A data set of 27 different numbers has a mean of 33 and a median of 33. A new data...

1. TRANSLATE the problem information

• Given information:
  • Original data set: 27 different numbers
  • \(\mathrm{Mean = 33}\), \(\mathrm{Median = 33}\)
  • New data set created by: adding 7 to numbers \(\mathrm{\gt 33}\), subtracting 7 from numbers \(\mathrm{\lt 33}\)

• What this tells us: Since there are 27 numbers (odd), the median is the 14th number when ordered from least to greatest. This means 13 numbers are below 33, 1 number equals 33, and 13 numbers are above 33.

2. INFER how the transformation affects each measure

• Strategy: Analyze what happens to median, mean, sum, and standard deviation when we spread the data
• Key insight: We're moving the lower half farther down and the upper half farther up, while the middle stays the same

3. Analyze the median

• The 14th number (middle value) stays at 33 - it's neither above nor below the median, so unchanged
• Numbers below 33 become even smaller, numbers above 33 become even larger
• Median remains 33 ✓

4. INFER the effect on mean and sum

• Original sum = \(\mathrm{27 \times 33 = 891}\)
• Change from transformation: 13 numbers gain 7 each, 13 numbers lose 7 each
• Net change = \(\mathrm{13(+7) + 13(-7) = +91 - 91 = 0}\)
• New sum = 891, new mean = \(\mathrm{891 \div 27 = 33}\) ✓

5. INFER the effect on standard deviation

• Standard deviation measures how spread out the data is from the mean
• Our transformation takes numbers that were close to the mean (33) and moves them 7 units farther away
• Greater spread from mean = larger standard deviation ✗

Answer: D. Standard deviation

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning about standard deviation: Students often think that since the mean stays the same, the standard deviation should also stay the same. They fail to recognize that standard deviation depends on how far individual data points are from the mean, not just the mean value itself.

This may lead them to select Choice A, B, or C (the measures that actually do stay the same) or causes confusion and random guessing.

Second Most Common Error:

Inadequate TRANSLATE of the transformation: Students may misunderstand which numbers get modified. They might think all numbers change, or they might not realize that the median value itself (33) doesn't change since it's neither greater than nor less than the median.

This leads to incorrect calculations of the new mean and sum, potentially causing them to select Choice B (Mean) or Choice C (Sum of the numbers).

The Bottom Line:

This problem tests whether students truly understand what standard deviation measures. The key insight is that spreading data points farther from their mean (even when the mean itself doesn't change) increases the standard deviation.