Construct the frequency distribution table from the following cumulative frequency table:
`{:("Ages" ,"No of students"),("Below" 4,0),("Below" 7,85),("Below" 10,140),("Below" 13,243),("Below" 16,300):}`
State the age group which has the least number of students

Answer

To construct the frequency distribution table from the given cumulative frequency table, we will follow these steps:

### Step 1: Identify the Class Intervals
The cumulative frequency table provides us with the upper limits of the age groups. We will derive the class intervals based on these limits.

- Below 4: 0 students (no interval)
- Below 7: 85 students (4 to 7)
- Below 10: 140 students (7 to 10)
- Below 13: 243 students (10 to 13)
- Below 16: 300 students (13 to 16)

### Step 2: Calculate the Frequencies
To find the frequency for each class interval, we will subtract the cumulative frequency of the previous interval from the cumulative frequency of the current interval.

1. **For the interval 4 to 7:**
- Frequency = Cumulative frequency (below 7) - Cumulative frequency (below 4)
- Frequency = 85 - 0 = 85

2. **For the interval 7 to 10:**
- Frequency = Cumulative frequency (below 10) - Cumulative frequency (below 7)
- Frequency = 140 - 85 = 55

3. **For the interval 10 to 13:**
- Frequency = Cumulative frequency (below 13) - Cumulative frequency (below 10)
- Frequency = 243 - 140 = 103

4. **For the interval 13 to 16:**
- Frequency = Cumulative frequency (below 16) - Cumulative frequency (below 13)
- Frequency = 300 - 243 = 57

### Step 3: Construct the Frequency Distribution Table
Now we can summarize the class intervals and their corresponding frequencies in a table format:

| Age Group | No of Students |
|-----------|----------------|
| 4 - 7 | 85 |
| 7 - 10 | 55 |
| 10 - 13 | 103 |
| 13 - 16 | 57 |

### Step 4: Identify the Age Group with the Least Number of Students
From the frequency distribution table, we can see that the age group with the least number of students is 7 - 10, with a frequency of 55.

### Final Answer
The age group which has the least number of students is **7 to 10**.

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