NCERT Solutions for Class 9 Maths Chapter 14 Statistics Exercise 14.2

Understanding Data Representation & Frequency Distribution Tables

When we collect data, it is initially in the form of raw data, which is unsorted and difficult to analyze. To make this data readable, we organize it. There are two primary ways to represent data in tabular form:

1. Ungrouped Frequency Distribution Table: Used when the number of observations is small and values do not vary widely. We list each unique value alongside its frequency (the number of times it appears).
2. Grouped Frequency Distribution Table: Used when the data set is very large. We group the values into ranges called Class Intervals (e.g., 0-10, 10-20).

Key terminology to remember includes:

• Lower Class Limit: The lowest value in a class interval.
• Upper Class Limit: The highest value in a class interval.
• Class Width/Size: The difference between the upper and lower limits (e.g., class size of 10-20 is 10).
• Range: The difference between the maximum and minimum values in the raw dataset.

How to Construct a Grouped Frequency Distribution Table

1. Step 1: Identify Range and Class Size — Find the maximum and minimum observations in your raw data. Subtract the minimum from the maximum to find the range. Decide on a class width (usually 5 or 10, as specified in your NCERT problem).
2. Step 2: Define Class Intervals — Set up class intervals starting from a round number lower than or equal to your minimum value. For example, if the lowest value is 3 and class size is 5, begin with 0-5, 5-10, 10-15, etc.
3. Step 3: Record Frequencies using Tally Marks — Go through each raw data value. Draw a vertical tally bar '|' for each occurrence in the corresponding class interval. For every fifth occurrence, draw a diagonal strike-through line '||||' through the four vertical bars to make groups of 5.
4. Step 4: Total the Frequencies — Count the tally marks for each interval and record them under the 'Frequency' column. Add up all frequencies; the total must equal the number of observations given in the problem.

Crucial Exam Tip: The Upper Limit Rule

A very common mistake students make is deciding which interval a boundary value belongs to. In the exclusive grouping method (like 0-10, 10-20), the upper limit is always excluded from the interval and included in the next interval.

• Example: The number 10 belongs to the class 10-20, NOT 0-10.
• Rule of thumb: An interval $[a, b)$ includes all values from $a$ up to, but not including, $b$. Always double-check your counts for numbers that match class boundaries to prevent losing marks!

Practice Questions with Solutions

• Q: The blood groups of 30 students of Class IX are recorded as follows: A, B, O, O, AB, O, A, O, B, A, O, B, A, O, O, A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O. Represent this data in the form of a frequency distribution table. Which is the most common and which is the rarest blood group? A: Step 1: List the distinct blood groups: A, B, AB, O. Step 2: Count the frequency of each group using tally marks: - Blood Group A: 9 students - Blood Group B: 6 students - Blood Group AB: 3 students - Blood Group O: 12 students Step 3: Sum the frequencies: 9 + 6 + 3 + 12 = 30 (matches the total student count). Step 4: Identify rarest and most common: - Most common (highest frequency): Blood Group O (12) - Rarest (lowest frequency): Blood Group AB (3) Final answer: Blood Group O is the most common, and Blood Group AB is the rarest.
• Q: The heights (in cm) of 12 students are: 150, 161, 154, 165, 168, 150, 153, 154, 162, 150, 151, 162. Construct a grouped frequency distribution table taking class intervals as 150-155, 155-160, 160-165, 165-170. A: Step 1: Set up the class intervals: 150-155, 155-160, 160-165, 165-170. Step 2: Sort the heights into their corresponding intervals, applying the upper limit exclusion rule: - 150-155 (includes 150, 154, 150, 153, 154, 150, 151): 7 students - 155-160 (no values in this range): 0 students - 160-165 (includes 161, 162, 162): 3 students - 165-170 (includes 165, 168): 2 students Step 3: Compile into a table and verify total frequency: 7 + 0 + 3 + 2 = 12. Final answer: The table shows intervals 150-155 (Freq: 7), 155-160 (Freq: 0), 160-165 (Freq: 3), and 165-170 (Freq: 2).
• Q: The relative humidity (in %) of a certain city for a month of 30 days was as follows: 98.1, 98.6, 99.2, 90.3, 86.5, 95.3, 92.9, 96.3, 94.2, 95.1, 89.2, 92.3, 97.1, 93.5, 92.7, 95.1, 97.2, 93.3, 95.2, 97.3, 96.2, 92.1, 84.9, 90.2, 95.7, 98.3, 97.3, 96.1, 92.1, 89.0. Find the Range of this data. A: Step 1: Identify the maximum relative humidity value from the given data. Max value = 99.2%. Step 2: Identify the minimum relative humidity value from the given data. Min value = 84.9%. Step 3: Apply the formula for Range: Range = Maximum Value - Minimum Value. Step 4: Calculate: Range = 99.2 - 84.9 = 14.3%. Final answer: The range of the data is 14.3%.
• Q: Thirty children were asked about the number of hours they watched TV programs in the previous week. The results were: 1, 6, 2, 3, 5, 12, 5, 8, 4, 8, 10, 3, 4, 12, 2, 8, 15, 1, 17, 6, 3, 2, 8, 5, 9, 6, 8, 7, 14, 12. Construct a grouped frequency table with class width 5, starting with interval 0-5. How many children watched television for 15 or more hours a week? A: Step 1: Create class intervals of width 5: 0-5, 5-10, 10-15, 15-20. Step 2: Distribute the values into intervals: - 0-5 (includes 1, 2, 3, 4, etc. - excluding 5): 10 children - 5-10 (includes 5, 6, 7, 8, 9 - excluding 10): 13 children - 10-15 (includes 10, 12, 14 - excluding 15): 5 children - 15-20 (includes 15, 17): 2 children Step 3: Sum the frequencies: 10 + 13 + 5 + 2 = 30 children. Step 4: Find children watching TV for 15 or more hours. This corresponds to the class interval 15-20, which has 2 children. Final answer: 2 children watched television for 15 or more hours a week.

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