A student records quiz scores for five math quizzes. The table shows the original data set of the student's quiz...

1. TRANSLATE the problem information

• Given information:
  • Original data set: 84, 88, 92, 76, 80 (5 scores)
  • New data set: adds score of 66 (6 scores total)
  • Need to compare the means of both data sets

2. SIMPLIFY to find the original mean

• Mean of original = \((84 + 88 + 92 + 76 + 80) \div 5\)
• Sum: \(84 + 88 + 92 + 76 + 80 = 420\)
• Mean of original = \(420 \div 5 = 84\)

3. SIMPLIFY to find the new mean

• Mean of new = \((84 + 88 + 92 + 76 + 80 + 66) \div 6\)
• Sum: \(420 + 66 = 486\)
• Mean of new = \(486 \div 6 = 81\)

4. INFER the comparison and select answer

• Compare: \(84 \gt 81\)
• Therefore: Original mean > New mean
• This matches choice (A)

Answer: (A) The mean of the original data set is greater than the mean of the new data set.

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students may misunderstand what "compare the means" requires, thinking they need to find the difference between means rather than determining which is larger.

This leads to calculating \(84 - 81 = 3\), but then getting confused about how to interpret this difference in relation to the answer choices. This causes them to get stuck and randomly select an answer.

Second Most Common Error:

Weak INFER skill: Students correctly calculate both means but fail to recognize the logical relationship that adding a score (66) significantly below the current mean (84) will pull the overall average down.

Instead, they might think the new mean should be higher because "more data points usually increase averages," leading them to select Choice (B) (original mean is less than new mean).

The Bottom Line:

This problem tests whether students understand how individual data points affect the mean. The key insight is recognizing that adding a value below the current mean will decrease the overall mean - a fundamental concept in descriptive statistics.