Question
Statistics Unit Test
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Question
Of the mean, median, and mode, which measure of center is most affected by outliers? Explain your reasoning including an example with at least 10 data points.(4 points)
Math Short Answer Rubric (4 points)
This table shows the criteria for awarding points and the corresponding point values.
Points Awarded Criteria
4
The answer is clear and correct.
The answer shows that the student used problem-solving skills and reasoning.
All work is shown or explained.
3
The answer is mostly correct. The student made small errors while solving.
The answer shows that the student used some problem-solving skills and reasoning.
Most work is shown or explained.
2
The answer is partially correct. The student made errors while solving.
The answer shows that the student used some problem-solving skills or reasoning.
Some work is shown or explained.
1
The answer is incorrect. The student made errors while solving.
The answer shows little problem-solving skills or reasoning.
Little or no work is shown.
0 The question is not answered.
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Answers
GPT-4o mini
The measure of center that is most affected by outliers is the **mean**. The mean is calculated by adding all data points together and dividing by the number of data points. Because it considers every value, a single outlier can skew the mean significantly.
### Example:
Consider the following data set of 10 test scores:
\[ 85, 87, 90, 88, 92, 86, 95, 84, 91, 1000 \]
1. **Calculating the Mean:**
- Sum of the data points: \( 85 + 87 + 90 + 88 + 92 + 86 + 95 + 84 + 91 + 1000 = 1718 \)
- Number of data points: \( 10 \)
- Mean: \( \frac{1718}{10} = 171.8 \)
2. **Calculating the Median:**
- Ordered data set: \( 84, 85, 86, 87, 88, 90, 91, 92, 95, 1000 \)
- Since there are 10 values (even), the median is the average of the 5th and 6th values:
- Median: \( \frac{88 + 90}{2} = 89 \)
3. **Calculating the Mode:**
- Mode: There is no repeating number in this set, so there is **no mode**.
### Analysis:
- The mean (171.8) is heavily influenced by the outlier (1000), which is much larger than the other scores.
- The median (89), however, is much less affected by this outlier because it is only concerned with the middle values of the ordered data set.
- The mode is not applicable here due to the lack of repeating numbers.
In conclusion, the mean is the measure of central tendency that is most affected by outliers, as illustrated by the significant difference between the mean and median in this example.
### Example:
Consider the following data set of 10 test scores:
\[ 85, 87, 90, 88, 92, 86, 95, 84, 91, 1000 \]
1. **Calculating the Mean:**
- Sum of the data points: \( 85 + 87 + 90 + 88 + 92 + 86 + 95 + 84 + 91 + 1000 = 1718 \)
- Number of data points: \( 10 \)
- Mean: \( \frac{1718}{10} = 171.8 \)
2. **Calculating the Median:**
- Ordered data set: \( 84, 85, 86, 87, 88, 90, 91, 92, 95, 1000 \)
- Since there are 10 values (even), the median is the average of the 5th and 6th values:
- Median: \( \frac{88 + 90}{2} = 89 \)
3. **Calculating the Mode:**
- Mode: There is no repeating number in this set, so there is **no mode**.
### Analysis:
- The mean (171.8) is heavily influenced by the outlier (1000), which is much larger than the other scores.
- The median (89), however, is much less affected by this outlier because it is only concerned with the middle values of the ordered data set.
- The mode is not applicable here due to the lack of repeating numbers.
In conclusion, the mean is the measure of central tendency that is most affected by outliers, as illustrated by the significant difference between the mean and median in this example.