Data set F consists of 55 integers between 170 and 290. Data set G consists of all the integers in...

1. TRANSLATE the problem information

• Given information:
  • Dataset F: 55 integers between 170 and 290
  • Dataset G: All integers from F plus the integer 10
  • Need to determine what must be less for F than for G
• What this tells us: G has one more value than F, and that additional value (10) is much smaller than any value in F

2. INFER the strategic approach

• Key insight: When we add an extreme value to a dataset, it pulls the mean and median toward that extreme value
• Since 10 is much smaller than any value in F (which are all \(\geq 170\)), adding 10 should decrease both the mean and median
• We need to analyze each measure separately

3. CONSIDER ALL CASES for the mean

• Mean of F = \(\frac{\mathrm{sum\ of\ F's\ 55\ values}}{55}\)
• Mean of G = \(\frac{\mathrm{sum\ of\ F's\ 55\ values} + 10}{56}\)
• Since we're adding a small value (10) and increasing the count, the mean of G will be smaller than the mean of F
• Therefore: \(\mathrm{Mean\ of\ G} \lt \mathrm{Mean\ of\ F}\)
• This means the mean is NOT less for F than for G

4. CONSIDER ALL CASES for the median

• F has 55 values (odd number), so median of F = 28th value when arranged in ascending order
• G has 56 values (even number), so median of G = average of 28th and 29th values when arranged in ascending order
• When 10 is added to create G, it becomes the smallest value in the ordered list
• In G's ascending arrangement: 10, then all of F's values in their original order
• So G's 28th value = F's 27th value, and G's 29th value = F's 28th value (original median)
• Median of G = \(\frac{\mathrm{F's\ 27th\ value} + \mathrm{F's\ 28th\ value}}{2}\)
• Since \(\mathrm{F's\ 27th\ value} \lt \mathrm{F's\ 28th\ value}\), we get: \(\mathrm{Median\ of\ G} \lt \mathrm{Median\ of\ F}\)
• Therefore: the median is NOT less for F than for G

Answer: D. Neither I nor II

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students incorrectly assume that because G has 'more data,' its mean and median must be larger than F's measures.

They think: 'G has all of F's values plus an extra one, so G should have bigger statistics.' This backwards reasoning ignores the fact that the additional value (10) is an extreme outlier that pulls the measures downward. This may lead them to select Choice C (I and II) or get confused about the direction of the comparison.

Second Most Common Error:

Inadequate CONSIDER ALL CASES execution: Students correctly analyze one measure (usually the mean) but struggle with the median calculation for even vs. odd number of data points.

They might correctly determine that the mean decreases when 10 is added, but then assume the median behaves the same way without working through the positional analysis. This incomplete reasoning may lead them to select Choice A (I only) instead of recognizing that both measures behave similarly.

The Bottom Line:

This problem tests whether students can think beyond 'more data = bigger numbers' to understand how extreme values actually affect measures of central tendency. The key insight is recognizing that adding an outlier pulls statistics toward the outlier, regardless of dataset size.