Out of line

This simple chart showing life expectancies in 10 countries raises one's eyebrows.

The first curiosity is the deliberate placement of Pakistan behind India and China. Every nation is sorted from lowest to highest, except for Pakistan. Is the reason politics? I have no idea. If you have an explanation, please leave a comment.

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This graphic is an example of data visualization that does not actually show the data.

The positions of the flags do not in fact encode the data! For example, the Indian flag is closer to the Chinese flag than to the Pakistani flag even though the gap between India and China (7) is more than double the gap between India and Pakistan (3).

Here is what it looks like if the gaps encode the data. With this selection of countries, Pakistan and India are separated from the rest. 

In the original chart, the readers must read the data labels to understand it, and resist intepreting the visual elements.

I removed the flag poles because they have the unintended consequence of establishing a zero level (where the cartoon characters stand) but the positions of the flags don't reflect a start-at-zero posture.

***

Returning to our first topic for a second. If the message of the chart is to single out Pakistan, it actually works! If all other countries are sorted by value, with Pakistan inserted out of order, it draws our attention.

In a conventional layout, Pakistan is shoved to the left side in the bottom corner. See below:

 

 

Hammock plots

Prof. Matthias Schonlau gave a presentation about "hammock plots" in New York recently.

Here is an example of a hammock plot that shows the progression of different rounds of voting during the 1903 papal conclave. (These are taken at the event and thus a little askew.)

The chart shows how Cardinal Sarto beat the early favorite Rampolla during later rounds of voting. The chart traces the movement of votes from one round to the next. The Vatican destroys voting records, and apparently, records were unexpectedly retained for this particular conclave.

The dataset has several features that brings out the strengths of such a plot.

There is a fixed number of votes, and a fixed number of candidates. At each stage, the votes are distributed across the subset of candidates. From stage to stage, the support levels for candidate shift. The chart brings out the evolution of the vote.

From the "marginals", i.e. the stacked columns shown at each time point, we learn the relative strengths of the candidates, as they evolve from vote to vote.

The links between the column blocks display the evolution of support from one vote to the next. We can see which candidate received more votes, as well as where the additional votes came from (or, to whom some voters have drifted).

The data are neatly arranged in successive stages, resulting in discrete time steps.

Because the total number of votes are fixed, the relative sizes of the marginals are nicely constrained.

The chart is made much more readable because of binning. Only the top three candidates are shown individually with all the others combined into a single category. This chart would have been quite a mess if it showed, say, 10 candidates.

How precisely we can show the intra-stage movement depends on how the data records were kept. If we have the votes for each person in each round, then it should be simple to execute the above! If we only have the marginals (the vote distribution by candidate) at each round, then we are forced to make some assumptions about which voters switched their votes. We'd likely have to rule out unlikely scenarios, such as that in which all of the previous voters for candidate X switched to someone other candidates while another set of voters switched their votes to candidate X.

***

Matthias also showed examples of hammock plots applied to different types of datasets.

The following chart displays data from course evaluations. Unlike the conclave example, the variables tied to questions on the survey are neither ordered nor sequential. Therefore, there is no natural sorting available for the vertical axes.

Time is a highly useful organizing element for this type of charts. Without such an organizing element, the designer manually customizes an order.

The vertical axes correspond to specific questions on the course evaluation. Students are aggregated into groups based on the "profile" of grades given for the whole set of questions. It's quite easy to see that opinions are most aligned on the "workload" question while most of the scores are skewed high.

Missing values are handled by plotting them as a new category at the bottom of each vertical axis.

This example is similar to the conclave example in that each survey response is categorical, one of five values (plus missing). Matthias also showed examples of hammock plots in which some or all of the variables are numeric data.

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Some of you will see some resemblance of the hammock plot with various similar charts, such as the profile chart, the alluvial chart, the parallel coordinates chart, and Sankey diagrams. Matthias discussed all those as well.

Matthias has a book out called "Applied Statistical Learning" (link).

Also, there is a Python package for the hammock plot on github.

Patiently looking

Voronoi (aka Visual Economist) made this map about service times at emergency rooms around the U.S.

 

This map shows why one shouldn’t just stick state-level data into a state-level map by default.

The data are median service times, defined as the duration of the visit from the moment a patients arrive to the moment they leave. For reasons to be explained below, I don’t like this metric. The data are in terms of hours and minutes, and encoded in the color scale.

As with any choropleth, the dominant features of this map are the shapes and sizes of various pieces but these don’t carry any data. The eastern seaboard contains many states that are small in area but dense in population, and always produces a messy, crowded smorgasbord of labels and guiding lines.

The color scale is progressive (continuous) making it even harder to gain an appreciation of the spatial pattern. For the sake of argument, imagine a truly continuous color scale tuned to the median service times in number of minutes. There would be as many shades as there are unique number of values on the map. For example, the state with 2 hr 12 min median time would receive a different shade than the one with 2 hr 11 min. Looking at the dataset, I found 43 unique values of median service time in the 52 states and territories. Thus, almost every state would wear its unique shade, making it hard to answer such common questions as: which cluster of states have high/medium/low median service times?

(As the underlying software may only be capable of printing a finite number of shades so in reality, there aren’t any true continuous scales. A continuous scale is just a discrete scale with many levels of shades. For this map, I’d group the states into at most five categories, requiring five shades.)

***

We’re now reaching the D corner of the Trifecta Checkup (link).

I’d transform the data to relative values, such as an index against the median or average in the nation. The colors now indicate how much higher or lower is the state’s median service time than that of the nation. With this transformed data, it makes more sense to use a bidirectional color scale so that there are different colors for higher vs lower than average.

Lastly, I’m not sure about the use of median service time, as opposed to average (mean) service time. I suspect that the distribution is heavily skewed toward longer values so that the median service time falls below the mean service time. If, however, the service time distribution is roughly symmetric around the median, then the mean and median service times will be very similar, and thus the metric selection doesn’t matter.

Imagine you're the healthcare provider and your bonus is based on managing median service times. You have an incentive to let a small number of patients wait an extraordinary amount of time, while serving a bunch of patients who require relatively simple procedures. If it's a mean service time, the values of the extreme outliers will be spread over all the patients while the median service time is affected by the number of such outliers but not their magnitudes.

When I pulled down the publicly available data (link), I found additional data fields. The emergency room visits are further broken into four categories (low, medium, high, very high), and a median is reported within each category. Thus, we have a little idea how extreme the top values can be.

The following dotplot shows this:

A chart like this is still challenging to read since there are 52 territories, ordered by the value on a metric. If the analyst can say what are interesting questions, e.g. breaking up the territories into regions, then a grouping can be applied to the above chart to aid comprehension.

 

Simple presentations

In the previous post, I looked at this chart that shows the distributions of four subgroups found in a dataset:

This chart takes quite some effort to decipher, as does another version I featured.

The key messages appear to be: (i) most English words are of Germanic origin, (ii) the most popular English words are even more skewed towards Germanic origin, (iii) words of French origin started showing up around rank 50, those of Latin origin around rank 250.

***

If we are making a graphic for presentation, we can simplify the visual clutter tremendously by - hmmm - a set of pie charts.

For those allergic to pies, here's a stacked column chart:

Both of these can be thought of as "samples" from the original chart, selected to highlight shifts in the relative proportions.

I also reversed the direction of the horizontal axis as I think the story is better told starting from the whole dataset and honing in on subsets.

 

P.S. [1/10/2025] A reader who has expertise in this subject also suggested a stacked column chart with reversed axis in a comment, so my recommendation here is confirmed.

the wtf moment

You're reading some article that contains a standard chart. You're busy looking for the author's message on the chart. And then, the wtf moment strikes.

It's the moment when you discover that the chart designer has done something unexpected, something that changes how you should read the chart. It's when you learn that time is running right to left, for example. It's when you realize that negative numbers are displayed up top. It's when you notice that the columns are ordered by descending y-value despite time being on the x-axis.

Tell me about your best wtf moments!

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The latest case of the wtf moment occurred to me when I was reading Rajiv Sethi's blog post on his theory that Kennedy voters crowded out Cheney voters in the 2024 Presidential election (link). Was the strategy to cosy up to Cheney and push out Kennedy wise?

In the post, Rajiv has included this chart from Pew:

The chart is actually about the public's confidence in scientists. Rajiv summarizes the message as: 'Public confidence in scientists has fallen sharply since the early days of the pandemic, especially among Republicans. There has also been a shift among Democrats, but of a slightly different kind—the proportion with “a great deal” of trust in scientists to act in our best interests rose during the first few months of the pandemic but has since fallen back.'

Pew produced a stacked column chart, with three levels for each demographic segment and month of the survey. The question about confidence in scientists admits three answers: a great deal, a fair amount, and not too much/None at all. [It's also possible that they offered 4 responses, with the bottom two collapsed as one level in the visual display.]

As I scan around the chart understanding the data, suddenly I realized that the three responses were not listed in the expected order. The top (light blue) section is the middling response of "a fair amount", while the middle (dark blue) section is the "a great deal" answer.

wtf?

***

Looking more closely, this stacked column chart has bells and whistles, indicating that the person who made it expended quite a bit of effort. Whether it's worthwhile effort, it's for us readers to decide.

By placing "a great deal" right above the horizon, the designer made it easier to see the trend in the proportion responding with "a great deal". It's also easy to read the trend of those picking the "negative" response because of how the columns are anchored. In effect, the designer is expressing the opinion that the middle group (which is also the most popular answer) is just background, and readers should not pay much attention to it.

The designer expects readers to care about one other trend, that of the "top 2 box" proportion. This is why sitting atop the columns are the data labels called "NET" which is the sum of those responding "a great deal" or "a fair amount".

***

For me, it's interesting to know whether the prior believers in science who lost faith in science went down one notch or two. Looking at the Republicans, the proportion of "a great deal" roughly went down by 10 percentage points while the proportion saying "Not too much/None at all" went up about 13%. Thus, the shift in the middle segment wasn't enough to explain all of the jump in negative sentiment; a good portion went from believer to skeptic during the pandemic.

As for Democrats, the proportion of believers also dropped by about 10 percentage points while the proportion saying "a fair amount" went up by almost 10 percent, accounting for most of the shift. The proportion of skeptics increased by about 2 percent.

So, for Democrats, I'm imagining a gentle slide in confidence that applies to the whole distribution while for Republicans, if someone loses confidence, it's likely straight to the bottom.

If I'm interested in the trends of all three responses, it's more effective to show the data in a panel like this:

***

Remember to leave a comment when you hit your wtf moment next time!

 

Dizziness

Statista uses side-by-side stacked column charts to show the size of different religious groups in the world:

It's hard to know where to look. It's so colorful and even the middle section is filled in whereas the typical such chart would only show guiding lines.

What's more, the chart includes gridlines, as well as axis labels.

The axis labels compete with the column section labels, the former being cumulative while the latter isn't.

The religious groups are arranged horizontally in two rows at the top while they are stacked up from bottom to top inside the columns.

The overall effect is dizzying.

***

The key question this chart purportedly address is the change in the importance of religions over the time frame depicted.

Look at the green sections in the middle of the chart, signifying "Unaffiliated" people. The change between the two time points is 16 vs 13 which is -3 percent.

Where is this -3 percent encoded?

It's in the difference in height between the two green blocks. On this design, that's a calculation readers have to do themselves.

One might take the slope of the guiding line that links the tops of the green blocks as indicative of the change, but it's not. In fact the top guiding line slopes upwards, implying an increase over time. That increase is associated with the cumulative total of the top three religious groups, not the share of the Unaffiliated group.

So, if we use those guiding lines, we have to take the difference of two lines, not just the top line. The line linking the bottoms of the green blocks is also relevant. However, the top and bottom lines will in general not be parallel, so readers have to somehow infer from the parallelogram bounded by the guiding lines and vertical block edges that the change in the Unaffiliated group is 3 percent.

Ouch.

***

I generally like to use Bumps charts (also called slopegraphs) to show change across two points in time:

What's sacrificed is the cumulation of percentages. I also am pleased that Christian and Muslim, where the movements are greatest, are found at the top of the chart. (There isn't a need to use so many colors; I just inherited them from the original chart.)

Tidying up the details

This column chart caught my attention because of the color labels.

Well, it also concerns me that the chart takes longer to take in than you'd think.

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The color labels say "FY2123", "FY2022", and "FY1921". It's possible but unlikely that the author is making comparisons across centuries. The year 2123 hasn't yet passed, so such an interpretation would map the three categories to long-ago past, present and far-into-the-future.

Perhaps hyphens were inadvertently left off so "FY2123" means "FY2021 - FY2023". It's odd to report financial metrics in multi-year aggregations. I rule this out because the three categories would then also overlap.

Here's what I think the mistake is: somehow the prefix is rolled forward when it is applied to the years. "FY23", "FY22", "FY21" got turned into "FY[21]23", "FY[20]22", "FY[19]21" instead of putting 20 in all three slots.

The chart appeared in an annual financial report, and the comparisons were mostly about the reporting year versus the year before so I'm pretty confident the last two digits are accurately represented.

Please let me know if you have another key to this puzzle.

In the following, I'm going to assume that the three colors represent the most recent three fiscal years.

***

A few details conspire to blow up our perception time.

There was no extra spacing between groups of columns.

The columns are arranged in reverse time order, with the most recent year shown on the left. (This confuses those of us that use the left-to-right convention.)

The colors are not ordered. If asked to sort the three colors, you will probably suggest what is described as "intuitive" below:

The intuitive order aligns with the amount of black added to a base color (hue). But this isn't the order assigned to the three years on the original chart.

***

Some of the other details on the chart are well done. For example, I like the handling of the gridlines and the axes.

The following revision tidies up some of the details mentioned above, without changing the key features of the chart:

 

Several tips for visualizing matrices

Continuing my review of charts that were spammed to my inbox, today I look at the following visualization of a matrix of numbers:

The matrix shows pairwise correlations between the returns of 16 investment asset classes. Correlation is a number between -1 and 1. It is a symmetric scale around 0. It embeds two dimensions: the magnitude of the correlation, and its direction (positive or negative).

The correlation matrix is a special type of matrix: a bit easier to deal with as the data already come “standardized”. As with the other charts in this series, there is a good number of errors in the chart's execution.

I’ll leave the details maybe for a future post. Just check two key properties of a correlation matrix: the diagonal consisting of self-correlations should contain all 1s; and the matrix should be symmetric across that diagonal.

***

For this post, I want to cover nuances of visualizing matrices. The chart designer knows exactly what the message of the chart is - that the asset class called "art" is attractive because it has little correlation with other popular asset classes. Regardless of the chart's errors, it’s hard for the reader to find the message in the matrix shown above.

That's because the specific data carrying the message sit in the bottom row (and the rightmost column). The cells in this row (and column) has a light purple color, which has been co-opted by the even lighter gray color used for the diagonal cells. These diagonal cells pop out of the chart despite being the least informative (they have the same values for all correlation matrices!)

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Several tactics can be deployed to push the message to the fore.

First, let's bring the key data to the prime location on the chart - this is the top row and left column (for cultures which read top to bottom, left to right).

For all the drafts in this post, I have dropped the text descriptions of the asset classes, and replaced them with numbers so that it's easier to follow the changes. (For those who're paying attention, I also edited the data to make the matrix symmetric.)

Second, let's look at the color choice. Here, the designer made a wise choice of restricting the number of color levels to three (dark, medium and light). I retained that decision in the above revision - actually, I used four colors but there are no values in one of the four sections, therefore, effectively, only three colors appear. But let's look at what happens when the number of color levels is increased.

The more levels of color, the more strain it puts on our processing... with little reward.

Third, and most importantly, the order of the categories affects perception majorly. I have no idea what the designer used as the sorting criterion. In step one of the fix, I moved the art category to the front but left all the other categories in the original order.

The next chart has the asset classes organized from lowest to highest average correlation. Conveniently, using this sorting metric leaves the art category in its prime spot.

Notice that the appearance has completely changed. The new version brings out clusters in the data much more effectively. Most of the assets in the bottom of the chart have high correlation with each other.

Finally, because the correlation matrix is symmetric across the diagonal of self-correlations, the two halves are mirror images and thus redundant. The following removes one of the mirrored halves, and also removes the diagonal, leading to a much cleaner look.

Next time you visualize a matrix, think about how you sort the rows/columns, how you choose the color scale, and whether to plot the mirrored image and the diagonal.

 

 

 

Two metrics in-fighting

The Wall Street Journal shows the following chart which pits two metrics against each other:

The primary metric is the change in median yearly salary between the two periods of time. We presume it's primary because of its presence in the chart title, and the blue bars being more readable than the green bubbles. The secondary metric is the median yearly salary in the later period.

That, I believe, was the intended design. When I saw this chart, my eyes went to the numbers inside the green bubbles. Perhaps it's because I didn't read the chart title first, and the horizontal axis wasn't labelled so it wasn't obvious what the blue bars coded.

As with most bubble charts, the data labels exist to cover up the inadequacy of circular areas. The self-sufficiency test - removing the data labels - shows this well:

It's simply impossible to know what values should be in each bubble, or to perceive the relative sizes of those bubbles.

***

Reversing the order of the blue bars also helps:

The original order is one of the more annoying features in most visualization packages. Because internally, the categories are numbered 1, 2, 3, ..., and because the convention is to have values run higher as they run up the vertical axis, these packages would place the top-ranked item at the bottom of the chart.

Most people read top to bottom, which means that they read the least important item first, and the most important item last!

In most visualization packages, it takes only 1 click or 1 action to reverse the order of the items. Please do it!

***

For change over time, I like using a Bumps chart, otherwise called a slope graph:

An elaborate data vessel

I recently came across the following dataviz showing global oil production (link).

This is an ambitious graphic that addresses several questions of composition.

The raw data show the amount of production by country adding up to the global total. The countries are then grouped by region. Further, the graph presents an oil-and-gas specific grouping, as indicated by the legend shown just below the chart title. This grouping is indicated by the color of the circumference of the circle containing the flag of the country.

This chart form is popular in modern online graphics programs. It is like an elaborate data vessel. Because the countries are lined up around the barrel, a space has been created on three sides to admit labels and text annotations. This is a strength of this chart form.

***

The chart conveys little information about the underlying data. Each country is given a unique odd shaped polygon, making it impossible to compare sizes. It’s definitely possible to pick out U.S., Russia, Saudi Arabia as the top producers. But in presenting the ranks of the data, this chart form pales in comparison to a straightforward data table, or a bar chart. The less said about presenting values, the better.

Indeed, our self-sufficiency test exposes the inability of these polygons to convey the data. This is precisely why almost all values of the dataset are present on the chart.

***

The dataviz subtly presumes some knowledge on the part of the readers.

The regions are not directly labeled. The readers must know that Saudi Arabia is in the Middle East, U.S. is part of North America, etc. Admittedly this is not a big ask, but it is an ask.

It is also assumed that readers know their flags, especially those of smaller countries. Some of the small polygons have no space left for country names and they are labeled with just flags.

In addition, knowing country acronyms is required for smaller countries as well. For example, in Africa, we find AGO, COG and GAB.

For this chart form the designer treats each country according to the space it has on the chart (except those countries that found themselves on the edges of the barrel). Font sizes, icons, labels, acronyms, data labels, etc. vary.

The readers are assumed to know the significance of OPEC and OPEC+. This grouping is given second fiddle, and can be found via the color of the circumference of the flag icons.

I’d have not assigned a color to the non-OPEC countries, and just use the yellow and blue for OPEC and OPEC+. This is a little edit but makes the search for the edges more efficient.

***

Let’s now return to the perception of composition.

In exactly the same manner as individual countries, the larger regions are represented by polygons that have arbitrary shapes. One can strain to compile the rank order of regions but it’s impossible to compare the relative values of production across regions. Perhaps this explains the presence of another chart at the bottom that addresses this regional comparison.

The situation is worse for the OPEC/OPEC+ grouping. Now, the readers must find all flag icons with edges of a specific color, then mentally piece together these arbitrarily shaped polygons, then realizing that they won’t fit together nicely, and so must now mentally morph the shapes in an area-preserving manner, in order to complete this puzzle.

This is why I said earlier this is an elaborate data vessel. It’s nice to look at but it doesn’t convey information about composition as readers might expect it to.