Neither the forest nor the trees

On the NYT's twitter feed, they featured an article titled "These Seven Tech Stocks are Driving the Market". The first sentence of the article reads: "The S&P 500 is at an all-time high, and investors have just a handful of stocks to thank for it."

Without having seen any data, I'd surmise from that line that (a) the S&P 500 index has gone up recently, and (b) most if not all of the gain in the index can be attributed to gains in the tech stocks mentioned in the headline. (For purists, a handful is five, not seven.)

The chart accompanying the tweet is a treemap:

Nyt_magnificentseven

The treemap is possibly the most overhyped chart type of the modern era. Its use here is tangential to the story of surging market value. That's because the treemap presents a snapshot of the composition of the index, but contains nothing about the trend (change over time) of the average index value or of its components.

***

Even in representing composition, the treemap is inferior to, gasp, a pie chart. Of course, we can only use a pie chart for small numbers of components. The following illustration takes the data from the NYT chart on the Magnificent Seven tech stocks, and compares a treemap versus a pie chart side by side:

Junkcharts_redo_nyt_magnificent7

The reason why the treemap is worse is that both the width and the height of the boxes are changing while only the radius (or angle) of the pie slices is varying. (Not saying use a pie chart, just saying the treemap is worse.)

There is a reason why the designer appended data labels to each of the seven boxes. The effect of not having those labels is readily felt when our eyes reach the next set of stocks – which carry company names but not their market values. What is the market value of Berkshire Hathaway?

Even more so, what proportion of the total is the market value of Berkshire Hathaway? Indeed, if the designer did not write down 29%, it would take a bit of work to figure out the aggregate value of yellow boxes relative to the entire box!

This design sucessfully draws our attention to the structural importance of various components of the whole. There are three layers - the yellow boxes (Magnificent Seven), the gray boxes with company names, and the other gray boxes. I also like how they positioned the text on the right column.

***

Going inside the NYT article itself, we find two line charts that convey the story as told.

Here's the first one:

Nyt_magnificent7_linechart1

They are comparing the most recent stock prices with those from October 12 2022, which is identified as the previous "low". (I'm actually confused by how the most recent "low" is defined, but that's a different subject.)

This chart carries a lot of good information, even though it does not plot "all the data", as in each of the 500 S&P components individually. Over the period under analysis, the average index value has gone up about 35% while the Magnificent Seven's value have skyrocketed by 65% in aggregate. The latter accounted for 30% of the total value at the most recent time point.

If we set the S&P 500 index value in 2024 as 100, then the M7 value in 2024 is 30. After unwinding the 65% growth, the M7 value in October 2022 was 18; the S&P 500 in October 2022 was 74. Thus, the weight of M7 was 24% (18/74) in October 2022, compared to 30% now. Consequently, the weight of the other 473 stocks declined from 76% to 70%.

This isn't even the full story because most of the action within the M7 is in Nvidia, the stock most tightly associated with the current AI hype, as shown in the other line chart.

Nyt_magnificent7_linechart2

Nvidia's value jumped by 430% in that time window. From the treemap, the total current value of M7 is $12.3 b while Nvidia's value is $1.4 b, thus Nvidia is 11.4% of M7 currently. Since M7 is 29% of the total S&P 500, Nvidia is 11.4%*29% = 3% of the S&P. Thus, in 2024, against 100 for the S&P, Nvidia's share is 3. After unwinding the 430% growth, Nvidia's share in October 2022 was 0.6, about 0.8% of 74. Its weight tripled during this period of time.


The cult of raw unadjusted data

Long-time reader Aleks came across the following chart on Facebook:

Unadjusted temp data fgfU4-ia fb post from aleks

The author attached a message: "Let's look at raw, unadjusted temperature data from remote US thermometers. What story do they tell?"

I suppose this post came from a climate change skeptic, and the story we're expected to take away from the chart is that there is nothing to see here.

***

What are we looking at, really?

"Nothing to see" probably refers to the patch of blue squares that cover the entire plot area, as time runs left to right from the 1910s to the present.

But we can't really see what's going on in the middle of the patch. So, "nothing to see" is effectively only about the top-to-bottom range of roughly 29.8 to 82.0. What does that range signify?

The blue patch is subdivided into vertical lines consisting of blue squares. Each line is a year's worth of temperature measurements. Each square is the average temperature on a specific day. The vertical range is the difference between the maximum and minimum daily temperatures in a given year. These are extreme values that say almost nothing about the temperatures in the other ~363 days of the year.

We know quite a bit more about the density of squares along each vertical line. They are broken up roughly by seasons. Those values near the top came from summers while the values near the bottom came from winters. The density is the highest near the middle, where the overplotting is so severe that we can barely see anything.

Within each vertical line, the data are not ordered chronologically. This is a very key observation. From left to right, the data are ordered from earliest to latest but not from top to bottom! Therefore, it is impossible for the human eye to trace the entire trajectory of the daily temperature readings from this chart. At best, you can trace the yearly average temperature – but only extremely roughly by eyeballing where the annual averages are inside the blue patch.

Indeed, there is "nothing to see" on this chart because its design has pulverized the data.

***

_numbersense_bookcoverIn Numbersense (link), I wrote "not adjusting the raw data is to knowingly publish bad information. It is analogous to a restaurant's chef knowingly sending out spoilt fish."

It's a fallacy to think that "raw unadjusted" data are the best kind of data. It's actually the opposite. Adjustments are designed to correct biases or other problems in the data. Of course, adjustments can be subverted to introduce biases in the data as well. It is subversive to presume that all adjustments are of the subversive kind.

What kinds of adjustments are of interest in this temperature dataset?

Foremost is the seasonal adjustment. See my old post here. If we want to learn whether temperatures have risen over these decades, we can't do so without separating out the seasons.

The whole dataset can be simplified by drawing the smoothed annual average temperature grouped by season of the year, and when that is done, the trend of rising temperatures is obvious.

***

The following chart by the EPA roughly implements the above:

Epa-seasonal-temperature_2022

The original can be found here. They made one adjustment which isn't the one I expected.

Note the vertical scale is titled "temperature anomaly". So, they are not plotting the actual recorded average temperatures, but the "anomalies", i.e. the difference between the recorded temperatures and some kind of "expected" temperature. This is a type of data adjustment as well. The purpose is to focus attention on the relative rather than absolute values. Think of this formula: recorded value = expected value + anomaly. The chart shows how many degrees above or below expectation, rather than how many degrees.

For a chart like this, there should be a required footnote that defines what "anomaly" is. Specifically, the reader should know about the model behind the "expectation". Typically, it's a kind of long-term average value.

For me, this adjustment is not necessary. Without the adjustment, the four panels can be combined into one panel with four lines. That's because the data nicely fit into four levels based on seasons.

The further adjustment I'd have liked to see is "smoothing". Each line above has a "smooth" trend, as well as some variability around this trend. The latter is not a big part of the story.

***

It's weird to push back on climate change advocacy by attacking data adjustments. The more productive direction, in my view, is to ask whether the observed trend is caused by human activities or part of some long-term up-and-down cycle. That is a very challenging question to answer.


The choice to encode data using colors

NBC News published the following heatmap that shows inflation by product category in the last year or so:

Nbcnews_inflationtracker

The general story might be that inflation was rampant in airfare and electricity prices about a year ago but these prices have moderated recently, especially in airfare. Gas prices appear to have inflated far less than overall inflation during these months.

***

Now, if you're someone who cares about the magnitude of differences, not just the direction, then revisit the above statements, and you'll feel a sense of inadequacy.

When we choose to encode data in colors, we're giving up on showing magnitudes or precision. The color scale shown up top sends the message that the continuous nature of the number line is being displayed but it really isn't.

The largest value of the chart is found on the left side of the airfare row:

Nbcnews_inflationtracker_highest

The value is about 36% which strangely enough is far larger than the maximum value shown in the legend above. Even if those values align, it is still impossible to guess what values the different colors and shades in the cells map to from the legend.

***

The following small-multiples chart shows the underlying values more precisely:

Redo_junkcharts_nbcnewsinflation

I have transformed the data differently. In these line charts, the data are indexed to the first month (100) so each chart shows the cumulative change in prices from that month to the current month, for each category, compared to the overall.

The two most interesting categories are airfare and gas. Airfare has recently decreased quite drastically relative to September 2022, and thus the line is far below the overall inflation trend. Gas prices moved in reverse: they dropped in the last quarter of 2022 but have steadily risen over 2023, and in the most recent month, is tracking overall inflation.

 

 


Chartjunk as marketing copy

I got some spam marketing message last week. How exciting. They even use a subject line that has absolutely nothing to do with its content, baiting me to open it. And open I did, to some data graphics horrors.

The marketer promises a whole series of charts to prove that art is a great asset class for investment returns.

The very first chart already caught my full attention. It's this one:

Masterworks_chart1

It's a simple bar chart, with four values. Looks innocuous.

I'm unable to appreciate the recent trend to align bars in the middle, rather than at their bases. So I converted it to the canonical form:

Redo_masterworks_1_barchart

Do you see the problem?

The second value ($1.7 trillion) is exactly half the size of the first value ($3.4 trillion) and yet the second bar is two-thirds of the length of the first bar. So, the size of the second bar is exaggerated relative to its label – and that’s the bar displaying the market size for “art,” which is what the spammer is pitching.

The bottom pair of values share the same relationship: $0.8 trillion is exactly half of $1.6 trillion. Again, the relative lengths of those two bars are not 50% but slightly over 60%.

Redo_masterworks_1_barchart_excess

Did the designer think that the bar lengths could be customized to whatever s/he desires? This one is hard to crack.

***

The sixth chart in the series is a different kind of puzzle:

Masterworks_chart6

All three lines have the exact same labels but show different values over time.

***

And they have pie charts, of course. Take a look:

Masterworks_chart

Something went wrong here too. I'll leave it to my readers who can certainly figure it out :)

***

These charts were probably spammed to at least thousands.

 


Two metrics in-fighting

The Wall Street Journal shows the following chart which pits two metrics against each other:

Wsj_salaries25to29

The primary metric is the change in median yearly salary between the two periods of time. We presume it's primary because of its presence in the chart title, and the blue bars being more readable than the green bubbles. The secondary metric is the median yearly salary in the later period.

That, I believe, was the intended design. When I saw this chart, my eyes went to the numbers inside the green bubbles. Perhaps it's because I didn't read the chart title first, and the horizontal axis wasn't labelled so it wasn't obvious what the blue bars coded.

As with most bubble charts, the data labels exist to cover up the inadequacy of circular areas. The self-sufficiency test - removing the data labels - shows this well:

Redo_wsj_salaries25to29

It's simply impossible to know what values should be in each bubble, or to perceive the relative sizes of those bubbles.

***

Reversing the order of the blue bars also helps:

Redo_wsjsalaries25to29_2

The original order is one of the more annoying features in most visualization packages. Because internally, the categories are numbered 1, 2, 3, ..., and because the convention is to have values run higher as they run up the vertical axis, these packages would place the top-ranked item at the bottom of the chart.

Most people read top to bottom, which means that they read the least important item first, and the most important item last!

In most visualization packages, it takes only 1 click or 1 action to reverse the order of the items. Please do it!

***

For change over time, I like using a Bumps chart, otherwise called a slope graph:

Redo_wsjsalaries25to29_3


When words speak louder than pictures

I've been staring at this chart from the Wall Street Journal (link) about U.S. workers working remotely:

Wsj_remotework_byyear

It's one of those offerings I think on which the designer spent a lot of effort, but ultimately didn't realize that the reader would spend equal if not more effort deciphering.

However, the following paragraph lifted straight from the article says exactly what needs to be said:

Workers overall spent an average of 5 hours and 25 minutes a day working from home in 2022. That is about two hours more than in 2019, the year before Covid-19 sent millions of workers scrambling to set up home oces, and down just 12 minutes from 2021, according to the Labor Department’s American Time Use Survey.

***

Why is the chart so hard to read?

_trifectacheckup_imageIt's mostly because the visual is fighting the message. In the Trifecta Checkup (link), this is represented by a disconnect between the Q(uestion) and the V(isual) corners - note the green arrow between these two corners.

The message concentrates on two comparisons: first, the increase in amount of remote work after the pandemic; and second, the mild decrease in 2022 relative to 2021.

On the chart, the elements that grab my attention are (a) the green and orange columns (b) the shading in the bottom part of those green and orange columns (c) the thick black line that runs across the chart (d) the indication on the left side that tells me one unit is an hour.

None of those visual elements directly addresses the comparisons. The first comparison - before and after the pandemic - is found by how much the green column spikes above the thick black line. Our comprehension is retarded by the decision to forego the typical axis labels in favor of chopping columns into one-hour blocks.

The second comparison - between 2022 and 2021 - is found in the white space above the top of the orange column.

So, in reality, the text labels that say exactly what needs to be said are carrying a lot of weight. A slight edit to the pointers helps connect those descriptions to the visual depiction, like this:

Redo_junkcharts_wsj_remotework

I've essentially flipped the tactics used in the various pointers. For the average level of remote work pre-pandemic, I dispense of any pointers while I'm using double-headed arrows to indicate differences across time.

Nevertheless, this modified chart is still too complex.

***

Here is a version that aligns the visual to the message:

Redo_junkcharts_wsj_remotework_2

It's a bit awkward because the 2 hour 48 minutes calculation is the 2021 number minus the average of 2015-19, skipping the 2020 year.

 


Why some dataviz fail

Maxim Lisnic's recent post should delight my readers (link). Thanks Alek for the tip. Maxim argues that charts "deceive" not merely by using visual tricks but by a variety of other non-visual means.

This is also the reasoning behind my Trifecta Checkup framework which looks at a data visualization project holistically. There are lots of charts that are well designed and constructed but fail for other reasons. So I am in agreement with Maxim.

He analyzed "10,000 Twitter posts with data visualizations about COVID-19", and found that 84% are "misleading" while only 11% of the 84% "violate common design guidelines". I presume he created some kind of computer program to evaluate these 10,000 charts, and he compiled some fixed set of guidelines that are regarded as "common" practice.

***

Let's review Maxim's examples in the context of the Trifecta Checkup.

_trifectacheckup_image

The first chart shows Covid cases in the U.S. in July and August of 2021 (presumably the time when the chart was published) compared to a year ago (prior to the vaccination campaign).

Maxim_section1

Maxim calls this cherry-picking. He's right - and this is a pet peeve of mine, even with all the peer-reviewed scientific research. In my paper on problems with observational studies (link), my coauthors and I call for a new way forward: researchers should put their model calculations up on a website which is updated as new data arrive, so that we can be sure that the conclusions they published apply generally to all periods of time, not just the time window chosen for the publication.

Looking at the pair of line charts, readers can quickly discover its purpose, so it does well on the Q(uestion) corner of the Trifecta. The cherry-picking relates to the link between the Question and the Data, showing that this chart suffers from subpar analysis.

In addition, I find that the chart also misleads visually - the two vertical scales are completely different: the scale on the left chart spans about 60,000 cases while on the right, it's double the amount.

Thus, I'd call this a Type DV chart, offering opportunities to improve in two of the three corners.

***

The second chart cited by Maxim plots a time series of all-cause mortality rates (per 100,000 people) from 1999 to 2020 as columns.

The designer does a good job drawing our attention to one part of the data - that the average increase in all-cause mortality rate in 2020 over the previous five years was 15%. I also like the use of a different color for the pandemic year.

Then, the designer lost the plot. Instead of drawing a conclusion based on the highlighted part of the data, s/he pushed a story that the 2020 rate was about the same as the 2003 rate. If that was the main message, then instead of computing a 15% increase relative to the past five years, s/he should have shown how the 2003 and 2020 levels are the same!

On a closer look, there is a dashed teal line on the chart but the red line and text completely dominate our attention.

This chart is also Type DV. The intention of the designer is clear: the question is to put the jump in all-cause mortality rate in a historical context. The problem lies again with subpar analysis. In fact, if we take the two insights from the data, they both show how serious a problem Covid was at the time.

When the rate returned to the level of 2003, we have effectively gave up all the gains made over 17 years in a few months.

Besides, a jump in 15% from year to year is highly significant if we look at all other year-to-year changes shown on the chart.

***

The next section concerns a common misuse of charts to suggest causality when the data could only indicate correlation (and where the causal interpretation appears to be dubious). I may write a separate post about this vast topic in the future. Today, I just want to point out that this problem is acute with any Covid-19 research, including official ones.

***

I find the fourth section of Maxim's post to be less convincing. In the following example, the tweet includes two charts, one showing proportion of people vaccinated, and the other showing the case rate, in Iceland and Nigeria.

Maxim_section4

This data visualization is poor even on the V(isual) corner. The first chart includes lots of countries that are irrelevant to the comparison. It includes the unnecessary detail of fully versus partially vaccinated, unnecessary because the two countries selected are at two ends of the scale. The color coding is off sync between the two charts.

Maxim's critique is:

The user fails to account, however, for the fact that Iceland had a much higher testing rate—roughly 200 times as high at the time of posting—making it unreasonable to compare the two countries.

And the section is titled "Issues with Data Validity". It's really not that simple.

First, while the differential testing rate is one factor that should be considered, this factor alone does not account for the entire gap. Second, this issue alone does not disqualify the data. Third, if testing rate differences should be used to invalidate this set of data, then all of the analyses put out by official sources lauding the success of vaccination should also be thrown out since there are vast differences in testing rates across all countries (and also across different time periods for the same country).

One typical workaround for differential testing rate is to look at deaths rather than cases. For the period of time plotted on the case curve, Nigeria's cumulative death per million is about 1/8th that of Iceland. The real problem is again in the Data analysis, and it is about how to interpret this data casually.

This example is yet another Type DV chart. I'd classify it under problems with "Casual Inference". "Data Validity" is definitely a real concern; I just don't find this example convincing.

***

The next section, titled "Failure to account for statistical nuance," is a strange one. The example is a chart that the CDC puts out showing the emergence of cases in a specific county, with cases classified by vaccination status. The chart shows that the vast majority of cases were found in people who were fully vaccinated. The person who tweeted concluded that vaccinated people are the "superspreaders". Maxim's objection to this interpretation is that most cases are in the fully vaccinated because most people are fully vaccinated.

I don't think it's right to criticize the original tweeter in this case. If by superspreader, we mean people who are infected and out there spreading the virus to others through contacts, then what the data say is exactly that most such people are fully vaccinated. In fact, one should be very surprised if the opposite were true.

Indeed, this insight has major public health implications. If the vaccine is indeed 90% effective at stopping cases, we should not be seeing that level of cases. And if the vaccine is only moderately effective, then we may not be able to achieve "herd immunity" status, as was the plan originally.

I'd be reluctant to complain about this specific data visualization. It seems that the data allow different interpretations - some of which are contradictory but all of which are needed to draw a measured conclusion.

***
The last section on "misrepresentation of scientific results" could use a better example. I certainly agree with the message: that people have confirmation bias. I have been calling this "story-first thinking": people with a set story visualize only the data that support their preconception.

However, the example given is not that. The example shows a tweet that contains a chart from a scientific paper that apparently concludes that hydroxychloroquine helps treat Covid-19. Maxim adds this study was subsequently retracted. If the tweet was sent prior to the retraction, then I don't think we can grumble about someone citing a peer reviewed study published in Lancet.

***

Overall, I like Maxim's message. In some cases, I think there are better examples.

 

 


Some chart designs bring out more information than others

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

Nbwa_beer_index

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:

Junkcharts_redo_nbwabeerpurchasersindex

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.

Ctmirror_highschools

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.)

Redo_junkcharts_ctmirrorhighschoolsgraduation_1

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.

Redo_junkcharts_ctmirrorhighschoolsgraduation_2

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:

Danielzvinca_dualaxes_linecolumn

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.

Danielzvinca_dualaxes_estimatedy

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.

Forecastvsactualinflationchart

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.