Calculate the median and sixty first percentile from the following data of marks obtained by 10 students in an examination. 22, 26, 30, 14, 35, 11, 18, 12, 32, 41.

Answer

To solve the problem of calculating the median and the 61st percentile from the given data of marks obtained by 10 students, we will follow these steps:

### Step 1: Arrange the Data in Ascending Order
The given marks are: 22, 26, 30, 14, 35, 11, 18, 12, 32, 41.

First, we need to arrange these marks in ascending order:
- **Sorted Data**: 11, 12, 14, 18, 22, 26, 30, 32, 35, 41

### Step 2: Calculate the Median
The median is the middle value of a data set. Since we have an even number of observations (n = 10), the median is calculated as the average of the n/2-th term and the (n/2 + 1)-th term.

1. **Find n**: Here, n = 10.
2. **Calculate n/2**: n/2 = 10/2 = 5.
3. **Identify the 5th and 6th Terms**:
- 5th term = 22
- 6th term = 26
4. **Calculate the Median**:
\[ \text{Median} = \frac{\text{5th term} + \text{6th term}}{2} = \frac{22 + 26}{2} = \frac{48}{2} = 24 \]

### Step 3: Calculate the 61st Percentile
The formula for calculating the k-th percentile (P_k) is given by:
\[ P_k = \frac{k \times (n + 1)}{100} \]

1. **Substitute k and n**:
- k = 61
- n = 10
\[ P_{61} = \frac{61 \times (10 + 1)}{100} = \frac{61 \times 11}{100} = \frac{671}{100} = 6.71 \]

2. **Identify the 6th and 7th Terms**:
- 6th term = 26
- 7th term = 30

3. **Calculate the 61st Percentile**:
\[ P_{61} = \text{6th term} + (0.71 \times (\text{7th term} - \text{6th term})) \]
\[ P_{61} = 26 + (0.71 \times (30 - 26)) = 26 + (0.71 \times 4) = 26 + 2.84 = 28.84 \]

### Final Answers
- **Median**: 24
- **61st Percentile (P61)**: 28.84