For which of the following data sets is the mean greater than the median?

1. INFER the key insight about median location

• Given information: Four datasets, each with exactly 9 values arranged in order
• Key insight: For any dataset with 9 values, the median is always the 5th (middle) value
• Strategy: Find median first (easy), then calculate mean and compare

2. INFER which datasets to prioritize

• Choices A and B look symmetric - these likely have \(\mathrm{mean} = \mathrm{median}\)
• Choice C has small values on left, huge values on right - this looks skewed right
• Choice D has one small value, then larger values - needs calculation
• Focus on C and D as most likely candidates

3. SIMPLIFY the calculations systematically

Choice A: All 5's

• Median: 5 (5th value)
• Mean: Obviously 5 (all values identical)
• Result: \(\mathrm{Mean} = \mathrm{Median}\)

Choice B: Evenly spaced 0 to 80

• Median: 40 (5th value)
• Mean: This is symmetric, so \(\mathrm{mean} = 40\)
• Result: \(\mathrm{Mean} = \mathrm{Median}\)

Choice C: Powers of 2 pattern

• Median: 32 (5th value)
• Mean: \(\frac{2+4+8+16+32+64+128+256+512}{9} = \frac{1022}{9}\) (use calculator) \(\approx 113.56\)
• Result: \(\mathrm{Mean} \gt \mathrm{Median}\) ✓ This could be our answer!

Choice D: Mixed values

• Median: 207 (5th value)
• Mean: \(\frac{7+107+107+207+207+207+307+307+307}{9} = \frac{1763}{9}\) (use calculator) \(\approx 195.89\)
• Result: \(\mathrm{Mean} \lt \mathrm{Median}\)

4. INFER the final comparison

• Only Choice C has \(\mathrm{mean} \gt \mathrm{median}\)
• The extreme right-skewness (large values 128, 256, 512) pulls the mean well above the median

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students calculate means and medians correctly but fail to systematically compare them across all choices, often stopping after finding one where \(\mathrm{mean} \neq \mathrm{median}\).

They might calculate Choice A, see \(\mathrm{mean} = \mathrm{median}\), then jump to Choice C, calculate correctly, and immediately select C without verifying that it's the ONLY choice where \(\mathrm{mean} \gt \mathrm{median}\). If they happened to check Choice D first and found \(\mathrm{mean} \lt \mathrm{median}\) there, they might incorrectly eliminate it as 'close enough' or get confused about the direction of the inequality.

This leads to confusion and potentially guessing between multiple choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make arithmetic errors when calculating means, especially for Choices C and D with larger numbers.

For Choice C, they might miscalculate \(\frac{1022}{9}\) or make addition errors with the large values (128, 256, 512). For Choice D, the repeated values (three 207's, three 307's) create opportunities for counting mistakes. These calculation errors lead to wrong mean values and incorrect comparisons.

This may lead them to select Choice D if they miscalculate its mean as greater than 207.

The Bottom Line:

This problem requires both computational accuracy with larger numbers AND systematic comparison across all four choices. Students must resist the urge to stop early and must maintain precision when the numbers get unwieldy.