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To solve the problem, we need to calculate the 40th percentile of the given data set. Here's how we can do it:

### Step-by-Step Approach:
1. **List the scores in ascending order**:
   The given scores are:
   ```
   207, 193, 185, 221, 241, 216, 257, 233, 214, 197,
   188, 202, 237, 227, 233, 268, 186, 192, 203, 207
   ```

   Arranging them in ascending order:
   ```
   185, 186, 188, 192, 193, 197, 202, 203, 207, 207, 
   214, 216, 221, 227, 233, 233, 237, 241, 257, 268
   ```

2. **Calculate the rank for the 40th percentile**:
   The formula for finding the position (rank) for the \( P^{th} \) percentile is:
   \[
   \text{Rank} = \frac{P}{100} \times (n + 1)
   \]
   where \( P \) is the desired percentile and \( n \) is the number of data points.

   Here, \( P = 40 \) and \( n = 20 \) (since there are 20 data points).

   Substituting the values:
   \[
   \text{Rank} = \frac{40}{100} \times (20 + 1) = 0.40 \times 21 = 8.4
   \]

   So, the 40th percentile corresponds to the value between the 8th and 9th data points in the ordered list.

3. **Interpolate between the 8th and 9th data points**:
   The 8th data point is 203 and the 9th data point is 207. Since the rank is 8.4, we can use interpolation to estimate the value.

   \[
   \text{Percentile value} = 203 + 0.4 \times (207 - 203) = 203 + 0.4 \times 4 = 203 + 1.6 = 204.6
   \]

Thus, the value at the 40th percentile is approximately **204.6**.

Would you like further clarification on this, or more detailed calculations? Here are some related questions:

1. How do we calculate percentiles for larger data sets?
2. What is the difference between interpolation and exact percentile calculation?
3. Can we use a different method for finding percentiles if we have grouped data?
4. How does the percentile calculation change for a continuous data set?
5. Why do we add 1 to the number of data points when calculating the percentile rank?

**Tip:** Always sort your data before calculating percentiles. It ensures accurate results.

Find the 40th and 96th percentiles for scores of the given data set.

Learn how to calculate the 40th and 96th percentiles of a given data set. This problem covers key concepts like percentile rank, data ordering, and interpolation. Suitable for high school or college-level students.

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