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.