Visualizing the impossible

Note [July 6, 2022]: Typepad's image loader is broken yet again. There is no way for me to fix the images right now. They are not showing despite being loaded properly yesterday. I also cannot load new images. Apologies!

Note 2: Manually worked around the automated image loader.

Note 3: Thanks Glenn for letting me about the image loading problem. It turns out the comment approval function is also broken, so I am not able to approve the comment.

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A twitter user sent me this chart:

It's, hmm, mystifying. It performs magic, as I explain below.

What's the purpose of the gridlines and axis labels? Even if there is a rationale for printing those numbers, they make it harder, not easier, for readers to understand the chart!

I think the following chart shows the main message of this poll result. Democrats are much more likely to think of immigration as a positive compared to Republicans, with Independents situated in between.

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The axis title gives a hint as to what the chart designer was aiming for with the unconventional axis. It reads "Overall Percentage for All Participants". It appears that the total length of the stacked bar is the weighted aggregate response rate. Roughly 17% of Americans thought this development to be "very positive" which include 8% of Republicans, 27% of Democrats and 12% of Independents. Since the three segments are not equal in size, 17% is a weighted average of the three proportions.

Within each of the three political affiliations, the data labels add to 100%. These numbers therefore are unweighted response rates for each segment. (If weighted, they should add up to the proportion of each segment.)

This sets up an impossible math problem. The three segments within each bar then represent the sum of three proportions, each unweighted within its segment. Adding these unweighted proportions does not yield the desired weighted average response rate. To get the weighted average response rate, we need to sum the weighted segment response rates instead.

This impossible math problem somehow got resolved visually. We can see that each bar segment faithfully represent the unweighted response rates shown in the respective data labels. Summing them would not yield the aggregate response rates as shown on the axis title. The difference is not a simple multiplicative constant because each segment must be weighted by a different multiplier. So, your guess is as good as mine: what is the magic that makes the impossible possible?

[P.S. Another way to see this inconsistency. The sum of all the data labels is 300% because the proportions of each segment add up to 100%. At the same time, the axis title implies that the sum of the lengths of all five bars should be 100%. So, the chart asserts that 300% = 100%.]

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This poll question is a perfect classroom fodder to discuss how wording of poll questions affects responses (something called "response bias"). Look at the following variants of the same questions. Are we likely to get answers consistent with the above question?

As you know, the demographic makeup of America is changing and becoming more diverse, while the U.S. Census estimates that white people will still be the largest race in approximately 25 years. Generally speaking, do you find these changes to be very positive, somewhat positive, somewhat negative or very negative?

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As you know, the demographic makeup of America is changing and becoming more diverse, with the U.S. Census estimating that black people will still be a minority in approximately 25 years. Generally speaking, do you find these changes to be very positive, somewhat positive, somewhat negative or very negative?

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As you know, the demographic makeup of America is changing and becoming more diverse, with the U.S. Census estimating that Hispanic, black, Asian and other non-white people together will be a majority in approximately 25 years. Generally speaking, do you find these changes to be very positive, somewhat positive, somewhat negative or very negative?

What is also amusing is that in the world described by the pollster in 25 years, every race will qualify as a "minority". There will be no longer majority since no race will constitute at least 50% of the U.S. population. So at that time, the word "minority" will  have lost meaning.

Best chart I have seen this year

Marvelling at this chart:

 

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The credit ultimately goes to a Reddit user (account deleted). I first saw it in this nice piece of data journalism by my friends at System 2 (link). They linked to Visual Capitalism (link).

There are so many things on this one chart that makes me smile.

The animation. The message of the story is aging population. Average age is moving up. This uptrend is clear from the chart, as the bulge of the population pyramid is migrating up.

The trend happens to be slow, and that gives the movement a mesmerizing, soothing effect.

Other items on the chart are synced to the time evolution. The year label on the top but also the year labels on the right side of the chart, plus the counts of total population at the bottom.

OMG, it even gives me average age, and life expectancy, and how those statistics are moving up as well.

Even better, the designer adds useful context to the data: look at the names of the generations paired with the birth years.

This chart is also an example of dual axes that work. Age, birth year and current year are connected to each other, and given two of the three, the third is fixed. So even though there are two vertical axes, there is only one scale.

The only thing I'm not entirely convinced about is placing the scroll bar on the very top. It's a redundant piece that belongs to a less prominent part of the chart.

Think twice before you spiral

After Nathan at FlowingData sang praises of the following chart, a debate ensued on Twitter as others dislike it.

The chart was printed in an opinion column in the New York Times (link).

I have found few uses for spiral charts, and this example has not changed my mind.

The canonical time-series chart is like this:

 

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The area chart takes no effort to understand. We can see when the peaks occurred. We notice that the current surge is already double the last peak seen a year ago.

It's instructive to trace how one gets from the simple area chart to the spiral chart.

Step 1 is to center the area on the zero baseline, instead of having the zero baseline as the baseline. While this technique frequently makes for a more pleasant visual (because of our preference for symmetry), it actually makes it harder to see the trend over time. Effectively, any change is split in half, which is why the envelope of the area is less sharp.

In Step 2, I massively compress the vertical scale. That's because when you plot a spiral, you are forced to fit each cycle of data into a much shorter range. Such compression causes the year on year doubling of cases to appear less dramatic. (Actually, the aspect ratio is devastated because while the vertical scale is hugely compressed, the horizontal scale is dramatically stretched out due to the curled up design)

Step 3 may elude your attention. If you simply curl up the compressed, centered area chart, you don't get the spiral chart. The key is to ask about the radius of the spiral. As best I can tell, the radius has no meaning; it is gradually increased so that each year of data has its own "orbit". What would the change in radius translate to on our non-circular chart? It should mean that the center of the area is gradually lifted away from the zero line. On the right chart, I mimic this effect (I only measured the change in radius every 3 months so the change is more angular than displayed in the spiral chart.) The problem I have with this Step is that it serves no purpose, while it complicates cognition,

In Step 4, just curl up the object into a ball based on aligning months of the year.

This is the point when I realized I missed a Step 2B. I carefully aligned the scales of both charts so that the 150K cases shown in the legend on the right have the same vertical representation as on the left. This exposes a severe horizontal rescaling. The length of the horizontal axis on the left chart is many times smaller than the circumference of the spiral! That's why earlier, I said one of the biggest feature of this spiral chart is that it imposes a dubious aspect ratio, that is extremely wide and extremely short.

As usual, think twice before you spiral.

 

 

To explain or to eliminate, that is the question

Today, I take a look at another project from Ray Vella's class at NYU.

(The above image is a honeypot for "smart" algorithms that don't know how to handle image dimensions which don't fit their shadow "requirement". Human beings should proceed to the full image below.)

As explained in this post, the students visualized data about regional average incomes in a selection of countries. It turns out that remarkable differences persist in regional income disparity between countries, almost all of which are more advanced economies.

The graphic is by Danielle Curran.

I noticed two smart decisions.

First, she came up with a different main metric for gauging regional disparity, landing on a metric that is simple to grasp.

Based on hints given on the chart, I surmised that Danielle computed the change in per-capita income in the richest and poorest regions separately for each country between 2000 and 2015. These regional income growth values are expressed in currency, not indiced. Then, she computed the ratio of these growth rates, for each country. The end result is a simple metric for each country that describes how fast income has been growing in the richest region relative to the poorest region.

One of the challenges of this dataset is the complex indexing scheme (discussed here). Carlos' solution keeps the indices but uses design to facilitate comparisons. Danielle avoids the indices altogether.

The reader is relieved of the need to make comparisons, and so can focus on differences in magnitude. We see clearly that regional disparity is by far the highest in the U.K.

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The second smart decision Danielle made is organizing the countries into clusters. She took advantage of the horizontal axis which does not encode any data. The branching structure places different clusters of countries along the axis, making it simple to navigate. The locations of these clusters are cleverly aligned to the map below.

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Danielle's effort is stronger on communications while Carlos' effort provides more information. The key is to understand who your readers are. What proportion of your readers would want to know the values for each country, each region and each year?

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A couple of suggestions

a) The reference line should be set at 1, not 0, for a ratio scale. The value of 1 happens when the richest region and the poorest region have identical per-capita incomes.

b) The vertical scale should be fixed.

Displaying convoluted indices

I reviewed another batch of projects from Ray Vella's class at NYU. The following piece by Carlos Lasso made an impression on me. There are no pyrotechnics but he made one decision that added a lot of clarity to the graphic.

The underlying dataset gauges the income disparity of regions within nine countries. The richest and the poorest regions are selected for each country. Two time points are shown. Altogether, there are 9x2x2 = 36 numbers.

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Let's take a deeper look at these numbers. Notice they are not in dollars, or any kind of currency, despite being about incomes. The numbers are index values, relative to 100. What does the reference level of 100 represent?

The value of 100 crosses every bar of the chart so that 100 has meaning in each country and each year. In fact, there are 18 definitions of 100 in this chart with 36 numbers, one for each country-year pair. The average national income is set to 100 for each country in each year. This is a highly convoluted indexing strategy.

The following chart is a re-visualization of the bottom part of Carlos' infographic.

I shifted the scale of the horizontal axis. The value of zero does not hold special meaning in Carlos' chart. I subtracted 100 from the relative regional income indices, thus all regions with income above the average have positive values while those below the national average have negative values. (There are other challenges with the ratio scale, which I'll skip over in this post. The minimum value is -100 while the maximum value can be very large.)

The rescaling is not really the point of this post. To see what Carlos did, we have to look at the example shown in class. The graphic which the students were asked to improve has the following structure:

This one-column structure places four bars beside each country, grouped by year. Carlos pulled the year dimension out, and showed the same dataset in two columns.

This small change makes a great difference in ease of comprehension. Carlos' version unpacks the two key types of comparisons one might want to make: trend within a given country (horizontal comparison) and contrast between countries in a given year (vertical comparison).

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I always try to avoid convoluted indexing. The cost of using such indices is the big how-to-read-this box.

Asymmetry and orientation

An author in Significance claims that a single season of Premier League football without live spectators is enough to prove that the so-called home field advantage is really a live-spectator advantage.

The following chart depicts the data going back many seasons:

I find this bar chart challenging.

It plots the ratio of home wins to away wins using an odds scale, which is not intuitive. The odds scale (probability of success divided by probability of failure) runs from 0 to positive infinity, with 1 being a special value indicating equal odds. But all the values for which away wins exceed home wins are squeezed into the interval between 0 and 1 while the values for which home wins exceed away wins are laid out between 1 and infinity. So it's an inherently asymmetric graphic for a symmetric formula.

The section labeled "more away wins than home wins" are filled with red bars for all those seasons with positive home field advantage while the most recent season, the outlier, has a shorter bar in that section than the rest.

Here's an alternative view:

I have incorporated dual axes here - but both axes are different only by scaling. There are 380 games in a Premier League season so the percentage scale is just a re-expression of the counts.

 

 

Ridings, polls, elections, O Canada

Stephen Taylor reached out to me about his work to visualize Canadian elections data. I took a look. I appreciate the labor of love behind this project.

He led with a streamgraph, which presents a quick overview of relative party strengths over time.

I am no Canadian election expert, and I did a bare minimum of research in writing this blog. From this chart, I learn that:

• the Canadians have an irregular election schedule
• Canada has a two party plus breadcrumbs system
• The two dominant parties are Liberals and Conservatives. The Liberals currently hold just less than half of the seats. The Conservatives have more than half of the seats not held by Liberals
• The Conservative party (maybe) rebranded as "progressive conservative" for several decades. The Reform/Alliance party was (maybe) a splinter movement within the Conservatives as well.
• Since the "width" of the entire stream increased over time, I'm guessing the number of seats has expanded

That's quite a bit of information obtained at a glance. This shows the power of data visualization. Notice Stephen didn't even have to include a "how to read this" box.

The streamgraph form has its limitations.

The feature that makes it more attractive than an area chart is its middle anchoring, resulting in a form of symmetry. The same feature produces erroneous intuition - the red patch draws out a declining trend; the reader must fight the urge to interpret the lines and focus on the areas.

The breadcrumbs are well hidden. The legend below discloses that the Green Party holds 3 seats currently. The party has never held enough seats to appear on the streamgraph though.

The bars showing proportions in the legend is a very nice touch. (The numbers appear messed up - I have to ask Stephen whether the seats shown are current values, or some kind of historical average.) I am a big fan of informative legends.

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The next featured chart is a dot plot of polling results since 2020.

One can see a three-tier system: the two main parties, then the NDP (yellow) is the clear majority of the minority, and finally you have a host of parties that don't poll over 10%.

It looks like the polls are favoring the Conservatives over the Liberals in this election but it may be an election-day toss-up.

The purple dots represent "PPC" which is a party not found elsewhere on the page.

This chart is clear as crystal because of the structure of the underlying data. It just amazes me that the polls are so highly correlated. For example, across all these polls, the NDP has never once polled better than either the Liberals or the Conservatives, and in addition, it has never polled worse than any of the small parties.

What I'd like to see is a chart that merges the two datasets, addressing the question of how well these polls predicted the actual election outcomes.

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The project goes very deep as Stephen provides charts for individual "ridings" (perhaps similar to U.S. precincts).

Here we see population pyramids for Vancouver Center, versus British Columbia (Province), versus Canada.

This riding has a large surplus of younger people in their twenties and thirties. Be careful about the changing scales though. The relative difference in proportions are more drastic than visually displayed because the maximum values (5%) on the Province and Canada charts are half that on the Riding chart (10%). Imagine squashing the Province and Canada charts to half their widths.

Analyses of income and rent/own status are also provided.

This part of the dashboard exhibits a problem common in most dashboards - they present each dimension of the data separately and miss out on the more interesting stuff: the correlation between dimensions. Do people in their twenties and thirties favor specific parties? Do richer people vote for certain parties?

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The riding-level maps are the least polished part of the site. This is where I'm looking for a "how to read it" box.

It took me a while to realize that the colors represent the parties. If I haven't come in from the front page, I'd have been totally lost.

Next, I got confused by the use of the word "poll". Clicking on any of the subdivisions bring up details of an actual race, with party colors, candidates and a donut chart showing proportions. The title gives a "poll id" and the name of the riding in parentheses. Since the poll id changes as I mouse over different subdivisions, I'm wondering whether a "poll" is the term for a subdivision of a riding. A quick wiki search indicates otherwise.

My best guess is the subdivisions are indicated by the numbers.

Back to the donut charts, I prefer a different sorting of the candidates. For this chart, the two most logical orderings are (a) order by overall popularity of the parties, fixed for all ridings and (b) order by popularity of the candidate, variable for each riding.

The map shown above gives the winner in each subdivision. This type of visualization dumps a lot of information. Stephen tackles this issue by offering a small multiples view of each party. Here is the Liberals in Vancouver.

Again, we encounter ambiguity about the color scheme. Liberals have been associated with a red color but we are faced with abundant yellow. After clicking on the other parties, you get the idea that he has switched to a divergent continuous color scale (red - yellow - green). Is red or green the higher value? (The answer is red.)

I'd suggest using a gray scale for these charts. The hardest decision is going to be the encoding between values and shading. Should each gray scale be different for each riding and each party?

If I were to take a guess, Stephen must have spent weeks if not months creating these maps (depending on whether he's full-time or part-time). What he has published here is a great start. Fine-tuning the issues I've mentioned may take more weeks or months more.

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Stephen is brave and smart to send this project for review. For one thing, he's got some free consulting. More importantly, we should always send work around for feedback; other readers can tell us where our blind spots are.

To read more, start with this post by Stephen in which he introduces his project.

Ranking data provide context but can also confuse

This dataviz from the Economist had me spending a lot of time clicking around - which means it is a success.

The graphic presents four measures of wellbeing in society - life expectancy, infant mortality rate, murder rate and prison population. The primary goal is to compare nations across those metrics. The focus is on comparing how certain nations (or subgroups) rank against each other, as indicated by the relative vertical position.

The Economist staff has a particular story to tell about racial division in the US. The dotted bars represent the U.S. average. The colored bars are the averages for Hispanic, white and black Americans. The wider the gap between the colored bars, the more variant is the experiences between American races.

The chart shows that the racial gap of life expectancy is the widest. For prison population, the U.S. and its racial subgroups occupy many of the lowest (i.e. least desirable) ranks, with the smallest gap in ranking.

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The primary element of interactivity is hovering on a bar, which then highlights the four bars corresponding to the particular nation selected. Here is the picture for Thailand:

According to this view of the world, Thailand is a close cousin of the U.S. On each metric, the Thai value clings pretty near the U.S. average and sits within the range by racial groups. I'm surprised to learn that the prison population in Thailand is among the highest in the world.

Unfortunately, this chart form doesn't facilitate comparing Thailand to a country other than the U.S as one can highlight only one country at a time.

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While the main focus of the chart is on relative comparison through ranking, the reader can extract absolute difference by reading the lengths of the bars.

This is a close-up of the bottom of the prison population metric:

The length of each bar displays the numeric data. The red line is an outlier in this dataset. Black Americans suffer an incarceration rate that is almost three times the national average. Even white Americans (blue line) is imprisoned at a rate higher than most countries around the world.

As noted above, the prison population metric exhibits the smallest gap between racial subgroups. This chart is a great example of why ranking data frequently hide important information. The small gap in ranking masks the extraordinary absolute difference in incareration rates between white and black America.

The difference between rank #1 and rank #2 is enormous.

The opposite situation appears for life expectancy. The life expectancy values are bunched up especially at the top of the scale. The absolute difference between Hispanic and black America is 82 - 75 = 7 years, which looks small because the axis starts at zero. On a ranking scale, Hispanic is roughly in the top 15% while black America is just above the median. The relative difference is huge.

For life expectancy, ranking conveys the view that even a 7-year difference is a big deal because the countries are tightly bunched together. For prison population, ranking shows the view that a multiple fold difference is "unimportant" because a 20-0 blowout and a 10-0 blowout are both heavy defeats.

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Whenever you transform numeric data to ranks, remember that you are artificially treating the gap between each value and the next value as a constant, even when the underlying numeric gaps show wide variance.

 

 

 

 

 

Start at zero improves this chart but only slightly

The following chart was forwarded to me recently:

It's a good illustration of why the "start at zero" rule exists for column charts. The poor Indian lady looks extremely short in this women's club. Is the average Indian woman really half as tall as the average South African woman? (Surely not!)

The problem is only superficially fixed by starting the vertical axis at zero. Doing so highlights the fact that the difference in average heights is but a fraction of the average heights themselves. The intra-country differences are squashed in such a representation - which works against the primary goal of the data visualization itself.

Recall the Trifecta Checkup. At the top of the trifecta is the Question. The designer obviously wants to focus our attention on the difference of the averages. A column chart showing average heights fails the job!

This "proper" column chart sends the message that the difference in average heights is noise, unworthy of our attention. But this is a bad take of the underlying data. The range of average heights across countries isn't that wide, by virtue of large population sizes.

According to Wikipedia, they range from 4 feet 10.5 to 5 feet 6 (I'm ignoring several entries in the table based on non representative small samples.) How do we know that the difference of 2 inches between averages of South Africa and India is actually a sizable difference? The Wikipedia table has the average heights for most of the world's countries. There are perhaps 200 values. These values are sprinkled inside the range of about 8 inches top to bottom. If we divide the full range into 10 equal bins, that's roughly 0.8 inches per bin. So if we have two numbers that are 2 inches apart, they almost span 2 bins. If the data were evenly distributed, that's a huge shift.

(In reality, the data should be normally distributed, bell-shaped, with much more at the center than on the edges. That makes a difference of 2 inches even more significant if these are normal values near the center but less significant if these are extreme values on the tails. Stats students should be able to articulate why we are sure the data are normally distributed without having to plot the data.)

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The original chart has further problems.

Another source of distortion comes from the scaling of the stick figures. The aspect ratio is being preserved, which means the area is being scaled. Given that the heights are scaled as per the data, the data are encoded twice, the second time in the widths. This means that the sizes of these figures grow at the rate of the square of the heights. (Contrast this with the scaling discussed in my earlier post this week which preserves the relative areas.)

At the end of that last post, I discuss why adding colors to a chart when the colors do not encode any data is a distraction to the reader. And this average height chart is an example.

From the Data corner of the Trifecta Checkup, I'm intrigued by the choice of countries. Why is Scotland highlighted instead of the U.K.? Why Latvia? According to Wikipedia, the Latvia estimate is based on a 1% sample of only 19 year olds.

Some of the data appear to be incorrect (or the designer used a different data source). Wikipedia lists the average height of Latvian women as 5 ft 6.5 while the chart shows 5 ft 5 in. Peru's average height of females is listed as 4 ft 11.5 and of males as 5 ft 4.5. The chart shows 5 ft 4 in.

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Lest we think only amateurs make this type of chart, here is an example of a similar chart in a scientific research journal:

(link to original)

I have seen many versions of the above column charts with error bars, and the vertical axes not starting at zero. In every case, the heights (and areas) of these columns do not scale with the underlying data.

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I tried a variant of the stem-and-leaf plot:

The scale is chosen to reflect the full range of average heights given in Wikipedia. The chart works better with more countries to fill out the distribution. It shows India is on the short end of the scale but not quite the lowest. (As mentioned above, Peru actually should be placed close to the lower edge.)

 

Distorting perception versus distorting the data

This chart appears in the latest ("last print issue") of Schwab's On Investing magazine:

I know I don't like triangular charts, and in this post, I attempt to verbalize why.

It's not the usual complaint of distorting the data. When the base of the triangle is fixed, and only the height is varied, then the area is proportional to the height and thus nothing is distorted.

Nevertheless, my ability to compare those triangles pales in comparison to the following columns.

This phenomenon is not limited to triangles. One can take columns and start varying the width, and achieve a similar effect:

It's really the aspect ratio - the relationship between the height and the width that's the issue.

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Interestingly, with an appropriately narrow base, even the triangular shape can be saved.

In a sense, we can think of the width of these shapes as noise, a distraction - because the width is constant, and not encoding any data.

It's like varying colors for no reason at all. It introduces a pointless dimension.

It may be prettier but the colors also interfere with our perception of the changing heights.