Some chart designs bring out more information than others

I forgot where I found this chart but here it is:

The designer realizes the flaw of the design, which is why the number 50 is placed in a red box, and there is another big red box  placed right in our faces telling us that any number above 50 represents growing, while all below 50 shrinking.

The real culprit is the column chart design, which treats zero as the baseline, not 50. Thus, the real solution is to move away from a column chart design.

There are many possibilities. Here's one using the Bumps chart form:

There are several interesting insights buried in that column chart!

First we learn that almost all segments were contracting in both years.

Next, there are some clustering of segments. The Premium Regular and Cider segments were moving in sync. Craft, FMB/SEltzer and Below Premium were similar in 2022; intriguingly, Below Premium diverged from the other two segments.

In fact, Below Premium has distinguished itself as the only segment that experienced an improved index relative to 2022!

 

 

 

 

Finding the story in complex datasets

In CT Mirror's feature about Connecticut, which I wrote about in the previous post, there is one graphic that did not rise to the same level as the others.

This section deals with graduation rates of the state's high school districts. The above chart focuses on exactly five districts. The line charts are organized in a stack. No year labels are provided. The time window is 11 years from 2010 to 2021. The column of numbers show the difference in graduation rates over the entire time window.

The five lines look basically the same, if we ignore what looks to be noisy year-to-year fluctuations. This is due to the weird aspect ratio imposed by stacking.

Why are those five districts chosen? Upon investigation, we learn that these are the five districts with the biggest improvement in graduation rates during the 11-year time window.

The same five schools also had some of the lowest graduation rates at the start of the analysis window (2010). This must be so because if a school graduated 90% of its class in 2010, it would be mathematically impossible for it to attain a 35% percent point improvement! This is a dissatisfactory feature of the dataviz.

***

In preparing an alternative version, I start by imagining how readers might want to utilize a visualization of this dataset. I assume that the readers may have certain school(s) they are particularly invested in, and want to see its/their graduation performance over these 11 years.

How does having the entire dataset help? For one thing, it provides context. What kind of context is relevant? As discussed above, it's futile to compare a school at the top of the ranking to one that is near the bottom. So I created groups of schools. Each school is compared to other schools that had comparable graduation rates at the start of the analysis period.

Amistad School District, which takes pole position in the original dataviz, graduated only 58% of its pupils in 2010 but vastly improved its graduation rate by 35% over the decade. In the chart below (left panel), I plotted all of the schools that had graduation rates between 50 and 74% in 2010. The chart shows that while Amistad is a standout, almost all schools in this group experienced steady improvements. (Whether this phenomenon represents true improvement, or just grade inflation, we can't tell from this dataset alone.)

The right panel shows the group of schools with the next higher level of graduation rates in 2010. This group of schools too increased their graduation rates almost always. The rate of improvement in this group is lower than in the previous group of schools.

The next set of charts show school districts that already achieved excellent graduation rates (over 85%) by 2010. The most interesting group of schools consists of those with 85-89% rates in 2010. Their performance in 2021 is the most unpredictable of all the school groups. The majority of districts did even better while others regressed.

Overall, there is less variability than I'd expect in the top two school groups. They generally appeared to have been able to raise or maintain their already-high graduation rates. (Note that the scale of each chart is different, and many of the lines in the second set of charts are moving within a few percentages.)

One more note about the charts: The trend lines are "smoothed" to focus on the trends rather than the year to year variability. Because of smoothing, there is some awkward-looking imprecision e.g. the end-to-end differences read from the curves versus the observed differences in the data. These discrepancies can easily be fixed if these charts were to be published.

Thoughts on Daniel's fix for dual-axes charts

I've taken a little time to ponder Daniel Z's proposed "fix" for dual-axes charts (link). The example he used is this:

In that long post, Daniel explained why he preferred to mix a line with columns, rather than using the more common dual lines construction: to prevent readers from falsely attributing meaning to crisscrossing lines. There are many issues with dual-axes charts, which I won't repeat in this post; one of their most dissatisfying features is the lack of connection between the two vertical scales, and thus, it's pretty easy to manufacture an image of correlation when it doesn't exist. As shown in this old post, one can expand or restrict one of the vertical axes and shift the line up and down to "match" the other vertical axis.

Daniel's proposed fix retains the dual axes, and he even restores the dual lines construction.

How is this chart different from the typical dual-axes chart, like the first graph in this post?

Recall that the problem with using two axes is that the designer could squeeze, expand or shift one of the axes in any number of ways to manufacture many realities. What Daniel effectively did here is selecting one specific way to transform the "New Customers" axis (shown in gray).

His idea is to run a simple linear regression between the two time series. Think of fitting a "trendline" in Excel between Revenues and New Customers. Then, use the resulting regression equation to compute an "estimated" revenues based on the New Customers series. The coefficients of this regression equation then determines the degree of squeezing/expansion and shifting applied to the New Customers axis.

The main advantage of this "fix" is to eliminate the freedom to manufacture multiple realities. There is exactly one way to transform the New Customers axis.

The chart itself takes a bit of time to get used to. The actual values plotted in the gray line are "estimated revenues" from the regression model, thus the blue axis values on the left apply to the gray line as well. The gray axis shows the respective customer values. Because we performed a linear fit, each value of estimated revenues correspond to a particular customer value. The gray line is thus a squeezed/expanded/shifted replica of the New Customers line (shown in orange in the first graph). The gray line can then be interpreted on two connected scales, and both the blue and gray labels are relevant.

***

What are we staring at?

The blue line shows the observed revenues while the gray line displays the estimated revenues (predicted by the regression line). Thus, the vertical gaps between the two lines are the "residuals" of the regression model, i.e. the estimation errors. If you have studied Statistics 101, you may remember that the residuals are the components that make up the R-squared, which measures the quality of fit of the regression model. R-squared is the square of r, which stands for the correlation between Customers and the observed revenues. Thus the higher the (linear) correlation between the two time series, the higher the R-squared, the better the regression fit, the smaller the gaps between the two lines.

***

There is some value to this chart, although it'd be challenging to explain to someone who has not taken Statistics 101.

While I like that this linear regression approach is "principled", I wonder why this transformation should be preferred to all others. I don't have an answer to this question yet.

***

Daniel's fix reminds me of a different, but very common, chart.

This chart shows actual vs forecasted inflation rates. This chart has two lines but only needs one axis since both lines represent inflation rates in the same range.

We can think of the "estimated revenues" line above as forecasted or expected revenues, based on the actual number of new customers. In particular, this forecast is based on a specific model: one that assumes that revenues is linearly related to the number of new customers. The "residuals" are forecasting errors.

In this sense, I think Daniel's solution amounts to rephrasing the question of the chart from "how closely are revenues and new customers correlated?" to "given the trend in new customers, are we over- or under-performing on revenues?"

Instead of using the dual-axes chart with two different scales, I'd prefer to answer the question by showing this expected vs actual revenues chart with one scale.

This does not eliminate the question about the "principle" behind the estimated revenues, but it makes clear that the challenge is to justify why revenues is a linear function of new customers, and no other variables.

Unlike the dual-axes chart, the actual vs forecasted chart is independent of the forecasting method. One can produce forecasted revenues based on a complicated function of new customers, existing customers, and any other factors. A different model just changes the shape of the forecasted revenues line. We still have two comparable lines on one scale.

 

 

 

 

 

All about Connecticut

This dataviz project by CT Mirror is excellent. The project walks through key statistics of the state of Connecticut.

Here are a few charts I enjoyed.

The first one shows the industries employing the most CT residents. The left and right arrows are perfect, much better than the usual dot plots.

The industries are sorted by decreasing size from top to bottom, based on employment in 2019. The chosen scale is absolute, showing the number of employees. The relative change is shown next to the arrow heads in percentages.

The inclusion of both absolute and relative scales may be a source of confusion as the lengths of the arrows encode the absolute differences, not the relative differences indicated by the data labels. This type of decision is always difficult for the designer. Selecting one of the two scales may improve clarity but induce loss aversion.

***

The next example is a bumps chart showing the growth in residents with at least a bachelor's degree.

This is more like a slopegraph as it appears to draw straight lines between two time points 9 years apart, omitting the intervening years. Each line represents a state. Connecticut's line is shown in red. The message is clear. Connecticut is among the most highly educated out of the 50 states. It maintained this advantage throughout the period.

I'd prefer to use solid lines for the background states, and the axis labels can be sparser.

It's a little odd that pretty much every line has the same slope. I'm suspecting that the numbers came out of a regression model, with varying slopes by state, but the inter-state variance is low.

In the online presentation, one can click on each line to see the values.

***

The final example is a two-sided bar chart:

This shows migration in and out of the state. The red bars represent the number of people who moved out, while the green bars represent those who moved into the state. The states are arranged from the most number of in-migrants to the least.

I have clipped the bottom of the chart as it extends to 50 states, and the bottom half is barely visible since the absolute numbers are so small.

I'd suggest showing the top 10 states. Then group the rest of the states by region, and plot them as regions. This change makes the chart more compact, as well as more useful.

***

There are many other charts, and I encourage you to visit and support this data journalism.

 

 

 

Area chart is not the solution

A reader left a link to a Wiki chart, which is ghastly:

This chart concerns the trend of relative proportions of House representatives in the U.S. Congress by state, and can be found at this Wikipedia entry. The U.S. House is composed of Representatives, and the number of representatives is roughly proportional to each state's population. This scheme actually gives small states disporportional representation, since the lowest number of representatives is 1 while the total number of representatives is fixed at 435.

We can do a quick calculation: 1/435 = 0.23% so any state that has less than 0.23% of the population is over-represented in the House. Alaska, Vermont and Wyoming are all close to that level. The primary way in which small states get larger representation is via the Senate, which sits two senators per state no matter the size. (If you've wondered about Nate Silver's website: 435 Representatives + 100 Senators + 3 for DC = 538 electoral votes for U.S. Presidental elections.)

***

So many things have gone wrong with this chart. There are 50 colors for 50 states. The legend arranges the states by the appropriate metric (good) but in ascending order (bad). This is a stacked area chart, which makes it very hard to figure out the values other than the few at the bottom of the chart.

A nice way to plot this data is a tile map with line charts. I found a nice example that my friend Xan put together in 2018:

A tile map is a conceptual representation of the U.S. map in which each state is represented by equal-sized squares. The coordinates of the states are distorted in order to line up the tiles. A tile map is a small-multiples setup in which each square contains a chart of the same design to faciliate inter-state comparisons.

In the above map, Xan also takes advantage of the foregrounding concept. Each chart actually contains all 50 lines for every state, all shown in gray while the line for the specific state is bolded and shown in red.

***

A chart with 50 lines looks very different from one with 50 areas stacked on each other. California, the most populous state, has 12% of the total population so the line chart has 50 lines that will look like spaghetti. Thus, the fore/backgrounding is important to make sure it's readable.

I suspect that the designer chose a stacked area chart because the line chart looked like spaghetti. But that's the wrong solution. While the lines no longer overlap each other, it is a real challenge to figure out the state-level trends - one has to focus on the heights of the areas, rather than the boundary lines.

[P.S. 2/27/2023] As we like to say, a picture is worth a thousand words. Twitter reader with the handle LHZGJG made the tile map I described above. It looks like this:

You can pick out the states with the key changes really fast. California, Texas, Florida on the upswing, and New York, Pennsylvania going down. I like the fact that the state names are spelled out. Little tweaks are possible but this is a great starting point. Thanks LHZGJG! ]

 

Getting simple charts right

Ian K. submitted this chart on Twitter:

The chart comes from a video embedded in this report (link) about Chicago cops leaving their jobs.

Let's start with the basics. This is an example of a simple line chart illustrating a time series of five observations. The vertical axis starts at 10,000 instead of 0. With this choice, the designer wants to focus on the point-to-point change in values, rather than its relation to the initial value.

Every graph has add-ons that assist cognition. On this chart, we have axis labels, gridlines and data labels. Every add-on increases reading time so we should be sparing.

First consider the gridlines. In the following chart, I conduct a self-sufficiency test by removing the data labels from the chart:

You can see that the last three values present no problems. The first two, especially the first value, are hard to read - because the top gridline is missing! The next chart restores the bounding gridline, so you can see the difference that one small detail can make:

***

Next, let's compare the following versions of the chart. The left one contains data labels without gridlines and axis labels. The right one has the gridlines and axis labels but no data labels.

The left chart prints the entire dataset onto the chart. The reader in essence is reading the raw data. That appears to be the intention of the chart designer as the data labels are in large size, placed inside shiny white boxes. The level of the boxes determines the reader's perception as those catch more of our attention than the dots that actually represent the data.

The right chart highlights the dots and the lines between them. The gridlines are way too thick and heavy so as to distract rather than abet. This chart presumes that the reader isn't that interested in the precise numbers as she is in the trend.

***

As Ian pointed out, one of the biggest problems with this chart is the appearance of even time intervals when all except one of the date values are January. This seemingly innocent detail destroys the chart. The line segments of the chart encodes the pre-post change in the staffing numbers. For most of the line segments, the metric is year-on-year change but the last two line segments on the right show something else: a 19-month change, followed by a 5-month change.

I did the following analysis to understand how big of a staffing problem CPD faces.

First I restored the January 2022 time value, while shifting the Aug 2022 value to its rightful place on the time axis. Next, I added the dashed brown line, which represents a linear extension of the trend seen between January 2020-2021, before the sudden dip. We don't know what the true January 2022 value is but the projected value based on past trend is around 12,200. By August, the projected value is around 11,923, about 300 above the actual value of 11,611. By January 2023, the projected value is almost exactly the same as the actual value.

This linear trending analysis is likely too simplistic but it offers a baseline to start thinking about what the story is. The long-term trend is still down but the apparent dip in 2022 may not be meaningful.

 

 

Dual axes: a favorite of tricksters

Twitter readers directed me to this abomination from the St. Louis Fed (link).

This chart is designed to paint the picture that China is this grave threat because it's been ramping up military expenditure so much so that it exceeded U.S. spending since the 2000s.

Sadly, this is not what the data are suggesting at all! This story is constructed by manipulating the dual axes. Someone has already fixed it. Here's the same data plotted with a single axis:

(There are two set of axis labels but they have the same scale and both start at zero, so there is only one axis.)

Certainly, China has been ramping up military spending. Nevertheless, China's current level of spending is about one-third of America's. Also, imagine the cumulative spending excess over the 30 years shown on the chart.

Note also, the growth line of U.S. military spending in this period is actually similarly steep as China's.

***

Apparently, the St. Louis Fed is intent on misleading its readers. Even though on Twitter, they acknowledged people's feedback, they decided not to alter the chart.

If you click through to the article, you'll find the same flawed chart as before so I'm not sure how they "listened". I went to Wayback Machine to check the first version of this page, and I notice no difference.

***

If one must make a dual axes chart, it is the responsibility of the chart designer to make it clear to readers that different lines on the chart use different axes. In this case, since the only line that uses the right hand side axis is the U.S. line, which is blue, they should have colored the right hand axis blue. Doing that does not solve the visualization problem; it merely reduces the chance of not noticing the dual axes.

***

I have written about dual axes a lot in the past. Here's a McKinsey chart from 2006 that offends.

People flooded this chart presented without comment with lots of comments

The recent election in Italy has resulted in some dubious visual analytics. A reader sent me this Excel chart:

In brief, an Italian politician (trained as a PhD economist) used the graph above to make a point that support of the populist Five Star party (M5S) is highly correlated with poverty - the number of people on RDC (basic income). "Senza commento" - no comment needed.

Except a lot of people noticed the idiocy of the chart, and ridiculed it.

The chart appeals to those readers who don't spend time understanding what's being plotted. They notice two lines that show similar "trends" which is a signal for high correlation.

It turns out the signal in the chart isn't found in the peaks and valleys of the "trends".  It is tempting to observe that when the blue line peaks (Campania, Sicilia, Lazio, Piedmonte, Lombardia), the orange line also pops.

But look at the vertical axis. He's plotting the number of people, rather than the proportion of people. Population varies widely between Italian provinces. The five mentioned above all have over 4 million residents, while the smaller ones such as Umbira, Molise, and Basilicata have under 1 million. Thus, so long as the number of people, not the proportion, is plotted, no matter what demographic metric is highlighted, we will see peaks in the most populous provinces.

***

The other issue with this line chart is that the "peaks" are completely contrived. That's because the items on the horizontal axis do not admit a natural order. This is NOT a time-series chart, for which there is a canonical order. The horizontal axis contains a set of provinces, which can be ordered in whatever way the designer wants.

The following shows how the appearance of the lines changes as I select different metrics by which to sort the provinces:

This is the reason why many chart purists frown on people who use connected lines with categorical data. I don't like this hard rule, as my readers know. In this case, I have to agree the line chart is not appropriate.

***

So, where is the signal on the line chart? It's in the ratio of the heights of the two values for each province.

Here, we find something counter-intuitive. I've highlighted two of the peaks. In Sicilia, about the same number of people voted for Five Star as there are people who receive basic income. In Lombardia, more than twice the number of people voted for Five Star as there are people who receive basic income. 

Now, Lombardy is where Milan is, essentially the richest province in Italy while Sicily is one of the poorest. Could it be that Five Star actually outperformed their demographics in the richer provinces?

***

Let's approach the politician's question systematically. He's trying to say that the Five Star moement appeals especially to poorer people. He's chosen basic income as a proxy for poverty (this is like people on welfare in the U.S.). Thus, he's divided the population into two groups: those on welfare, and those not.

What he needs is the relative proportions of votes for Five Star among these two subgroups. Say, Five Star garnered 30% of the votes among people on welfare, and 15% of the votes among people not on welfare, then we have a piece of evidence that Five Star differentially appeals to people on welfare. If the vote share is the same among these two subgroups, then Five Star's appeal does not vary with welfare.

The following diagram shows the analytical framework:

What's the problem? He doesn't have the data needed to establish his thesis. He has the total number of Five Star voters (which is the sum of the two yellow boxes) and he has the total number of people on RDC (which is the dark orange box).

As shown above, another intervening factor is the proportion of people who voted. It is conceivable that the propensity to vote also depends on one's wealth.

So, in this case, fixing the visual will not fix the problem. Finding better data is key.

What do I think about spirals?

A twitter user asked how I feel about this latest effort (from NASA) to illustrate global warming. To see the entire video, go to their website.

This video hides the lede so be patient or jump ahead to 0:56 and watch till the end.

Let's first describe what we are seeing.

The dataset consists of monthly average global temperature "anomalies" from 1880 to 2021 - an "anomaly" is the deviation of the average temperature that month from a reference level (seems like this is fixed at the average temperatures by month between 1951 and 1980).

A simple visualization of the dataset is this:

We see a gradual rise in temperature from the 1980s to today. The front half of this curve is harder to interpret. The negative values suggest that the average temperatures prior to 1951 are generally lower than the temperature in the reference period. Other than 1880-1910, temperatures have generally been rising.

Now imagine chopping up the above chart into yearly increments, 12 months per year. Then wrap each year's line into a circle, and place all these lines onto the following polar grid system.

Close but not quite there. The circles in the NASA video look much smoother. Two possibilities here. First is the aspect ratio. Note that the polar grid stretches the time axis to the full circle while the vertical axis is squashed. Not enough to explain the smoothness, as seen below.

The second possibility is additional smoothing between months.

The end result is certainly pretty:

***

Is it a good piece of scientific communications?

What is the chart saying?

I see red rings on the outside, white rings in the middle, and blue rings near the center. Red presumably means hotter, blue cooler.

The gridlines are painted over. The 0 degree (green) line is printed over again and again.

The biggest red circles are just beyond the 1 degree line with the excess happening in the January-March months. In making that statement, I'm inferring meaning to excess above 1 degree. This inference is purely based on where the 1-degree line is placed.

I also see in the months of December and January, there may have been "cooling", as the blue circles edge toward the -1 degree gridline. Drawing this inference actually refutes my previous claim. I had said that the bulge beyond the +1 degree line is informative because the designer placed the +1 degree line there. If I applied the same logic, then the location of the -1 degree line implies that only values more negative than -1 matter, which excludes the blue bulge!

Now what years are represented by these circles? Test your intuition. Are you tempted to think that the red lines are the most recent years, and the blue lines are the oldest years? If you think so, like I do, then we fall into a trap. We have now imputed two meanings to color -- temperature and recency, when the color coding can only hold one.

The only way to find out for sure is to rewind the tape and watch from the start. The year dimension is pushed to the background in this spiral chart. Instead, the month dimension takes precedence. Recall that at the start, the circles are white. The bluer circles appear in the middle of the date range.

This dimensional flip flop is a key difference between the spiral chart and the line chart (shown again for comparison).

In the line chart, the year dimension is primary while the month dimension is pushed to the background.

Now, we have to decide what the message of the chart should be. For me, the key message is that on a time scale of decades, the world has experienced a significant warming to the tune of about 1.5 degrees Celsius (35 F2.7 F). The warming has been more pronounced in the last 40 years. The warming is observed in all twelve months of the year.

Because the spiral chart hides the year dimension, it does not convey the above messages.

The spiral chart shares the same weakness as the energy demand chart discussed recently (link). Our eyes tend to focus on the outer and inner envelopes of these circles, which by definition are extreme values. Those values do not necessarily represent the bulk of the data. The spiral chart in fact tells us that there is not much to learn from grouping the data by month. 

The appeal of a spiral chart for periodic data is similar to a map for spatial data. I don't recommend using maps unless the spatial dimension is where the signal lies. Similarly, the spiral chart is appropriate if there are important deviations from a seasonal pattern.

 

 

Dots, lines, and 2D histograms

Daniel Z. tweeted about my post from last week. In particular, he took a deeper look at the chart of energy demand that put all hourly data onto the same plot, originally published at the StackOverflow blog:

I noted that this is not a great chart particularly since what catches our eyes are not the key features of the underlying data. Daniel made a clearly better chart:

This is a dot plot, rather than a line chart. The dots are painted in light gray, pushed to the background, because readers should be looking at the orange line. (I'm not sure what is going on with the horizontal scale as I could not get the peaks to line up on the two charts.)

What is this orange line? It's supposed to prove the point that the apparent dark band seen in the line chart does not represent the most frequently occurring values, as one might presume.

Looking closer, we see that the gray dots do not show all the hourly data but binned values.

We see vertical columns of dots, each representing a bin of values. The size of the dots represents the frequency of values of each bin. The orange line connects the bins with the highest number of values.

Daniel commented that

"The visual aggregation doesn't in fact map to the most frequently occurring values. That is because the ink of almost vertical lines fills in all the space between start and end."

Xan Gregg investigated further, and made a gif to show this effect better. Here is a screenshot of it (see this tweet):

The top chart is a true dot plot so that the darker areas are denser as the dots overlap. The bottom chart is the line chart that has the see-saw pattern. As Xan noted, the values shown are strangely very well behaved (aggregated? modeled?) - with each day, it appears that the values sweep up and down consistently.  This means the values are somewhat evenly spaced on the underlying trendline, so I think this dataset is not the best one to illustrate Daniel's excellent point.

It's usually not a good idea to connect lots of dots with a single line.

 

[P.S. 3/21/2022: Daniel clarified what the orange line shows: "In the posted chart, the orange line encodes the daily demand average (the mean of the daily distribution), rounded, for displaying purposes, to the closed bin. Bin size = 1000. Orange could have encode the daily median as well."]