Junk Charts

Cram it like Koby

You have to gradually build up your gut by eating larger and larger amounts of food, and then be sure to work it all off so body fat doesn't put a squeeze on the expansion of your stomach in competition  -- Takeru Kobayashi, six-time champion of the Coney Island hot dog eating contest

Kobayashi is a phenom.  He can stuff 60 hot dogs or 100 burgers in ten or twelve minutes and show no consequences.  Ordinary people can't hope to emulate these feats.

Junk Charts sees Kobayashi as a hero; an anti-hero really.  We are ordinary people; we can't hope to cram it like Koby.  A message we keep repeating here is: too much data sinks a chart.

Not long after this chart showed up in the Economist, several readers urged us to take a look.  It's a well-nourished chart indeed, one to challenge Kobayashi, but for all that it contains, the reader has to try very hard to find insights.  What with the multiple colors, iron-fisted gridlines, above-and-below boxes, dotted and solid lines, and a legend with nine pieces split in two spots?  Besides, the U.S. boxes grab all the attention by virtue of them being wider (country being more partisan).

The key to unraveling this chart is to identify the relevant comparisons:

• UK average vs US average
• UK left vs US left
• UK right vs US right
• UK independent vs US independent

And then for the gluttonous:

• UK right vs US left
• UK left vs independent vs right
• US left vs independent vs right

In the junkchart version, we address these comparisons sequentially.

(Apologies for the tiny font.)

We are again using a small multiples approach that places four comparisons next to each other: average, left, independent, right. Consistently, the British is to the left of Americans.  The only places where the two cultures meet are where liberals agree on "ideology" and "military action".

Also note that we use a symmetric horizontal scale centered at 0.  There are too many charts out there where the center is not at the center!

A similar presentation addresses the other three comparisons.  Democrats in the U.S. are miles to the right of Tories in terms of "religion".  In the UK, Labor and Tories are not much different except on "ideology".  In the US, Independents lean closer to Democrats.

Joining the lines (I hear the grumbles) helps bring out the gap between the groups being compared.  Without lines, the chart would look like this.

It is often hard to keep track of which dot is which as they trade order from issue to issue.

PS. Anyone knows what is being measured on the horizontal axis?  The original graph mysteriously stated "respondents' views".

References: 

Eric Talmadge: "Pigout champion Kobayashi limbers up for hot dog gold" June 25, 2004

"Anglo-Saxon Attitudes", Economist, Mar 27 2008.

Chart cleanup

Anna E. submitted this great example from Yahoo! Green.  A well-meaning chart but stuffed with redundancy.

Much appear to be going on and yet the entire chart contains 15 data points, Boston's ranks on each of 15 categories.  The bar lengths convey the same information as the data labels.  The legend provides a catchy name for different levels of ranks (0-10 = "leader"; 10-20 = "advances"; etc.).  The colors merely reiterate the catchy titles.  Similarly, the colored squares repeat the information in the bars.

In the name of green, we cleaned up this chart:

As a standalone graph, the categories should be ordered by Boston's ranks.  Here, we assume that cross-referencing cities is needed so we leave the order unchanged.

Football rankings 1

The Times' sports pages made wise use of graphics in a series of NFL articles recently.  Here is a rank plot (below left) comparing Jaguars quarterback David Garrard to seven other quarterbacks who started the weekend of January 5.

Simple and effective, this chart does not fuss around in showing us where Garrard ranks relative to the others. 

The junkart revision (below right) plays with a different scale: the spacing between the tick marks represent proportional differences in the underlying metric.  This gives us a little more: for example, Garrard's second rank in completion percentage is less remarkable than first thought as he essentially tied with the 3rd and 4th best while the top six were bunched between 60 and 65 percent.

But Garrard's touchdown to interception ratio stands out as the next best quarterback attained only about half his ratio.  (Todd Collins who had not thrown an interception until that time was omitted; he also had only started four games.)

References: "Two Dreams (One Big, One Tiny) Come True", New York Times, Jan 4 2008; ESPN statistics.

Hits and misses

In this NYT article, we are told that "the most likely result when a policeman discharges a gun is that he or she will miss the target completely."  That's a shocker for those of us conditioned by Hollywood movies to think anyone who picks up a gun for the first time hits the villain right on the temple.  The following graphic attempts to tell the story.

The one hit here is how the distances are visually presented.  The elliptical lines remind us of the neglected variable of direction; it also means the scale is correct only along one direction.

The dot matrix construct highlights the absolute numbers of shots, hits and misses but barely addresses the key issue of hit rates (accuracy). Specifically, this data set was presumably collected to explore the relationship between hit rates and distances from the target.  The use of different widths clouds our judgement of proportions.  To wit, it is not obvious that the 10-wide block and the 40-wide block shown left depict roughly equal hit rates (23%, 29%).

The junkart version adopts a different approach.  This is the Lorenz curve, often used to show income inequality (see also here and here).  Here, the shots were ordered from closest to furthest from target, then summed up by distance segments.  For example, shots from 0 to 6 feet accounted for 60% of all shots but 72% of all hits.

If distance does not affect hit rates, we'd expect 60% of all shots to result in 60% of all hits.  This data point would show up on the 45-degree diagonal on the chart, labelled "totally unpredictable".  Any data appearing above the diagonal indicates that closer shots are more accurate, accounting for more than their fair share of hits.

Comparing the fitted blue line and the diagonal, one sees that distance is a weak predictor of hit rate.  The police commissioner explains this in the article; many other variables also affect accuracy, including "the adrenaline flow, the movement of the target, the movement of the shooter, the officer, the lighting conditions, the weather..."

Note that the shots with "unknown" distances were removed from the analysis.  Also, the categories of 21-45 and 45-above were combined: the rates were similar and with only three hits, it does not make sense to treat these as separate categories.

Of course, this version would not work well in the mass media.  For that, one can just plot hit rates against the distance categories.

Source: "A Hail of Bullets, a Heap of Uncertainty", New York Times, Dec 9 2007; New York Firearms Discharge Report 2006.

Sense of proportion

[I'm back from vacation.  Will provide my reaction to the responses to the Gelman challenge, and for those who have sent me email, I will work through them soon.]

The NYT commented on a trend among marketers to shift their advertising spending from so-called "measured" media like print and TV to so-called "unmeasured" media like product placements, contests, etc.  The following chart accompanied the article:

This construct is akin to a population pyramid; it's great for comparing two groups along one metric, say age groups between males and females.  Here, the two halves aren't comparable groups but two different metrics.  The main metric, that is, the proportion of unmeasured, is not directly depicted: the reader must figure out mentally how much of each bar the black part covers.  Also, the companies are sorted by unmeasured media spending but this leaves the measured spending with a jagged profile, confusing matters.

As for the little white slits on the gray bars, they are admittedly cute but it is difficult to compare the detailed breakdown between print, TV and other media among companies.

The following dot plot gives the two halves equal weight.  (Pink dots are measured, blue unmeasured.) It's not a very interesting graphic though. The sense of proportion is still missing.

I settled on a scatter plot which relates the proportion spent on unmeasured to the total amount of spending.  It appears that the largest advertisers had the lowest proportional unmeasured spend while the smallest (among the majors) had the highest.  (It's only a weak correlation: a linear fit yields only 16% R-squared.)

Source: "The New Advertising Outlet: Your Life", New York Times, Oct 14, 2007.

Structuring a chart

This chart from the NYT was intended to show how the EPA has moved the bar on vehicle mileage ratings: 2008 estimates were lower than 2007 estimates across the board, regardless of manufacturer, model and city/highway.

The chart was built from one basic component, repeated for each model.  I like the discreet gridlines (the white ticks) which enable readers to count off the mileage ratings.

The data is rich: ratings were given along three dimensions (model, year of estimate and city/highway).  Readers can benefit from a stronger guidance in where to look for the most pertinent information.  As the chart stands, it is merely a container for the data.  It fails our self-sufficiency test: all the data were printed on the chart, and the bars add little.

In the junkart version, I use knowledge of the data to structure the chart. First, noting that sedans, hybrids and trucks/SUVs/minvans have different levels of mileage ratings, I clustered the models into three groups.  Secondly, the city and highway ratings were separated into two columns as I consider the between-model comparisons more important than city-highway comparisons.  The chart is a dot plot, with a vertical tick for 2007 estimates and a dot for 2008 estimates.  It's easy to see that all dots sit to the left of vertical ticks.

More subtly, we can also see that the hybrids appeared to have been penalized more.  Or perhaps, the higher the rating, the larger the downward adjustment...

Source: "Mileage Ratings Are Still Estimates, Though Closer to Reality", New York Times, Sept 16 2007.

Less is more

Derek pointed me to the style.org site which also parses political speeches.  Their preferred graphic is not the tag cloud but a labeled bar chart.

From top to bottom, each bar represents a sentence; the length of each bar is the length of each sentence.  Further, the user can specify word pairs for comparison.  Here the red bars are sentences containing the word "freedom"; the blue bars, "security".

It's a good illustration of the "small multiples" principle in constructing comparative graphics.

However, the choice of dimensions is perplexing.  I'd be much more interested in the timing of mentions of those words, rather than which sentence they appeared in.  I also find the length of each sentence to be irrelevant.

Here's one concept that brings out the point better.  It uses less space and voluntarily gives up some of the data (the sentence structure).

March mildness

The Times published this great graphic to show 2007 was an upset-starved year in the recent history of the NCAA Basketball tournament, which is on-going.

Each box contains the number of upsets in a given year of a given pairing, e.g. in 1998, there was one case of a 9-seed beating an 8-seed.  An upset is defined as a lower seed beating a higher seed although the editorial comment argued that 9 beating 8 is "rarely considered an upset".

The rightmost column (which sums across a row) tells us that the number of upsets fluctuates wildly between the years, ranging from 3 to 13.  (That's why people bet on NCAA pools.)

A couple of improvements will make this chart even more effective:

• Include a row showing the average number of upsets for each pairing;
• Include a column of zeroes for 16-1 pairings.
  

This second point cannot be emphasized more.  The fact that no 1-seed has ever lost to a 16-seed should not be relegated to a footnote.  Think of it this way: if the results for 15-2 and 16-1 were reversed so that no 15-seed had ever beaten a 2-seed but one 1-seed had lost to a 16-seed, nobody would omit the 15-2 column! 

In his seminal work, The Visual Display of Quantitative Information, Tufte discussed the Challenger disaster at considerable length.  A key learning was that non-events (things not happening) contain important information, and should never be dropped from an analysis without unassailable logic.

The mildly improved chart would look like this. What then to make of the comment that "9 beating 8 is rarely an upset"?  For one thing, 9-8 upsets happen about as frequently as 10-7 upsets so if the comment refers to the surprise factor, then even 10-7 upsets should be excluded.

But the comment also underlines a deeper issue, which is hindsight.  Obviously, the seeding committee felt, and predicted, that the 8 seed would beat the 9 seed.  It was only after the fact that we found out 9 had beaten 8.  Instead of denying the 9-8 upset, would it make more sense to ask if there was a seeding error?

Reference: "March Mildness", New York Times, March 17, 2007, p.D2.

Horrid stuff 2

Jon P took my comment on negative correlation and explored it further.  Given the large ranges of values cited in the original Economist chart, Jon concluded that there wasn't enough evidence to make a judgement.

I agree to a large extent.  Apart from the high variability of individual measurements, we also face the tiny sample of 5 cities.  In his chart, he made an implicit assumption that the correlation of two factors is related to the product of the ranges (variability) of each factor by plotting the rectangles.

A different way of looking at it is to plot only the mid-range values (i.e. ignoring the within-city variability).  The graph on the left hand side shows very little pattern.

Resorting to the formula, I found that the correlation = -0.03.  So barely detectable negative correlation.  Lets visualize this. 

On the right graph, I added the mean lines for both variables.  This divides the graph into four quadrants; dots that fall into the lower right and upper left quadrants make the correlation value negative.  There were three of those versus two in the positive quadrants; hence, the tiny negative correlation. 

Convenience charting

Statisticians have long riled against "convenience sampling", that is, the practice of selecting samples based on what's easily available, not at random.  Say picking your friends.

Dustin J sent in this example of what can only be called "convenience charting".  Dustin said he had no clue what this chart is saying, and I am not surprised. 

The chart plots a statistical object known as the "survival function".  It is likely that "survival analysis" was done, after which the chart creator  picked up the resulting statistical object and dumped it onto this "convenience chart".

If we take the top line on the "child survival" graph, it shows the probability of one child surviving up to a certain age, if the child belonged to a family with 1-3 kids.  The chance is about 92.5% that the child will survive through age 2, and 88% that the child will survive through age 18.  The difference between those percentages is due to the chance that the child may die between ages 2 and 18.

A slight transformation of the data will make this point much clearer.  What is the probability of a child dying by a certain age?  Using the example, a child has 12% chance to die by age 18, and 7.5% chance of dying between ages 0-2.

The junkart chart depicts this probability.  (I reverse-engineered the data which explains why the distances between the line segments look strange.)

What this chart doesn't address is how we are to interpret the probability of "a child dying" in a family with more than one child.  Is it a random child dying?  At least one child dying?  Exactly one child dying (the other X-1 surviving)? 

The original chart also committed a number of standard errors.  The child survival function represent probabilities, not percentages.  The third category should be 8-11 kids, not 7-11.  If we are picky, then we would also like to see "confidence intervals" because there must have been many fewer families in the 12+ sample than the 1-3 sample.  In the second chart (which I don't have space to discuss), some data labels are missing, which indicates a presumption that all readers have seen the first chart.

Reference:  "Child, Parents Drive Each Other to Early Graves", Washington Post, Jan 14, 2007.