A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to persons having age 18 years onwards but less than 60 years.
`{:("Age (in years)","Number of policy holders","Age (in years)","Number of policy holders"),("Below 20"," "2,"Below 45"," "89),("Below 25"," "6,"Below 50"," "92),("Below 30"," "24,"Below 55"," "98),("Below 35"," "45,"Below 60"," "100),("Below 40"," "78,,):}`

Answer

To calculate the median age of the policyholders based on the given data, we will follow these steps:

### Step 1: Organize the Data into a Frequency Table
We will create a frequency table based on the age intervals provided in the question.

| Age Interval (Years) | Number of Policy Holders |
|----------------------|--------------------------|
| Below 20 | 2 |
| Below 25 | 6 |
| Below 30 | 24 |
| Below 35 | 45 |
| Below 40 | 78 |
| Below 45 | 89 |
| Below 50 | 92 |
| Below 55 | 98 |
| Below 60 | 100 |

### Step 2: Calculate the Frequency for Each Age Interval
We will calculate the frequency for each age interval by subtracting the cumulative frequencies.

| Age Interval (Years) | Frequency |
|----------------------|-----------|
| 18-20 | 2 |
| 20-25 | 4 |
| 25-30 | 18 |
| 30-35 | 21 |
| 35-40 | 33 |
| 40-45 | 11 |
| 45-50 | 3 |
| 50-55 | 6 |
| 55-60 | 2 |

### Step 3: Create the Cumulative Frequency Table
Next, we will create a cumulative frequency table.

| Age Interval (Years) | Frequency | Cumulative Frequency |
|----------------------|-----------|----------------------|
| 18-20 | 2 | 2 |
| 20-25 | 4 | 6 |
| 25-30 | 18 | 24 |
| 30-35 | 21 | 45 |
| 35-40 | 33 | 78 |
| 40-45 | 11 | 89 |
| 45-50 | 3 | 92 |
| 50-55 | 6 | 98 |
| 55-60 | 2 | 100 |

### Step 4: Identify the Median Class
To find the median, we first need to find \( n \) (the total number of policyholders), which is 100. Then, we calculate \( n/2 \):

\[ n/2 = 100/2 = 50 \]

Now, we look for the cumulative frequency that is just greater than or equal to 50. From the cumulative frequency table, we see that the median class is **35-40 years** (since the cumulative frequency is 78).

### Step 5: Apply the Median Formula
The formula for the median is:

\[ \text{Median} = L + \left( \frac{n/2 - CF}{F} \right) \times H \]

Where:
- \( L \) = lower limit of the median class = 35
- \( n \) = total frequency = 100
- \( CF \) = cumulative frequency of the class before the median class = 45
- \( F \) = frequency of the median class = 33
- \( H \) = width of the median class = 5

Substituting the values into the formula:

\[ \text{Median} = 35 + \left( \frac{50 - 45}{33} \right) \times 5 \]

Calculating the values:

\[ = 35 + \left( \frac{5}{33} \right) \times 5 \]

\[ = 35 + \frac{25}{33} \]

Calculating \( \frac{25}{33} \approx 0.76 \):

\[ \text{Median} \approx 35 + 0.76 \approx 35.76 \]

### Final Answer
The median age of the policyholders is approximately **35.76 years**.

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