Data set A: 5, 5, 5, 5, 5, 5, 5, 5, 5Data set B: 5, 5, 5, 5, 5, 5,...

1. TRANSLATE the problem information

• Given information:
  • Data set A: 5, 5, 5, 5, 5, 5, 5, 5, 5 (nine identical values)
  • Data set B: 5, 5, 5, 5, 5, 5, 5, 5, 100 (eight 5's and one 100)
  • Need to compare means and medians of both sets

2. INFER the approach

• To answer this question, I need to:
  • Calculate the mean of each data set
  • Find the median of each data set
  • Compare both statistics to see which (if any) are different

3. SIMPLIFY calculations for Data Set A

• Mean of A = \((5+5+5+5+5+5+5+5+5) \div 9 = 45 \div 9 = 5\)
• Median of A: Since all 9 values are identical (5), the median is 5

4. SIMPLIFY calculations for Data Set B

• Mean of B = \((5+5+5+5+5+5+5+5+100) \div 9 = 140 \div 9 \approx 15.56\)
• Median of B: Order the values: 5, 5, 5, 5, 5, 5, 5, 5, 100
  Since there are 9 values, the median is the 5th value = 5

5. INFER by comparing results

• Means: Data Set A = 5, Data Set B \(\approx 15.56\) → Different
• Medians: Data Set A = 5, Data Set B = 5 → Same
• Therefore: Only the means are different

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about median calculation: Students may incorrectly think that because data set B has a much larger value (100), the median must also be different. They don't recognize that median depends on position, not the magnitude of extreme values.

This may lead them to select Choice C (Both the means and the medians are different)

Second Most Common Error:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating the mean, particularly for data set B where they might incorrectly add the values or divide by the wrong number of data points.

This leads to confusion about whether the means are actually different, causing them to guess between the answer choices.

The Bottom Line:

This problem tests understanding that mean is sensitive to extreme values while median is resistant to them. The key insight is recognizing that adding one large value changes the mean significantly but may not affect the median at all.