Visual Displays of Data: Cautions and Critical Evaluations

The primary learning goal for evaluating graphics is to develop the ability to critically analyze visual data and identify common methods used to create misleading or deceptive representations.
Understanding these cautions is essential for accurate statistical reasoning in everyday life.

Many graphics are constructed in a manner that distorts the viewer's natural perception of the data.
The Length vs. Area Distortion: This is one of the most common types of visual deception. Visual elements may use length to represent a value, but the human eye tends to focus on the overall area of the graphic.
Example: Figure 3.20 (The Declining Dollar):
- Dollar-shaped bars are utilized to illustrate the declining value of the dollar over a specific period.
- In this graphic, the value of the dollar is intended to be represented by the length of the dollar bill image.
- However, because the width of the bill is scaled proportionally with the length, the area of the dollar bill decreases much more rapidly than the length.
- This leads to a perception that the dollar's value shrank significantly more than the actual numerical data suggests.

A change in the scale of an axis can fundamentally alter the interpretation of the data.
Deceptive Vertical Scales: If a vertical axis does not begin at a baseline of $0$ or does not cover the full possible range (e.g., $0\%$ to $100\%$ ), it can exaggerate small differences.
Example: Figure 3.21 (Women in College):
- This graph tracks the percentage of college students who have been women since the year $1910$ .
- In Figure 3.21a, the percentage appears to grow by a "huge margin" after approximately $1950$ .
- Critical analysis reveals the vertical axis does not start at $0$ and does not end at $100\%$ .
- While Figure 3.21a and Figure 3.21b show identical data from the National Center for Education Statistics, they look drastically different because their vertical scales utilize different ranges.

Nonlinear Scales: These are scales where each increment on the axis does not represent the same fixed change in value.
Exponential (Logarithmic) Scales: A specific type of nonlinear scale where the scale grows by powers (frequently powers of $10$ ). These are often identified by the use of exponents.
Comparison of Linear vs. Exponential Scales (Figure 3.22):
- In Figure 3.22, computer speeds are compared using two different y-axis formats.
- The left graph uses an exponential scale, while the right graph uses a linear scale.
Predictive Utility:
- Based on Figure 3.22a (exponential), it is often easier to predict future trends, such as the speed of the fastest computers in the year $2030$ , because exponential growth often appears as a straight line on a logarithmic scale.
- Making the same prediction with Figure 3.22b (linear) is difficult because the data points may appear to curve upward so sharply that they become almost vertical, masking the underlying trend.

Graphs showing percentage change are extremely common in economic reporting but require careful interpretation to avoid being misled.
The "Rate of Change" vs. "Actual Value" Trap: A downward trend on a percentage change graph does not necessarily mean the actual value is decreasing; it may simply mean the value is increasing at a slower rate.
Example: Figure 3.24 (College Tuition Trends):
- Figure 3.24a shows the annual percentage change in tuition. It peaked in the mid- $2000s$ and then showed a downward slope.
- A casual observer might conclude that college costs became less expensive after the peak.
- However, because the data values remain positive (above $0\%$ ), tuition was still increasing every year—it was just increasing by smaller amounts.
- Figure 3.24b shows the actual tuition values, confirming that costs rose every single year throughout the period.

Pictographs: These are graphs that have been embellished with artwork or illustrations to make them more visually appealing.
Distraction and Misleading Elements: While artwork adds impact, it often introduces flaws:
- Decorative Data: In many pictographs, such as Figure 3.25 (World Population), elements like a "line of people" are purely decorative and carry no actual statistical information.
- Non-linear Horizontal Scales: Pictographs often ignore standard scaling on the horizontal axis. In Figure 3.25, the time intervals between points on the x-axis are not consistent (the time gaps are not the same), which can distort the viewer's understanding of how fast a trend (like population growth) is occurring.
- Data source for population examples: United Nations Population Division.