The exception to the rule against dual axes

Dual axes are almost always a bad idea. But there is one situation under which I'd use it.


Last week, Alberto Cairo (link) engaged in a Twitter/blogging debate about a chart that first appeared in Reuters concerning the state of the woman CEO in the Fortune 500 companies. Here is the chart under discussion:


This chart already is cleaner and more useful than the original original, which came from a research report from Catalyst (link):


Jonathan Keller re-made the Reuters chart as follows:



Cairo Jorge Camões contributed this version:


The Voila blog (link) has yet another take:


Then Chris Moore, responding to Cairo, created this view and also left some insightful comments:



What's at stake here? There are really three related topics of discussion.

First, there is the matter of the upper limit of the vertical axis. Three solutions were suggested: 100 percent, 50 percent, and 4 percent. (Cairo at one point suggested 25 percent, which can be wrapped into the 50 percent bucket.) In reality, this is an argument over which of two key messages should be emphasized. The first message is that women still comprises a pathetically small proportion of Fortune 500 CEOs. The second message is more hopeful, that the growth in this proportion has been quite rapid since 1995.

All versions of the chart actually display both messages. In the Reuters chart (as well as Moore and Cairo), the message about the absolute proportion of women is given as an annotation while the Keller and Voila versions extend the vertical axis, thus encoding this message directly to the chart. Conversely, the Keller and Voila versions deemphasize the growth in proportions, and so I'd have preferred to see a note about that growth when using their versions.

Voila selectes a 50% upper limit because the 50/50 split has an intuitive meaning in the context of gender balance. Because the resulting chart is so visually arresting, and so biased to one of the two key messages, I'd only consider it if the point of the display is to draw attention to the female deficit.


The second disagreement is in using absolute counts versus relative proportions. Moore chose absolute counts. I am in this camp as well. This is primarily because we are talking about Fortune 500 and the 500 number is an idee fixe. In Moore's version, I find the data labels distracting since all the numbers are small and insignificant.

Finally, the linkage between the absolute and the relative numbers also produces multiple solutions. Cairo's post pinpoints this issue. His solution is to include an inset pie chart with an arrow to explicitly link the two views. Moore likes the inset idea, but experimented with a donut chart or a partition in place of the pie chart. He also removes the explicit guiding arrow.


It turns out this dataset is perfectly made for the dual axes. The absolute counts and relative proportions are in one to one correspondence because it's really only one data series expressed twice. This happy situation leads to one line that can be cross-referenced on two axes, one side showing counts and the other side showing proportions. This is shown in my version below (the orange line).


In addition to having two axes, I have plotted two related data series. The second series (in red) shows the incremental change in the number of women CEOs from the previous year (also shown in both counts and proportions).

The first series (the same one everyone plotted) draws attention to the first message, that the growth rate of women CEOs is quite strong since 1995. The second series is a bit of a downer on that message, suggesting that from the absolute count perspective, the progress (only one or two additions per year) has been painfully slow, and not that impressive.

Thanks again to Alberto for making me aware of this discussion. This has been fun!


PS. I have left out the other chart and may return to it in a future post.

What's in a cronut? Let me find out

Analyticsseo_gaReader Ross S. did not join the line for this cronut, illustrating the popularity of different makers of tracking software on 1.3 million websites.

Original by Analytics SEO is here.


The biggest beef I have with this cronut is the quality of the data. As I read their description of the underlying data, I see several red flags.

The analysis is hobbled by ignoring the competitive landscape in tracking software. Google Analytics carves out a huge share of the market by virtue of offering a richly featured product for free. (They justify this by establishing a gigantic spying operation on unsuspecting users.) However, industry insiders know that Omniture (owned by Adobe) is the heavyweight enterprise solution, with a complete feature set.

In other words, most of the 670,000 "customers" of Google Analytics are tiny websites; in addition, a lot of large websites also maintain Google Analytics in addition to Omniture since the former is free. It would be great if the researcher gives us one of two alternative views of market share: the share of revenues in the tracking software market; and the share of e-commerce revenues represented by the customers of each tracking software vendor. These two views give a fuller picture of the competitive landscape.

You'll notice this is the same game Google is playing in the mobile universe. Android has the most users but Apple makes the bulk of revenues.


The SEO agency says the chart is "based on 1.3 million e-commerce websites in May 2013". Are there really 1.3 million websites out there selling us stuff? How do they define e-commerce? Is an e-commerce website, for example? Or for that matter?

In the summary, they made a pretty startling claim--that "a large number of websites have no tracking software at all". The only problem is readers can't find out what proportion of websites don't track users. The data in the cronut excluded sites without tracking, which is a big problem.


Here is the link to the annual Top 500 Retailers report by Internet Retailer magazine. In Sep 2011, they found that 217 out of the top 500 use Omniture, 161 use Google Analytics, and 103 use Coremetrics (now owned by IBM).

Another place to look for corroborating evidence is Google Trends, which measures the popularity of search keywords. The relative order of the major vendors (excluding Google Analytics) does not match well with the data shown by Analytics SEO.


Compared to:


Coremetrics is way down in the list compiled by Analytics SEO.

English donuts rival Spanish donuts

On my holiday travel, I found a disguised donut chart in the Delta Sky Magazine (Dec 2010), talking about manufacturing jobs in the U.S. Then, flipping through the Spanish section at the back of the same magazine, I found the translated article, plus a translated chart. To my surprise, they look different:


Surprise No. 1: the sizes of the cog wheels are different. Even though the color is still mapped to year in the same way, somehow one of these authors decided to take liberty with the relative size. The suspect is the Spanish author who decided to make 2009 much larger and Jan to Dec 2012 much smaller.

Surprise No. 2: the use of commas within a number, and the format of dates differ by culture. That explains why the Spanish author removed the commas from the numbers, making it harder for me (English-speaking) to comprehend. Also, the swap from "01/12-09/12" to "Sep. 2012" suggests that Spanish speakers don't like the month/year formatting of dates. It also suggests that the Spanish readers have no trouble inferring that the "Sep. 2012" data point refers to "Jan. 2012 to Sep. 2012".

Surprise No. 3: The Spanish author improved the chart in one way. He grouped the annual data together via overlapping, leaving the 2012 partial-year data point by itself.


There are some problems with both charts. The most serious is the failure to project the 2012 jobs number. The chart seems to indicate that 2012 is a lackluster year, at best level with the previous years but in fact, the number of jobs in three quarters has already exceeded the full-year count of 2011, 2010 and 2009. Unless the fourth quarter is a particularly bad quarter for manufacturing jobs, it would seem that the message should be that 2012 is a great year of recovery. You can't tell from these charts: in particular, the Spanish author decided to shrink the 2012 cog wheel into insignificance.

The issue here is providing context for comparison. Even if the projected 2012 full-year number is provided, that may not be enough to judge whether manufacturing is healthy. Other useful context can be the growth rate of manufacturing versus other sectors of the economy; and the growth rate of jobs in relation to the population/work force growth rate.

As usual, a simple line chart displays the time-series data more clearly. (I simply linearly extrapolated the 2012 full-year number, which is probably an over-estimate. In practice, you can look up the data and figure out the ratio of Jan-Sept jobs to full-year jobs on average and inflate the number that way.)





Gelman joins in the fun

The great Andrew Gelman did a Junk Charts style post today, and very well indeed.

The offending Economist plot is the donut chart, which is a favorite of that magazine.  I commented on this type of chart before.


Andrew created two alternatives, one is a line chart (profile chart) which is often a better option (despite the data being categorical), the other is more creative, and the better of the two.




Some of Gelman's readers complained that he arbitrarily "standardized" the data by indexing against the average of the countries depicted; one can further grumble that a 50% "excess" may sound impressive but it would be equivalent to less than an hour, perhaps not as startling. These types of complaints are fair but do realize that blog posts like these are primarily concerned with how data is best visualized. If one prefers a different indexing method, or a different set of countries, or a different color for the lines, etc., one can easily revise the chart to reflect those preferences.

The easiest way to see why the third chart is better than the first is that the strongest message coming off the first chart is that there are no material differences between these six countries in terms of time usage but in the third chart, the designer (here, it's Gelman) is asserting that there are interesting differences.

Have data graphics progressed in the last century?

Received a wonderful link via reader Lonnie P. to this website that presents a historical reconstruction of W.E.B. DuBois's exhibit of the "American negro" at the 1900 Paris Expo. Amusingly, DuBois presented a large series of data graphics to educate the world on the state (plight) of blacks in America over a century ago.

You can really spend a whole afternoon examining these charts (and more); too bad the charts have poor resolution and it is often hard to make out the details.


Judging from this evidence, we must face up to the fact that data graphics have made little progress during these eleven decades. Ideas, good or bad, get reinvented. Disappointingly, we haven't learned from the worst ones.

Exhibit A 


(see discussion here)

Exhibit B


 (see discussion here)

Exhibit C 


(See discussion here.)

Exhibit D

 (see the Vampire chart here)

Exhibit E

(see the discussion here.)

Exhibit F

(see discussion here.)

Detached in time and space

A reader sent in this "pie chart" (better called a "donut chart") which summarizes the results of this survey.


My dislike of donut charts has been well documented. Click here.


What I want to discuss is the use of interactivity, a feature of this chart but something that backfires. The underlying data is a 5-level rating of "corporate sentiment" by industry, by country, and over time. That would be 4 dimensions jostling for space on a surface. Obviously, some decisions have to be made as to which dimension to highlight and which to push to the background.

This chart highlights the 5-level ratings using the donut device. All other dimensions are well hidden by the interactive feature. Pressing on the forward/backward buttons reveals the industry dimension. Pressing on the arrow on the top left corner reveals the time dimension. Pressing on the map reveals the country dimension.

The problem with this level of detachment is that readers are obstructed from viewing multiple dimensions at once. For instance, it is very hard to understand the differences in sentiment between different industries, or between different countries, or the change in sentiment over time.


Redo_asiasentiment The version on the right shows, for instance, the distribution of ratings by industry for Q3 2010, and for all Asia combined. This is a rough sketch, and one would want to fix quite a few things: making the sector labels horizontal, reducing the distance between the columns, labeling the ratings 1 as "very positive", ordering the sectors from most positive to least positive, etc.

A chart of ratings by country (aggregate of all industry sectors) would follow the same format. Similarly, one can compare ratings across countries, for a given sector... and this can be replicated 11 times for each sector. Similarly, ratings across industries for any given country.

For comparisons across time, I'd suggest using average ratings rather than keeping track of five proportions. This reduces a lot of clutter that does not improve readers' comprehension of the trends. A line chart would be preferred.


A better way to organize the chart is to start with the types of questions that the reader is likely to want to answer. Clicking on each question (say, compare ratings across industries within a country) would reveal one of the above collections of charts.


Another improvement is to add annotations. For instance, one wonders whether the airlines colluded to all give a 2 rating. It is always a great idea to direct readers' attention to the most salient parts of a chart, especially if it contains a lot of data.





Hoisted from the archives: a revolution

In October 2007, I wrote about the "canvass" metaphor for graphing software. This was what I said:

With the advent of AJAX and other interactive technologies, one can only hope that new graphing software will use the "canvass" metaphor.  If we want to reduce the spacing between bars, we should be able to grab the bars and move them together.  If we want to change the ordering, we should be able to mouse over some menu and select a pre-defined ordering scheme, or to drag and move bars around as we please. etc. etc.

To push this metaphor further, this kind of software should facilitate the "exploratory" stage of graph-making. I blogged about this stage of making sketches before. One longs for software that allows one to flip through many different chart types quickly, to settle on the desired type, and then to make the nitty-gritty changes to the axes, colors, dots, etc.

The revolution has arrived in the form of JMP's Graph Builder function. It is not perfect yet, as even the example I use will show, but I'm excited because we are getting closer to that "canvass" metaphor.


Spam_donutsI'm going to re-make this inedible pair of donuts from an otherwise quite nice infographics on the growth and nature of spam in the last 10 years. (New Scientist)

I have pointed out the biggest shortcoming of donut charts often: the fact that the most important clue to the size of each sector of the underlying pie chart, that is, the angle at the center of the pie, has been cut off from the chart, and often, as in here, obscured by a number.

There are dramatic shifts in proportions of spam types during the last decade but the effect is underwhelming as depicted.

In the Graph Builder, I can push around the data and create different chart types.  First, I made a small-multiples bar chart.


By clicking on the word "Year" and dragging it to a box called "Overlay", I made a paired bar chart:

Paired bars

What about a dot plot instead? This change requires a right click but easy enough:


Here's where I encountered a little inconvenience. It's probably ignorance on my part since I didn't read the manual. I couldn't figure out how to increase the dot size for all dots at once, only one at a time.

In any case, I'm still searching.  I want to do a small-multiples line chart. For this, I drag the word "Year" into the bottom of the chart labelled "X", and then right-click to add a line to the dot chart.


This is close to a desired chart type for this data.  The change from year to year is highly apparent, and the increased and decreased spam types are also obvious. I would color the increases differently from the decreases if I have the time.

I had a very difficult time (and failed in) getting the year labels to say 1999 and 2009 which are the logical points for this data. JMP seems to have a mind of its own.

Since it takes no time, I experimented some more.  By moving "Category" to "Wrap", I reproduced the above chart but in a matrix form:


Finally, I made the "Category" an "overlay" which resulted in this chart.  This is kind of like the Bumps chart but obviously a bad idea for this data: (I'm not even showing the really ugly legend).


So, my dream toy -- the "canvass" style graph maker -- is here! It only takes a few minutes to move the data around this canvass, and see these different chart types.

I indicated that this goes a long way but isn't perfect. Right now, sketching and exploring is easy but refining and detailing is not as easy.

What I would like to see: once the general form of the chart is chosen, maybe a second canvass is needed, with Photoshop as a metaphor, in which we can chisel out the nitty-gritty details, like the axis labels, dot sizes, line widths and so on.

Also, the number of chart types can, and I presume will, be increased over time. For instance, I don't think the current version allows a profile chart; it seems to adhere to the overly-rigid rule that a categorical data series should not be connected by a line.

(I should say that in the current release, one way to accomplish this is to save the resulting graph-sketch as a "JMP script" and then go into the code and change things around. But since we are doing point and click, and visual interaction, why not go all the way?)

Most existing graphing software fall into two extremes: the Excel style which is super-rigid, or the R style which allows minute control over every little thing. This, I think, is the third way.


Serving donuts

David Leonhardt's article on the graduation rates of public universities caught my attention for both graphical and statistical reasons.

Nyt_gradrate David gave a partial review of a new book "Crossing The Finish Line", focusing on their conclusion that public universities must improve their 4-year graduation rates in order for education in the U.S. to achieve progress.  This conclusion was arrived at through statistical analysis of detailed longitudinal data (collected since 1999).

This chart is used to illustrate this conclusion.  We will come to the graphical offering later but first I want to fill in some details omitted from David's article by walking through how a statistician would look at this matter, what it means by "controlling for" something.

The question at hand is whether public universities, especially less selective ones, have "caused" students to lag behind in graduation rate.  A first-order analysis would immediately find that the overall graduation rate at less selective public universities to be lower, about 20% lower, than at more selective public universities.  

A doubter appears, and suggests that less selective schools are saddled with lower-ability students, and that would be the "cause" of lower graduation rates, as opposed to anything the schools actually do to students.  Not so fast, the statistician now disaggregates the data and look at the graduation rates within subgroups of students with comparable ability (in this instance, the researchers used GPA and SAT scores as indicators of ability).  This is known as "controlling for the ability level".  The data now shows that at every ability level, the same gap of about 20% exists: about 20% fewer students graduate at the less selective colleges than at the more selective ones.  This eliminates the mix of abilities as a viable "cause" of lower graduation rates.

The researchers now conclude that conditions of the schools (I think they blame the administrators) "caused" the lower graduation rates.  Note, however, that this does not preclude factors other than mix of abilities and school conditions from being the real "cause" of lower graduation rates.  But as far as this analysis goes, it sounds pretty convincing to me.

That is, if I ignore the fact that graduation rates are really artifacts of how much the administrators want to graduate students.  As the book review article pointed out, at the less selective colleges, they may want to reduce graduation rates in order to save money since juniors and seniors are more expensive to support due to smaller class sizes and so on.  On the other hand, the most selective colleges have an incentive to maintain a near-perfect graduation rates since the US News and other organizations typically use this metric in their rankings -- if you were the administrator, what would you do?  (You didn't hear it from here.)

Back to the chart, or shall we say the delivery of 16 donuts?

First, it fails the self-sufficiency principle.  If we remove the graphical bits, nothing much is lost from the chart.  Both are equally impenetrable.

A far better alternative is shown below, using a type of profile chart.


Finally, I must mention that in this particular case, there is no need to draw all four lines.  Since the finding of a 20% gap essentially holds for all subgroups, no information is lost by collapsing the subgroups and reporting the average line instead (with a note explaining that the same effect affected every subgroup).  

By the way, that is the difference between the statistical grapher - who is always looking to simplify the data - and the information grapher - who is aiming for fidelity. 

Reference: "Colleges are lagging in graduation rates", New York Times, Sept 9, 2009; "Book review: (Not) Crossing the Finish Line", Inside Higher Education, Sept 9 2009.

A shocking failure to communicate

So said a reader, Stephen B., of the following graphic (note: pdf) in the London Times concerning Andy Murray's recent tennis triumphs.


How can we disagree?  Shocking?  Yes.  Failure?  Definitely.  Failing to communicate?  No doubt.

Lt_murray_a Let's first start with the five tennis balls at the bottom.  It fails the self-sufficiency test.  It makes no difference whether the balls (bubbles) are the same size, or different sizes.  Readers will look at the data and ignore the bubbles.

Amazingly, the caption said that "Murray has one of the best returns of serve in the game."  And yet, the graphic showed the five players who were better than Murray, and nobody worse!  For those unfamiliar with tennis statistics, it does not provide any helpful statistics like averages, medians, etc. to help us understand the data.

But that is only the beginning.

Take a look at these two donuts.

(The color scheme from light to dark: first, second, third, fourth round of tournament)

So we're told: the 75% of first-serve points won in the fourth round was 25.6% of the sum of the percentages of first-serve points won from first to fourth rounds (75%+70%+71%+76%).  What does this mean?  Why should we care?

The challenge with these two statistics is that they are correlated and have to be interpreted together.  If a first-serve is won, then there would be no second serve, etc.  Here's one attempt at it, using statistics from the Soderling-Federer match.  It's clear that Federer was better on both serves.


Reference: "Murray's march to the last eight", London Times.