NCERT Solutions for Class 10 Maths Chapter 14 Exercise 14.2

Welcome, Class 10 students, to your definitive guide on NCERT Class 10 Maths Chapter 14, Exercise 14.2. In this exercise, we focus on learning how to calculate the Mode of Grouped Data. In earlier classes, you learned that the mode is simply the observation that occurs most frequently. However, when data is grouped into class intervals, we cannot determine the mode simply by looking at the frequencies. Instead, we must locate the Modal Class—the class interval with the highest frequency—and use a specific formula to estimate the precise mode. Mastering this concept is critical for your CBSE board exams, as mode-based questions are highly scoring and frequently asked in Section C and D. YoLearn AI's interactive sketchpad and step-by-step tutorials will help you visualize each step, manage calculations with precision, and avoid common algebraic mistakes. Let's dive in and master the modal class formula together!

Understanding Mode of Grouped Data

In statistics, the mode represents the value that has the maximum frequency of occurrence. For ungrouped data, identifying the mode is a simple visual task of counting repetitions. However, in grouped data (which is presented in class intervals like 0-10, 10-20, etc.), we can only determine the 'Modal Class' by identifying which interval has the maximum frequency. To find the actual numeric mode within that class interval, we rely on interpolation using the standard formula: Mode = l + ((f1 - f0) / (2f1 - f0 - f2)) * h. In this formula, l is the lower limit of the modal class, h is the width of the class interval (assuming all classes are of equal width), f1 is the frequency of the modal class, f0 is the frequency of the class preceding the modal class, and f2 is the frequency of the class succeeding the modal class. Note that the mode always lies inside the boundary of the modal class. If your final calculated mode falls outside the modal class interval, it is a clear indicator of an arithmetic error.

The Step-by-Step Method to Calculate Mode

  1. Locate the Modal Class — Scan the frequency (f) column of your grouped frequency distribution table. The class interval corresponding to the highest frequency is your Modal Class.
  2. Identify the Five Crucial Variables — From the modal class, write down: l (its lower limit), h (class width, upper limit - lower limit), f1 (its frequency), f0 (frequency of the preceding class), and f2 (frequency of the succeeding class).
  3. Substitute Variables into the Mode Formula — Write down the formula: Mode = l + ((f1 - f0) / (2f1 - f0 - f2)) * h. Substitute your values carefully into the fraction.
  4. Solve the Fraction first, then Add — Calculate the numerator (f1 - f0) and denominator (2f1 - f0 - f2). Multiply this quotient by h, then finally add the lower limit l. Ensure your final answer lies within the modal class boundaries.

Avoid Board Exam Calculation Traps

  1. Order of Operations (BODMAS): The most frequent error is adding l to the fraction before multiplying by h. Always compute the entire term ((f1 - f0) / (2f1 - f0 - f2)) * h first, and add l at the very end.
  1. Sign Mistakes in Denominator: The denominator is 2f1 - f0 - f2. Be careful with subtraction: subtract both f0 and f2 from 2f1.
  1. Sanity Check: Your calculated Mode must lie between the lower limit (l) and upper limit of your modal class. If your modal class is 30-40, your mode must be a value between 30 and 40. If it is 29.5 or 41, recheck your steps immediately!

Practice Questions with Solutions

  • Q: The following table shows the ages of patients admitted to a hospital during a year: Age (in years): 5-15, 15-25, 25-35, 35-45, 45-55, 55-65. Number of patients: 6, 11, 21, 23, 14, 5. Find the Mode of the data. A: Step 1: Identify the modal class. Looking at the frequencies, the maximum frequency is 23, which belongs to the class interval 35-45. Therefore, the modal class is 35-45. Step 2: Identify the variables: l (lower limit of modal class) = 35 h (class size) = 10 f1 (frequency of modal class) = 23 f0 (frequency of preceding class) = 21 f2 (frequency of succeeding class) = 14 Step 3: Substitute values into the formula: Mode = l + ((f1 - f0) / (2f1 - f0 - f2)) h Mode = 35 + ((23 - 21) / (2(23) - 21 - 14)) 10 Mode = 35 + (2 / (46 - 35)) 10 Mode = 35 + (2 / 11) 10 = 35 + 1.81 Step 4: Calculate the final value: Mode = 36.81 years (approx). Final answer: The mode of the given data is approximately 36.81 years.
  • Q: Calculate the mode of the following distribution: Class Interval: 0-20, 20-40, 40-60, 60-80, 80-100, 100-120. Frequency: 10, 35, 52, 61, 38, 29. A: Step 1: Find the class interval with the highest frequency. The highest frequency is 61, which belongs to the class interval 60-80. Thus, the modal class is 60-80. Step 2: Identify the parameters: l = 60 h = 20 f1 = 61 f0 = 52 f2 = 38 Step 3: Use the formula: Mode = l + ((f1 - f0) / (2f1 - f0 - f2)) h Mode = 60 + ((61 - 52) / (2(61) - 52 - 38)) 20 Mode = 60 + (9 / (122 - 90)) 20 Mode = 60 + (9 / 32) 20 Mode = 60 + 5.625 Step 4: Calculate final answer: Mode = 65.625. Final answer: The mode is 65.625.
  • Q: Find the mode of the following data: Marks: 10-20, 20-30, 30-40, 40-50, 50-60. Number of students: 4, 8, 10, 12, 6. A: Step 1: The maximum frequency is 12, which corresponds to the class interval 40-50. This is our modal class. Step 2: Extract variables: l = 40 h = 10 f1 = 12 f0 = 10 f2 = 6 Step 3: Solve using formula: Mode = 40 + ((12 - 10) / (2(12) - 10 - 6)) 10 Mode = 40 + (2 / (24 - 16)) 10 Mode = 40 + (2 / 8) * 10 Mode = 40 + 2.5 Step 4: Calculate: Mode = 42.5 marks. Final answer: The mode of the given marks data is 42.5.
  • Q: Find the mode of the following car traffic distribution: Cars: 0-10, 10-20, 20-30, 30-40, 40-50, 50-60, 60-70, 70-80. Frequency: 7, 14, 13, 12, 20, 11, 15, 8. A: Step 1: The highest frequency is 20, belonging to the class interval 40-50. The modal class is 40-50. Step 2: Identify parameters: l = 40 h = 10 f1 = 20 f0 = 12 f2 = 11 Step 3: Apply the Mode formula: Mode = l + ((f1 - f0) / (2f1 - f0 - f2)) h Mode = 40 + ((20 - 12) / (2(20) - 12 - 11)) 10 Mode = 40 + (8 / (40 - 23)) 10 Mode = 40 + (8 / 17) 10 Mode = 40 + 4.7 Step 4: Final calculation: Mode = 44.7 cars. Final answer: The mode of the distribution is approximately 44.7 cars.

Frequently Asked Questions

Can the mode be calculated if class intervals are not continuous?

No, class intervals must be continuous to apply the mode formula. If they are discontinuous (like 1-10, 11-20), you must convert them into continuous intervals (e.g., 0.5-10.5, 10.5-20.5) first.

What happens if there are two class intervals with the same maximum frequency?

Such data is called bi-modal. In the standard Class 10 CBSE curriculum, you will only be asked questions with a single, unique modal class to keep calculations straightforward.

Can the value of the mode be larger than the upper limit of the modal class?

No, mathematically the mode must always lie within the boundaries of the modal class. If your answer falls outside, recheck your addition and multiplication steps immediately.