Sorry to be all post-y today but reader Ryan Bavetta sent in a hot tip and I had to jump on it before Drudge did. Here’s Obama delivering his State of the Union address. Ryan says, “I don’t think they got the sizes of the circles right.”
So I go all Woodward and Bernstein with my compass and protractor. I measure off the diameters.
The ratio of the diameters is 2.45, which means the ratio of the areas is going to be (2.45)^2 or 6.00. But the ratio of America’s GDP to China’s GDP (14.6T/5.7T) is only 2.56! The US circle is too big! What’s the progressive propaganda machine trying to sell us here?!
Here’s how it should have looked:
For Classroom Use
I think you have to get rid of one of the quantities, ask the students to determine it, show them the full SOTU screenshot, and then encourage them to marvel at the difference. You can give them Obama’s circle and ask them to tell you what that should make the GDP.
Or you can give them the GDP and ask what that should make the circle.
I don’t know how to get excited about the difference when Jon Stewart’s probably trying to call my booking agent right now.
The problem archive, including:
- the original image,
- the image without the GDP,
- the image without the circle,
- video from the speech itself,
- an extension problem.
2011 Feb 16: Updated to add higher-resolution images, video from the speech itself, and an extension problem.
LysmekahJanuary 25, 2011 - 8:36 pm -
Great demonstration. It will be great to use with my grade 8 unit on graphing and misrepresentation! Thanks for sharing.
RhettJanuary 25, 2011 - 8:44 pm -
Awesome (and quick).
However, what if those are not circles, but spheres? I won’t give away the answer in case someone wants to use it in their class. They do look spherically.
AmandaJanuary 25, 2011 - 8:57 pm -
Love it! Well caught too.
I did miss that part of the SOTU address (although it’s interesting to Canadians, dinner was more pressing this evening), the part that I did enjoy the most (for part cynical hilarity and part thankfulness) was when the President said the following:
“Become a teacher. Your country needs you.”
Cynical hilarity because, in my neck of the woods at least, there are too many teachers and not enough jobs due to budget cuts and declining enrollment. Thankfulness for having the importance of teachers and the job they do put in the (inter)national spotlight for just a moment.
I’ll be making use of the circles next semester I’m sure, so thanks.
DavidJanuary 25, 2011 - 9:10 pm -
It is clearly a non-linear relationship, but it seems consistent at least. If you look at Germany and France relative to China they seem to share a similar ration as that between China and US.
DavidJanuary 25, 2011 - 9:23 pm -
Actually doing a quick measurement of my own (I got slightly different numbers), it looks like they simply scaled the diameter in proportion to the GDP, which of course created an exponential relationship between the areas.
Not strictly accurate in a mathematical sense, but the eye does have a hard time telling the difference in areas of circles, so this is probably better at making the point than a mathematically accurate chart would be.
GregJanuary 25, 2011 - 9:25 pm -
At issue here is the axiom that people interpret such graphs by perceiving the area of the various figures (rather than the diameter, say). I happen to believe this axiom, but I have no way of justifying it to someone who doesn’t … have any studies been done on this?
Tom DavisJanuary 25, 2011 - 9:38 pm -
It seems that *all* of the disks relate to the GDPs by their diameter and not by their area. If you alter the US disk so that the sizes of the US GDP and China’s GDP are represented by area, you should also change the sizes of the disks representing French and German GDPs.
Of course, as Rhett suggested, those might not be disks. They could be spheres or domes, in which case the relations might be based on volume or surface area, though clearly if left as published by the White House, the relationships are based on the diameters.
My point is that if you are looking for clarity, you will need to replace the blue round-looking-things-whatever-they-are with bars with one axis the same for every GDP shown and the other axis varying, or remove them all together.
Jason BuellJanuary 25, 2011 - 10:07 pm -
I’m looking forward to your future interview on fox news….
ErickJanuary 25, 2011 - 11:41 pm -
These definitely look Spherical… But if one were to argue the accuracy of the spheres. Would they be measuring
Surface Area, Volume, or even Mass. Mass could be argued to the point where the spheres are physically proportionate… That is to say density was involved in the calculations; then there would have to be a Legend explaining each Sphere’s properties.
I would hate to assume that this slide will be used by foreign countries as a motivational tool…, “This is a prime example why Americans fail in mathematical academics”.
But, I bet the designer of the graphic hadn’t thought this deep… Considering they were a probably a state worker on a paycut and returning from their 3rd furlough in one month.
Dan AndersonJanuary 26, 2011 - 3:34 am -
Or maybe they made the mistake on purpose so that Obama could spotlight the need for better math/science education! That’s totally gotta be it. Right?
jgJanuary 26, 2011 - 4:23 am -
Greg: “At issue here is the axiom that people interpret such graphs by perceiving the area of the various figures (rather than the diameter, say). I happen to believe this axiom, but I have no way of justifying it to someone who doesn’t … have any studies been done on this?”
I’m betting the opposite. Exactly because this happens all the time (infographics in USA Today, Newsweek, etc. on stuff like this or drink sizes, etc.) – designers consistently represent sizes by linear dimensions while trying to represent them with areas or volumes – I say that people perceive the size, not the area. I don’t think that the designers know the score and are trying to trick the public; I think they’re in the same boat.
Jason DyerJanuary 26, 2011 - 5:51 am -
Bad circle area graphs are fun! Here’s another one: Bad Circle Area Graph Alert
kevinJanuary 26, 2011 - 6:31 am -
Well this is all a step in the development of the brain, right? It is called the “conservation of quantities,” where if a child sees a wide glass of water poured into a narrow glass he/she will assume the narrow glass has “more” although one would think they would know it must be the same amount. Piaget says we figure this out prior to six years-old. I am not so optimistic.
What is a trillion anyway?
gasstationwithoutpumpsJanuary 26, 2011 - 7:42 am -
“I don’t think that the designers know the score and are trying to trick the public; I think they’re in the same boat.”
Most trained graphics designers doing infographics have (or should have) read Tufte’s “The Visual Display of Quantitative Information”. That means that they either they do know the score and are being deceitful, or that they have not had even minimal training in their craft (that is, that the White House is hiring people known to be incompetent). What you believe here depends on what type of cynicism you have about government. Personally, I’ve found incompetence to be far more likely than conspiracy, though deliberate hiring of incompetents could be viewed as conspiracy also.
Even more likely is that the infographics was given to an untrained intern to do who was a poli-sci undergrad with no education in anything quantitative. I’m hoping we see a correction from the White House, but I’m doubtful that we will.
DavidJanuary 26, 2011 - 9:12 am -
“Most trained graphics designers doing infographics have (or should have) read Tufte’s “The Visual Display of Quantitative Information”. That means that they either they do know the score and are being deceitful, or that they have not had even minimal training in their craft (that is, that the White House is hiring people known to be incompetent).”
Or perhaps they are their own intelligent individual and simply (gasp) disagree with Tufte.
Tufte says you shouldn’t use 2D objects to represent 1D quantities, but in practice it is often clearer to use these sorts of graphics. In the end it is all about conveying the information.
DaveJanuary 26, 2011 - 9:58 am -
As someone who does a fair share of work with Adobe Illustrator, I will bet anything that most designers approach infographics like this:
1) Select the “Ellipse” tool.
2) Click on the canvas, which brings up an “Ellipse” dialogue window with two blanks: height and width.
3) Look at the raw data values, and enter the value into both the height and width fields.
I hate to be stereotypical, but based the designers I’ve worked with would be bored to tears by this discussion (and they definitely haven’t read or heard of Tufte’s book). If you explained the value-as-diameter error to them, I would expect at least half to openly contradict you and say that the aesthetic gain from over-amplifying the differences in the values outweighs the loss in accuracty.
RMJanuary 26, 2011 - 9:59 am -
I love the irony of this and Obama’s talking points about education. This one little graphic can create a great discussion for any math class. Thanks for sharing!
Chris SearsJanuary 26, 2011 - 10:58 am -
I watched the SOTU on PBS with no graphics at all. I don’t get cable, but if I did I still wouldn’t watch cable news. I can’t stand all the blinking graphics, news crawls, and screaming people.
HerbertJanuary 26, 2011 - 11:03 am -
This is so mathtastic OMG!
ChrisJanuary 26, 2011 - 11:38 am -
Re-watched the SOTU on youtube (http://www.youtube.com/watch?v=nZ5DMpNznlA), and I didn’t see this graphic. Which station did it come from? Sounds like a TV station mistake not a white house mistake.
Adam CavnarJanuary 26, 2011 - 11:41 am -
While the point about the size of the circles is certainly interesting, it is dwarfed in importance by this question:
Are the numbers accurate ?
erJanuary 26, 2011 - 12:09 pm -
as david said. it’s the diameter of the circle is proportional, not the area
Ihab IsmailJanuary 26, 2011 - 12:16 pm -
From a biological perspective, one has to consider osmosis as a potential path for inflating the circle: The heavy ink surrounding the circle would penetrate through and into the lighter ink inside the circle. We may have started with true dimensions and later on ended as presented here.
As I stirred on the circle, the small circle inflated too. :) Give it some time and the optical/mental illusion combined with cellular reorganization of eye nerve cells may place the presentation on entirely different path.
BUT, that is just a thought at 3:12 p.m. I may be just not thinking mathematically.
Dan MeyerJanuary 26, 2011 - 12:20 pm -
@Chris, that graphic comes from the enhanced SOTU put out by the White House.
CatherineJanuary 26, 2011 - 12:21 pm -
Actually, this is the truly correct way, in the data visualization research world, to size comparative spheres/circles. (They should be proportional by radius, NOT area.)
The reason for this is a matter of how we perceive size. For example, a circle that has twice the AREA of another, does not look twice as large – it looks only about 1/3 larger. However, if a circle has twice the diameter, it is perceived by the viewer as having twice the volume.
Sometimes to effectively show data, you don’t always have to follow the mathematical rules. In any case, comparing circular sizes should generally be avoided because of this dichotomy.
(Main source for this is in the excellent Wall Street Journal guide to data visualizations:
MarkJanuary 26, 2011 - 1:14 pm -
I agree with David and many others as well. They are scaled according to diameter.
I liked the post and while you can argue that area is more appropriate (which I agree with), your two examples would be “correctly” answered just like the original. There are 4 pieces of data and you fill in the last according to the pattern.
Again, the pattern might be “wrong” on some level, but the answer to your questions would be the White House answer.
Daniel SchabenJanuary 26, 2011 - 1:48 pm -
I showed students your original slide with the US GDP blocked out and had them guess what the value in the black box should be. One of my groups did make a circle the size of China’s circle and then duplicate it four times placing these neatly in the US circle and come up with a value of 5 X china’s GDP. What a great Graphic!! More fuel to the fire of why every citizen in the US needs to know mathematics if we are going to keep our democracy alive.
Created some great discussion.
Used it as my warm up. Instead of the Normal ACT PREP Question.
Great discussion followed.
ShinobiJanuary 26, 2011 - 3:10 pm -
I get that these bubbles are cool looking, but what is wrong with an old fashioned bar chart? They are easy to read, you can create one axis, label it and then you don’t have to label all of the data points. The relationships aren’t confusing and are not open to interpretation.
BRING BACK BAR CHARTS!
JeannieJanuary 27, 2011 - 6:06 am -
The area principle states that “the area occupied by a part of the graph should correspond to the magnitude of the value it represents. Violations of the area principle are a common way to lie (or, since most mistakes are unitentional, we should say err) with Statistics.” Taken from Stats, Modeling the World by Bock, Belleman, DeVeaux, 3rd edition.
Trevor LohrbeerJanuary 27, 2011 - 9:12 am -
@Catherine I’m not sure where you read that. In my copy of the Wall Street Journal Guide to Information Graphics, on page 81, it clearly states that the sizes of circles should be based on their surface area and that it is incorrect to use the radius (or diameter, which produces the same result since it’s just the radius times 2).
A nice demonstration of this is at Simple Visual Perception Experiment and Robert Kosara has a detailed article on it at Linear vs. Quadratic Change.
On the spherical issues, this is irrelevant. At issue is not the mathematical differences between the circles, but the perceived differences. It doesn’t matter if they are circles or spheres, the perceived differences remain the same. We naturally can perceive areas based on how much of our visual field is taken up by an object, but we can’t naturally perceive volume. We need to extrapolate volume based on the area (or use stereoscopic vision to triangulate).
On the incompetence versus deception issue, I propose incompetence/ignorance. Reason being is that the other circle chart in the State of the Union underrepresented the differences rather than overrepresenting the differences.
For another view of this chart being corrected, as well as the corrected version of the Tax Cuts for the Wealthiest Americans chart, check out 2011 State of the Union Visualizations: Charts, Graphs & Infographics.
For those who want to see this chart in the enhanced video speech, you can start watching at the point this chart appears by going to this link on YouTube.
Eric WestJanuary 27, 2011 - 9:50 am -
Here’s a similar infographic that accompanies a Roanoke Times feature on children with autism spectrum disorders:
And my forum post pointing out the inaccuracy:
I hope that what Catherine says isn’t true, because these circles are very misleading. The misrepresentation is not helpful to the average citizen in the way that, say, the London Underground map is to travelers.
jctokpyJanuary 27, 2011 - 9:58 am -
If you calculate the area of the circles, there is error as the formula to estimate this area is a non-liner function of the radio (http://www.mathgoodies.com/lessons/vol2/circle_area.html).
ChrisJanuary 27, 2011 - 10:03 am -
Thanks for the clarification. I will continue to make fun of the White House now.
ErickJanuary 27, 2011 - 12:10 pm -
This is great!
AlexJanuary 27, 2011 - 12:27 pm -
Way to go!
Dave inNHJanuary 27, 2011 - 1:32 pm -
Thanks for clearing that up, for a while I thought 14.6 trillion in debt was a real large amount. I feel so much better now that diameter of the circle is 64 pixels and not 98 pixels.
R. WrightJanuary 27, 2011 - 7:31 pm -
Isn’t the first “for classroom use” suggestion just a kind of pseudocontext?
Dan MeyerJanuary 27, 2011 - 7:39 pm -
Interesting. You might be right. But the point of the exercise is to immediately point out the difference between what the erroneous circle implies for the GDP and what the GDP actually is, which maybe inoculates it, maybe it doesn’t. I’m trying to pull an example from the archives where the context is flatly untrue but the point of the problem is to prove its untruth. I’m struggling.
Jay WarnerJanuary 27, 2011 - 10:07 pm -
Years ago (1983) Edward Tufte wrote a book, _The Visual Display of Quantitative Information_, in which he showed that graphs using 2-D objects tend to be read by _area_, not by diameter. Hence, 2-D objects proportional by diameter exaggerate differences. Whether Obama’s staff, or he himself, noticed the distortion, or chose to keep it in because it apparently made their point better, is unclear. People not versed in the use & abuse of graphics, including many/most non statistics-using professions, fall into this trap frequently, but it has been known for a long time.
Jason MackenzieJanuary 28, 2011 - 8:55 am -
I think the real question is why didn’t they use a bar chart? It is simpler and doesn’t have the area being a non-linear relationship problem that circles do.
We need a new “Sputnik” to energize our educational system.
JonathanJanuary 29, 2011 - 7:22 am -
To be fair, Obama didn’t botch it, the designers of the graphic did. However as others have pointed out, the radii are correct and while technically it’s the area that is important, it’s become such a common mistake outside of proper statistical graphics that it’s become acceptable (like a lot of grammatical errors).
Also, I’d have used a bar chart.
But… the facts are correct. The graphic debatably isn’t. But that’s not Obama’s fault.
CathyFebruary 2, 2011 - 9:04 am -
I am excited to be using this example with teachers tomorrow:-) The ideas here match with several grades of our junior high math curriculum, particularly in the area of circle geometry and misrepresenting statistics. I will be sharing the infographic without the circle representing the US GDP and see what size they predict it should be. Then look at what was actually shared in the SOTU.
Another question: Why were these particular countries chosen. Does the choice of country exaggerate the large US GDP? I don’t know enough about this topic to determine whether these are appropriate comparisons. Can someone answer this?
Rick HarringtonFebruary 2, 2011 - 4:16 pm -
What a great “catch” on your part. Typical use of “eye candy” to exaggerate a point. As others have said, this will make a great warm-up lesson for my middle school math students:)
KateFebruary 5, 2011 - 6:22 am -
This is awesome!
MattFebruary 10, 2011 - 9:08 am -
We should take pity on the person responsible for that SOTU chart, they must have taken AP Stats back in high school, as this chart from college board would suggest (they did the same thing!)
College Board AP Math and Science Improvement
For Science, I get the ratio of the number of students getting 3’s or better in ’01 compared to ’10 to be 0.467, and for the circles I get the ratio of the diameters to be 0.471, but the ratio of the areas would be ~0.22
I’m but a lowly physics teacher, but now Dan has me seeing the world in ‘math.’
adogFebruary 25, 2011 - 8:28 pm -
Thank you for the post! I dig the bit on classroom use!