Class 8 — Mathematics (NCERT)
Data Handling
Class 8
- ✓By the end of this lesson students will be able to organise raw data into frequency distribution tables, including grouped frequency distributions.
- ✓By the end of this lesson students will be able to construct and interpret pie charts (circle graphs) to represent data.
- ✓By the end of this lesson students will be able to understand the basic concepts of probability, including experiment, outcome, and event.
- ✓By the end of this lesson students will be able to calculate the probability of simple events.
Key concepts
Data is a collection of numerical facts or information gathered for a specific purpose. Raw data is data in its original form, as collected. Grouped data is data organised into classes or groups to make it more manageable and understandable.
The frequency of an observation is the number of times that particular observation occurs in the given data. It tells us how often a specific value appears.
A frequency distribution table is a systematic way of organising raw data by listing each observation or class interval and its corresponding frequency. It typically includes columns for 'Observations/Class Intervals', 'Tally Marks', and 'Frequency'. Tally marks are used to count the occurrences of each observation.
When the number of observations is very large, it becomes difficult to make a frequency distribution table for each individual observation. In such cases, we group the data into class intervals. Each group is called a class interval. The lower class limit is the smallest value in a class interval, and the upper class limit is the largest value. The class size or width is the difference between the upper and lower class limits of a class interval (or the difference between the lower limits of two consecutive classes).
A pie chart or a circle graph shows the relationship between a whole and its parts. The whole circle represents the sum of all categories, and the size of each sector (part) is proportional to the value it represents. The central angle for each sector is calculated to represent its share of the total.
Probability is a measure of the likelihood or chance that an event will occur. An experiment is an operation which can produce some well-defined outcomes. An outcome is a possible result of an experiment. An event is a collection of one or more outcomes of an experiment. Outcomes are equally likely if each outcome has the same chance of occurring.
Key facts to remember
- 1Data is a collection of numerical facts or information.
- 2Frequency is the number of times an observation occurs.
- 3A frequency distribution table organises data with observations, tally marks, and frequencies.
- 4Grouped frequency distributions are used for large datasets, using class intervals.
- 5A pie chart (circle graph) represents parts of a whole, where the sum of all central angles is 360°.
- 6The formula for the central angle in a pie chart is (Value of component / Total value) × 360°.
- 7Probability measures the chance of an event occurring.
- 8The probability of an event E is P(E) = (Number of favourable outcomes) / (Total number of possible outcomes).
Worked examples
Example 1
The marks obtained by 20 students in a Mathematics test (out of 10) are given below:\n8, 7, 9, 6, 8, 7, 5, 9, 10, 8, 6, 7, 8, 9, 7, 6, 5, 8, 7, 9.\nConstruct a frequency distribution table for this data.
Answer
The frequency distribution table is as follows:\nMarks Obtained | Tally Marks | Frequency\n--------------|-------------|----------\n5 | || | 2\n6 | ||| | 3\n7 | ||||| | 5\n8 | ||||| | 5\n9 | |||| | 4\n10 | | | 1\n--------------|-------------|----------\nTotal | | 20
Always double-check the total frequency to ensure it matches the total number of observations given in the problem.
Example 2
A survey was conducted to find the type of music that a group of 100 young people liked in a city. The results are shown in the following table:\nType of Music | Number of People\n--------------|-----------------\nClassical | 10\nSemi Classical| 20\nFolk | 40\nLight | 30\nDraw a pie chart to represent this data.
Answer
The central angles for the pie chart are:\n- Classical: 36°\n- Semi Classical: 72°\n- Folk: 144°\n- Light: 108°\n\nTo draw the pie chart, first draw a circle. Then, using a protractor, draw sectors with these calculated angles, starting from a radius and proceeding clockwise or anti-clockwise. Label each sector with its corresponding music type.
Accuracy in measuring angles with a protractor is crucial for a correct pie chart. Always check that the sum of central angles is 360°.
Example 3
A bag contains 3 red balls and 5 blue balls. A ball is drawn at random from the bag. What is the probability of drawing:\n(i) a red ball?\n(ii) a blue ball?
Answer
(i) The probability of drawing a red ball is 3/8.\n(ii) The probability of drawing a blue ball is 5/8.
Probabilities are always expressed as fractions between 0 and 1 (inclusive). Always simplify the fraction to its lowest terms if possible.
Common mistakes
- ✗Incorrectly counting tally marks, especially when dealing with large numbers or complex patterns.
- ✗Errors in calculating central angles for pie charts, often by using incorrect total values or making calculation mistakes.
- ✗Not ensuring that the sum of central angles in a pie chart equals 360°.
- ✗Confusing favourable outcomes with total possible outcomes when calculating probability.
- ✗Not simplifying the probability fraction to its lowest terms, which is generally expected in final answers.
Exam tips
- ★Read the question carefully to understand whether a simple frequency table or a grouped frequency distribution is required.
- ★For pie charts, always show the calculation of central angles step-by-step. Use a compass for drawing the circle and a protractor for accurate angles.
- ★In probability questions, clearly identify the total number of possible outcomes and the number of favourable outcomes before applying the formula.
- ★Double-check all calculations, especially sums and divisions, to avoid arithmetic errors.
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