Let us analyze these sets and their properties, focusing on comparisons such as averages, medians, ranges, and possible distributions.

1. Details of Each Set:

Set 1: {8, 9, 10, 11, 12}

• Mean (average): $\text{Mean} = \frac{8 + 9 + 10 + 11 + 12}{5} = \frac{50}{5} = 10$
• Median: The middle value (since the set is ordered) is 10.
• Range: $\text{Range} = 12 - 8 = 4$

Set 2: {8, 9, 10, 11, 100}

• Mean (average): $\text{Mean} = \frac{8 + 9 + 10 + 11 + 100}{5} = \frac{138}{5} = 27.6$
• Median: The middle value is still 10.
• Range: $\text{Range} = 100 - 8 = 92$

Set 3: {8, 9, 10, 11, 1000}

• Mean (average): $\text{Mean} = \frac{8 + 9 + 10 + 11 + 1000}{5} = \frac{1038}{5} = 207.6$
• Median: The middle value is still 10.
• Range: $\text{Range} = 1000 - 8 = 992$

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2. Observations:

• As the largest value increases significantly in Set 2 and Set 3, the mean is heavily influenced, shifting far from the central tendency of most other values.
• The median remains unchanged across all sets because it represents the middle value, unaffected by extreme values (outliers).
• The range expands significantly in Sets 2 and 3 due to the addition of extreme values (100 and 1000, respectively).

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3. Key Insights:

• The median is more robust to outliers, representing the central tendency better in such cases.
• The mean becomes less representative of most data points as extreme values are added.
• The range increases dramatically, highlighting the influence of outliers on data spread.

Would you like me to explore any specific analysis or explain further?

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5 Related Questions:

1. What would happen if the largest value in each set were replaced with another number closer to the median?
2. How does standard deviation differ across these sets, given the increased spread?
3. Could the trimmed mean (ignoring outliers) be a better measure of central tendency for these sets?
4. How would a boxplot visualization represent the differences among these sets?
5. In real-world scenarios, when is the median more useful than the mean?

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Tip:

When analyzing datasets with outliers, consider using robust statistics like the median or interquartile range to better represent the data.