Owning the Dataset
Let's start by understanding what we're working with in this table. We have participant scores from a competition, along with their squared differences from the mean:
| Participant | Score | Square of Difference from Mean |
| Sauer | 41 | 3481 |
| Lasek | 26 | 3025 |
| Hough | 23 | 1936 |
| Fournier | 19 | 1681 |
Key insight: The "Square of Difference from Mean" column is particularly important for understanding standard deviation (\(\mathrm{SD}\)). This value directly shows how much each participant contributes to the overall standard deviation of the dataset.
Analyzing the Statements
Let's sort the table by "Square of Difference from Mean" in descending order to immediately reveal the most influential participants:
After sorting, we can instantly see:
- Sauer: 3481 (largest squared difference)
- Lasek: 3025 (second largest)
- Hough: 1936 (third largest)
- Fournier: 1681 (fourth largest)
Statement 1 Analysis
Statement 1: "Sauer's disqualification decreases \(\mathrm{SD}\) more than any other single disqualification."
When we remove a data point from a set, the standard deviation is affected based on how extreme that value is compared to the mean. The participant with the largest squared difference from the mean will have the greatest impact on standard deviation when removed.
Looking at our sorted data, Sauer has the largest squared difference (3481), which is significantly higher than any other participant. This means removing Sauer will decrease the standard deviation more than removing any other single participant.
Therefore, Statement 1 is True.
Statement 2 Analysis
Statement 2: "Lasek's disqualification decreases \(\mathrm{SD}\) more than any other single disqualification."
From our sorted table, we can immediately see that Lasek has the second largest squared difference (3025), which is less than Sauer's (3481). This means removing Lasek will not decrease the standard deviation as much as removing Sauer would.
Therefore, Statement 2 is False.
Statement 3 Analysis
Statement 3: "If Fournier and Sauer were both disqualified, \(\mathrm{SD}\) would decrease."
Removing extreme values (those far from the mean) always decreases the standard deviation of a dataset.
From our sorted table, we know:
- Sauer has the largest squared difference (3481)
- Fournier has the fourth largest squared difference (1681)
Both of these values are significant contributors to the standard deviation. Removing both of them would eliminate two extreme values from the dataset, which will definitely decrease the standard deviation.
Therefore, Statement 3 is True.
Final Answer
Based on our analysis:
- Statement 1: True
- Statement 2: False
- Statement 3: True
The correct answer pattern is: True, False, True.
