When the pie chart is more complex than the data

The trading house, Charles Schwab, included the following graphic in a recent article:

This graphic is more complicated than the story that it illustrates. The author describes a simple scenario in which an investor divides his investments into stocks, bonds and cash. After a stock crash, the value of the portfolio declines.

The graphic is a 3-D pie chart, in which the data are encoded twice, first in the areas of the sectors and then in the heights of the part-cylinders.

As readers, we perceive the relative volumes of the part-cylinders. Volume is the cross-sectional area (i.e. of the base) multipled by the height. Since each component holds the data, the volumes are proportional to the squares of the data.

Here is a different view of the same data:

This "bumps chart" (also called a slopegraph) shows clearly the only thing that drives the change is the drop in stock prices. Because the author assumes no change in bonds or cash, the drop in the entire portfolio is completely accounted for by the decline in stocks. Of course, this scenario seems patently unrealistic - different investment asset classes tend to be correlated.

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A cardinal rule of data visualization is that the visual should be less complex than the data.

Designs of two variables: map, dot plot, line chart, table

The New York Times found evidence that the richest segments of New Yorkers, presumably those with second or multiple homes, have exited the Big Apple during the early months of the pandemic. The article (link) is amply assisted by a variety of data graphics.

The first few charts represent different attempts to express the headline message. Their appearance in the same article allows us to assess the relative merits of different chart forms.

First up is the always-popular map.

The advantage of a map is its ease of comprehension. We can immediately see which neighborhoods experienced the greater exoduses. Clearly, Manhattan has cleared out a lot more than outer boroughs.

The limitation of the map is also in view. With the color gradient dedicated to the proportions of residents gone on May 1st, there isn't room to express which neighborhoods are richer. We have to rely on outside knowledge to make the correlation ourselves.

The second attempt is a dot plot.

We may have to take a moment to digest the horizontal axis. It's not time moving left to right but income percentiles. The poorest neighborhoods are to the left and the richest to the right. I'm assuming that these percentiles describe the distribution of median incomes in neighborhoods. Typically, when we see income percentiles, they are based on households, regardless of neighborhoods. (The former are equal-sized segments, unlike the latter.)

This data graphic has the reverse features of the map. It does a great job correlating the drop in proportion of residents at home with the income distribution but it does not convey any spatial information. The message is clear: The residents in the top 10% of New York neighborhoods are much more likely to have left town.

In the following chart, I attempted a different labeling of both axes. It cuts out the need for readers to reverse being home to not being home, and 90th percentile to top 10%.

The third attempt to convey the income--exit relationship is the most successful in my mind. This is a line chart, with time on the horizontal axis.

The addition of lines relegates the dots to the background. The lines show the trend more clearly. If directly translated from the dot plot, this line chart should have 100 lines, one for each percentile. However, the closeness of the top two lines suggests that no meaningful difference in behavior exists between the 20th and 80th percentiles. This can be conveyed to readers through a short note. Instead of displaying all 100 percentiles, the line chart selectively includes only the 99th , 95th, 90th, 80th and 20th percentiles. This is a design choice that adds by subtraction.

Along the time axis, the line chart provides more granularity than either the map or the dot plot. The exit occurred roughly over the last two weeks of March and the first week of April. The start coincided with New York's stay-at-home advisory.

This third chart is a statistical graphic. It does not bring out the raw data but features aggregated and smoothed data designed to reveal a key message.

I encourage you to also study the annotated table later in the article. It shows the power of a well-designed table.

[P.S. 6/4/2020. On the book blog, I have just published a post about the underlying surveillance data for this type of analysis.]

 

 

Graph literacy, in a sense

Ben Jones tweeted out this chart, which has an unusual feature:

What's unusual is that time runs in both directions. Usually, the rule is that time runs left to right (except, of course, in right-to-left cultures). Here, the purple area chart follows that convention while the yellow area chart inverts it.

On the one hand, this is quite cute. Lines meeting in the middle. Converging. I get it.

On the other hand, every time a designer defies conventions, the reader has to recognize it, and to rationalize it.

In this particular graphic, I'm not convinced. There are four numbers only. The trend on either side looks linear so the story is simple. Why complicate it using unusual visual design?

Here is an entirely conventional bumps-like chart that tells the story:

I've done a couple of things here that might be considered controversial.

First, I completely straightened out the lines. I don't see what additional precision is bringing to the chart.

Second, despite having just four numbers, I added the year 1996 and vertical gridlines indicating decades. A Tufte purist will surely object.

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Related blog post: "The Return on Effort in Data Graphics" (link)

Pulling the multi-national story out, step by step

Reader Aleksander B. found this Economist chart difficult to understand.

Given the chart title, the reader is looking for a story about multinationals producing lower return on equity than local firms. The first item displayed indicates that multinationals out-performed local firms in the technology sector.

The pie charts on the right column provide additional information about the share of each sector by the type of firms. Is there a correlation between the share of multinationals, and their performance differential relative to local firms?

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We can clean up the presentation. The first changes include using dots in place of pipes, removing the vertical gridlines, and pushing the zero line to the background:

The horizontal gridlines attached to the zero line can also be removed:

Now, we re-order the rows. Start with the aggregate "All sectors". Then, order sectors from the largest under-performance by multinationals to the smallest.

The pie charts focus only on the share of multinationals. Taking away the remainders speeds up our perception:

Help the reader understand the data by dividing the sectors into groups, organized by the performance differential:

For what it's worth, re-sort the sectors from largest to smallest share of multinationals:

Having created groups of sectors by share of multinationals, I simplify further by showing the average pie chart within each group:

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To recap all the edits, here is an animated gif: (if it doesn't play automatically, click on it)

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Judging from the last graphic, I am not sure there is much correlation between share of multinationals and the performance differentials. It's interesting that in aggregate, local firms and multinationals performed the same. The average hides the variability by sector: in some sectors, local firms out-performed multinationals, as the original chart title asserted.

Clearing a forest of labels

This chart by the Financial Times has a strong message, and I like a lot about it:

The countries are by and large aligned along a diagonal, with the poorer countries growing strongly between 2007-2019 while the richer countries suffered negative growth.

A small issue with the chart is the thick forest of text - redundant text. The sub-title, the axis titles, the quadrant labels, and the left-right-half labels all repeat the same things. In the following chart, I simplify the text:

Typically, I don't put axis titles as a sub-header (or, header of the graphic) but as this may be part of the FT style, I respected it.

Seeking simplicity in complex data: Bloomberg's dataviz on UK gender pay gap

Bloomberg featured a thought-provoking dataviz that illustrates the pay gap by gender in the U.K. The dataset underlying this effort is complex, and the designers did a good job simplifying the data for ease of comprehension.

U.K. companies are required to submit data on salaries and bonuses by gender, and by pay quartiles. The dataset is incomplete, since some companies are slow to report, and the analyst decided not to merge companies that changed names.

Companies are classified into industry groups. Readers who read Chapter 3 of Numbers Rule Your World (link) should ask whether these group differences are meaningful by themselves, without controlling for seniority, job titles, etc. The chapter features one method used by the educational testing industry to take a more nuanced analysis of group differences.

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The Bloomberg visualization has two sections. In the top section, each company is represented by the percent difference between average female pay and average male pay. Then the companies within a given industry is shown in a histogram. The histograms provide a view of the disparity between companies within a given industry. The black line represents the relative proportion of companies in a given industry that have no gender pay gap but it’s the weight of the histogram on either side of the black line that carries the graphic’s message.

This is the histogram for arts, entertainment and recreation.

The spread within this industry is very wide, especially on the left side of the black line. A large proportion of these companies pay women less on average than men, and how much less is highly variable. There is one extreme positive value: Chelsea FC Foundation that pays the average female about 40% more than the average male.

This is the histogram for the public sector.

It is a much tighter distribution, meaning that the pay gaps vary less from organization to organization (this statement ignores the possibility that there are outliers not visible on this graphic). Again, the vast majority of entities in this sector pay women less than men on average.

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The second part of the visualization look at the quartile data. The employees of each company are divided into four equal-sized groups, based on their wages. Think of these groups as the Top 25% Earners, the Second 25%, etc. Within each group, the analyst looks at the proportion of women. If gender is independent of pay, then we should expect the proportions of women to be about the same for all four quartiles. (This analysis considers gender to be the only explainer for pay gaps. This is a problem I've called xyopia, that frames a complex multivariate issue as a bivariate problem involving one outcome and one explanatory variable. Chapter 3 of Numbers Rule Your World (link) discusses how statisticians approach this issue.)

On the right is the chart for the public sector. This is a pie chart used as a container. Every pie has four equal-sized slices representing the four quartiles of pay.

The female proportion is encoded in both the size and color of the pie slices. The size encoding is more precise while the color encoding has only 4 levels so it provides a “binned” summary view of the same data.

For the public sector, the lighter-colored slice shows the top 25% earners, and its light color means the proportion of women in the top 25% earners group is between 30 and 50 percent. As we move clockwise around the pie, the slices represent the 2nd, 3rd and bottom 25% earners, and women form 50 to 70 percent of each of those three quartiles.

To read this chart properly, the reader must first do one calculation. Women represent about 60% of the top 25% earners in the public sector. Is that good or bad? This depends on the overall representation of women in the public sector. If the sector employs 75 percent women overall, then the 60 percent does not look good but if it employs 40 percent women, then the same value of 60% tells us that the female employees are disproportionately found in the top 25% earners.

That means the reader must compare each value in the pie chart against the overall proportion of women, which is learned from the average of the four quartiles.

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In the chart below, I make this relative comparison explicit. The overall proportion of women in each industry is shown using an open dot. Then the graphic displays two bars, one for the Top 25% earners, and one for the Bottom 25% earners. The bars show the gap between those quartiles and the overall female proportion. For the top earners, the size of the red bars shows the degree of under-representation of women while for the bottom earners, the size of the gray bars shows the degree of over-representation of women.

The net sum of the bar lengths is a plausible measure of gender inequality.

The industries are sorted from the ones employing fewer women (at the top) to the ones employing the most women (at the bottom). An alternative is to sort by total bar lengths. In the original Bloomberg chart - the small multiples of pie charts, the industries are sorted by the proportion of women in the bottom 25% pay quartile, from smallest to largest.

In making this dataviz, I elected to ignore the middle 50%. This is not a problem since any quartile above the average must be compensated by a different quartile below the average.

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The challenge of complex datasets is discovering simple ways to convey the underlying message. This usually requires quite a bit of upfront analytics, data transformation, and lots of sketching.

 

 

An exercise in decluttering

My friend Xan found the following chart by Pew hard to understand. Why is the chart so taxing to look at? 

It's packing too much.

I first notice the shaded areas. Shading usually signifies "look here". On this chart, the shading is highlighting the least important part of the data. Since the top line shows applicants and the bottom line admitted students, the shaded gap displays the rejections.

The numbers printed on the chart are growth rates but they confusingly do not sync with the slopes of the lines because the vertical axis plots absolute numbers, not rates. 

The vertical axis presents the total number of applicants, and the total number of admitted students, in each "bucket" of colleges, grouped by their admission rate in 2017. On the right, I drew in two lines, both growth rates of 100%, from 500K to 1 million, and from 1 to 2 million. The slopes are not the same even though the rates of growth are.

Therefore, the growth rates printed on the chart must be read as extraneous data unrelated to other parts of the chart. Attempts to connect those rates to the slopes of the corresponding lines are frustrated.

Another lurking factor is the unequal sizes of the buckets of colleges. There are fewer than 10 colleges in the most selective bucket, and over 300 colleges in the largest bucket. We are unable to interpret properly the total number of applicants (or admissions). The quantity of applications in a bucket depends not just on the popularity of the colleges but also the number of colleges in each bucket.

The solution isn't to resize the buckets but to select a more appropriate metric: the number of applicants per enrolled student. The most selective colleges are attracting about 20 applicants per enrolled student while the least selective colleges (those that accept almost everyone) are getting 4 applicants per enrolled student, in 2017.

As the following chart shows, the number of applicants has doubled across the board in 15 years. This raises an intriguing question: why would a college that accepts pretty much all applicants need more applicants than enrolled students?

Depending on whether you are a school administrator or a student, a virtuous (or vicious) cycle has been realized. For the top four most selective groups of colleges, they have been able to progressively attract more applicants. Since class size did not expand appreciably, more applicants result in ever-lower admit rate. Lower admit rate reduces the chance of getting admitted, which causes prospective students to apply to even more colleges, which further suppresses admit rate. 

 

 

 

The Economist on the Economist: must read now

A visual data journalist at the Economist takes a critical eye on charts published by the Economist (link). This is a must read! (Hat tip: Fernando)

Here are some of my commentary on past Economist charts.

NYT hits the trifecta with this market correction chart

Yesterday, in the front page of the Business section, the New York Times published a pair of charts that perfectly captures the story of the ongoing turbulence in the stock market.

Here is the first chart:

Most market observers are very concerned about the S&P entering "correction" territory, which the industry arbitrarily defines as a drop of 10% or more from a peak. This corresponds to the shortest line on the above chart.

The chart promotes a longer-term reflection on the recent turbulence, using two reference points: the index has returned to the level even with that at the start of 2018, and about 16 percent higher since the beginning of 2017.

This is all done tastefully in a clear, understandable graphic.

Then, in a bit of a rhetorical flourish, the bottom of the page makes another point:

When viewed back to a 10-year period, this chart shows that the S&P has exploded by 300% since 2009.

A connection is made between the two charts via the color of the lines, plus the simple, effective annotation "Chart above".

The second chart adds even more context, through vertical bands indicating previous corrections (drops of at least 10%). These moments are connected to the first graphic via the beige color. The extra material conveys the message that the market has survived multiple corrections during this long bull period.

Together, the pair of charts addresses a pressing current issue, and presents a direct, insightful answer in a simple, effective visual design, so it hits the Trifecta!

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There are a couple of interesting challenges related to connecting plots within a multiple-plot framework.

While the beige color connects the concept of "market correction" in the top and bottom charts, it can also be a source of confusion. The orientation and the visual interpretation of those bands differ. The first chart uses one horizontal band while the chart below shows multiple vertical bands. In the first chart, the horizontal band refers to a definition of correction while in the second chart, the vertical bands indicate experienced corrections.

Is there a solution in which the bands have the same orientation and same meaning?

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These graphs solve a visual problem concerning the visualization of growth over time. Growth rates are anchored to some starting time. A ten-percent reduction means nothing unless you are told ten-percent of what.

Using different starting times as reference points, one gets different values of growth rates. With highly variable series of data like stock prices, picking starting times even a day apart can lead to vastly different growth rates.

The designer here picked several obvious reference times, and superimposes multiple lines on the same plotting canvass. Instead of having four lines on one chart, we have three lines on one, and four lines on the other. This limits the number of messages per chart, which speeds up cognition.

The first chart depicts this visual challenge well. Look at the start of 2018. This second line appears as if you can just reset the start point to 0, and drag the remaining portion of the line down. The part of the top line (to the right of Jan 2018) looks just like the second line that starts at Jan 2018.

However, a closer look reveals that the shape may be the same but the magnitude isn't. There is a subtle re-scaling in addition to the re-set to zero.

The same thing happens at the starting moment of the third line. You can't just drag the portion of the first or second line down - there is also a needed re-scaling.

A second take on the rural-urban election chart

Yesterday, I looked at the following pictograms used by Business Insider in an article about the rural-urban divide in American politics:

The layout of this diagram suggests that the comparison of 2010 to 2018 is a key purpose.

The following alternate directly plots the change between 2010 and 2018, reducing the number of plots from 4 to 2.

The 2018 results are emphasized. Then, for each party, there can be a net add or loss of seats.

The key trends are:

• a net loss in seats in "Pure rural" districts, split by party;
• a net gain of 3 seats in "rural-suburban" districts;
• a loss of 10 Democratic seats balanced by a gain of 13 Republican seats.