Each of the following frequency tables represents a data set. Which data set has the greatest mean?

1. TRANSLATE the problem information

• Given information:
  • Four frequency tables with values 70, 80, 90, 100
  • Need to determine which has the greatest mean
• What this tells us: We need to calculate or compare the means of four different data sets

2. INFER the approach

• We have two strategic options:
  • Calculate the mean for each data set using the frequency formula
  • Look for patterns in the frequency distributions that reveal which mean is largest
• Since we need the greatest mean, let's look for patterns first, then calculate if needed

3. INFER patterns in the frequency distributions

• Choice B: All frequencies are equal (6, 6, 6, 6) → perfectly uniform
• Choice C: Frequencies are 7, 6, 6, 7 → symmetric (outer values same, inner values same)
• Choice D: Frequencies are 8, 5, 5, 8 → symmetric (outer values same, inner values same)
• Choice A: Frequencies are 4, 5, 6, 7 → NOT symmetric, increasing pattern

4. INFER what symmetry means for the mean

• When frequencies are symmetric around the center, the mean equals the middle value
• For symmetric patterns with values 70, 80, 90, 100: \(\mathrm{mean = (70+100)/2 = (80+90)/2 = 85}\)
• Choices B, C, and D all have symmetric patterns → all have mean = 85

5. INFER what Choice A's pattern means

• Choice A has frequencies 4, 5, 6, 7 (increasing)
• Higher frequencies on larger values (90, 100) pull the mean above the center
• This means Choice A's \(\mathrm{mean \gt 85}\)

6. SIMPLIFY to verify Choice A (optional)

• \(\mathrm{Mean = (70×4 + 80×5 + 90×6 + 100×7) ÷ (4+5+6+7)}\)
• \(\mathrm{Mean = (280 + 400 + 540 + 700) ÷ 22}\)
  \(\mathrm{Mean = 1920 ÷ 22}\)
  \(\mathrm{Mean ≈ 87.3}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skills: Students attempt to calculate all four means without recognizing the symmetry patterns, leading to unnecessary calculation and increased chance of arithmetic errors.

They might make calculation mistakes like:

• Forgetting to multiply values by frequencies
• Adding frequencies incorrectly
• Making division errors

This may lead them to select any incorrect choice based on their calculation errors, or get overwhelmed and guess.

Second Most Common Error:

Incomplete TRANSLATE reasoning: Students misunderstand what "frequency" means and try to find the mean by simply averaging the four values (70, 80, 90, 100) without considering how often each appears.

This gives them a mean of 85 for all choices, leading to confusion about how to distinguish between them. This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem rewards pattern recognition over brute-force calculation. Students who can spot the symmetry in choices B, C, and D immediately know these all equal 85, making Choice A the clear winner without extensive arithmetic.