Question

An SBI health insurance agent found the following data for distribution of ages of 100 policy holders. Find the modal age and median age of the policy holders.

Hint:

Median class is the first class whose cumulative frequency is greater than or equal to \( N/2 \).

Answer 1

Step 1: Understanding the Concept:
We use the grouped data formulas for Mode and Median.
Step 2: Key Formula or Approach:
Mode \( = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h \).
Median \( = l + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h \).
Step 3: Detailed Explanation:
Part 1: Calculating Mode
Highest frequency is 33 in the class interval 35 - 40.
\( l = 35 \), \( f_1 = 33 \), \( f_0 = 21 \), \( f_2 = 11 \), \( h = 5 \).
\[ \text{Mode} = 35 + \left( \frac{33 - 21}{66 - 21 - 11} \right) \times 5 \]
\[ \text{Mode} = 35 + \left( \frac{12}{34} \right) \times 5 = 35 + 1.764... \approx 36.76 \]
Part 2: Calculating Median
Total Frequency \( N = 100 \). \( N/2 = 50 \).
Cumulative Frequency (CF) table:
15-20: 2
20-25: 6
25-30: 24
30-35: 45
35-40: 78 (Median class)
For class 35-40: \( l = 35 \), \( cf \text{ (previous)} = 45 \), \( f = 33 \), \( h = 5 \).
\[ \text{Median} = 35 + \left( \frac{50 - 45}{33} \right) \times 5 \]
\[ \text{Median} = 35 + \frac{25}{33} = 35 + 0.757... \approx 35.76 \]
Step 4: Final Answer:
The modal age is approx 36.76 years and the median age is approx 35.76 years.