The box plot above shows the distribution of lengths, in millimeters, of 50 manufactured parts measured during quality control. Due...

1. TRANSLATE the box plot information

Looking at the box plot carefully:

• Q1 (first quartile): The left edge of the gray box = 22 mm
• Q3 (third quartile): The right edge of the gray box = 28 mm
• Median: The vertical line inside the box ≈ 25 mm

The problem tells us: "each measurement was 3 mm less than the actual length"

This means: Actual length = Measured length + 3

2. SIMPLIFY to find the measured IQR

Using the IQR formula:

• IQR (measured) = Q3 - Q1
• IQR (measured) = 28 - 22 = 6 mm

3. INFER the key insight about transformations

Here's the critical question: When we add 3 mm to every measurement, what happens to the IQR?

Let's find the actual quartiles:

• Q1_actual = 22 + 3 = 25 mm
• Q3_actual = 28 + 3 = 31 mm

Now calculate the actual IQR:

• IQR (actual) = Q3_actual - Q1_actual
• IQR (actual) = 31 - 25 = 6 mm

Key insight: Both quartiles shifted up by 3 mm, so their difference stayed the same!

Mathematical principle: When you add (or subtract) the same constant to every value in a dataset, the spread doesn't change—only the center shifts. The IQR measures spread, so it remains unchanged.

Answer: B (6 mm)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students incorrectly think that since each individual measurement needs to be adjusted by 3 mm, the IQR should also be adjusted by 3 mm.

This faulty reasoning goes: "The measurements are off by 3 mm, so I need to add 3 to the IQR: 6 + 3 = 9 mm."

This error comes from not understanding what IQR represents. IQR is a measure of spread (the distance between Q3 and Q1), not a measure of center (like the mean or median). When you add a constant to all values:

• Measures of center shift by that constant (mean and median both increase by 3)
• Measures of spread stay the same (IQR, range, and standard deviation don't change)

This may lead them to select Choice C (9).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misread the box plot and incorrectly identify the quartiles, perhaps confusing the median line with Q1 or Q3, or misreading the scale.

For example, if they think Q1 = 20 and Q3 = 28, they'd calculate IQR = 8 mm. Then they might either:

• Keep 8 mm as their answer (not matching any choice, leading to confusion)
• Add 3 to get 11 mm (also not matching, causing guessing)

This confusion about reading the box plot correctly causes them to get stuck and guess among the choices.

The Bottom Line:

This problem tests two things simultaneously: (1) your ability to read a box plot accurately, and (2) your understanding of how transformations affect measures of spread versus measures of center. The "adjustment by 3 mm" is designed to trap students who don't recognize that IQR measures spread, which is invariant under constant shifts.