Misleading Graphical Displays of Data: How Graph Construction Affects Interpretation

Chapter 3: Constructing Graphical and Tabular Displays of Data (3.5)

3.5 Misleading Graphical Displays of Data

Graphical displays are powerful tools for summarizing and communicating data. However, the way graphs are constructed can sometimes mislead viewers, either intentionally or unintentionally. This section explores common ways in which graphical displays can be misleading and provides guidance on how to interpret them critically.

How Histogram Class Width Can Be Misleading

• Definition: A histogram is a graphical representation of the distribution of numerical data, where data are grouped into ranges (classes) and represented by bars.

• Class Width: The size of the intervals (bins) used to group data in a histogram.

• Key Point: The choice of class width can significantly affect the appearance and interpretation of a histogram.

Example: Consider daily approval ratings of a president in 2009. Two histograms are constructed: one with a small class width (2 percentage points) and one with a large class width (5 percentage points).

• The histogram with the smaller class width reveals a bimodal distribution, indicating two distinct periods (high approval in the first half, low in the second half).

• The histogram with the larger class width appears unimodal, hiding the drop in approval ratings.

• Conclusion: Using a larger class width can obscure important features of the data, potentially misleading the viewer.

Formula for Class Width:

$\text{Class Width} = \frac{\text{Maximum Value} - \text{Minimum Value}}{\text{Number of Classes}}$

Starting Value of the Vertical Axis

• Definition: The vertical axis (y-axis) in a bar graph or time-series plot typically represents the variable being measured.

• Key Point: Starting the vertical axis at a nonzero value can de-emphasize or exaggerate differences between groups.

Impact of Starting a Vertical Axis at a Nonzero Value

• Bar graphs with a vertical axis starting at 0 show the absolute sizes of the values.

• Bar graphs with a vertical axis starting at a higher value (e.g., 35%) emphasize small differences between groups, making them appear larger than they are in absolute terms.

Example: Comparing the percentage of social media users who visit platforms several times a day:

Platform	Percent (Approx.)
Facebook	~50%
Snapchat	~48%
Instagram	~38%

• If a company wants to emphasize its superiority, it may use a graph that starts at a higher value to exaggerate differences.

• If a company wants to minimize differences, it may use a graph that starts at 0 to show bars of similar height.

• Estimating values is easier when the axis increments are small and start at 0.

Impact of Starting a Vertical Axis at Zero in Time-Series Plots

• Time-series plots show how a variable changes over time.

• Starting the vertical axis at 0 emphasizes the absolute increase in values over time.

• Starting the vertical axis at a higher value (e.g., $38,000) emphasizes relative changes and can make increases appear more dramatic.

Example: Northwestern University tuition over several years:

Year	Tuition (thousands of dollars)
2010	~38
2017	~50

• If the goal is to de-emphasize tuition increases, use a plot with the vertical axis starting at 0.

• If the goal is to emphasize tuition increases, use a plot with the vertical axis starting at a higher value.

Summary Table: How Graph Construction Can Mislead

Graph Feature	Potential Effect	How It Misleads
Class Width (Histogram)	Obscures or reveals modes	Large width hides detail; small width shows more structure
Vertical Axis Starting Value	Emphasizes or de-emphasizes differences	Nonzero start exaggerates small differences
Axis Increments	Affects ease of estimation	Large increments make estimation harder

Best Practices for Interpreting Graphs

• Always check the starting value and increments of axes.

• Be cautious of graphs that use nonzero baselines or unusual class widths.

• Consider the context and what the graph is trying to emphasize or hide.

Additional info: These principles apply to all types of graphical data displays, including bar graphs, histograms, and time-series plots. Critical evaluation of graphs is essential for accurate data interpretation in statistics.