# How does the 45 degree banking rule apply when plotting multiple data series in one chart?

Cleveland's rule of thumb that the average slope of a line should be 45 degrees. But what about when a chart plots multiple data series? Is there a generalization of Cleveland's rule, or do you set the chart's aspect ratio by the "Eh, that looks good" method?

• Bill Cleveland did spectacularly good work on graphics, but this rule of thumb long predates him. It's in Fisher's Statistical methods for research workers, in essence, to name only one antecedent. But it was always about the typical slope, and that principle just extends when there are several variables as well as several up- or down- segments in a single series. Commented Sep 30, 2014 at 19:07

The Visweek 2012 paper, An Empirical Model of Slope Ratio Comparisons [PDF] by Justin Talbot, John Gerth, and Pat Hanrahan, attempts to generalize the question of optimal banking. Excerpt from introduction:

Despite the practical success of this guideline, its perceptual underpinnings remain unclear. Cleveland et al. justified the guideline with an experiment that showed that placing the mid-angle of two lines (the angle halfway between them) at 45° minimizes errors made in judging the ratio of their slopes. However, examination of their experimental design suggests that this conclusion might not be generally applicable. ... This paper seeks to improve our understanding of slope ratio estimation in line plots through empirical modeling and experimentation.

The experiments suggest the optimal median angle may be around 30° and that flatter is better for perceiving angle differences. Excerpt of a graph supporting the former point (observed error in black; predicted error in red):

However, like seemingly every other research paper, the bottom line is:

It is still unclear if the results derived in our studies for pairwise discrete comparisons will transfer to real plots. ... there remains substantial work to be done to build a solid understanding of aspect ratio selection.

Related papers:

• Arc Length-Based Aspect Ratio Selection [PDF], Justin Talbot, John Gerth, and Pat Hanrahan
• Crowdsourcing Graphical Perception: Using Mechanical Turk to Assess Visualization Design [PDF], Jeffrey Heer and Michael Bostock
• Nice answer! Another similar paper on testing Cleveland's perceptual hierarchy is Heer and Bostock vis.stanford.edu/files/2010-MTurk-CHI.pdf. For time series it is important to plot on several different aspect ratios, to see long term trend vs seasonal trend. For multivariate data scatterplots should have a square aspect ratio to perceive association better. Show your data on several different aspect ratios in your plots, like you would use several different binwidths when making histograms. Commented Oct 2, 2014 at 13:14
• Thanks Di, I added a link to that paper and another I found. I like your other points. You might expand them into a separate answer or appended them to this answer.
– xan
Commented Oct 3, 2014 at 12:36