📊 Statistics Problem Solver
Understanding Median in Grouped Data - Life Insurance Policy Holders
🤔 Background Knowledge Check
Before we solve this problem, let's test your understanding of basic concepts!
Question 1: What is the median of a dataset?
Question 2: In grouped data, what does "Below 25" mean?
Question 3: If we have 100 values, which position represents the median?
📋 Problem Statement
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 |
|---|---|
| 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 |
📝 What is Given?
Given Information:
- ✅ Total policy holders = 100
- ✅ Age restriction: 18 years onwards, less than 60
- ✅ Data type: Cumulative frequency
🔍 Important Observation:
The data is cumulative. "Below 25" (6 holders) includes the "Below 20" (2 holders).
🎯 What do we need to Find?
We need to find: MEDIAN AGE
The middle value when all 100 policy holders are arranged by age.
For 100 policy holders:
Median position = n2 = 1002 = 50th value
🔧 Step-by-Step Solution
Step 1: Create Class Intervals & Frequencies
| Class Interval | Frequency (f) | Cumulative (CF) |
|---|---|---|
| 18-20 | 2 | 2 |
| 20-25 | 6-2 = 4 | 6 |
| 25-30 | 24-6 = 18 | 24 |
| 30-35 | 45-24 = 21 | 45 |
| 35-40 | 78-45 = 33 | 78 |
| 40-45 | 89-78 = 11 | 89 |
| 45-50 | 92-89 = 3 | 92 |
| 50-55 | 98-92 = 6 | 98 |
| 55-60 | 100-98 = 2 | 100 |
Step 2: Find the Median Class
The 50th value falls in the class whose cumulative frequency is just greater than 50. This is the 35-40 class (CF = 78).
Step 3: Apply Median Formula
Median = L + [ (n/2) - CFf ] × h
- L (lower limit of median class) = 35
- n/2 = 50
- CF (of class before median class) = 45
- f (of median class) = 33
- h (class width) = 5
Step 4: Calculate
= 35 + [ 50 - 4533 ] × 5
= 35 + (5/33) × 5
= 35 + 0.757... ≈ 35.76 years
🎉 Final Answer & Conclusion
📊 MEDIAN AGE = 35.76 years
🔍 What this means:
Half the policy holders are younger than 35.76, and half are older.
🤔 Why is this useful?
The median age is a key metric for insurance companies for risk assessment and setting premium rates.