To solve this problem, we will use the ordered list of the number of professor visits made by the sales staff. Here are the given data points:

$38, 40, 41, 45, 48, 48, 50, 50, 51, 51, 52, 52, 53, 54, 55, 55, 55, 56, 56, 57, 59, 59, 59, 62, 62, 62, 63, 64, 65, 66, 66, 67, 67, 69, 69, 71, 77, 78, 79, 79$

a. Median:

The median is the middle value in an ordered data set. Since there are 40 numbers, the median will be the average of the 20th and 21st values.

• The 20th value is 57.
• The 21st value is 59.

Thus, the median is: $\text{Median} = \frac{57 + 59}{2} = 58$

b. First and Third Quartiles:

The first quartile $Q1$ is the median of the first half of the data (20 values), and the third quartile $Q3$ is the median of the second half of the data (20 values).

• First half: $38, 40, 41, 45, 48, 48, 50, 50, 51, 51, 52, 52, 53, 54, 55, 55, 55, 56, 56, 57$

  • $Q1$ is the average of the 10th and 11th values.
  • 10th value: 51
  • 11th value: 52
  • $Q1 = \frac{51 + 52}{2} = 51.5$

• Second half: $59, 59, 59, 62, 62, 62, 63, 64, 65, 66, 66, 67, 67, 69, 69, 71, 77, 78, 79, 79$

  • $Q3$ is the average of the 10th and 11th values.
  • 10th value: 66
  • 11th value: 66
  • $Q3 = \frac{66 + 66}{2} = 66$

c. First and Ninth Deciles:

The first decile $D1$ is the 10th percentile, and the ninth decile $D9$ is the 90th percentile.

• $D1$: $\frac{10}{100} \times 40 = 4$th value.

  • 4th value is 45.

• $D9$: $\frac{90}{100} \times 40 = 36$th value.

  • 36th value is 71.

d. 33rd Percentile:

The 33rd percentile $P33$ is the value below which 33% of the data fall.

$P_{33} = \left( \frac{33}{100} \times (n - 1) \right) + 1$

Here, $n = 40$.

$P_{33} = \left( \frac{33}{100} \times 39 \right) + 1$ $P_{33} = (12.87) + 1 = 13.87$

This means $P_{33}$ lies between the 13th and 14th values.

• 13th value: 53
• 14th value: 54

Using linear interpolation: $P_{33} = 53 + (0.87 \times (54 - 53))$ $P_{33} = 53 + 0.87 = 53.87$

Therefore, the answers are:

• Median: 58
• Q1: 51.50
• Q3: 66.00
• D1: 45.0
• D9: 71.0
• P33: 53.87

Would you like more details on these calculations, or do you have any questions? Here are some related questions you might find interesting:

1. What is the difference between the mean and median in a dataset?
2. How do outliers affect the quartiles?
3. What are some real-world applications of percentiles?
4. How can we calculate other deciles in this dataset?
5. What is the significance of the interquartile range?
6. How can we visualize this data to better understand its distribution?
7. What methods are used to determine if data is normally distributed?
8. What is the role of standard deviation in summarizing data?

Tip: When working with large datasets, consider using statistical software to automate calculations and reduce errors.