NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 13: Statistics and Probability - Exercise 13.3

Question 17

Step 1: Identify the modal class.

The modal class is the class interval with the highest frequency (the largest number of families). Looking at the table, the highest frequency is 41 in the interval 10000–15000. So, the modal class = 10000–15000.

Step 2: Write down the formula for Mode (grouped data):

\[ \text{Mode} = l + \dfrac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h \]

• \(l\) = lower limit of the modal class
• \(h\) = class width
• \(f_1\) = frequency of the modal class
• \(f_0\) = frequency of the class before modal class
• \(f_2\) = frequency of the class after modal class

Step 3: Write down the values:

• Modal class = 10000–15000
• \(l = 10000\)
• \(h = 5000\) (difference between 15000 and 10000)
• \(f_1 = 41\) (frequency of modal class)
• \(f_0 = 26\) (frequency of class before it: 5000–10000)
• \(f_2 = 16\) (frequency of class after it: 15000–20000)

Step 4: Substitute values in the formula:

\[ \text{Mode} = 10000 + \dfrac{(41 - 26)}{(2 \times 41 - 26 - 16)} \times 5000 \]

Step 5: Simplify step by step:

\(41 - 26 = 15\)

Denominator = \(2 \times 41 - 26 - 16 = 82 - 42 = 40\)

So fraction = \(\tfrac{15}{40}\)

Multiply with 5000: \(\tfrac{15}{40} \times 5000 = 1875\)

Step 6: Add to lower limit:

\(10000 + 1875 = 11875\)

Final Answer: The modal income is ≈ Rs 11,875.