. **Question:** The marks obtained by 50 students in a Commerce exam are given below. Calculate the 3rd decile.
- Marks: 15, 22, 29, 35, 38, 41, 42, 45, 47, 48, 50, 51, 53, 55, 56, 58, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 74, 75, 76, 78, 80, 81, 83, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100.

To calculate the 3rd decile (D₃) from the given data, we follow these steps:

### Step 1: Arrange the Data in Ascending Order
The marks are already provided in ascending order:

\[ 15, 22, 29, 35, 38, 41, 42, 45, 47, 48, 50, 51, 53, 55, 56, 58, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 74, 75, 76, 78, 80, 81, 83, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100 \]

### Step 2: Calculate the Position of the 3rd Decile
The position of a decile \( D_k \) in a sorted list is calculated using the formula:

\[
D_k = \left( \frac{k \times (N + 1)}{10} \right)
\]

Where:
- \( k = 3 \) (since we are calculating the 3rd decile),
- \( N = 50 \) (the total number of students).

So, the position of \( D_3 \) is:

\[
D_3 = \left( \frac{3 \times (50 + 1)}{10} \right) = \left( \frac{3 \times 51}{10} \right) = \frac{153}{10} = 15.3
\]

### Step 3: Interpret the Position
Since the position is 15.3, \( D_3 \) will be between the 15th and 16th data points.

### Step 4: Calculate the Value of the 3rd Decile
The 15th and 16th data points are:

- 15th value: 56
- 16th value: 58

Since the position is 15.3, the value of \( D_3 \) can be interpolated as:

\[
D_3 = 56 + 0.3 \times (58 - 56) = 56 + 0.3 \times 2 = 56 + 0.6 = 56.6
\]

### Answer
The 3rd decile \( D_3 \) is **56.6**.

Would you like more details on how this was calculated or any other questions?

### Related Questions

1. How do you calculate the 1st and 9th deciles for the same dataset?
2. What is the difference between deciles and percentiles?
3. How do you interpret the meaning of deciles in a real-world context?
4. What are the advantages of using deciles over other statistical measures?
5. Can you explain how interpolation is used when the decile position is not a whole number?
6. How do quartiles differ from deciles?
7. How would the 3rd decile change if the dataset had more or fewer students?
8. Can you explain how to calculate the median using a similar approach?

### Tip:
When working with large datasets, consider using software tools like Excel or statistical packages to automate the calculation of deciles, percentiles, and other statistical measures.

Learn how to calculate the 3rd decile from a dataset of student exam marks using statistical methods. This problem involves understanding deciles, percentiles, and interpolation in statistics. Suitable for advanced high school students.

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