9.2 Range, Quartiles, Interquartile Range (IQR), and Comparing Parallel Box Plots

Two classes can share the exact same mean and median of $65\%$ on an end-of-semester exam while having completely different teaching needs: in Class A, every student scored between $60\%$ and $70\%$, whereas in Class B, scores ranged from $25\%$ to $100\%$. To describe how tightly clustered or widely dispersed a dataset is, educators pair a measure of centre with a measure of spread (variability).

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Measuring Spread: Range vs. Interquartile Range (IQR)

1. The Range

The range is the difference between the single highest observation and the single lowest observation:

$$\text{Range} = \text{Maximum} - \text{Minimum}$$

Although easy to compute, the range depends entirely on the two most extreme values. If one student in a class of $30$ leaves half the paper blank and scores $12\%$ while the remaining $29$ students score between $68\%$ and $85\%$, the range jumps from $17$ marks to $73$ marks, overstating the variability of the class as a whole.

2. Quartiles and the Interquartile Range (IQR)

To measure spread without distortion from extreme tails, statisticians divide the sorted dataset into four equal-sized quarters ($25\%$ each) using three quartiles:

• First (Lower) Quartile ($Q_1$): The $25\text{th}$ percentile, found as the median of the lower half of the sorted data. Exactly $25\%$ of scores lie below $Q_1$, and $75\%$ lie above $Q_1$.
• Second Quartile ($Q_2$): The $50\text{th}$ percentile, which is the overall median of the dataset.
• Third (Upper) Quartile ($Q_3$): The $75\text{th}$ percentile, found as the median of the upper half of the sorted data. Exactly $75\%$ of scores lie below $Q_3$, and $25\%$ lie above $Q_3$.

The interquartile range (IQR) is the distance between the upper and lower quartiles:

$$\text{IQR} = Q_3 - Q_1$$

Because the IQR measures the spread of the middle $50\%$ of observations, it ignores the lowest $25\%$ and highest $25\%$ of scores and is resistant to outliers.

Worked Example: Finding the Five-Number Summary and IQR

• Odd sample size ($n = 11$): Consider $11$ sorted Year 6 mathematics test scores: $42, 48, 51, 55, 58, \mathbf{62}, 66, 70, 74, 79, 95$.
  • $\text{Minimum} = 42$ and $\text{Maximum} = 95$, so $\text{Range} = 95 - 42 = 53$.
  • The overall median ($Q_2$) is the $6\text{th}$ value: $Q_2 = 62$.
  • Exclude the middle value $62$ to form the lower half of $5$ scores ($42, 48, \mathbf{51}, 55, 58$) and the upper half of $5$ scores ($66, 70, \mathbf{74}, 79, 95$).
  • The median of the lower half is $Q_1 = 51$, and the median of the upper half is $Q_3 = 74$.
  • Therefore, $\text{IQR} = Q_3 - Q_1 = 74 - 51 = 23$.
• Even sample size ($n = 10$): For $10$ sorted scores $40, 46, 50, 54, 58, 62, 66, 72, 78, 84$, the median sits between the $5\text{th}$ and $6\text{th}$ scores: $Q_2 = \frac{58 + 62}{2} = 60$. The lower $5$ scores ($40, 46, \mathbf{50}, 54, 58$) have median $Q_1 = 50$, and the upper $5$ scores ($62, 66, \mathbf{72}, 78, 84$) have median $Q_3 = 72$, giving $\text{IQR} = 72 - 50 = 22$.

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Anatomy of a Box-and-Whisker Plot and the Width Trap

A box-and-whisker plot (box plot) displays the five-number summary ($\text{Minimum}, Q_1, \text{Median } [Q_2], Q_3, \text{Maximum}$) along a number line. Vertical lines mark $\text{Min}$, $Q_1$, $Q_2$, $Q_3$, and $\text{Max}$, with a rectangular box drawn from $Q_1$ to $Q_3$ and whiskers extending to the minimum and maximum:

1. Bottom/Left Whisker ($\text{Min}$ to $Q_1$): Contains the lowest $25\%$ of students ($0\text{th}$ to $25\text{th}$ percentile).
2. Lower/Left Half of Box ($Q_1$ to $Q_2$): Contains the second $25\%$ of students ($25\text{th}$ to $50\text{th}$ percentile).
3. Upper/Right Half of Box ($Q_2$ to $Q_3$): Contains the third $25\%$ of students ($50\text{th}$ to $75\text{th}$ percentile).
4. Top/Right Whisker ($Q_3$ to $\text{Max}$): Contains the highest $25\%$ of students ($75\text{th}$ to $100\text{th}$ percentile).

The Critical Box-Plot Width Trap: Candidates frequently assume that a wider half-box or a longer whisker contains more students than a short whisker or narrow half-box. This is false. Each of the four segments on a single box plot always contains exactly $25\%$ of that cohort. A narrow segment (e.g., $Q_1 = 62$ to $Q_2 = 66$, width $4$) means $25\%$ of students are tightly clustered within $4$ marks (high score density), whereas a wide segment (e.g., $Q_2 = 66$ to $Q_3 = 88$, width $22$) means the same $25\%$ share of students is spread out across $22$ marks (low score density).

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Comparing Parallel Box Plots Across Classes and Year Levels

When parallel box plots are stacked on the same scale, compare cohorts systematically using the table below.

Worked Comparison: Year 9 Numeracy Diagnostic ($\text{Max Possible} = 100$)

• Central tendency (Medians): Class 9A has a higher median ($72$) than Class 9B ($64$), indicating higher typical achievement in Class 9A.
• Consistency (IQR and Range): Class 9A has a much smaller IQR ($16$ vs. $26$) and smaller range ($46$ vs. $58$), showing that Class 9A's middle $50\%$ (and overall cohort) achieved much more consistent scores than Class 9B.
• Cross-cohort landmark comparisons: Notice that Class 9A's lower quartile ($Q_1 = 64$) equals Class 9B's median ($Q_2 = 64$). Therefore, $75\%$ of Class 9A scored $64$ or higher, whereas only $50\%$ of Class 9B scored $64$ or higher.
• Converting percentages to headcounts: Because the two classes have different sizes ($N = 28$ vs. $N = 32$), always multiply the quartile percentage by that specific cohort's $N$: the number of students scoring $64$ or higher is $0.75 \times 28 = 21$ students in Class 9A, compared with $0.50 \times 32 = 16$ students in Class 9B.