Misleading graph

In statistics, a misleading graph, also known as a distorted graph, is a graph which misrepresents data, constituting a misuse of statistics and with the result that an incorrect conclusion may be derived from it. Graphs may be misleading through being excessively complex or poorly constructed. Even when well-constructed to accurately display the characteristics of their data, graphs can be subject to different interpretation.

Misleading graphs may be created intentionally to hinder the proper interpretation of data, but can be also created accidentally by users for a variety of reasons including unfamiliarity with the graphing software, the misinterpretation of the data, or because the data cannot be accurately conveyed. Misleading graphs are often used in false advertising. One of the first authors to write about misleading graphs was Darrell Huff, who published the best-selling book How to Lie With Statistics in 1954. It is still in print.

The field of data visualization describes ways to present information that avoids creating misleading graphs.

Misleading graph methods

It [a misleading graph] is vastly more effective, however, because it contains no adjectives or adverbs to spoil the illusion of objectivity. There's nothing anyone can pin on you.

--How to Lie with Statistics (1954)

There are numerous ways in which a misleading graph may be constructed.

Excessive usage

The use of graphs where they are not needed can lead to unnecessary confusion/interpretation. Generally, the more explanation a graph needs, the less the graph itself is needed. Graphs do not always convey information better than tables.^{[citation needed]}

Biased labeling

The use of biased or loaded words in the graph's title, axis labels, or caption may inappropriately prime the reader.

Pie chart

Main article: Pie chart

• Comparing pie charts of different sizes could be misleading as people cannot accurately read the comparative area of circles.
• The usage of thin slices which are hard to discern may be difficult to interpret.
• The usage of percentages as labels on a pie chart can be misleading when the sample size is small.^{[citation needed]}
• Making a pie chart 3D or adding a slant will make interpretation difficult due distorted effect of perspective. Bar-charted pie graphs in which the height of the slices is varied may confuse the reader.

3D Pie chart slice perspective

A perspective (3D) pie chart is used to give the chart a 3D look. Often used for aesthetic reasons, the third dimension does not improve the reading of the data; on the contrary, these plots are difficult to interpret because of the distorted effect of perspective associated with the third dimension. The use of superfluous dimensions not used to display the data of interest is discouraged for charts in general, not only for pie charts. In a 3D pie chart, the slices that are closer to the reader appear to be larger than those in the back due to the angle at which they're presented.

Misleading Pie Chart Comparison

Misleading Pie Chart	Regular Pie Chart
In the misleading pie chart, Item C appears to be at least as large as Item A, whereas in actuality, it is less than half as large.

Edward Tufte, a prominent American statistician noted why tables may be preferred to pie charts in The Visual Display of Quantitative Information:

Tables are preferable to graphics for many small data sets. A table is nearly always better than a dumb pie chart; the only thing worse than a pie chart is several of them, for then the viewer is asked to compare quantities located in spatial disarray both within and between pies - Given their low data-density and failure to order numbers along a visual dimension, pie charts should never be used.

Improper scaling

When using pictogram in bar graphs, they should not be scaled uniformly as this creates a perceptually misleading comparison. The area of the pictogram is interpreted instead of only its height or width. This causes the scaling to make the difference appear to be squared.

Improper scaling of 2D pictogram in bar graph

Improper Scaling	Regular	Comparison
Note how in the improperly scaled pictogram bar graph, the image for B is actually 9 times larger than A.

2D shape scaling comparison

Square	Circle	Triangle
Note how the perceived size increases when scaling.

The effect of improper scaling of pictogram is further exemplified when the pictogram has 3 dimensions, in which case the effect is cubed.

Note how the usage of improper scaling of a three-dimensional pictogram in this fictitious graph. It appears that home sales have gone up significantly in 2001 over the previous year. Additionally, because no frequency axis is supplied, readers are unable to quantify the change, and are only left with a misleading perception of the change. The scaling, which is 2x, causes the change to appear to be $2^3$ or 8 times larger.

Additionally, an improperly scaled pictogram may leave the reader with the sense that the item itself has actually changed in size.

Misleading	Regular
Assuming the pictures represent equivalent quantities, note how in the misleading graph, there appears to be more bananas because the bananas occupy the most area and are furthest to the right.

Truncated graph

A truncated (also known as a torn or gee-whiz) graph has a y-axis that does not start at 0. These graphs can create the impression of important change where there is relatively little change.

Truncated graphs are useful in illustrating small differences. Graphs may also be truncated to save space. Commercial software such as MS Excel will tend to truncate graphs by default if the values are all within a narrow range, as in this example.

Truncated bar graph

Truncated bar graph	Regular bar graph
Note that both of these graphs display identical data; however, in the truncated bar graph on the left, the data appear to show significant differences, whereas in the regular bar graph on the right, these differences are hardly visible.

Indicating a y-axis break

There are several ways to indicate a y-axis break.

Axis changes

Changing y-axis maximum

Original graph	Smaller maximum	Larger maximum
Changing the y-axis maximum affects how the graph appears. A higher maximum will cause the graph to appear to have less-volatility, less-growth and a less steep line than a lower maximum.

Changing ratio of graph dimensions

Original graph	Half width, twice height	Twice width, half height
Changing the ratio of a graph's dimensions will affect how the graph appears.

No scale

The scales of a graph are often used to exaggerate or minimize differences.

Misleading bar graph with no scale

Less difference	More difference
Note the lack of a starting value for the y-axis, which makes it unclear if the graph is truncated. Additionally, note the lack of tick marks which prevents the reader from determining if the graph bars are properly scaled. Without a scale, the visual difference between the bars can be easily manipulated.

Misleading line graph with no scale

Volatility	Steady, fast growth	Slow growth
Though all three graphs share the same data, and hence the actual slope of the (x,y) data is the same, the way that the data is plotted can change the visual appearance of the angle made by the line on the graph. This is because each plot has different scale on its vertical axis. Because the scale is not shown, these graphs can be misleading.

Improper intervals/units

The intervals and units used in a graph may be manipulated to create or mitigate the expression of change.

Omitting data

Graphs created with omitted data remove information from which to base a conclusion.

Scatter plot with missing categories

Scatter plot with missing categories	Regular scatter plot
Note how in the scatter plot with missing categories on the left, the growth appears to be more linear with less variation.

In financial reports, negative returns, or data which does not correlate a positive outlook may be excluded to create a more favorable visual impression.

In engineering applications, the omission of data can be fatal. In the Space Shuttle Challenger disaster, engineers failed to properly display data.

Improper extraction

Graphs based on other graphs should be representative in their presentation.

Extraction has valid uses when searching for anomalies.

Original Graph	Extracted Graph
Note how the extracted graph does not accurately represent the original graph.

3D

The use of a superfluous third dimension which does not contain information is strongly discouraged as it may confuse the reader.

• 

  The third dimension may confuse readers.

• 

  Note how the blue bar in the front appears larger than the green bar in the back, despite having the same value, due to perspective.

• 

  When scaling in three dimensions, the effect of the change is cubed.

Complexity

Graphs are designed to allow for easier interpretation of statistical data. However, graphs with excessive complexity can obfuscate the data and make interpretation difficult.

Poor construction

Poorly constructed graphs can make data difficult to discern and thus interpret.

Measuring distortion

Several methods have been developed to determine whether graphs are distorted and to quantify this distortion.

Lie factor

$\text{Lie factor}=\frac{\text{size of effect shown in graphic}}{\text{size of effect shown in data}}$

where

$\text{size of effect}=\left| \frac{\text{second value − first value}}{\text{first value}} \right|$

A graph with a high lie factor (>1) would exaggerate change in the data it represents, while one with a small lie factor (>0, <1) would obscure change in the data. A perfectly accurate graph would exhibit a lie factor of 1.0.

Graph discrepancy index

$\text{graph discrepancy index}=100 \left(\frac{a}{b} - 1\right)$

where

$a=\text{percentage change depicted in graph}$

$b=\text{percentage change in data}$

The graph discrepancy index also known as the graph distortion index (GDI) was originally proposed by Paul John Steinbart in 1998. GDI is calculated as a percentage ranging from -100% to positive infinity with zero percent indicating that the graph has been properly constructed and anything outside the ±5% margin is considered to be distorted. Research into the usage of GDI as a measure of graphics distortion has found it to be inconsistent and discontinuous making the usage of GDI as a measurement for comparisons difficult.

Data-ink ratio

See also: Chartjunk

$\text{data-ink ratio}=\frac{\text{`ink' used to display the data}}{\text{total `ink' used to display the graphic}}$

The data-ink ratio should be relatively high, otherwise the chart may have unnecessary graphics.

Data density

$\text{data density}=\frac{\text{number of entries in data matrix}}{\text{area of data graphic}}$

The data density should be relatively high, otherwise a table may be better suited for displaying the data.

Usage in finance and corporate reports

Graphs are useful in the summary and interpretation of financial data. Graphs allow for trends in large data sets to be seen while also allowing the data to be interpreted by non-specialists.

Graphs are often used in corporate annual reports as a form of impression management. In the United States, graphs do not have to be audited as they fall under AU Section 550 Other Information in Documents Containing Audited Financial Statements.

Several published studies have looked at the usage of graphs in corporate reports for different corporations in different countries and have found frequent usage of improper design, selectivity, and measurement distortion within these reports. The presence of misleading graphs in annual reports have led to requests for standards to be set.

Research has found that while readers with poor levels of financial understanding have a greater chance of being misinformed by misleading graphs, even those with financial understanding, such as loan officers, may be misled.

Academia

The perception of graphs is studied in psychophysics, cognitive psychology, and computational visions.

See also

• Impression management
• Misuse of statistics
• How to Lie with Statistics

References

Books

• Huff, Darrell (1954). How to lie with statistics. pictures by Irving Geis (1st ed. ed.). New York: Norton. ISBN 0393052648. 
• Hurley, Patrick J. (2000). A Concise Introduction to Logic. Wadsworth Publishing. ISBN 9780534520069. 
• Keller, Gerald. Statistics for Management and Economics, abbreviated (9th ed., abbreviated ed. ed.). Mason, OH: South-Western. ISBN 1111527326. 
• Kirk, Roger E. (23 February 2007). Statistics: An Introduction. Cengage Learning. ISBN 978-0-534-56478-0. Retrieved 28 June 2012. 
• Nolan, Susan; Heinzen, Thomas (1 February 2011). Statistics for the Behavioral Sciences. Macmillan. ISBN 978-1-4292-3265-4. Retrieved 28 June 2012. 
• Rumsey, Deborah (17 May 2010). Statistics Essentials For Dummies. John Wiley & Sons. ISBN 978-0-470-61839-4. Retrieved 28 June 2012. 
• Weiss, Neil A. (1993). Elementary statistics. Addison-Wesley. ISBN 978-0-201-56640-6. Retrieved 28 June 2012. 
• Tufte, Edward (1997). Visual Explanations: Images and Quantities, Evidence and Narrative. Cheshire, CT: Graphics Press. 
• Utts, Jessica M. (2005). Seeing through statistics (3rd ed. ed.). Belmont: Thomson, Brooks/Cole. ISBN 9780534394028. 
• Wainer, Howard (1 July 2000). Visual Revelations: Graphical Tales of Fate and Deception From Napoleon Bonaparte To Ross Perot. Psychology Press. ISBN 978-0-8058-3878-7. Retrieved 19 July 2012. 
• Whitbread, David (2001). The design manual (2nd ed. ed.). Sydney: University of New South Wales Press. ISBN 0868406589. 

Further reading

• A discussion of misleading graphs, Mark Harbison, Sacramento City College
• Robbins, Naomi B. (2005). Creating more effective graphs. Hoboken, N.J.: Wiley-Interscience. ISBN 9780471698180. 
• Durbin CG, Jr (2004 Oct). "Effective use of tables and figures in abstracts, presentations, and papers.". Respiratory care 49 (10): 1233–7. PMID 15447809. 
• Goundar, Nadesa (2009). "Impression Management in Financial Reports Surrounding CEO Turnover". Masters Dissertation. Unitec Institute of Technology. Retrieved 9 July 2012. 
• Huff, Darrell; Geis, Irving (17 October 1993). How to Lie With Statistics. W. W. Norton & Company. ISBN 978-0-393-31072-6. Retrieved 28 June 2012. 
• Bracey, Gerald (2003). "Seeing Through Graphs". Understanding and using education statistics : it's easier than you think. Educational Research Service. ISBN 9781931762267. 
• Harvey, J. Motulsky (June 2009). "The Use and Abuse of Logarithmic Axes". GraphPad Software Inc. 
• Chandar, N.; Collier, D.; Miranti, P. (15 February 2012). "Graph standardization and management accounting at AT&T during the 1920s". Accounting History 17 (1): 35–62. doi:10.1177/1032373211424889. 
• Mather, Paul; Ramsay, Alan; Steen, Adam (1 January 2000). "The use and representational faithfulness of graphs in Australian IPO prospectuses". Accounting, Auditing & Accountability Journal 13 (1): 65–83. doi:10.1108/09513570010316144. 
• Beattie, Vivien; Jones, Michale John (1996). Financial graphs in corporate annual reports : a review of practice in six countries. London: Institute of Chartered Accounants in England and Wales. ISBN 9781853557071. 
• Galliat, Tobias (Summer 2005). "Visualisierung von Informationsräumen". Fachhochschule Köln, University of Applied Sciences Cologne. Archived from the original on 2006-01-04. Retrieved 9 July 2012. 
• Carvalho, Clark R.; McMillan, Michael D. (September 1992). "Graphic Representation in Managerial Decision Making: The Effect of Scale Break on the Dependent Axis". AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OH. 
• Johnson, R. Rice; Roemmich, R. (October 1980). "Pictures that Lie: The Abuse of Graphs in Annual Reports". Management Accounting: 50–56. 
• Davis, Alan J. (1 August 1999). "Bad graphs, good lessons". ACM SIGGRAPH Computer Graphics 33 (3): 35–38. doi:10.1145/330572.330586. 
• Louwers, T.; Radtke, R; Pitman, M. (May–June 1999). "Please Pass the Salt: A Look at Creative Reporting in Annual Reports". Today's CPA: 20–23. 
• Beattie, Vivien; Jones, Michael John (May 2001). "A six-country comparison of the use of graphs in annual reports". The International Journal of Accounting 36 (2): 195–222. doi:10.1016/S0020-7063(01)00094-2. 
• Wainer, Howard (1984). "How to Display Data Badly". The American Statistician 38 (2): 137–147. 
• Lane, David M.; Sándor, Anikó (1 January 2009). "Designing better graphs by including distributional information and integrating words, numbers, and images.". Psychological Methods 14 (3): 239–257. doi:10.1037/a0016620. 
• Campbell, Mary Pat (Feb). "Spreadsheet Issues: Pitfalls, Best Practices, and Practical Tips". Actuarial Practice Forum. 
• Arocha, Carlos (May 2011). "Words or Graphs?". The Stepping Stone. 
• Raschke, Robyn L.; Steinbart, Paul John (1 September 2008). "Mitigating the Effects of Misleading Graphs on Decisions by Educating Users about the Principles of Graph Design". Journal of Information Systems 22 (2): 23–52. doi:10.2308/jis.2008.22.2.23. 

External links

• Gallery of Data Visualization The Best and Worst of Statistical Graphics, York University