Two groups of students, Group P and Group Q, participated in a fundraising event, and each group consists of 19...

1. TRANSLATE the problem information

• Given information:
  • Two groups, each with 19 students
  • Frequency table showing distribution of items sold (10, 20, 30, 40, 50)
  • Need to compare standard deviations between groups

• What this tells us: We need to analyze the spread of data in each distribution

2. INFER the most efficient approach

• Since we're comparing standard deviations, we should first check if the means are equal
• Both distributions appear symmetric around 30, suggesting equal means
• If means are equal, we can focus directly on comparing spread patterns

3. SIMPLIFY to find the means

• Group P: \((10\times1 + 20\times4 + 30\times9 + 40\times4 + 50\times1)/19 = 570/19 = 30\)
• Group Q: \((10\times4 + 20\times3 + 30\times5 + 40\times3 + 50\times4)/19 = 570/19 = 30\)

Both means equal 30, confirming our expectation.

4. INFER the spread patterns

• Group P: Most students (9 out of 19) cluster right at the mean value of 30, with very few at extremes (only 1 each at 10 and 50)
• Group Q: Fewer students at the mean (5 out of 19), but more students at extreme values (4 each at 10 and 50)

This pattern suggests Group Q has greater spread, meaning higher standard deviation.

5. SIMPLIFY to calculate standard deviations for verification

Group P variance:

\([(10-30)^2\times1 + (20-30)^2\times4 + (30-30)^2\times9 + (40-30)^2\times4 + (50-30)^2\times1]/19\)

\(= [400 + 400 + 0 + 400 + 400]/19 = 1,600/19 \approx 84.2\)

Standard deviation \(\approx \sqrt{84.2} \approx 9.2\) (use calculator)

Group Q variance:

\([400\times4 + 100\times3 + 0\times5 + 100\times3 + 400\times4]/19\)

\(= 3,800/19 = 200\)

Standard deviation \(= \sqrt{200} \approx 14.1\) (use calculator)

6. APPLY CONSTRAINTS to select the correct comparison

• Group P standard deviation (≈9.2) < Group Q standard deviation (≈14.1)
• This confirms our conceptual analysis

Answer: B) The standard deviation for Group P is less than the standard deviation for Group Q.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often get overwhelmed by the calculation requirement and miss the key insight that more concentration at the mean indicates lower standard deviation.

Many students see the frequency table and immediately jump into complex calculations without first analyzing the distribution patterns. They might calculate correctly but take much longer, or make arithmetic errors in the process. Some students incorrectly assume that having more students at extreme values somehow balances out and creates equal standard deviations.

This conceptual confusion may lead them to select Choice C (equal standard deviations) or abandon systematic analysis and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors when computing variance, particularly with the squared differences and frequency weighting.

Common mistakes include forgetting to multiply by frequencies, incorrectly squaring the differences \((x - \mu)\), or making errors when taking square roots. These calculation errors can flip the comparison result entirely.

This may lead them to select Choice A (Group P greater than Group Q) due to reversed calculations.

The Bottom Line:

This problem rewards students who recognize that standard deviation comparison can often be done conceptually by analyzing spread patterns, with calculations serving as verification rather than the primary method. The key insight is understanding what the frequency distributions tell us about data concentration versus dispersion.