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.

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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.

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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.

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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.

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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.

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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.

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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.

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

 

 

 

Lay off bubbles

Wall Street Journal says that the scale of layoffs in the tech industry recently is worse than those caused by the pandemic lockdown. Here is the chart:

It's the dreaded bubble chart, complete with overlapping circles. Each bubble represents the total number of employees laid off in the U.S. in a given month.

The above isn't really the chart you find in the Journal. I have removed the two data labels from the chart. Look at the highlighted months of April 2020 and November 2022. Can you guess how much larger is the number of laid-off employees in November 2022 relative to April 2020?

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If you guessed it's 100% - that the larger bubble is twice the size of the smaller one, then you're much better than I at reading bubble charts. Here is the published chart with the data labels:

I like to run this exercise - removing data labels - in order to reveal whether the graphical elements on the page are sufficient to convey the underlying data. Bubbles are typically not great at this. (This is what I call the self-sufficiency test.)

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Another problem with bubble charts is that the sizes of the bubbles are arbitrary. This allows the designer to convey different messages with the same data.

Take a look at these two bubble charts:

The first one has huge bubbles, and lots of overlapping while the second one is roughly the same as the WSJ chart (I pulled a different dataset so the numbers may not be exactly the same).

Both charts are made from exactly the same data! In the second chart, the smallest bubbles are made very small while in the first chart, the smallest bubbles are still quite large.

Think twice before you make a bubble chart.