About Data Handling Ex 5.1 Class 8 NCERT — Concepts & Solutions

Understanding Data and Grouping Concepts

To handle data effectively, we must first learn how to organize it. Raw data is often chaotic and hard to interpret. We use tally marks to count the occurrence of each data point, which gives us its frequency (the number of times a particular value occurs). When the dataset is very large, listing individual frequencies becomes tedious. To solve this, we group data into convenient groups called class intervals (e.g., 0–10, 10–20, 20–30). The lower value in an interval is the lower class limit, and the higher value is the upper class limit. The difference between the upper limit and the lower limit is the class size or width of the interval. For instance, in the interval 10–20, 10 is the lower limit, 20 is the upper limit, and the class size is 10.

Step-by-Step: Constructing a Grouped Frequency Table

1. Analyze the Range of Data — Find the minimum and maximum values in your raw dataset to decide where your intervals should begin and end.
2. Choose Appropriate Class Intervals — Divide the range into equal, non-overlapping intervals such as 100-110, 110-120, etc. Ensure the class size is uniform.
3. Record Tally Marks — Go through your raw data item by item. Place a vertical bar (|) in the respective interval column. For every fifth count, draw a diagonal line across the four bars (||||\) to represent a bundle of 5.
4. Calculate and Write Frequencies — Count the tally bundles and write the total numerical frequency for each class interval. Sum up all frequencies to verify they match the total number of data points.

When to Use a Histogram vs. Bar Graph

Crucial Exam Tips & Common Pitfalls

1. The Boundary Value Dilemma: If a data point falls exactly on a boundary (for example, the number 20 when intervals are 10–20 and 20–30), always place it in the higher class interval. So, 20 belongs to 20–30, not 10–20.
2. Tally Marking Mistake: Students often count raw values multiple times or miss some. Always tick or cross out raw data points as you register them in your tally column to avoid duplication or omission.
3. Continuous Intervals: Remember that a histogram can only be drawn when data is divided into continuous numerical intervals. You cannot draw a histogram for categories like 'Brands of cars sold' because there are no numerical intervals between categories.

Practice Questions with Solutions

• Q: For which of the following would you use a histogram to show the data? (a) The number of letters for different areas in a postman's bag. (b) The height of competitors in an athletics meet. (c) The number of cassettes produced by 5 companies. A: Step 1: Analyze option (a). 'Different areas' are non-numerical, discrete categories. There are no continuous intervals, so we cannot use a histogram. Step 2: Analyze option (b). 'Height of competitors' is continuous numerical data that can be grouped into intervals (like 150-160 cm, 160-170 cm, etc.). Hence, a histogram can be used. Step 3: Analyze option (c). '5 companies' are separate, categorical entities. No continuous intervals are possible, so a histogram cannot be used. Final answer: Option (b) is the correct scenario for using a histogram.
• Q: The weekly wages (in Rs) of 15 workers in a factory are: 830, 835, 890, 810, 835, 836, 869, 845, 898, 890, 820, 860, 832, 833, 855. Construct a frequency table with intervals like 800-810, 810-820, and so on. A: Step 1: Identify the intervals required based on the range (minimum value is 810, maximum is 898). The intervals will range from 800-810 to 890-900. Step 2: Place each wage into its corresponding group. Remember, 890 goes into 890-900, not 880-890. Step 3: Count and build the distribution: - 810-820: 810 (1 tally) -> Frequency = 1 - 820-830: 820 (1 tally) -> Frequency = 1 - 830-840: 830, 835, 835, 836, 832, 833 (6 tallies) -> Frequency = 6 - 840-850: 845 (1 tally) -> Frequency = 1 - 850-860: 855 (1 tally) -> Frequency = 1 - 860-870: 869, 860 (2 tallies) -> Frequency = 2 - 870-880: No values -> Frequency = 0 - 880-890: No values -> Frequency = 0 - 890-900: 890, 898, 890 (3 tallies) -> Frequency = 3 Step 4: Verify the total frequency: 1 + 1 + 6 + 1 + 1 + 2 + 0 + 0 + 3 = 15 workers. Final answer: The constructed frequency table has classes from 810 to 900 with respective frequencies: [810-820: 1], [820-830: 1], [830-840: 6], [840-850: 1], [850-860: 1], [860-870: 2], [890-900: 3].
• Q: Consider the class interval 250-275. Find its lower limit, upper limit, and class size. A: Step 1: The lower limit of a class interval is the smaller numerical value marking the start of the interval. Here, it is 250. Step 2: The upper limit is the larger value marking the end of the interval. Here, it is 275. Step 3: Calculate the class size (or width) by subtracting the lower limit from the upper limit. Class Size = Upper Limit - Lower Limit Class Size = 275 - 250 = 25. Final answer: Lower limit = 250, Upper limit = 275, Class size = 25.
• Q: A group of 20 students spent the following hours playing video games in a week: 5, 12, 15, 8, 2, 6, 7, 18, 11, 9, 4, 14, 13, 16, 3, 5, 8, 10, 19, 1. How many students played video games for 10 hours or more? A: Step 1: Identify the condition given in the problem: "10 hours or more". Step 2: Filter the data points that are greater than or equal to 10 from the raw list: The values are: 12, 15, 18, 11, 14, 13, 16, 10, 19. Step 3: Count the number of filtered elements. There are 9 values in total (12, 15, 18, 11, 14, 13, 16, 10, 19). Final answer: 9 students played video games for 10 hours or more.

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