Exercise 12.2: Information Handling - Box & Whisker Plots, IQR & Correlation Analysis

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Chapter Overview & SLOs

How do you construct a box and whisker plot from a five-number summary? A Box and Whisker plot (Boxplot) is a visual data representation that maps a dataset using five key milestones: Minimum, First Quartile ($Q_1$), Median ($Q_2$), Third Quartile ($Q_3$), and Maximum.

Boxplot construction steps:

1. Draw the central box from $Q_1$ to $Q_3$. The horizontal length of this box equals the Interquartile Range: $\text{IQR} = Q_3 - Q_1$ (capturing the middle 50% of data).
2. Mark the median with a vertical line inside the box at the $Q_2$ position.
3. Draw the whiskers as horizontal lines extending from the box edges ($Q_1$ and $Q_3$) to the minimum and maximum values.

What is the Interquartile Range (IQR) and how is it calculated? The IQR measures the spread of the middle 50% of data.

• Formula: $\text{IQR} = Q_3 - Q_1$
• Important: The IQR is always a positive value. If $Q_3 < Q_1$, take the absolute value: $\text{IQR} = |Q_3 - Q_1|$

How do you interpret correlation trends using boxplots? Correlation describes how two variables behave relative to each other.

Example: Car Mass vs Fuel Efficiency

• Mass quartiles: $Q_1 = 1200$ kg, $Q_3 = 1700$ kg → $\text{IQR}_{\text{Mass}} = 1700 - 1200 = 500$ kg
• Efficiency quartiles: $Q_1 = 20$ km/l, $Q_3 = 12$ km/l → $\text{IQR}_{\text{Efficiency}} = |12 - 20| = 8$ km/l

Negative Correlation: As car mass increases, fuel efficiency decreases. This inverse relationship is called negative correlation.

Engineering Conclusion: Lighter vehicles are significantly more fuel-efficient than heavier ones.

How do you analyze percentile groups using quartile boundaries? Quartiles divide data into four equal parts, each containing 25% of observations.

• Q1 (25th percentile): 25% of data at or below
• Q2 (50th percentile/Median): 50% of data at or below
• Q3 (75th percentile): 75% of data at or below
• Above Q3: 100% - 75% = 25% of data in top quartile

Example: If $Q_3 = 85$ on a test, 75% of students scored at or below 85, leaving exactly 25% scoring above 85.

What key concepts are tested in FBISE Exercise 12.2? This exercise tests:

• Box and whisker plot construction from five-number summary
• Interquartile Range calculation (IQR = $Q_3 - Q_1$)
• Absolute IQR magnitude to avoid negative dispersion values
• Positive vs negative correlation interpretation
• Percentile group percentages using quartile boundaries
• Engineering conclusions from correlation trends

These notes are strictly aligned with the Student Learning Outcomes (SLOs) for the FBISE 2026 annual examination.

• How do you construct accurate box and whisker plots on a scaled number line using a five-number data summary (minimum, Q1, median, Q3, maximum)? Students will draw the central box, mark median, and extend whiskers correctly.
• How do you calculate absolute interquartile ranges (IQR) across varying dataset scales using IQR = |Q3 - Q1| to evaluate data spread while avoiding negative dispersion values?
• How do you identify positive, negative, or neutral correlation trends between paired engineering variables? Students will analyze relationships like car mass vs fuel efficiency.
• How do you deduce percentile group percentages and student ranking blocks using quartile boundaries? Students will determine that Q3 = 75th percentile and 25% of data lies above Q3.

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