14.2 Univariate Displays (Histograms, Boxplots, Stemplots) & Distribution Shapes

14.2 Univariate Displays (Histograms, Boxplots, Stemplots) & Distribution Shapes

Graphical Displays for Univariate Quantitative Data

Graphical displays transform raw quantitative data into visual representations that reveal patterns in center, spread, shape, and unusual features (such as gaps, clusters, or outliers). Depending on sample size and analytic goals, several core displays are utilized:

1. Dot Plots: A dot plot positions individual data points as dots along a single continuous horizontal axis. When duplicate values occur, dots stack vertically. Dot plots are highly effective for small datasets ($n < 30$), clearly identifying exact values, local clusters, and gaps. However, for large datasets, dot plots become visually cluttered and unwieldy.

2. Stem-and-Leaf Plots (Stemplots): A stem-and-leaf display separates each numerical observation into a leading "stem" (the leading digit or digits) and a trailing "leaf" (the final trailing digit).

• Data Preservation: Unlike histograms, stemplots preserve the raw quantitative values while simultaneously displaying distribution shape. Every stemplot must include an explanatory key (e.g., $4 \mid 2 = 42$).
• Split Stems: When data cluster tightly within a few stems, each stem can be split into two lines (leaves 0–4 on the first line, 5–9 on the second line) or five lines (leaves 0–1, 2–3, 4–5, 6–7, 8–9) to reveal finer distribution geometry.
• Back-to-Back Stemplots: By placing a common set of stems in a central vertical column with leaves extending to the left for one group and to the right for another, back-to-back stemplots provide direct pairwise graphical comparisons between two related distributions.

1. Histograms: A histogram divides the continuous range of data into adjacent, non-overlapping intervals called bins or classes (typically defined as $[a, b)$, closed on the left and open on the right). Vertical rectangles are constructed over each bin:

• Frequency Histogram: The height of each rectangle represents the raw count (frequency) of observations falling within that bin.
• Relative Frequency Histogram: The height represents the proportion (relative frequency = $\frac{\text{frequency}}{n}$) or percentage of total observations.
• Contiguity: Unlike bar charts for categorical data (which feature gaps between bars), histogram bars must be contiguous with zero gaps between adjacent classes unless an intermediate bin has a frequency of zero (representing a true data gap). Bin width selection governs visual resolution: bins that are too wide obscure modal peaks, while bins that are too narrow create noisy, erratic spikes.

1. Cumulative Frequency Ogives: An ogive (cumulative frequency polygon) plots the cumulative relative frequency (from 0% to 100%) against the upper boundary of each class interval. Connecting these points with line segments produces a non-decreasing S-shaped curve that allows direct graphical interpolation of percentiles, deciles, and the median (the 50% horizontal line).

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The Five-Number Summary and Modified Boxplots

The five-number summary provides a concise, resistant numerical description of a quantitative dataset: $\{\text{Minimum}, Q_1, \text{Median } (M), Q_3, \text{Maximum}\}$ These five values partition an ordered dataset into four equal segments, each containing approximately 25% of the observations.

• $Q_1$ (First Quartile): The 25th percentile, separating the lowest 25% of data from the upper 75%.
• $\text{Median } (M)$: The 50th percentile, bisecting the distribution.
• $Q_3$ (Third Quartile): The 75th percentile, separating the lowest 75% of data from the top 25%.
• Interquartile Range: $IQR = Q_3 - Q_1$, the width of the central box.

A box-and-whisker plot graphs the five-number summary:

• A central rectangular box spans from $Q_1$ to $Q_3$, representing the middle 50% of the distribution.
• A vertical line segment inside the box marks the location of the median $M$.
• In a standard boxplot, horizontal whiskers extend outward to the minimum and maximum observations.

Modified Boxplots and Whiskers

In advanced statistical practice and on the FTCE examination, modified boxplots are used to isolate outliers:

• Whiskers extend outward only to the most extreme data values that fall within the calculated outlier fences.
• Any observation falling beyond the outlier fences is plotted individually as an isolated symbol (an asterisk, dot, or open circle). Whiskers never extend to an outlier.

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Tukey's $1.5 \times IQR$ Outlier Rule

John Tukey formalized a robust mathematical criterion for flagging potential outliers using the interquartile range:

1. Calculate the interquartile range: $IQR = Q_3 - Q_1$
2. Compute the Lower Fence ($LF$) and Upper Fence ($UF$): $LF = Q_1 - 1.5 \times IQR$ $UF = Q_3 + 1.5 \times IQR$
3. Outlier Identification:
   • Any observation $x < LF$ is flagged as a low outlier.
   • Any observation $x > UF$ is flagged as a high outlier.
4. Extreme Outliers: Boundaries using $3.0 \times IQR$ ($Q_1 - 3 \times IQR$ and $Q_3 + 3 \times IQR$) delineate extreme outliers from mild outliers.

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Distribution Shape, Skewness, and Modality

The geometric profile of a univariate distribution reveals fundamental properties of the underlying population:

1. Symmetric Distributions: The left and right halves of the distribution are approximate mirror images reflected across the center:

• Bell-Shaped (Mound-Shaped): Unimodal with density thinning symmetrically into both tails (e.g., normal distributions). Here, $\text{Mean} \approx \text{Median} \approx \text{Mode}$.
• Uniform (Rectangular): Constant frequency across the entire support domain.

1. Skewed Right (Positively Skewed):

• The distribution extends an elongated tail toward larger, positive values on the right, while the primary cluster of observations concentrates on the left.
• Because extreme high values pull the non-resistant mean rightward while the median resists this pull: $\text{Mean} > \text{Median} > \text{Mode}$
• In a boxplot, positive skewness manifests as a median line positioned toward the left of the central box, a longer right whisker, and high outliers plotted to the right.

1. Skewed Left (Negatively Skewed):

• The distribution extends an elongated tail toward smaller, negative values on the left, while the primary cluster concentrates on the right.
• Extreme low values pull the arithmetic mean leftward: $\text{Mean} < \text{Median} < \text{Mode}$
• In a boxplot, negative skewness manifests as a median line positioned toward the right of the central box, a longer left whisker, and low outliers plotted to the left.

1. Bimodal and Multimodal Distributions:

• Distributions exhibiting two or more distinct peaks indicate the presence of distinct sub-populations (e.g., combined heights of adult males and females).
• Critical Limitation of Boxplots: A boxplot cannot detect multimodality. A bimodal distribution and a unimodal distribution with identical five-number summaries produce identical boxplots. Histograms, dot plots, and stemplots are required to identify peaks and valleys.

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Distribution Shape and Display Comparison Table

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Worked Exemplar: Five-Number Summary and Outlier Determination

Problem: A sports statistician records the recovery times (in days) for a sample of 16 professional athletes rehabilitating from a joint injury: $24, 18, 29, 31, 22, 19, 27, 33, 26, 25, 30, 28, 21, 62, 23, 12$ Determine the five-number summary, calculate the interquartile range, determine the lower and upper fences for outliers using Tukey's $1.5 \times IQR$ rule, identify any outliers, and specify the termination values for the whiskers in a modified boxplot.

• Step 1: Sort the Raw Data. Arrange the $n = 16$ observations in ascending order: $12, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 62$

• Step 2: Calculate the Five-Number Summary.

  • Minimum: $x_{(1)} = 12$
  • Maximum: $x_{(16)} = 62$
  • Median ($M$): Since $n = 16$ is even, average the 8th and 9th ordered values: $M = \frac{x_{(8)} + x_{(9)}}{2} = \frac{25 + 26}{2} = 25.5$
  • First Quartile ($Q_1$): The lower half contains the first 8 observations: ${12, 18, 19, 21, 22, 23, 24, 25}$. The median of this lower half is the average of the 4th and 5th values: $Q_1 = \frac{21 + 22}{2} = 21.5$
  • Third Quartile ($Q_3$): The upper half contains the last 8 observations: ${26, 27, 28, 29, 30, 31, 33, 62}$. The median of this upper half is the average of the 12th and 13th values: $Q_3 = \frac{29 + 30}{2} = 29.5$
  • Summary Vector: ${\text{Min} = 12, Q_1 = 21.5, M = 25.5, Q_3 = 29.5, \text{Max} = 62}$

• Step 3: Calculate the Interquartile Range ($IQR$). $IQR = Q_3 - Q_1 = 29.5 - 21.5 = 8.0$

• Step 4: Compute Tukey's Outlier Fences. $1.5 \times IQR = 1.5(8.0) = 12.0$ $\text{Lower Fence } (LF) = Q_1 - 1.5 \times IQR = 21.5 - 12.0 = 9.5$ $\text{Upper Fence } (UF) = Q_3 + 1.5 \times IQR = 29.5 + 12.0 = 41.5$

• Step 5: Identify Outliers and Whisker Termination Values.

  • Comparing data points against the fences:
    • Minimum value is $12 > 9.5$, so there are no low outliers.
    • The value $62 > 41.5$, so $62$ is an upper outlier.
  • In a modified boxplot:
    • The left whisker terminates at the minimum non-outlier value: 12.
    • The right whisker terminates at the largest data value within the upper fence: 33 (since 33 is the largest value $\le 41.5$).
    • The outlier 62 is plotted as an isolated symbol at $x = 62$.