## A reader's guide to a New York Times graphic

##### Mar 11, 2012

The New York Times chose to present the poll results from Super Tuesday in the following chart (link):

It took me a bit of time to take in what this chart has to offer. To save your troubles, I've drawn up a reader's guide:

The graphic is a disguised scatter plot with one axis being Romney's share minus Santorum's share and the other axis being the total share of all other candidates. This is an "uneven canvass" in the sense that the data are much more likely to fall into a small part of the chart area (the orange shaded region).

If the reader just wants to know which segments of the electorate favors Romney v. Santorum, the chart is pretty effective at pointing to the answer. It is quite challenging to learn much else about the data.

***

Here are the results for Ohio, plotted as a stacked bar chart, with three segments in each bar (Romney's share, Santorum's share and the share of all other candidates).

This more standard presentation conveys much more of the underlying information. The trade-off is that the reader has to try harder to figure out the answer for each segment of voters.

[PS: 3/13/2012]:

Thanks to several readers for your comments. I went back to look at the NYT graphic again, and can confirm that it is a ternary chart. The chart area is indeed an equilateral triangle with three equal sides.

What threw me off was the axis labels, particularly the Santorum and Romney labels which give the impression that there is a zero mid-point and some kind of share data along the  east-west axis. If this were true, then the chart could not be a ternary plot because Romney and Santorum shares are not mirror images.

In a ternary plot, we must identify Romney, Santorum, and "Other candidates" as the three vertices. The way this chart is labelled, it invites readers to drop a perpendicular line to the horizontal axis to read Santorum's share (e.g.). That doesn't work. Trying to fish the data out of a ternary plot is always challenging. You pick the vertex corresponding to the data series you want, say Romney's share. Then you take the side opposite that vertex. Now draw lines parallel to that side -- as you approach the Romney vertex, Romney's share goes from 0% to 100%. The following chart shows this:

For ternary plots, it's easier to go with the hand-waving principle that the closer you are to the vertex, the greater the weight of that vertex. So with the abortion data point, we see that it is much closer to the Santorum corner than the other two corners.

The vertical line for "other candidates" is also misleading. To read the share of votes that went to other candidates, one has to followS either the OR side or the SO side of the triangle. Basic geometry will show that going up the vertical line will not produce the share of "other candidates".

***

Lastly, here is a scatter plot representation of the data using the Romney-Santorum difference as the horizontal axis and the share of all others as the "other candidates":

The pattern of dots on this chart looks very similar to the ternary chart (that is one other reason why I thought the original graphic was a scatter plot.) However, the two plots are distinct entities. For the scatter plot, the horizontal axis goes from -100% to +100% while the vertical axis can only go from 0% to 100%.

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I think the plot is most often referred to as a Ternary plot

This reminds me of the horror that the Kolm Triangle would strike in the hearts of undergraduate economics students. The KT is the analogue of the famous Edgeworth Box for an economy with a public good.

Ternary plots plotting vote shares are sometimes called Nagayama triangles. See http://eprints.lse.ac.uk/4273/1/Is_Duverger's_Law_based_on_a_mistake_(LSERO).pdf for some examples.

I wish you'd read my post before making comments! The reason why it takes a bit of time to read this chart is precisely because it is NOT a ternary plot! I thought this post was about ternary plots, then I realize this is a scatter plot with two axes, while a ternary plot has three axes.

Kaiser, it seems to me that apart the triangle deformation and the misleading top axis label directions, a two axes scatterplot is the same thing as a ternary diagram.

http://i.imgur.com/tYWFy.png
In a ternary diagram, all the points along a line parallel to one side have the same percentage.
Look at the three points from top left to bottom right: they all have Romney 45%. Look at the three points from bottom left to top right: they all have Santorum 30%.
Look at the projections of Abortion point: it looks having 1/4 Romney and and 2/3 Santorum...

Antonio: thanks for the note! I see the light. The ternary plot and the scatter plot are different even though the pattern of the dots appears to be similar. See my additional comment above.

A bit confused this. If it is a ternary plot then going up the vertical line will absolutely produce the share of "other candidates". I think you need to read up on ternary plots.
Her is a simple introduction