The final marks in mathematics of `30` students are as follows:
`53,61,48,60,78,68,55,100,67,90,75,88,77,37,84, 58,60,48,62,56,44,58,52,64,98,59,70,39,50,60` Arrange these marks in the ascending order, `30` to `39` one group, `40` to `49` second group etc. Now answer the following: What is the highest score? What is the lowest score? What is the range? If `40` is the pass marks how many have failed? How many have scored `75` or more? Which observations between `50` and `60` have not actually appeared? How many have scored less than `50`?

Answer

To solve the problem step by step, we will first arrange the marks in ascending order and then create a frequency distribution table based on the specified groups. After that, we will answer each of the questions posed.

### Step 1: Arrange the Marks in Ascending Order
The given marks are:
`53, 61, 48, 60, 78, 68, 55, 100, 67, 90, 75, 88, 77, 37, 84, 58, 60, 48, 62, 56, 44, 58, 52, 64, 98, 59, 70, 39, 50, 60`

Arranging these marks in ascending order gives us:
`37, 39, 44, 48, 48, 50, 52, 53, 55, 56, 58, 58, 59, 60, 60, 60, 61, 62, 64, 67, 68, 70, 75, 77, 78, 84, 88, 90, 98, 100`

### Step 2: Create a Frequency Distribution Table
We will create groups based on the intervals provided:
- 30 to 39
- 40 to 49
- 50 to 59
- 60 to 69
- 70 to 79
- 80 to 89
- 90 to 99
- 100 to 109

Now, we will count how many marks fall into each group.

| Group | Marks in Group | Frequency |
|------------|------------------------|-----------|
| 30 - 39 | 37, 39 | 2 |
| 40 - 49 | 44, 48, 48 | 3 |
| 50 - 59 | 50, 52, 53, 55, 56, 58, 58, 59 | 8 |
| 60 - 69 | 60, 60, 60, 61, 62, 64, 67, 68 | 8 |
| 70 - 79 | 70, 75, 77, 78 | 4 |
| 80 - 89 | 84, 88 | 2 |
| 90 - 99 | 90, 98 | 2 |
| 100 - 109 | 100 | 1 |

### Step 3: Answer the Questions
1. **What is the highest score?**
- The highest score is **100**.

2. **What is the lowest score?**
- The lowest score is **37**.

3. **What is the range?**
- Range = Highest score - Lowest score = 100 - 37 = **63**.

4. **If 40 is the pass marks, how many have failed?**
- Marks less than 40: 37, 39.
- Number of students who failed = **2**.

5. **How many have scored 75 or more?**
- Scores 75 or more: 75, 77, 78, 84, 88, 90, 98, 100.
- Number of students scoring 75 or more = **8**.

6. **Which observations between 50 and 60 have not actually appeared?**
- The missing scores between 50 and 60 are **51, 54, and 57**.

7. **How many have scored less than 50?**
- Scores less than 50: 37, 39, 44, 48, 48.
- Number of students who scored less than 50 = **5**.

### Summary of Answers
- Highest score: 100
- Lowest score: 37
- Range: 63
- Students failed: 2
- Students scoring 75 or more: 8
- Missing observations between 50 and 60: 51, 54, 57
- Students scoring less than 50: 5