8.3 Measures of Dispersion: Range, Quartiles, IQR & Boxplots

8.3 Measures of Dispersion: Range, Quartiles, IQR & Boxplots

While measures of central tendency identify the center of a dataset, measures of dispersion (variation or spread) describe how spread out or tightly clustered the data values are. Two datasets can possess identical means and medians while having completely different spreads. On the CLEP College Mathematics exam, you will construct five-number summaries, compute the Interquartile Range (IQR), apply the $1.5 \times \text{IQR}$ outlier rule, and interpret box-and-whisker plots.

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1. The Range

The range is the simplest measure of dispersion, defined as the difference between the largest and smallest observations in a dataset.

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

Strengths and Severe Limitations of the Range

• Strength: Extremely fast and simple to calculate.
• Limitation: Highly unstable and sensitive to outliers. A single extreme outlier will artificially inflate the range, providing no information about how the remaining $98\%$ of data values are distributed.

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2. Quartiles & The Five-Number Summary

Quartiles are positional values that partition an ordered dataset into four equal quarters, each containing exactly $25\%$ of the observations.

+-----------------------------------------------------------------------------+
|                     THE FOUR QUARTERS OF AN ORDERED DATASET                 |
|                                                                             |
|   Minimum             Q_1                Q_2 (Median)           Q_3          Maximum  |
|      |---- 25% Data ---|---- 25% Data ----|---- 25% Data ----|--- 25% Data ---|       |
|     0th               25th               50th               75th            100th     |
|  Percentile        Percentile         Percentile         Percentile      Percentile   |
+-----------------------------------------------------------------------------+

The Five-Number Summary Components

1. Minimum ($\text{Min}$): Smallest data value in the dataset.
2. First Quartile ($Q_1$): The $25\text{th}$ percentile; the median of the lower half of the data.
3. Second Quartile / Median ($Q_2$): The $50\text{th}$ percentile; the overall median of the dataset.
4. Third Quartile ($Q_3$): The $75\text{th}$ percentile; the median of the upper half of the data.
5. Maximum ($\text{Max}$): Largest data value in the dataset.

Standard Protocol for Finding $Q_1$ and $Q_3$

1. Arrange the data in ascending order.
2. Find the overall median ($Q_2$).
3. If $n$ is odd: Exclude $Q_2$ from both halves. Find the median of the lower half ($Q_1$) and the median of the upper half ($Q_3$).
4. If $n$ is even: The median splits the data evenly into two halves. Find the median of the lower half ($Q_1$) and the median of the upper half ($Q_3$).

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3. The Interquartile Range (IQR)

The Interquartile Range (IQR) measures the spread of the middle $50\%$ of the dataset.

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

Why IQR is Superior to Range

• Outlier Resistance: Because the IQR completely ignores the lowest $25\%$ and highest $25\%$ of observations, it is unaffected by extreme outliers.
• It provides the standard mathematical foundation for identifying statistical outliers and constructing the central box in boxplots.

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4. Outlier Detection: Tukey's 1.5 × IQR Rule

An outlier is an observation that lies abnormally far from the main body of data. In standard statistics and on the CLEP exam, outliers are identified using inner fences:

+-----------------------------------------------------------------------------+
|                        1.5 x IQR OUTLIER FENCE PROTOCOL                     |
|                                                                             |
|   1. Compute IQR:              IQR = Q_3 - Q_1                              |
|   2. Compute Step Size:        Step = 1.5 * IQR                             |
|   3. Compute Lower Fence:      Lower Fence = Q_1 - 1.5 * IQR                |
|   4. Compute Upper Fence:      Upper Fence = Q_3 + 1.5 * IQR                |
|                                                                             |
|   OUTLIER CRITERION:                                                        |
|   - Any data point x < Lower Fence is a LOWER OUTLIER.                      |
|   - Any data point x > Upper Fence is an UPPER OUTLIER.                     |
+-----------------------------------------------------------------------------+

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5. Boxplots (Box-and-Whisker Plots)

A boxplot is a standardized graphical display that visually maps the Five-Number Summary onto a numerical scale.

+-----------------------------------------------------------------------------+
|                        ANATOMY OF A MODIFIED BOXPLOT                        |
|                                                                             |
|                     +-----------------+                                     |
|             |-------|        |        |-------|                 *           |
|             |       +-----------------+       |                             |
|            Min     Q_1     Median    Q_3     Max            Outlier         |
|                   (25th)   (50th)   (75th)  (non-outlier)                   |
|                     <----- IQR ----->                                       |
|             <---------- Range -------------->                               |
|   ---+------|-------+--------+--------+-------+-----------------+-------+-- |
|      0     10      20       30       40      50                70      80   |
+-----------------------------------------------------------------------------+

Structural Features of Boxplots

• The Central Box: Spans from $Q_1$ to $Q_3$, enclosing the middle $50\%$ of the data. Its width represents the $\text{IQR}$.
• The Median Line: A vertical line drawn inside the box at $Q_2$.
• Whiskers:
  • In a standard boxplot, whiskers extend to the absolute $\text{Min}$ and $\text{Max}$.
  • In a modified boxplot, whiskers extend only to the most extreme non-outlier data points within the fences, and outliers are plotted individually as standalone dots, stars, or asterisks ($*$).
• Reading Distribution Shapes from Boxplots:
  • Symmetric: Median line is centered in the box, and both whiskers are approximately equal in length.
  • Right-Skewed: Median line is closer to $Q_1$ (left side of box), and the right whisker is noticeably longer than the left whisker.
  • Left-Skewed: Median line is closer to $Q_3$ (right side of box), and the left whisker is noticeably longer than the right whisker.

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6. Step-by-Step Worked Examples

Worked Example 1: Full Five-Number Summary & Outlier Verification

Problem: A researcher records the number of minutes $11$ patients wait in a clinic:

$$\{8, \; 12, \; 15, \; 16, \; 18, \; 22, \; 25, \; 27, \; 31, \; 34, \; 68\}$$

1. Determine the Five-Number Summary.
2. Calculate the Range and Interquartile Range (IQR).
3. Apply the $1.5 \times \text{IQR}$ rule to determine if any wait times are statistical outliers.

Solution:

1. Five-Number Summary:
   • $\text{Minimum} = 8$
   • Dataset size $n = 11$ (odd). The overall median ($Q_2$) is the $6\text{th}$ value: $\mathbf{Q_2 = 22}$.
   • Lower half (first 5 values: $8, 12, 15, 16, 18$): Median is the $3\text{rd}$ value: $\mathbf{Q_1 = 15}$.
   • Upper half (last 5 values: $25, 27, 31, 34, 68$): Median is the $3\text{rd}$ value: $\mathbf{Q_3 = 31}$.
   • $\text{Maximum} = 68$
   • Five-Number Summary: $\{\text{Min} = 8, \; Q_1 = 15, \; Q_2 = 22, \; Q_3 = 31, \; \text{Max} = 68\}$.
2. Dispersion Metrics:
   • $\text{Range} = \text{Max} - \text{Min} = 68 - 8 = 60$
   • $\text{IQR} = Q_3 - Q_1 = 31 - 15 = 16$
3. Outlier Fences:
   • $\text{Step} = 1.5 \times \text{IQR} = 1.5 \times 16 = 24$
   • $\text{Lower Fence} = Q_1 - 24 = 15 - 24 = -9$
   • $\text{Upper Fence} = Q_3 + 24 = 31 + 24 = 55$
   • Evaluation: Since $68 > 55$, the value $68$ is a confirmed upper outlier.

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Worked Example 2: Comparing Parallel Boxplots

Problem: Two sections of College Algebra take the same midterm examination ($100$ points total). The resulting parallel boxplots yield:

• Section A: $\{\text{Min} = 50, \; Q_1 = 65, \; Q_2 = 78, \; Q_3 = 88, \; \text{Max} = 98\}$
• Section B: $\{\text{Min} = 60, \; Q_1 = 70, \; Q_2 = 78, \; Q_3 = 82, \; \text{Max} = 92\}$

Compare the medians and the dispersion of the middle $50\%$ of both classes.

Solution:

1. Central Tendency: Both sections have identical medians ($Q_2 = 78$). The typical student in either section performed equally well.
2. Dispersion of Middle 50% (IQR):
   • $\text{IQR}_A = 88 - 65 = 23$
   • $\text{IQR}_B = 82 - 70 = 12$
3. Conclusion: Although both classes have the same median score ($78$), Section A exhibits nearly twice the variability in its middle $50\%$ of students compared to Section B, indicating a far wider spread of student performance in Section A.

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7. Common CLEP Traps & Strategic Checkpoints

• Trap 1: Confusing Quartile Intervals with Data Counts: Each of the four intervals (Min to $Q_1$, $Q_1$ to $Q_2$, $Q_2$ to $Q_3$, $Q_3$ to Max) contains exactly $25\%$ of the data observations, regardless of how long or short the physical line segment appears on the boxplot.
• Trap 2: Including the Median in Quartile Halves for Odd $n$: When finding $Q_1$ and $Q_3$ in a dataset with odd $n$, do NOT include the overall median in either the lower or upper half.
• Trap 3: Subtracting Minimum from $Q_1$ for IQR: $\text{IQR}$ is always $Q_3 - Q_1$. Do not confuse it with range ($\text{Max} - \text{Min}$) or semi-interquartile range ($\frac{\text{IQR}}{2}$).
• Trap 4: Miscalculating Outlier Fences: Outlier fences add/subtract $1.5 \times \text{IQR}$ from the quartiles ($Q_1$ and $Q_3$), NOT from the median ($Q_2$).