Bubble charts, ratios and proportionality

A recent article in the Wall Street Journal about a challenger to the dominant weedkiller, Roundup, contains a nice selection of graphics. (Dicamba is the up-and-comer.)

The change in usage of three brands of weedkillers is rendered as a small-multiples of choropleth maps. This graphic displays geographical and time changes simultaneously.

The staircase chart shows weeds have become resistant to Roundup over time. This is considered a weakness in the Roundup business.

***

In this post, my focus is on the chart at the bottom, which shows complaints about Dicamba by state in 2019. This is a bubble chart, with the bubbles sorted along the horizontal axis by the acreage of farmland by state.

Below left is a more standard version of such a chart, in which the bubbles are allowed to overlap. (I only included the bubbles that were labeled in the original chart).

The WSJ’s twist is to use the vertical spacing to avoid overlapping bubbles. The vertical axis serves a design perogative and does not encode data.  

I’m going to stick with the more traditional overlapping bubbles here – I’m getting to a different matter.

***

The question being addressed by this chart is: which states have the most serious Dicamba problem, as revealed by the frequency of complaints? The designer recognizes that the amount of farmland matters. One should expect the more acres, the more complaints.

Let's consider computing directly the number of complaints per million acres.

The resulting chart (shown below right) – while retaining the design – gives a wholly different feeling. Arkansas now owns the largest bubble even though it has the least acreage among the included states. The huge Illinois bubble is still large but is no longer a loner.

Now return to the original design for a moment (the chart on the left). In theory, this should work in the following manner: if complaints grow purely as a function of acreage, then the bubbles should grow proportionally from left to right. The trouble is that proportional areas are not as easily detected as proportional lengths.

The pair of charts below depict made-up data in which all states have 30 complaints for each million acres of farmland. It’s not intuitive that the bubbles on the left chart are growing proportionally.

Now if you look at the right chart, which shows the relative metric of complaints per million acres, it’s impossible not to notice that all bubbles are the same size.

This Excel chart looks standard but gets everything wrong

The following CNBC chart (link) shows the trend of global car sales by region (or so we think).

This type of chart is quite common in finance/business circles, and has the fingerprint of Excel. After examining it, I nominate it for the Hall of Shame.

***

The chart has three major components vying for our attention: (1) the stacked columns, (2) the yellow line, and (3) the big red dashed arrow.

The easiest to interpret is the yellow line, which is labeled "Total" in the legend. It displays the annual growth rate of car sales around the globe. The data consist of annual percentage changes in car sales, so the slope of the yellow line represents a change of change, which is not particularly useful.

The big red arrow is making the point that the projected decline in global car sales in 2019 will return the world to the slowdown of 2008-9 after almost a decade of growth.

The stacked columns appear to provide a breakdown of the global growth rate by region. Looked at carefully, you'll soon learn that the visual form has hopelessly mangled the data.

What is the growth rate for Chinese car sales in 2006? Is it 2.5%, the top edge of China's part of the column? Between 1.5% and 2.5%, the extant of China's section? The answer is neither. Because of the stacking, China's growth rate is actually the height of the relevant section, that is to say, 1 percent. So the labels on the vertical axis are not directly useful to learning regional growth rates for most sections of the chart.

Can we read the vertical axis as global growth rate? That's not proper either. The different markets are not equal in size so growth rates cannot be aggregated by simple summing - they must be weighted by relative size.

The negative growth rates present another problem. Even if we agree to sum growth rates ignoring relative market sizes, we still can't get directly to the global growth rate. We would have to take the total of the positive rates and subtract the total of the negative rates.  

***

At this point, you may begin to question everything you thought you knew about this chart. Remember the yellow line, which we thought measures the global growth rate. Take a look at the 2006 column again.

The global growth rate is depicted as 2 percent. And yet every region experienced growth rates below 2 percent! No matter how you aggregate the regions, it's not possible for the world average to be larger than the value of each region.

For 2006, the regional growth rates are: China, 1%; Rest of the World, 1%; Western Europe, 0.1%; United States, -0.25%. A simple sum of those four rates yields 2%, which is shown on the yellow line.

But this number must be divided by four. If we give the four regions equal weight, each is worth a quarter of the total. So the overall average is the sum of each growth rate weighted by 1/4, which is 0.5%. [In reality, the weights of each region should be scaled to reflect its market size.]

***

tldr; The stacked column chart with a line overlay not only fails to communicate the contents of the car sales data but it also leads to misinterpretation.

I discussed several serious problems of this chart form: 

• stacking the columns make it hard to learn the regional data
  
  
• the trend by region takes a super effort to decipher
  
  
• column stacking promotes reading meaning into the height of the column but the total height is meaningless (because of the negative section) while the net height (positive minus negative) also misleads due to presumptive equal weighting
  
  
• the yellow line shows the sum of the regional data, which is four times the global growth rate that it purports to represent

 

***

PS. [12/4/2019: New post up with a different visualization.]

This chart tells you how rich is rich - if you can read it

Via twitter, John B. sent me the following YouGov chart (link) that he finds difficult to read:

The title is clear enough: the higher your income, the higher you set the bar.

When one then moves from the title to the chart, one gets misdirected. The horizontal axis shows pound values, so the axis naturally maps to "the higher your income". But it doesn't. Those pound values are the "cutoff" values - the line between "rich" and "not rich". Even after one realizes this detail, the axis  presents further challenges: the cutoff values are arbitrary numbers such as "45,001" sterling; and these continuous numbers are treated as discrete categories, with irregular intervals between each category.

There is some very interesting and hard to obtain data sitting behind this chart but the visual form suppresses them. The best way to understand this dataset is to first think about each income group. Say, people who make between 20 to 30 thousand sterling a year. Roughly 10% of these people think "rich" starts at 25,000. Forty percent of this income group think "rich" start at 40,000.

For each income group, we have data on Z percent think "rich" starts at X. I put all of these data points into a heatmap, like this:

Technical note: in order to restore the horizontal axis to a continuous scale, you can take the discrete data from the original chart, then fit a smoothed curve through those points, and finally compute the interpolated values for any income level using the smoothing model.

***

There are some concerns about the survey design. It's hard to get enough samples for higher-income people. This is probably why the highest income segment starts at 50,000. But notice that 50,ooo is around the level at which lower-income people consider "rich". So, this survey is primarily about how low-income people perceive "rich" people.

The curve for the highest income group is much straighter and smoother than the other lines - that's because it's really the average of a number of curves (for each 10,000 sterling segment).

 

P.S. The YouGov tweet that publicized the small-multiples chart shown above links to a page that no longer contains the chart. They may have replaced it due to feedback.

 

 

How to read this cost-benefit chart, and why it is so confusing

Long-time reader Antonio R. found today's chart hard to follow, and he isn't alone. It took two of us multiple emails and some Web searching before we think we "got it".

 

Antonio first encountered the chart in a book review (link) of Hal Harvey et. al, Designing Climate Solutions. It addresses the general topic of costs and benefits of various programs to abate CO2 emissions. The reviewer praised the "wealth of graphics [in the book] which present complex information in visually effective formats." He presented the above chart as evidence, and described its function as:

policy-makers can focus on the areas which make the most difference in emissions, while also being mindful of the cost issues that can be so important in getting political buy-in.

(This description is much more informative than the original chart title, which states "The policy cost curve shows the cost-effectiveness and emission reduction potential of different policies.")

Spend a little time with the chart now before you read the discussion below.

Warning: this is a long read but well worth it.

 

***

 

If your experience is anything like ours, scraps of information flew at you from different parts of the chart, and you had a hard time piecing together a story.

What are the reasons why this data graphic is so confusing?

Everyone recognizes that this is a column chart. For a column chart, we interpret the heights of the columns so we look first at the vertical axis. The axis title informs us that the height represents "cost effectiveness" measured in dollars per million metric tons of CO2. In a cost-benefit sense, that appears to mean the cost to society of obtaining the benefit of reducing CO2 by a given amount.

That's how far I went before hitting the first roadblock.

For environmental policies, opponents frequently object to the high price of implementation. For example, we can't have higher fuel efficiency in cars because it would raise the price of gasoline too much. Asking about cost-effectiveness makes sense: a cost-benefit trade-off analysis encapsulates the something-for-something principle. What doesn't follow is that the vertical scale sinks far into the negative. The chart depicts the majority of the emissions abatement programs as having negative cost effectiveness.

What does it mean to be negatively cost-effective? Does it mean society saves money (makes a profit) while also reducing CO2 emissions? Wouldn't those policies - more than half of the programs shown - be slam dunks? Who can object to programs that improve the environment at no cost?

I tabled that thought, and proceeded to the horizontal axis.

I noticed that this isn't a standard column chart, in which the width of the columns is fixed and uneventful. Here, the widths of the columns are varying.

***

In the meantime, my eyes are distracted by the constellation of text labels. The viewing area of this column chart is occupied - at least 50% - by text. These labels tell me that each column represents a program to reduce CO2 emissions.

The dominance of text labels is a feature of this design. For a conventional column chart, the labels are situated below each column. Since the width does not usually carry any data, we tend to keep the columns narrow - Tufte, ever the minimalist, has even advocated reducing columns to vertical lines. That leaves insufficient room for long labels. Have you noticed that government programs hold long titles? It's tough to capture even the outline of a program with fewer than three big words, e.g. "Renewable Portfolio Standard" (what?).

The design solution here is to let the column labels run horizontally. So the graphical element for each program is a vertical column coupled with a horizontal label that invades the territories of the next few programs. Like this:

The horror of this design constraint is fully realized in the following chart, a similar design produced for the state of Oregon (lifted from the Plan Washington webpage listed as a resource below):

In a re-design, horizontal labeling should be a priority.

 

***

Realizing that I've been distracted by the text labels, back to the horizontal axis I went.

This is where I encountered the next roadblock.

The axis title says "Average Annual Emissions Abatement" measured in millions metric tons. The unit matches the second part of the vertical scale, which is comforting. But how does one reconcile the widths of columns with a continuous scale? I was expecting each program to have a projected annual abatement benefit, and those would fall as dots on a line, like this:

Instead, we have line segments sitting on a line, like this:

Think of these bars as the bottom edges of the columns. These line segments can be better compared to each other if structured as a bar chart:

Instead, the design arranges these lines end-to-end.

To unravel this mystery, we go back to the objective of the chart, as announced by the book reviewer. Here it is again:

policy-makers can focus on the areas which make the most difference in emissions, while also being mindful of the cost issues that can be so important in getting political buy-in.

The primary goal of the chart is a decision-making tool for policy-makers who are evaluating programs. Each program has a cost and also a benefit. The cost is shown on the vertical axis and the benefit is shown on the horizontal. The decision-maker will select some subset of these programs based on the cost-benefit analysis. That subset of programs will have a projected total expected benefit (CO2 abatement) and a projected total cost.

By stacking the line segments end to end on top of the horizontal axis, the chart designer elevates the task of computing the total benefits of a subset of programs, relative to the task of learning the benefits of any individual program. Thus, the horizontal axis is better labeled "Cumulative annual emissions abatement".

 

Look at that axis again. Imagine you are required to learn the specific benefit of program titled "Fuel Economy Standards: Cars & SUVs".  

This is impossible to do without pulling out a ruler and a calculator. What the axis labels do tell us is that if all the programs to the left of Fuel Economy Standards: Cars & SUVs were adopted, the cumulative benefits would be 285 million metric tons of CO2 per year. And if Fuel Economy Standards: Cars & SUVs were also implemented, the cumulative benefits would rise to 375 million metric tons.

***

At long last, we have arrived at a reasonable interpretation of the cost-benefit chart.

Policy-makers are considering throwing their support behind specific programs aimed at abating CO2 emissions. Different organizations have come up with different ways to achieve this goal. This goal may even have specific benchmarks; the government may have committed to an international agreement, for example, to reduce emissions by some set amount by 2030. Each candidate abatement program is evaluated on both cost and benefit dimensions. Benefit is given by the amount of CO2 abated. Cost is measured as a "marginal cost," the amount of dollars required to achieve each million metric ton of abatement.

This "marginal abatement cost curve" aids the decision-making. It lines up the programs from the most cost-effective to the least cost-effective. The decision-maker is presumed to prefer a more cost-effective program than a less cost-effective program. The chart answers the following question: for any given subset of programs (so long as we select them left to right contiguously), we can read off the cumulative amount of CO2 abated.

***

There are still more limitations of the chart design.

• We can't directly read off the cumulative cost of the selected subset of programs because the vertical axis is not cumulative. The cumulative cost turns out to be the total area of all the columns that correspond to the selected programs. (Area is height x width, which is cost per benefit multiplied by benefit, which leaves us with the cost.) Unfortunately, it takes rulers and calculators to compute this total area.
  
  
• We have presumed that policy-makers will make the Go-No-go decision based on cost effectiveness alone. This point of view has already been contradicted. Remember the mystery around negatively cost-effective programs - their existence shows that some programs are stalled even when they reduce emissions in addition to making money!
  
  
• Since many, if not most, programs have negative cost-effectiveness (by the way they measured it), I'd flip the metric over and call it profitability (or return on investment). Doing so removes another barrier to our understanding. With the current cost-effectiveness metric, policy-makers are selecting the "negative" programs before the "positive" programs. It makes more sense to select the "positive" programs before the "negative" ones!

***

In a Trifecta Checkup (guide), I rate this chart Type V. The chart has a great purpose, and the design reveals a keen sense of the decision-making process. It's not a data dump for sure. In addition, an impressive amount of data gathering and analysis - and synthesis - went into preparing the two data series required to construct the chart. (Sure, for something so subjective and speculative, the analysis methodology will inevitably be challenged by wonks.) Those two data series are reasonable measures for the stated purpose of the chart.

The chart form, though, has various shortcomings, as shown here.  

***

In our email exchange, Antonio and I found the Plan Washington website useful. This is where we learned that this chart is called the marginal abatement cost curve.

Also, the consulting firm McKinsey is responsible for popularizing this chart form. They have published this long report that explains even more of the analysis behind constructing this chart, for those who want further details.

The rule governing which variable to put on which axis, served a la mode

When making a scatter plot, the two variables should not be placed arbitrarily. There is a rule governing this: the outcome variable should be shown on the vertical axis (also called y-axis), and the explanatory variable on the horizontal (or x-) axis.

This chart from the archives of the Economist has this reversed:

The title of the accompanying article is "Ice Cream and IQ"...

In a Trifecta Checkup (link), it's a Type DV chart. It's preposterous to claim eating ice cream makes one smarter without more careful studies. The chart also carries the xyopia fallacy: by showing just two variables, readers are unwittingly led to explain differences in "IQ" using differences in per-capita ice-cream consumption when lots of other stronger variables will explain any gaps in IQ.

In this post, I put aside my objections to the analysis, and focus on the issue of assigning variables to axes. Notice that this chart reverses the convention: the outcome variable (IQ) is shown on the horizontal, and the explanatory variable (ice cream) is shown on the vertical.

Here is a reconstruction of the above chart, showing only the dots that were labeled with country names. I fitted a straight regression line instead of a curve. (I don't understand why the red line in the original chart bends upwards when the data for Japan, South Korea, Singapore and Hong Kong should be dragging it down.)

Note that the interpretation of the regression line raises eyebrows because the presumed causality is reversed. For each 50 points increase in PISA score (IQ), this line says to expect ice cream consumption to raise by about 1-2 liters per person per year. So higher IQ makes people eat more ice cream.

***

If the convention is respected, then the following scatter plot results:

The first thing to note is that the regression analysis is different here from that shown in the previous chart. The blue regression line is not equivalent to the black regression line from the previous chart. You cannot reverse the roles of the x and y variables in a regression analysis, and so neither should you reverse the roles of the x and y variables in a scatter plot.

The blue regression line can be interpreted as having two sections, roughly, for countries consuming more than or less than 6 liters of ice cream per person per year. In the less-ice-cream countries, the correlation between ice cream and IQ is stronger (I don't endorse the causal interpretation of this statement).

***

When you make a scatter plot, you have two variables for which you want to analyze their correlation. In most cases, you are exploring a cause-effect relationship.

Higher income households cares more on politics.
Less educated citizens are more likely to not register to vote.
Companies with more diverse workforce has better business performance.

Frequently, the reverse correlation does not admit a causal interpretation:

Caring more about politics does not make one richer.
Not registering to vote does not make one less educated.
Making more profits does not lead to more diversity in hiring.

In each of these examples, it's clear that one variable is the outcome, the other variable is the explanatory factor. Always put the outcome in the vertical axis, and the explanation in the horizontal axis.

The justification is scientific. If you are going to add a regression line (what Excel calls a "trendline"), you must follow this convention, otherwise, your regression analysis will yield the wrong result, with an absurd interpretation!

 

[PS. 11/3/2019: The comments below contain different theories that link the two variables, including theories that treat PISA score ("IQ") as the explanatory variable and ice cream consumption as the outcome. Also, I elaborated that the rule does not dictate which variable is the outcome - the designer effectively signals to the reader which variable is regarded as the outcome by placing it in the vertical axis.]

Too much of a good thing

Several of us discussed this data visualization over twitter last week. The dataviz by Aero Data Lab is called “A Bird’s Eye View of Pharmaceutical Research and Development”. There is a separate discussion on STAT News.

Here is the top section of the chart:

We faced a number of hurdles in understanding this chart as there is so much going on. The size of the shapes is perhaps the first thing readers notice, followed by where the shapes are located along the horizontal (time) axis. After that, readers may see the color of the shapes, and finally, the different shapes (circles, triangles,...).

It would help to have a legend explaining the sizes, shapes and colors. These were explained within the text. The size encodes the number of test subjects in the clinical trials. The color encodes pharmaceutical companies, of which the graphic focuses on 10 major ones. Circles represent completed trials, crosses inside circles represent terminated trials, triangles represent trials that are still active and recruiting, and squares for other statuses.

The vertical axis presents another challenge. It shows the disease conditions being investigated. As a lay-person, I cannot comprehend the logic of the order. With over 800 conditions, it became impossible to find a particular condition. The search function on my browser skipped over the entire graphic. I believe the order is based on some established taxonomy.

***

In creating the alternative shown below, I stayed close to the original intent of the dataviz, retaining all the dimensions of the dataset. Instead of the fancy dot plot, I used an enhanced data table. The encoding methods reflect what I’d like my readers to notice first. The color shading reflects the size of each clinical trial. The pharmaceutical companies are represented by their first initials. The status of the trial is shown by a dot, a cross or a square.

Here is a sketch of this concept showing just the top 10 rows.

Certain conditions attracted much more investment. Certain pharmas are placing bets on cures for certain conditions. For example, Novartis is heavily into research on Meningnitis, meningococcal while GSK has spent quite a bit on researching "bacterial infections."

Wayward legend takes sides in a chart of two sides, plus data woes

Reader Chris P. submitted the following graph, found on Axios:

From a Trifecta Checkup perspective, the chart has a clear question: are consumers getting what they wanted to read in the news they are reading?

Nevertheless, the chart is a visual mess, and the underlying data analytics fail to convince. So, it’s a Type DV chart. (See this overview of the Trifecta Checkup for the taxonomy.)

***

The designer did something tricky with the axis but the trick went off the rails. The underlying data consist of two set of ranks, one for news people consumed and the other for news people wanted covered. With 14 topics included in the study, the two data series contain the same values, 1 to 14. The trick is to collapse both axes onto one. The trouble is that the same value occurs twice, and the reader must differentiate the plot symbols (triangle or circle) to figure out which is which.

It does not help that the lines look like arrows suggesting movement. Without first reading the text, readers may assume that topics change in rank between two periods of time. Some topics moved right, increasing in importance while others shifted left.

The design wisely separated the 14 topics into three logical groups. The blue group comprises news topics for which “want covered” ranking exceeds the “read” ranking. The orange group has the opposite disposition such that the data for “read” sit to the right side of the data for “want covered”. Unfortunately, the legend up top does more harm than good: it literally takes sides!

**

Here, I've put the data onto a scatter plot:

The two sets of ranks are basically uncorrelated, as the regression line is almost flat, with “R-squared” of 0.02.

The analyst tried to "rescue" the data in the following way. Draw the 45-degree line, and color the points above the diagonal blue, and those below the diagonal orange. Color the points on the line gray. Then, write stories about those three subgroups.

Further, the ranking of what was read came from Parse.ly, which appears to be surveillance data (“traffic analytics”) while the ranking of what people want covered came from an Axios/SurveyMonkey poll. As for as I could tell, there was no attempt to establish that the two populations are compatible and comparable.

 

 

 

 

 

Clarifying comparisons in censored cohort data: UK housing affordability

If you're pondering over the following chart for five minutes or more, don't be ashamed. I took longer than that.

The chart accompanied a Financial Times article about inter-generational fairness in the U.K. To cut to the chase, a recently released study found that younger generations are spending substantially higher proportions of their incomes to pay for housing costs. The FT article is here (behind paywall). FT actually slightly modified the original chart, which I pulled from the Home Affront report by the Intergenerational Commission.

One stumbling block is to figure out what is plotted on the horizontal axis. The label "Age" has gone missing. Even though I am familiar with cohort analysis (here, generational analysis), it took effort to understand why the lines are not uniformly growing in lengths. Typically, the older generation is observed for a longer period of time, and thus should have a longer line.

In particular, the orange line, representing people born before 1895 only shows up for a five-year range, from ages 70 to 75. This was confusing because surely these people have lived through ages 20 to 70. I'm assuming the "left censoring" (missing data on the left side) is because of non-existence of old records.

The dataset is also right-censored (missing data on the right side). This occurs with the younger generations (the top three lines) because those cohorts have not yet reached certain ages. The interpretation is further complicated by the range of birth years in each cohort but let me not go there.

TL;DR ... each line represents a generation of Britons, defined by their birth years. The generations are compared by how much of their incomes did they spend on housing costs. The twist is that we control for age, meaning that we compare these generations at the same age (i.e. at each life stage).

***

Here is my version of the same chart:

Here are some of the key edits:

• Vertical blocks are introduced to break up the analysis by life stage. These guide readers to compare the lines vertically i.e. across generations
• The generations are explicitly described as cohorts by birth years
• The labels for the generations are placed next to the lines
• Gridlines are pushed to the back
• The age axis is explicitly labeled
• Age labels are thinned
• A hierarchy on colors
• The line segments with incomplete records are dimmed

The harmful effect of colors can be seen below. This chart is the same as the one above, except for retaining the colors of the original chart:

 

 

Re-thinking a standard business chart of stock purchases and sales

Here is a typical business chart.

A possible story here: institutional investors are generally buying AMD stock, except in Q3 2018.

Let's give this chart a three-step treatment.

STEP 1: The Basics

Remove the data labels, which stand sideways awkwardly, and are redundant given the axis labels. If the audience includes people who want to take the underlying data, then supply a separate data table. It's easier to copy and paste from, and doing so removes clutter from the visual.

The value axis is probably created by an algorithm - hard to imagine someone deliberately placing axis labels  $262 million apart.

The gridlines are optional.

STEP 2: Intermediate

Simplify and re-organize the time axis labels; show the quarter and year structure. The years need not repeat.

Align the vocabulary on the chart. The legend mentions "inflows and outflows" while the chart title uses the words "buying and selling". Inflows is buying; outflows is selling.

STEP 3: Advanced

This type of data presents an interesting design challenge. Arguably the most important metric is the net purchases (or the net flow), i.e. inflows minus outflows. And yet, the chart form leaves this element in the gaps, visually.

The outflows are numerically opposite to inflows. The sign of the flow is encoded in the color scheme. An outflow still points upwards. This isn't a criticism, but rather a limitation of the chart form. If the red bars are made to point downwards to indicate negative flow, then the "net flow" is almost impossible to visually compute!

Putting the columns side by side allows the reader to visually compute the gap, but it is hard to visually compare gaps from quarter to quarter because each gap is hanging off a different baseline.

The following graphic solves this issue by focusing the chart on the net flows. The buying and selling are still plotted but are deliberately pushed to the side:

The structure of the data is such that the gray and pink sections are "symmetric" around the brown columns. A purist may consider removing one of these columns. In other words:

Here, the gray columns represent gross purchases while the brown columns display net purchases. The reader is then asked to infer the gross selling, which is the difference between the two column heights.

We are almost back to the original chart, except that the net buying is brought to the foreground while the gross selling is pushed to the background.

 

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.