https://file.aiquickdraw.com/mathbot/file/1727661272380zpqeqi73.jpg

Let's calculate the mean and median before and after the change to determine the correct answer.

### Data Set:
- Before the change: 75, 78, 81, 90, 96
- After the change: 85, 78, 81, 90, 96

### 1. Mean Calculation
- **Mean before the change:**
  \[
  \text{Mean} = \frac{75 + 78 + 81 + 90 + 96}{5} = \frac{420}{5} = 84
  \]

- **Mean after the change (75 becomes 85):**
  \[
  \text{Mean} = \frac{85 + 78 + 81 + 90 + 96}{5} = \frac{430}{5} = 86
  \]

### 2. Median Calculation
- **Before the change:** 
  The sorted data set is \(75, 78, 81, 90, 96\). The middle number is 81.
  \[
  \text{Median} = 81
  \]

- **After the change:** 
  The sorted data set is \(78, 81, 85, 90, 96\). The middle number is 85.
  \[
  \text{Median} = 85
  \]

### Conclusion:
- **Mean before the change:** 84
- **Mean after the change:** 86
- **Median before the change:** 81
- **Median after the change:** 85

Thus, the correct answer is:
- **Mean before the change = 84. Mean after the change = 86**
- **Median before the change = 81. Median after the change = 85**

This corresponds to the second option:
- "Mean before the change = 84. Mean after the change = 86. Median before the change = 81. Median after the change = 85."

Would you like to dive into more details or ask additional questions? Here are five related questions you might find helpful:

1. How does the mean change if a much larger value is introduced to the data set?
2. How does the median behave compared to the mean when outliers are introduced?
3. What is the mode, and how would it apply if some DVD counts were repeated?
4. Can we calculate the range or standard deviation for this data set?
5. How does changing only one value in a data set affect its variance?

**Tip:** The median is less sensitive to extreme values (outliers) compared to the mean, which is why it can be more representative of the central tendency in skewed distributions.

The data set below provides the number of DVD movies owned by 5 students: 75, 78, 81, 90, 96. Suppose that the number 75 from the data set below changed to 85. What is the mean before the change? After the change? What is the median before the change? After the change?

Learn how to calculate the mean and median before and after a change in a data set representing DVD movie ownership among students. This problem covers key statistical concepts and provides a step-by-step solution for understanding mean and median.

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