6, 6, 8, 8, 8, 10, 21Which of the following lists represents a data set that has the same median...

1. TRANSLATE the problem information

• Given information:
  • Original data set: 6, 6, 8, 8, 8, 10, 21
  • Need to find which answer choice has the same median

• What this tells us: We need to find the median of the original set, then find which choice has that same median

2. INFER the approach

• Since we're comparing medians, we need to:
  • Calculate the median of the original 7-value data set
  • Calculate the median of each 6-value answer choice
  • Find which answer choice matches the original median

• Key insight: Median calculation is different for odd vs even number of values

3. SIMPLIFY to find the original data set's median

• The original data set has 7 values (odd number)
• For odd numbers of values: median = middle value when arranged in order
• Since 6, 6, 8, 8, 8, 10, 21 is already in order, the median is the 4th value
• Original median = 8

4. SIMPLIFY to find each answer choice's median

• All answer choices have 6 values (even number)
• For even numbers of values: median = average of the two middle values (3rd and 4th values)

Choice A: 4, 6, 6, 6, 8, 8
• Middle values: 6 and 6 → Median = \(\mathrm{(6 + 6)/2 = 6}\)

Choice B: 6, 6, 8, 8, 10, 10
• Middle values: 8 and 8 → Median = \(\mathrm{(8 + 8)/2 = 8}\) ✓

Choice C: 6, 8, 10, 10, 10, 12
• Middle values: 10 and 10 → Median = \(\mathrm{(10 + 10)/2 = 10}\)

Choice D: 8, 8, 10, 10, 21, 21
• Middle values: 10 and 10 → Median = \(\mathrm{(10 + 10)/2 = 10}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding the difference between finding median with odd vs even number of values

Students might try to find the 'middle value' in the 6-value answer choices, picking the 3rd or 4th value instead of averaging them. For example, in Choice B (6, 6, 8, 8, 10, 10), they might say the median is just 8 (the 3rd value) rather than properly calculating \(\mathrm{(8 + 8)/2 = 8}\). While this happens to give the right answer for Choice B, it leads to wrong calculations for other choices.

This conceptual confusion may lead them to select the wrong answer or get stuck and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Miscounting positions or making arithmetic errors when averaging

Students might correctly understand they need to average two middle values but misidentify which values are in the middle positions, or make simple arithmetic errors when calculating the averages. For instance, they might think Choice D has middle values of 8 and 10 instead of 10 and 10.

This may lead them to select Choice C or D based on incorrect median calculations.

The Bottom Line:

This problem tests whether students truly understand how median calculation changes based on whether the data set has an odd or even number of values - a key distinction that many students overlook or apply inconsistently.