Reading this chart won't take as long as withdrawing troops from Afghanistan

Art sent me the following Economist chart, noting how hard it is to understand. I took a look, and agreed. It's an example of a visual representation that takes more time to comprehend than the underlying data.


The chart presents responses to 3 questions on a survey. For each question, the choices are Approve, Disapprove, and "Neither" (just picking a word since I haven't seen the actual survey question). The overall approval/disapproval rates are presented, and then broken into two subgroups (Democrats and Republicans).

The first hurdle is reading the scale. Because the section from 75% to 100% has been removed, we are left with labels 0, 25, 50, 75, which do not say percentages unless we've consumed the title and subtitle. The Economist style guide places the units of data in the subtitle instead of on
the axis itself.

Our attention is drawn to the thick lines, which represent the differences between approval and disapproval rates. These differences are signed: it matters whether the proportion approving is higher or lower than the proportion disapproving. This means the data are encoded in the order of the dots plus the length of the line segment between them.

The two bottom rows of the Afghanistan question demonstrates this mental challenge. Our brains have to process the following visual cues:

1) the two lines are about the same lengths

2) the Republican dots are shifted to the right by a little

3) the colors of the dots are flipped

What do they all mean?


A chart runs in trouble when you need a paragraph to explain how to read it.

It's sometimes alright to make complicated data visualization that illustrates complicated concepts. What justifies it is the payoff. I wrote about the concept of return on effort in data visualization here.

The payoff for this chart escaped me. Take the Democratic response to troop withdrawal. About 3/4 of Democrats approve while 15% disapprove. The thick line says 60% more Democrats approve than disapprove.


Here, I show the full axis, and add a 50% reference line


Small edits but they help visualize "half of", "three quarters of".


Next, I switch to the more conventional stacked bars.


This format reveals some of the hidden data on the chart - the proportion answering neither approve/disapprove, and neither yes/no.

On the stacked bars visual, the proportions are counted from both ends while in the dot plot above, the proportions are measured from the left end only.


Read all my posts about Economist charts here


Two commendable student projects, showing different standards of beauty

A few weeks ago, I did a guest lecture for Ray Vella's dataviz class at NYU, and discussed a particularly hairy dataset that he assigns to students.

I'm happy to see the work of the students, and there are two pieces in particular that show promise.

The following dot plot by Christina Barretto shows the disparities between the richest and poorest nations increasing between 2000 and 2015.

BARRETTO  Christina - RIch Gets Richer Homework - 2021-04-14

The underlying dataset has the average GDP per capita for the richest and the poor regions in each of nine countries, for two years (2000 and 2015). With each year, the data are indiced to the national average income (100). In the U.K., the gap increased from around 800 to 1,100 in the 15 years. It's evidence that the richer regions are getting richer, and the poorer regions are getting poorer.

(For those into interpreting data, you should notice that I didn't say the rich getting richer. During the lecture, I explain how to interpret regional averages.)

Christina's chart reflects the tidy, minimalist style advocated by Tufte. The countries are sorted by the 2000-to-2015 difference, with Britain showing up as an extreme outlier.


The next chart by Adrienne Umali is more infographic than Tufte.

Adrienne Umali_v2

It's great story-telling. The top graphic explains the underlying data. It shows the four numbers and how the gap between the richest and poorest regions is computed. Then, it summarizes these four numbers into a single metric, "gap increase". She chooses to measure the change as a ratio while Christina's chart uses the difference, encoded as a vertical line.

Adrienne's chart is successful because she filters our attention to a single country - the U.S. It's much too hard to drink data from nine countries in one gulp.

This then sets her up for the second graphic. Now, she presents the other eight countries. Because of the work she did in the first graphic, the reader understands what those red and green arrows mean, without having to know the underlying index values.

Two small suggestions: a) order the countries from greatest to smallest change; b) leave off the decimals. These are minor flaws in a brilliant piece of work.



Come si dice donut in italiano

One of my Italian readers sent me the following "horror chart". (Last I checked, it's not Halloween.)


I mean, people are selling these rainbow sunglasses.


The dataset behind the chart is the market share of steel production by country in 1992 and in 2014. The presumed story is how steel production has shifted from country to country over those 22 years.

Before anything else, readers must decipher the colors. This takes their eyes off the data and on to the color legend placed on the right column. The order of the color legend is different from that found in the nearest object, the 2014 donut. The following shows how our eyes roll while making sense of the donut chart.


It's easier to read the 1992 donut because of the order but now, our eyes must leapfrog the 2014 donut.


This is another example of a visualization that fails the self-sufficiency test. The entire dataset is actually printed around the two circles. If we delete the data labels, it becomes clear that readers are consuming the data labels, not the visual elements of the chart.


The chart is aimed at an Italian audience so they may have a patriotic interest in the data for Italia. What they find is disappointing. Italy apparently completely dropped out of steel production. It produced 3% of the world's steel in 1992 but zero in 2014.

Now I don't know if that is true because while reproducing the chart, I noticed that in the 2014 donut, there is a dark orange color that is not found in the legend. Is that Italy or a mysterious new entrant to steel production?

One alternative is a dot plot. This design accommodates arrows between the dots indicating growth versus decline.



A note to science journal editors: require better visuals

In reviewing a new small-scale study of the Moderna vaccine, I found this chart:


This style of charts is quite common in scientific papers. And they are horrible. It irks me to think that some authors are forced to adopt such styles.

The study's main goal is to compare two half doses to two full doses of the Moderna vaccine. (To understand the science, read the post on my book blog.) The participants were stratified by age group. The vaccine is expected to work better for younger people than for older people. The point of the study isn't to measure the difference by age group, and so the age-group dimension is secondary.

Upon recognizing that, I reduce the number of colors from 4 to 2:


Halving the number of colors presents no additional difficulty. The reader spends less time cross-referencing.

The existence of the Pbo (placebo) and Conv (convalescent plasma) columns on the sides is both unsightly and suboptimal. The "Conv" serves as a reference level for the amount of antibodies the vaccine stimulates in people. A better way to display reference levels is using reference lines.


The biggest problem with the chart is the log scale on the vertical axis. This isn't even a log-10 but a log-2. (Each tick is a doubling of value.)

Take the first set of columns as an example. The second column is clearly less than twice the height of the first column, and yet 25 is 3.5 times bigger than 7.  The third column is also visually less than double the size of the second column, and yet 189 is 7.5 times bigger than 25. The areas (heights) of the columns do not convey the right information about relative sizes of the underlying data.

Here's an amusing observation. The brown area shaded below is half of the entire area of the chart - if we reverted it to a linear scale. And yet there is not a single data point above 250 in the data so the brown area is entirely empty.


An effect of a log scale is to compress the larger values of a dataset. That's what you're seeing here.

I now revisualize using dotplots:


The version on the left retains the log scale while the right one (pun intended) reverts to the linear scale.

The biggest effect by far is the spike of antibodies between day 29 and 43 - which is after the second shot is administered. (For Moderna, the second shot is targeted for day 28.) In fact, it is during that window that the level of antibodies went from below the "conv" level (i.e. from natural infection) to far above.

The log-scale version buries this finding because it squeezes the large numbers on the chart. In addition, it artificially pulls the small numbers toward the "Conv" level. On the right chart, the second dot for 18-54, full doses is only at half the level of "Conv"  but it looks tantalizing close to the "Conv" level on the left chart.

The authors of the study also claim that there is negligible dropoff by 30 days after the second dose, i.e. between the third and fourth dots in each set. That may be so on the log-scale chart but on the linear chart, we see a moderate reduction. I don't believe the size of this study allows us to make a stronger conclusion but the claim of no dropoff is dubious.

The left chart also obscures the age-group differences. It appears as if all four sets show roughly the same pattern. With the linear scale, we notice that the vaccine clearly works better for the younger subgroup. As I discussed on the book blog, no one actually knows what level of antibodies constitutes "protection," and so I can't say whether that age-group difference has practical significance.


I recommend using log scales sparingly and carefully. They are a source of much mischief and misadventure.




Election visual 3: a strange, mash-up visualization

Continuing our review of FiveThirtyEight's election forecasting model visualization (link), I now look at their headline data visualization. (The previous posts in this series are here, and here.)


It's a set of 22 maps, each showing one election scenario, with one candidate winning. What chart form is this?

Small multiples may come to mind. A small-multiples chart is a grid in which every component graphic has the same form - same chart type, same color scheme, same scale, etc. The only variation from graphic to graphic is the data. The data are typically varied along a dimension of interest, for example, age groups, geographic regions, years. The following small-multiples chart, which I praised in the past (link), shows liquor consumption across the world.

image from

Each component graphic changes according to the data specific to a country. When we scan across the grid, we draw conclusions about country-to-country variations. As with convention, there are as many graphics as there are countries in the dataset. Sometimes, the designer includes only countries that are directly relevant to the chart's topic.


What is the variable FiveThirtyEight chose to vary from map to map? It's the scenario used in the election forecasting model.

This choice is unconventional. The 22 scenarios is a subset of the 40,000 scenarios from the simulation - we are left wondering how those 22 are chosen.

Returning to our question: what chart form is this?

Perhaps you're reminded of the dot plot from the previous post. On that dot plot, the designer summarized the results of 40,000 scenarios using 100 dots. Since Biden is the winner in 75 percent of all scenarios, the dot plot shows 75 blue dots (and 25 red).

The map is the new dot. The 75 blue dots become 16 blue maps (rounded down) while the 25 red dots become 6 red maps.

Is it a pictogram of maps? If we ignore the details on the maps, and focus on the counts of colors, then yes. It's just a bit challenging because of the hole in the middle, and the atypical number of maps.

As with the dot plot, the map details are a nice touch. It connects readers with the simulation model which can feel very abstract.

Oddly, if you're someone familiar with probabilities, this presentation is quite confusing.

With 40,000 scenarios reduced to 22 maps, each map should represent 1818 scenarios. On the dot plot, each dot should represent 400 scenarios. This follows the rule for creating pictograms. Each object in a pictogram - dot, map, figurine, etc. - should encode an equal amount of the data. For the 538 visualization, is it true that each of the six red maps represents 1818 scenarios? This may be the case but not likely.

Recall the dot plot where the most extreme red dot shows a scenario in which Trump wins 376 out of 538 electoral votes (margin = 214). Each dot should represent 400 scenarios. The visualization implies that there are 400 scenarios similar to the one on display. For the grid of maps, the following red map from the top left corner should, in theory, represent 1,818 similar scenarios. Could be, but I'm not sure.


Mathematically, each of the depicted scenario, including the blowout win above, occurs with 1/40,000 chance in the simulation. However, one expects few scenarios that look like the extreme scenario, and ample scenarios that look like the median scenario.  

So, the right way to read the 538 chart is to ignore the map details when reading the embedded pictogram, and then look at the small multiples of detailed maps bearing in mind that extreme scenarios are unique while median scenarios have many lookalikes.

(Come to think about it, the analogous situation in the liquor consumption chart is the relative population size of different countries. When comparing country to country, we tend to forget that the data apply to large numbers of people in populous countries, and small numbers in tiny countries.)


There's a small improvement that can be made to the detailed maps. As I compare one map to the next, I'm trying to pick out which states that have changed to change the vote margin. Conceptually, the number of states painted red should decrease as the winning margin decreases, and the states that shift colors should be the toss-up states.

So I'd draw the solid Republican (Democratic) states with a lighter shade, forming an easily identifiable bloc on all maps, while the toss-up states are shown with a heavier shade.


Here, I just added a darker shade to the states that disappear from the first red map to the second.

Election visuals 2: informative and playful

In yesterday's post, I reviewed one section of 538's visualization of its election forecasting model, specifically, the post focuses on the probability plot visualization.

The visualization, technically called  a pdf, is a mainstay of statistical graphics. While every one of 40,000 scenarios shows up on this chart, it doesn't offer a direct answer to our topline question. What is Nate's call at this point in time? Elsewhere in their post, we learn that the 538 model currently gives Biden a 75% chance of winning, thrice that of Trump's.


In graphical terms, the area to the right of the 270-line is three times the size of the left area (on the bottom chart). That's not apparent in the pdf representation. Addressing this, statisticians may convert the pdf into a cdf, which depicts the cumulative area as we sweep from the left to the right along the horizontal axis.  

The cdf visualization rarely leaves the pages of a scientific journal because it's not easy for a novice to understand. Not least because the relevant probability is 1 minus the cumulative probability. The cdf for the bottom chart will show 25% at the 270-line while the chance of Biden winning is 1 - 25% = 75%.

The cdf presentation is also wasteful for the election scenario. No one cares about any threshold other than the 270 votes needed to win, but the standard cdf shows every possible threshold.

The second graphical concept in the 538 post (link) is an attempt to solve this problem.


If you drop all the dots to an imaginary horizontal baseline, the above dotplot looks like this:


There is a recent trend toward centering dots to produce symmetry. It's actually harder to perceive the differences in heights of the band.

The secret sauce is to put down 100 dots, with a 75-25 blue-red split that conveys the 75% chance of a Biden win. Imposing the pdf line from the other visualization, I find that the density of dots roughly mimics the probability of outcomes.


It's easier to estimate the blue vs red areas using those dots than the lines.

The dots are stuffed toys. Clicking on each dot reveals a map showing one of the 40,000 scenarios. It displays which candidate wins which state. For example, the most extreme example of a Trump win is:


Here is a scenario of a razor-tight election won by Trump:


This presentation has a weakness as well. It gives the impression that each of the dots is equally important because they are the same size. In reality, the importance of each dot is proportional to the height of the band. Since the band is generally wider near the middle, the dots near the middle are more likely scenarios than the dots shown on the two edges.

On balance, I like this visualization that is both informative and playful.

As before, what strikes me about the simulation result is the flatness of the probability surface. This feature is obscured when we summarize the result as 75% chance of a Biden victory.

Putting vaccine trials in boxes

Bloomberg Businessweek has a special edition about vaccines, and I found this chart on the print edition:


The chart's got a lot of white space. Its structure is a series of simple "treemaps," one for each type of vaccine. Though simple, such a chart burns a few brain cells.

Here, I've extracted the largest block, which corresponds to vaccines that work with the virus's RNA/DNA. I applied a self-sufficiency test, removing the data from the boxes. 


What proportion of these projects have moved from pre-clinical to Phase 1?  To answer this question, we have to understand the relative areas of boxes, since that's how the data are encoded. How many yellow boxes can fit into the gray box?

It's not intuitive. We'd need a ruler to do this task properly.

Then, we learn that the gray box is exactly 8 times the size of the yellow box (72 projects are pre-clinical while 9 are in Phase I). We can cram eight yellows into the gray box. Imagine doing that, and it's pretty clear the visual elements fail to convey the meaning of the data.

Self-sufficiency is the idea that a data graphic should not rely on printed data to convey its meaning; the visual elements of a data graphic should bear much of the burden. Otherwise, use a data table. To test for self-sufficiency, cover up the printed data and see if the chart still works.


A key decision for the designer is the relative importance of (a) the number of projects reaching Phase III, versus (b) the number of projects utilizing specific vaccine strategies.

This next chart emphasizes the clinical phases:



Contrast this with the version shown in the online edition of Bloomberg (link), which emphasizes the vaccine strategies.


If any reader can figure out the logic of the ordering of the vaccine strategies, please leave a comment below.

This chart shows why the PR agency for the UK government deserves a Covid-19 bonus

The Economist illustrated some interesting consumer research with this chart (link):


The survey by Dalia Research asked people about the satisfaction with their country's response to the coronavirus crisis. The results are reduced to the "Top 2 Boxes", the proportion of people who rated their government response as "very well" or "somewhat well".

This dimension is laid out along the horizontal axis. The chart is a combo dot and bubble chart, arranged in rows by region of the world. Now what does the bubble size indicate?

It took me a while to find the legend as I was expecting it either in the header or the footer of the graphic. A larger bubble depicts a higher cumulative number of deaths up to June 15, 2020.

The key issue is the correlation between a country's death count and the people's evaluation of the government response.

Bivariate correlation is typically shown on a scatter plot. The following chart sets out the scatter plots in a small multiples format with each panel displaying a region of the world.


The death tolls in the Asian countries are low relative to the other regions, and yet the people's ratings vary widely. In particular, the Japanese people are pretty hard on their government.

In Europe, the people of Greece, Netherlands and Germany think highly of their government responses, which have suppressed deaths. The French, Spaniards and Italians are understandably unhappy. The British appears to be the most forgiving of their government, despite suffering a higher death toll than France, Spain or Italy. This speaks well of their PR operation.

Cumulative deaths should be adjusted by population size for a proper comparison across nations. When the same graphic is produced using deaths per million (shown on the right below), the general story is preserved while the pattern is clarified:


The right chart shows deaths per million while the left chart shows total deaths.


In the original Economist chart, what catches our attention first is the bubble size. Eventually, we notice the horizontal positioning of these bubbles. But the star of this chart ought to be the new survey data. I swapped those variables and obtained the following graphic:


Instead of using bubble size, I switched to using color to illustrate the deaths-per-million metric. If ratings of the pandemic response correlate tightly with deaths per million, then we expect the color of these dots to evolve from blue on the left side to red on the right side.

The peculiar loss of correlation in the U.K. stands out. Their PR firm deserves a bonus!

Working with multiple dimensions, an example from Germany

An anonymous reader submitted this mirrored bar chart about violent acts by extremists in the 16 German states.


At first glance, this looks like a standard design. On a second look, you might notice what the reader discovered- the chart used two different scales, one for each side. The left side (red) depicting left-wing extremism is artificially compressed relative to the right side (blue). Not sure if this reflects the political bias of the publication - but in any case, this distortion means the only way to consume this chart is to read the numbers.

Even after fixing the scales, this design is challenging for the reader. It's unnatural to compare two years by looking first below then above. It's not simple to compare across states, and even harder to compare left- and right-wing extremism (due to mirroring).

The chart feels busy because the entire dataset is printed on it. I appreciate not including a redundant horizontal axis. (I wonder if the designer first removed the axis, then edited the scale on one side, not realizing the distortion.) Another nice touch, hidden in the legend, is the country totals.

I present two alternatives.

The first is a small-multiples "bumps chart".


Each plot presents the entire picture within a state. You can see the general level of violence, the level of left- and right-wing extremism, and their year-on-year change. States can be compared holistically.

Several German state names are rather long, so I explored a horizontal orientation. In this case, a connected dot plot may be more appropriate.


The sign of a good multi-dimensional visual display is whether readers can easily learn complex relationships. Depending on the question of interest, the reader can mentally elevate parts of this chart. One can compare the set of blue arrows to the set of red arrows, or focus on just blue arrows pointing right, or red arrows pointing left, or all arrows for Berlin, etc.


[P.S. Anonymous reader said the original chart came from the Augsburger newspaper. This link in German contains more information.]

What is the price for objectivity

I knew I had to remake this chart.


The simple message of this chart is hidden behind layers of visual complexity. What the analyst wants readers to focus on (as discerned from the text on the right) is the red line, the seven-day moving average of new hospital admissions due to Covid-19 in Texas.

My eyes kept wandering away from the line. It's the sideway data labels on the columns. It's the columns that take up vastly more space than the red line. It's the sideway date labels on the horizontal axis. It's the redundant axis labels for hospitalizations when the entire data set has already been printed. It's the two hanging diamonds, for which the clues are filed away in the legend above.

Here's a version that brings out the message: after Phase 2 re-opening, the number of hospital admissions has been rising steadily.


Dots are used in place of columns, which push these details to the background. The line as well as periods of re-opening are directly labeled, removing the need for a legend.

Here's another visualization:


This chart plots the weekly average new hospital admissions, instead of the seven-day moving average. In the previous chart, the raggedness of moving average isn't transmitting any useful information to the average reader. I believe this weekly average metric is easier to grasp for many readers while retaining the general story.


On the original chart by TMC, the author said "the daily hospitalization trend shows an objective view of how COVID-19 impacts hospital systems." Objectivity is an impossible standard for any kind of data analysis or visualization. As seen above, the two metrics for measuring the trend in hospitalizations have pros and cons. Even if one insists on using a moving average, there are choices of averaging methods and window sizes.

Scientists are trained to believe in objectivity. It frequently disappoints when we discover that the rest of the world harbors no such notion. If you observe debates between politicians or businesspeople or social scientists, you rarely hear anyone claim one analysis is more objective - or less subjective - than another. The economist who predicts Dow to reach a new record, the business manager who argues for placing discounted products in the front not the back of the store, the sportscaster who maintains Messi is a better player than Ronaldo: do you ever hear these people describe their methods as objective?

Pursuing objectivity leads to the glorification of data dumps. The scientist proclaims disinterest in holding an opinion about the data. This is self-deception though. We clearly have opinions because when someone else  "misinterprets" the data, we express dismay. What is the point of pretending to hold no opinions when most of the world trades in opinions? By being "objective," we never shape the conversation, and forever play defense.