Question

An SBI health insurance agent found the following data for distribution of ages of 100 policy holders. The health insurance policies are given to persons of age 15 years and onwards, but less than 60 years. Find the modal age and median age of the policy holders.

Hint:

Always double-check that your calculated mode and median values lie within the respective modal and median class intervals.
Here, both \(36.76\) and \(35.76\) lie inside the class interval \(35 - 40\), confirming the calculation is logically consistent!

Answer 1

Step 1: Understanding the Question:
We are given a frequency distribution of the ages of 100 policy holders.
We need to calculate both the modal age and the median age of this data.

Step 2: Key Formula or Approach:
1. Modal Age Formula:
\[ \text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h \]
where \(l\) is the lower limit of the modal class, \(f_1\) is the frequency of the modal class, \(f_0\) is the preceding frequency, \(f_2\) is the succeeding frequency, and \(h\) is the class size.
2. Median Age Formula:
\[ \text{Median} = l + \left(\frac{\frac{N}{2} - cf}{f}\right) \times h \]
where \(l\) is the lower limit of the median class, \(cf\) is the cumulative frequency of the preceding class, \(f\) is the frequency of the median class, and \(h\) is the class size.

Step 3: Detailed Explanation:
1. Construct the Cumulative Frequency (\(cf\)) table:
- \(15 - 20\): \(f = 2\), \(cf = 2\)
- \(20 - 25\): \(f = 4\), \(cf = 6\)
- \(25 - 30\): \(f = 18\), \(cf = 24\)
- \(30 - 35\): \(f = 21\), \(cf = 45\)
- \(35 - 40\): \(f = 33\), \(cf = 78\)
- \(40 - 45\): \(f = 11\), \(cf = 89\)
- \(45 - 50\): \(f = 3\), \(cf = 92\)
- \(50 - 55\): \(f = 6\), \(cf = 98\)
- \(55 - 60\): \(f = 2\), \(cf = 100\)
2. Calculate the Modal Age:
- The maximum frequency is \(33\), which lies in the class \(35 - 40\). Thus, modal class is \(35 - 40\).
- Parameters: \(l = 35\), \(f_1 = 33\), \(f_0 = 21\), \(f_2 = 11\), \(h = 5\).
- Substitute values:
\[ \text{Mode} = 35 + \left(\frac{33 - 21}{2(33) - 21 - 11}\right) \times 5 \]
\[ \text{Mode} = 35 + \left(\frac{12}{66 - 32}\right) \times 5 = 35 + \left(\frac{12}{34}\right) \times 5 = 35 + 1.76 = 36.76\text{ years} \]
3. Calculate the Median Age:
- Total \(N = 100 \implies \frac{N}{2} = 50\).
- The cumulative frequency just greater than 50 is \(78\), which corresponds to the class \(35 - 40\). Thus, median class is \(35 - 40\).
- Parameters: \(l = 35\), \(cf = 45\), \(f = 33\), \(h = 5\).
- Substitute values:
\[ \text{Median} = 35 + \left(\frac{50 - 45}{33}\right) \times 5 \]
\[ \text{Median} = 35 + \left(\frac{5}{33}\right) \times 5 = 35 + \frac{25}{33} \approx 35 + 0.76 = 35.76\text{ years} \]

Step 4: Final Answer:
The modal age of the policy holders is \(36.76\text{ years}\) and the median age is \(35.76\text{ years}\).