Organisation of Data Class 11 NCERT - Statistics Chapter 3

Understanding Data Classification and Variables

Classification is the arrangement of data into groups or classes based on their common attributes. There are four major bases of classification: Geographical (by location), Chronological (by time), Qualitative (by attributes like gender, literacy, or religion), and Quantitative (by measurable numerical attributes like height, income, or marks). A vital building block here is the concept of a 'Variable'. A variable is a characteristic that can be measured and whose value changes. Variables are split into two categories: Discrete variables which increase by finite jumps (e.g., number of students in a class: 30, 31, 32) and Continuous variables which can assume any numerical value within a range (e.g., height of students: 155.5 cm, 156.2 cm).

How to Construct a Continuous Frequency Distribution

1. Find the Range — Locate the highest and lowest values in your raw dataset. Subtract the lowest value from the highest value to determine the Range.
2. Determine Class Intervals — Decide on a suitable number of classes (usually between 5 and 15) and calculate class width. Keep intervals uniform, such as 0-10, 10-20, 20-30.
3. Mark the Tallies — Go through your raw dataset. Record a tally mark (|) in the appropriate class interval for each data point. Bundle tallies in groups of five by drawing a diagonal line through four upright marks.
4. Sum up the Frequencies — Count the tally marks for each interval to get the frequency of each class, and ensure the sum of all frequencies matches the total number of observations.

Exam Tip: Converting Inclusive to Exclusive Intervals

Students often lose marks in exams by ignoring the distinction between inclusive and exclusive classes. In the Exclusive Method (e.g., 10-20, 20-30), the upper limit is excluded from the class (a value of 20 goes into 20-30). In the Inclusive Method (e.g., 10-19, 20-29), both limits are included.

Exam Trick: If you encounter inclusive classes in a problem, convert them to exclusive classes before performing calculations. To do this, find the difference between the upper limit of one class and the lower limit of the next (usually 1), divide it by 2 (0.5), subtract 0.5 from all lower limits, and add 0.5 to all upper limits.

Practice Questions with Solutions

• Q: Convert the following inclusive series into an exclusive series: 20-29, 30-39, 40-49, 50-59.
  A: Step 1: Find the difference between the upper limit of the first class (29) and the lower limit of the second class (30). Difference = 30 - 29 = 1.
  Step 2: Calculate the correction factor by dividing the difference by 2. Correction Factor = 1 / 2 = 0.5.
  Step 3: Subtract 0.5 from each lower limit and add 0.5 to each upper limit.

  • Lower limits: 20-0.5 = 19.5; 30-0.5 = 29.5; 40-0.5 = 39.5; 50-0.5 = 49.5
  • Upper limits: 29+0.5 = 29.5; 39+0.5 = 39.5; 49+0.5 = 49.5; 59+0.5 = 59.5

  Final answer: The converted exclusive series is 19.5-29.5, 29.5-39.5, 39.5-49.5, 49.5-59.5.

• Q: Determine the class mark (midpoint) and class width for the class interval 50 - 60.
  A: Step 1: Use the class mark formula: Class Mark = (Lower Limit + Upper Limit) / 2.
  Step 2: Substitute the values: Class Mark = (50 + 60) / 2 = 110 / 2 = 55.
  Step 3: Calculate the class width using: Width = Upper Limit - Lower Limit.
  Step 4: Substitute the values: Width = 60 - 50 = 10.
  Final answer: Class Mark is 55, and Class Width is 10.

• Q: Identify whether the following are discrete or continuous variables: (i) Number of active COVID-19 cases in a city, (ii) Weight of students in a class.
  A: Step 1: Analyze variable (i). The number of cases must be a whole integer (e.g., 100 cases, not 100.25 cases). Thus, it is a discrete variable.
  Step 2: Analyze variable (ii). Weight can take fractional and precise measurements (e.g., 52.4 kg, 60.15 kg). Thus, it is a continuous variable.
  Final answer: (i) Discrete variable, (ii) Continuous variable.

• Q: Organize the following raw marks of 12 students into a simple frequency table: 5, 8, 5, 10, 8, 5, 12, 10, 8, 12, 5, 8.
  A: Step 1: Identify all unique values in ascending order: 5, 8, 10, 12.
  Step 2: Tally and count the frequencies for each value:

  • Mark 5 appears 4 times.
  • Mark 8 appears 4 times.
  • Mark 10 appears 2 times.
  • Mark 12 appears 2 times.

  Step 3: Construct the frequency distribution table:

  • Marks 5: Frequency = 4
  • Marks 8: Frequency = 4
  • Marks 10: Frequency = 2
  • Marks 12: Frequency = 2

  Step 4: Sum all frequencies to verify against total students (4 + 4 + 2 + 2 = 12).
  Final answer: Marks: 5 (f=4), 8 (f=4), 10 (f=2), 12 (f=2). Total observations = 12.

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