The marks of 19 students in a test were as follows: 5, 6, 8, 9, 10, 11, 12, 13, 13, 14, 14, 15, 15, 15, 16, 16, 18, 19, 20. Calculate the median and mode.

Answer

To solve the problem of finding the median and mode of the given marks of 19 students, we can follow these steps:

### Step 1: List the marks in order
The marks of the 19 students are already provided in ascending order:
5, 6, 8, 9, 10, 11, 12, 13, 13, 14, 14, 15, 15, 15, 16, 16, 18, 19, 20.

### Step 2: Calculate the median
To find the median, we first need to determine the position of the median in the ordered list. Since there are 19 students (n = 19), which is an odd number, we can use the formula:

\[ \text{Median position} = \frac{n + 1}{2} \]

Substituting the value of n:

\[ \text{Median position} = \frac{19 + 1}{2} = \frac{20}{2} = 10 \]

Now, we find the 10th observation in the ordered list:

1. 5
2. 6
3. 8
4. 9
5. 10
6. 11
7. 12
8. 13
9. 13
10. **14** (10th term)

Thus, the median is **14**.

### Step 3: Calculate the mode
The mode is the value that appears most frequently in the data set. We will count the frequency of each mark:

- 5: 1 time
- 6: 1 time
- 8: 1 time
- 9: 1 time
- 10: 1 time
- 11: 1 time
- 12: 1 time
- 13: 2 times
- 14: 2 times
- 15: 3 times
- 16: 2 times
- 18: 1 time
- 19: 1 time
- 20: 1 time

From the counts, we see that:
- 15 appears **3 times** (the most frequent),
- 13, 14, and 16 each appear **2 times**.

Thus, the mode is **15**.

### Final Answers:
- **Median**: 14
- **Mode**: 15