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: Build the cumulative frequency table.
From the given class intervals of width 5 years and frequencies (total 100), the cumulative frequencies are:
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.

Step 2: Find the median class first (reversing the usual order).
Total $N=100$, so $\frac{N}{2}=50$. The first class whose cumulative frequency exceeds 50 is $35$-$40$ (cf=78), so this is the median class, with $l=35$, preceding cf$=45$, $f=33$, $h=5$:
\[ \text{Median} = 35+\left(\frac{50-45}{33}\right)\times5 = 35+\frac{25}{33} \approx 35.76\text{ years} \]

Step 3: Find the modal class.
The class with the highest frequency is also $35$-$40$ ($f_1=33$), with neighbouring frequencies $f_0=21$ and $f_2=11$, $l=35$, $h=5$:
\[ \text{Mode} = 35+\left(\frac{33-21}{66-21-11}\right)\times5 = 35+\left(\frac{12}{34}\right)\times5 \approx 36.76\text{ years} \]

Step 4: A quick graphical cross-check (as taught alongside the formula method).
The same mode can be located on a histogram of this data, by drawing diagonal lines from the top corners of the tallest bar (35-40) to the top corners of its two neighbouring bars, where these diagonals cross gives the mode. Plotting a "less than" cumulative frequency curve and marking where the horizontal line from $y=50$ meets it gives the median, both landing at the values found above.

Final Answer:
The median age is about $35.76$ years and the modal age is about $36.76$ years.
\[ \boxed{\text{Median} \approx 35.76\text{ years},\ \text{Mode} \approx 36.76\text{ years}} \]