Data set P consists of 30 values: six 10s, six 14s, six 18s, six 22s, and six 26s. Data set...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
Data set P consists of \(30\) values: six \(10\)s, six \(14\)s, six \(18\)s, six \(22\)s, and six \(26\)s. Data set Q is formed by changing four of the \(10\)s to \(26\)s and changing two of the \(14\)s to \(22\)s. Which of the following statements best compares the means of data sets P and Q?
The mean of Q is greater than the mean of P.
The mean of Q is less than the mean of P.
The mean of Q is equal to the mean of P.
There is not enough information to compare the means.
1. TRANSLATE the problem information
- Given information:
- Data set P: 30 values total with six 10s, six 14s, six 18s, six 22s, and six 26s
- Data set Q: formed by changing four 10s to 26s and two 14s to 22s
- What this tells us: We need to track how the data composition changes from P to Q
2. INFER the approach
- We can compare means in two ways: calculate both directly, or analyze how the changes affect the sum
- Since we need to determine which mean is larger, direct calculation will give us clear numerical comparison
3. SIMPLIFY to find the mean of P
- Mean of P = \(\mathrm{(6×10 + 6×14 + 6×18 + 6×22 + 6×26) ÷ 30}\)
- \(\mathrm{= (60 + 84 + 108 + 132 + 156) ÷ 30}\)
- \(\mathrm{= 540 ÷ 30}\)
- \(\mathrm{= 18}\)
4. TRANSLATE the changes to find Q's composition
- Starting from P's six of each value:
- Four 10s become 26s: now 2 tens and 10 twenty-sixes
- Two 14s become 22s: now 4 fourteens and 8 twenty-twos
- Unchanged: 6 eighteens
5. SIMPLIFY to find the mean of Q
- Mean of Q = \(\mathrm{(2×10 + 4×14 + 6×18 + 8×22 + 10×26) ÷ 30}\)
- \(\mathrm{= (20 + 56 + 108 + 176 + 260) ÷ 30}\)
- \(\mathrm{= 620 ÷ 30}\)
- \(\mathrm{= 20.67}\)
6. INFER the comparison
- Since \(\mathrm{20.67 > 18}\), the mean of Q is greater than the mean of P
Answer: (A) The mean of Q is greater than the mean of P
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misunderstand what "changing four of the 10s to 26s" means and incorrectly think this creates additional values rather than transforming existing ones, leading to incorrect totals for the number of each value in set Q.
For example, they might think Q has 6 tens + 10 twenty-sixes instead of 2 tens + 10 twenty-sixes. This inflates their calculation of Q's mean and could lead them to get an even larger difference or become confused about which dataset they're analyzing.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify the transformations but make arithmetic errors when calculating the means, especially when handling the multiple multiplications and additions required.
They might calculate one mean correctly but err on the other, potentially leading them to select Choice (B) (mean of Q less than mean of P) if they accidentally swap their calculated values.
The Bottom Line:
This problem tests whether students can carefully track data transformations and perform multi-step mean calculations. The key insight is recognizing that moving values from smaller numbers (10s and 14s) to larger numbers (26s and 22s) must increase the overall mean.
The mean of Q is greater than the mean of P.
The mean of Q is less than the mean of P.
The mean of Q is equal to the mean of P.
There is not enough information to compare the means.