (C) 180°
   (D) 360°

6. What is the probability of getting a prime number when a standard six-sided die is rolled once? (Prime numbers on a die: 2, 3, 5)
   (A) 1/6
   (B) 1/3
   (C) 1/2
   (D) 2/3

7. A double bar graph is used to:
   (A) Represent grouped data
   (B) Compare two sets of data simultaneously
   (C) Show parts of a whole
   (D) Represent data using pictures

8. Tally marks are used in frequency tables to:
   (A) Calculate the range
   (B) Determine the class size
   (C) Count observations quickly
   (D) Draw the histogram

9. The probability of an event that is certain to happen is:
   (A) 0
   (B) 1/2
   (C) 1
   (D) Greater than 1

10. Which graphical representation is most suitable for showing the proportion of expenditure on different items in a family's monthly budget?
    (A) Histogram
    (B) Bar Graph
    (C) Pictograph
    (D) Pie Chart

Answer Key:

1. (B) Frequency
2. (C) Histogram
3. (D) Values from 10 up to, but not including, 20
4. (D) 35 (Highest = 50, Lowest = 15, Range = 50 - 15 = 35)
5. (B) 90° (Calculation: (25/100) * 360° = (1/4) * 360° = 90°)
6. (C) 1/2 (Favourable outcomes {2, 3, 5} = 3; Total outcomes {1, 2, 3, 4, 5, 6} = 6; Probability = 3/6 = 1/2)
7. (B) Compare two sets of data simultaneously
8. (C) Count observations quickly
9. (C) 1
10. (D) Pie Chart

Study these notes thoroughly. Practice constructing these tables and graphs, and work through probability problems. Good luck!