A book critic reviewed 15 books and assigned each book a score on a scale from 1 (lowest) to 5...

1. TRANSLATE the problem information

• Given information:
  • 15 books were reviewed
  • Each book received a score from 1 to 5
  • Specific scores: 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5
  • Need to find: mean score

2. INFER the approach needed

• The mean requires using ALL individual scores, not just the unique values
• We must add every single score (all 15 numbers) and divide by the total count
• Strategy: Sum all scores → divide by 15

3. SIMPLIFY by adding all the scores

• Add each individual score:

\(1 + 2 + 2 + 2 + 2 + 3 + 3 + 3 + 3 + 3 + 4 + 4 + 4 + 4 + 5\)

• Group by frequency to make addition easier:

\(1 + (4×2) + (5×3) + (4×4) + 5 = 1 + 8 + 15 + 16 + 5 = 45\)

4. SIMPLIFY the final calculation

• Mean = Total sum ÷ Number of values
• \(\mathrm{Mean} = 45 ÷ 15 = 3\)

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to find the mean using only the distinct score values (1, 2, 3, 4, 5) instead of all individual scores.

They might calculate: \((1 + 2 + 3 + 4 + 5) ÷ 5 = 15 ÷ 5 = 3\). While this accidentally gives the correct answer in this case, it shows a fundamental misunderstanding of how mean works with repeated values.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when adding all 15 scores or when dividing by 15.

Common mistakes include miscounting the frequency of each score or making basic addition errors, leading to an incorrect sum and therefore an incorrect mean.

The Bottom Line:

This problem tests whether students truly understand that finding the mean requires using every single data point, not just the unique values. The key insight is recognizing that repeated scores must be counted multiple times in the calculation.