Hanging rootograms and viz. differences in time series

These two quick charting tips are based on the notion that comparing differences from a straight line are easier than comparing deviations from a curved line. The problems with comparing differences between curved lines are similar to the difference between comparings length and distance from a common baseline (so Cleveland’s work is applicable), but the task of comparing two curves comes up enough that it deserves some specific attention.

The first example is comparing differences between a histogram and an estimated distribution. For example, people often like to superimpose a distribution curve on a histogram, and here is an example SPSS chart.

I believe it was Tukey who suggested that instead of plotting the histogram bars at the zero upwards, you hang them from the expected value. What this does is that instead of comparing differences from a curved line, you are comparing differences to the straight reference line at zero.

Although it is usual to plot the bars to cover the entire bin, I sometimes find this distracting. So here is an alternative (in SPSS – with example code linked to at the end of the post) in which I only plot lines and dots and just note in text the bin widths are in-between the hash marks on the X axis.

The second example is taken from William Playfair’s atlas, and Cleveland uses it to show that comparing two curves can be misleading. (It took me forever to find this data already digitized, so thanks for the bissantz blog for posting it.)

Instead of comparing the two curves only in terms of vertical deviations from one another, we tend to compare the curves in terms of the nearest location. Here the visual error in the magnitude of differences is likely to occur in the area between 1760 and 1766, where they look very close to one another because of the upward slope for both time series in that period.

Here I like the default behavior of SPSS when plotting the differences as an interval element and it is easier to see this potential error (just compare the length of the bars). When using a continuous scale, SPSS plots the interval elements with zero area inside and only an exterior outline (which ends up being near equivalent to a edge element).

More frequently though, people suggest just to plot the differences, and here is a chart with all three (Imports, Exports and the difference) plotted on the same graph. Note the differences at 1763 (390) is actually larger than the difference and the start of the series, 280 at 1700.

You can do similar things to scatterplots, which Tukey calls detilting plots. Again, the lesson is it is easier to compare differences from a straight line than it is differences from a curve (or sloped line). Here I have posted the SPSS code to make the graphs (I slightly cheated though and post edited in the guidelines and labels in the graph editor).

Leave a comment


  1. Jon Peck

     /  August 28, 2013

    1) The code referred to in this note seems to have been eaten by trolls.
    2) The rootogram is typically plotted on a square root scale.


  1. Custom square root scale (with negative values) in ggplot2 (R) | Andrew Wheeler

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