Tuesday, December 9, 2008

Follow-Up on Differences in Test Score Variability

In an earlier post, I described a study by Hyde et al. in Science showing that the gender gap in math has disappeared, as measured by average scores on state accountability exams. However, in that article there was also evidence that boys are overrepresented at the high end of performance, due to greater variability in their scores.

When the Hyde et al. paper came out, the press did a good job of making the point that the scores of boys and girls are now equal on average. Most reporters ignored the overrepresentation of boys at the high end. Some may have wanted to tell a simple, positive story; some may have been influenced by Hyde et al.'s rhetorical strategies. (A reporter from Reuters was an exception, but I can't find her story to link to.)

Today I'm just following up because the variance ratio has now been observed in another study: "Global Sex Differences in Test Score Variability," by S. Machin and T. Pekkarinen, Science Vol. 322, 28 November 2008. Click here to go to the article (subscription required).

The new study, based on PISA results, is both weaker and stronger than the study by Hyde et al. It is stronger in that it includes more countries than just the U.S. But it is weaker in that it only looks at 15 year olds. (Hyde et al. considered students in 2nd through 11th grades.)

Also unlike Hyde et al., the study by Machin and Pekkarinen looks at both math and reading. Interestingly, the variance ratios turn out to be similar in the two subjects, but with different consequences or for different reasons. In reading, boys are overrepresented at the low end of performance, whereas in math boys are overrepresented at the high end.

As of this writing, I've only seen one news article on the study, a Washington Post article by Ruth Marcus. Her article is more about Larry Summers than about the research itself.

Sunday, November 9, 2008

A Historic Election

Have you outstript the rest? Are you the President?

As simplistic as my simulation was, I trusted it when it told me that Pennsylvania would be important. So I got in touch with the campaign and they assigned me to volunteer in Wilkes-Barre, Pennsylvania, about 3 hours' drive from New York City. I drove there on the Saturday night before the election, stayed in a hotel, and spent Sunday canvassing. I drove back to New York for work on Monday morning, and then on Monday night I drove back to Wilkes-Barre to get out the vote on election day.

On election night I celebrated with the campaign staff, then drove home at midnight for work on Wednesday. It was a tiring span of days, but a fascinating and thrilling experience.

The campaign headquarters in Wilkes-Barre looked like you'd expect it to look, with donut boxes here and there and campaign literature piled high in the corners. Cheap gray carpeting receded in the sickly light of exposed fluorescents. There were two inflatable mattresses in the office, and a lived-in smell that said they got used.

Two of the volunteers had British accents. This struck me as curious, and after playing it cool for most of the day, I finally asked, "So, how do you come to be in Wilkes-Barre?" It turned out they were Londoners. Two months earlier, they had relocated to small-town America specifically to help elect Barack Obama.

On Sunday there were as many as 20 out-of-state volunteers, most of us from Manhattan. On Tuesday, far fewer people took the day off work, and that left a lot of work to do on election day.


I was standing before a screen door with my knuckles poised to knock, but I wasn't knocking. First of all, the screen door had no screen, and this confused me momentarily; second of all, there was an argument going on inside the house, and I sensed I should wait for a lull before I knocked.

A lull. I knocked. The inner door swung open, and a woman turned to me and said (through the nonexistent screen):

"HE says he's going to raise taxes!"

The way she took the pronouns for granted, it was as if I'd been standing in the room all along, refereeing the argument like FactCheck.org.

I made firm eye contact (tricky; she was cross-eyed) and said, "Not on you and me, ma'am." Half-true anyway, I thought. And today I'm rounding up.

She was tall, chubby, blonde, young, gap-toothed, and pretty.

"HE says" (more playfully now) "he's a foreigner!"

The eye contact had evidently made an impression. (Maybe too much of an impression? How much of this was HE picking up on, I wondered?)

I shook my head slowly - slowly enough to maintain eye contact. "No ma'am. No ma'am. He's an American."

I felt she was ready to tip. I held out my campaign flyer. "Please vote today. Obama for America." She took it with a smile and shut the door.


The man on my list was out in the street working on a white Pontiac Trans Am, circa 1977. Nice car. A stereo on the porch played "Don't Stop Believin'." Awesome song. The upraised hood blocked my view of the man himself. As I walked into his field of view, he said, "Keep that stuff. Give it to somebody needs convincing. If he doesn't win, we're all doomed."

Maybe this state is going to come through after all.


Plains, Pennsylvania; county of Luzerne.

My canvassing partner looks as if he belongs in this neighborhood. He's wearing Carhartt pants, a flannel shirt, and a ball cap that says "No Beer, No Work." He has a rugged, sunburned complexion and a Carolinas accent.

This is not so much a neighborhood as it is a settlement. Or an outpost. At any rate, here we are, in the swampy foothills of the Appalachian mountain chain. All the names on my list are Irish. It's mid-afternoon, and kids are walking home from school, clambering over the slightly raised railroad track that evidently leads to a school building I never saw.

The town of Plains was settled as a "patch." Patches were small mining towns affiliated with particular coal mines. Sometimes patches were entirely owned by the mining company: general store, scrip, all of that. In Plains, we're standing in the heart of Pennsylvania's anthracite Coal Country. Mining operations declined precipitously in this region in the 1950's, partly due to the Knox Mine Disaster which took place in nearby Jenkins Township.

Patch towns were usually just three or four hundred people settled near the mine. Today, our assignment in Plains is twenty or thirty sagging wooden homes and a few grimy apartments lining three or four paved and unpaved roads. I have no idea what people do here for a living. The town is so small it resembles a movie set, and as my partner and I walk down the center of the street watching grey curtains drawn aside in the windows, we feel a little like we're walking towards high noon.

The kids have made their way home and mostly that's who we talk to. At one house, two little girls tell me excitedly that their mom has voted for Obama today. "Great!" I answer. Then I glance at the house next door, which is full of McCain-Palin signs. Two little boys, 9 or 10 years of age, are standing on the porch giving me the finger.

When I meet up with my partner to leave, he says, "Did you see the one in the tank top leaning out the window?" We turn the corner and there she is, waiting for us to come by again. She shouts and waves, and she's leaning out the window like she knows how to lean out a window. We wave and laugh and head back to the car. On the way to the car, I see a house with a green sign in the window: O'Bama 2008. The apostrophe is a shamrock.


The end of the night is a mad dash to drag people to the polls before they close at 8pm. It is long past dark, and we've been given tiny plastic flashlights to help us read our lists.

Most of the people I speak to have already voted, and some are seriously angry at being hounded yet again. A few simply close the door with a dour shake of their heads.

In some cases the people I'm speaking to don't support Obama, but instinct tells me they have stayed home today and not voted against him either. I'm beginning to feel optimistic.


By 8pm, everybody had returned to campaign headquarters in Wilke-Barre. We were standing in a group watching MSNBC on somebody's computer screen. By then, Obama had won only three electoral votes, and McCain was ahead. So far, no surprises. Then, just a few minutes after 8:00, NBC called Pennsylvania for Obama. We did it! A cheer went up, and everybody started hugging and even crying.

At that point I quickly set up my computer and re-ran my simulation to assess the new state of play. That morning, the probability of victory had stood at only 58% - down from the 70% figure a week earlier because of a tightening of the polls. Even worse, in the morning's simulation, the second most likely state scenario was actually a McCain victory. However, feeding in the new information about which states had been called for the two candidates, I computed the new probability of victory as 78%, with the nine most likely state scenarios being victories for Obama. Pennsylvania looked to be a dagger in the heart of the McCain campaign.

In the new simulation, Iowa emerged as the next crucial battleground, and interestingly, just a few minutes after Pennsylvania was called for Obama, the national campaign texted the Wilkes-Barre staff to say that we should now be calling voters in Iowa to get out the vote. (The polls in Iowa were open until 10pm.)

I leafed through the Iowa cities in my printout until I found Marshalltown, near where I used to live back in 2004. At that time, a lot of Mexican immigrants were living in Marshalltown, working on nearby farms and hog-processing plants. So to prioritize, I looked for Mexican names and started calling them. On my first call, a woman picked up and said:


Shit! She only speaks Spanish! "Ah. Si. Mmm, puedo, hablar, con Maria, por favor?"

"Si, momento"


"Uh, por favor, señora, por favor vote(?) usted por Obama, hoy? Antes del diez?"

(Laughing) "Si, gracias, ciao señor."

Fortunately, there wasn't much time to make calls to Iowa. Soon New Hampshire became the first Bush '04 state to flip, and when Ohio flipped it was pretty much over.

At that point, the group made its way to a neighborhood bar, where we celebrated the victory. I was told that the Wilkes-Barre team's mission had been not only to win the county, but to turn it decisively back to the Democratic party. This they did: the tally from Luzerne County was almost 70-30 for Obama. I may have been right about Obama doubters staying home.

At 11 or 12 o'clock I finally got back on the road. Driving through the night with the radio on, I listened to McCain's thoughtful concession speech, and then came Obama's victory speech. It was tremendous, and audacious, drawing a line across history through three points: Lincoln. King. Obama. Several times I enthusiastically took my hands off the wheel to clap, not even realizing what I was doing. My only disappointment was that Obama relied so heavily on the words of Lincoln and King, without introducing any memorable poetry of his own.

When I stopped at a gas station in a small Pennsylvania town, teenagers were zooming through the parking lot whooping and waving to me with shouts of "OBAMA!". When I returned to the city at 2am, Union Square was a giant party.


My last stop in Wilkes-Barre had been at the local Denny's, where I fueled up for the drive home with some steak and eggs. While eating, I spoke with Rebecca and a couple of friends who had worked hard for Obama's victory. I also texted with an African American friend in California. Americans en masse had finally made the ultimate overture to him, and I sensed rather than read, in the constrained messages filling up my phone's two-inch screen, that my friend's relationship to his country will never be the same. And that shift, multiplied a millionfold, in turn means that our country itself will never be the same.

Tuesday, November 4, 2008

Dispatch from Wilkes-Barre, November 4, 2008

AMERICA does not repel the past or what it has produced under its forms or amid other politics or the idea of castes or the old religions … accepts the lesson with calmness … is not so impatient as has been supposed that the slough still sticks to opinions and manners and literature while the life which served its requirements has passed into the new life of the new forms … perceives that the corpse is slowly borne from the eating and sleeping rooms of the house … perceives that it waits a little while in the door … that it was fittest for its days … that its action has descended to the stalwart and well shaped heir who approaches … and that he shall be fittest for his days.

More to say when I can - in the meantime, this from the Preface to Whitman's Leaves of Grass.

Tuesday, October 28, 2008

A Simple Election Simulation

As of today, Real Clear Politics lists Pennsylvania as "Strong Obama" with a +10.8 spread (51.4 to 40.6). However, those two percentages only add up to 92%, leaving 8% independents. If all of the independents go for McCain, then the race is much closer, 51.4 to 48.6. Recently there was an article in Salon arguing that independents will, in fact, break heavily for McCain. The article cites poll numbers and voting outcomes from a few recent elections to argue that for African American candidates, "what you see is what you get." In other words, if Obama is polling at 51.4% in Pennsylvania, then on election day he'll get very close to 51.4% of the vote. Hence, Obama needs to stay above 50% in key state polls, regardless of McCain's numbers.

If we take this perspective, then the race is much closer than the spread numbers would seem to indicate.

I did a quick simulation with numbers from Real Clear Politics and Pollster.com. Here's what I did and how it turned out.

I simulated the national election 10,000 times. For each iteration, I went through the 50 states (plus the District of Columbia) one by one, and for each state, I had the computer flip a biased coin to see which candidate would win the state's electoral votes on November 4. To bias each state's coin, I used a gaussian distribution with a mean equal to Obama's polling percentage and a standard deviation of 3 percentage points. So for example, in Pennsylvania Obama's chance of victory was governed by a gaussian distribution of mean 0.514 and standard deviation 0.03. If the number drawn from this distribution was greater than 0.5, I put the state in Obama's column; otherwise, McCain's.

The results were sobering. Obama wins the election in only 70% of my simulations.


Based on the simulations, state priorities for the two candidates emerge as follows.

Obama's Priorities. I identified these as all of the states that participate in more than 70% of Obama victories in the simulation and yet have poll numbers below 52% for Obama. New Mexico is not here because it only participates in 62% of Obama victories, Ohio is not here because it only participates in 59% of Obama victories, and Florida is not here because it only participates in 30% of Obama victories.

The first number is the percent of Obama victories in which the state goes for Obama. The second number is Obama's poll number for the state.

PA, 81.5, 51.7
IA, 80.3, 52.3
VA, 76.5, 51.7
WI, 76.5, 51.8
MN, 71.9, 51.3
NH, 71.2, 51.5
CO, 70.7, 51.3

These numbers may explain why Obama is circling back to Iowa this week.

McCain's Priorities. I identified these as all of the states which participate in more than 30% of McCain's victories in the simulation and yet have poll numbers below 52% for McCain (defined here as poll numbers above 48% for Obama). Iowa is not here because it only participates in 29% of McCain victories (just below the arbitrary threshhold I set), and Georgia is not here because Obama is only polling at 44.7% in Georgia (and 100-44.7 = 55.3, a comfortable victory for McCain). Florida is not here because Obama is only polling at 47.7%, which puts McCain above 52% according to the "what you see is what you get" thesis.

The first number is the percent of McCain victories in which the state goes for McCain. The second number is McCain's poll number for the state (here defined as 100 minus Obama's poll number).

NC 81.9 51.2
OH 76.8 50.1
RI 76.7 51.8
NV 67.3 51
PA 54.1 48.3
NM 45.9 49.3
MN 44.0 48.7
CO 42.6 48.7
VA 42.6 48.3
WI 38.2 48.2
NH 34.1 48.5

The true "battlegrounds" - the states on both candidates' lists - are:

New Hampshire

The output for each state is shown below.


SportsBook.com is giving odds of 4-1 for a bet on John McCain, i.e. a 20% chance of winning. Since my simulation gives John McCain a 30% chance of winning, I figure McCain is undervalued by the betting market. Maybe I should put ten bucks on the old gambler to win.


Nothing has been proved here, of course. The whole thing is much too complicated to compute. The only thing to do is fight until the end.

Even mathematically, within the assumptions of the model, the 70% figure is sensitive to the assumption of a 3 point margin of error in the polls. Here is the sensitivity analysis:

So my result would agree with the oddsmakers if I had assumed a margin of error of 2 points, not 3. To me that sounds like cutting it fine. The reason I chose 3 points is that many of the pools have about 1000 respondents, which makes the noise level (1 over the square root) about 3 percentage points. And systematic errors are bound to be at least this large.

Saturday, October 25, 2008

Git R Done

These are the worst economic times most of us have ever seen. In my own household, I have watched the market crash wipe out most of my daughter's college fund. Fortunately, college is still a long way off.

Right now, my wife and I believe the best investment in our daughter's future we can make is to repair the country she will inherit, and preserve the liberties we want her to enjoy long after we're gone. That's why we have just contributed $1,000.00 to help make sure that Barack Obama is elected president, and not John McCain (and not Sarah Palin).

Last month we contributed $500.00. But we think it's time to double down and get this thing done. The attacks on Obama are working in Florida and Ohio - two states with 47 electoral votes between them. McCain needs these votes desperately and will do anything to win them. The Obama-Biden campaign needs contributions to reinforce the message, get out the vote, and fight voter suppression on election day.

We can't afford to lose this fight. Let's win this goddam thing.

P.S. No pressure - but in case you want to add your own contribution, you can do so here.

Monday, October 13, 2008

The NACAC Report on Standardized Testing in College Admissions

The National Association for College Admission Counseling (NACAC) recently released a major report on the use and misuse of standardized tests in college admissions. The report has gotten a lot of press, including an editorial in the New York Times.

The major tests used nationwide for college admissions are the SAT and the ACT. I was dismayed by the extent to which the NACAC report treats these two tests as interchangeable. The two tests differ in a number of ways - for instance, in their approach to math. The ACT math test measures a specific set of math skills and techniques sampled from the early high school curriculum. The SAT math test, on the other hand, demands more of the kind of mental gymnastics we commonly call "reasoning."

(I think psychometricians agree with me that the two tests "measure different constructs." But honestly, all you have to do is work some practice tests and you'll see what I mean.)

The difference between the two exams is occasionally recognized in policy: the U.S. Department of Education has allowed Illinois to use the ACT for accountability purposes under NCLB (the test was tweaked to align better to Illinois's state standards), whereas to my knowledge the SAT has not been approved for state use. Even FairTest, no friend of either exam, admits that the ACT is more closely aligned with high school curricula than the SAT.

The NACAC report nevertheless treats the two exams as identical. Here's how the Executive Summary breaks down:
Number of times the acronym ACT appears in the Executive Summary in phrases like, "the ACT and the SAT": 13
Number of times the acronym ACT appears outside of such phrases: 0
Now let's compare this treatment with the research coverage in the bibliography of the report:
Number of titles in the bibliography: 82
Number of titles in the bibliography mentioning the acronym ACT: 3
Number of titles in the bibliography mentioning the SAT, PSAT, ETS, or College Board: 41
Half of the bibliography is specifically about the SAT and related programs; only 4% of the bibliography is specifically about the ACT. The heavy weighting of the evidence base towards the SAT does not seem to justify the drawing of identical conclusions about the two exams.

By the way, one of the three specific references to the ACT in the bibliography is a document entitled, "NACAC Commission on the Use of Standardized Tests: ACT's Response and Recommendations Regarding the Critical Issue Areas Identified by the Commission." Oddly, the author of this response is misidentified in the bibliography as NACAC itself. But in case anybody wants to read ACT's reply, the document can be found here. ACT actually affirms many of the report's recommendations. I have not seen this fact noted in press coverage of the commission's report.

There is a revealing article by Scott Jaschik at Inside Higher Ed about the NACAC report and the association's meeting in Seattle. Jaschik refers to the report as "the SAT report"; perhaps he read the bibliography too. At one point he writes of the goings-on at the association meeting:
Others who spoke at the forum and elsewhere at the meeting generally agreed. Some focused broadly on the SAT, while others had specific complaints - fees charged by the College Board, a new College Board policy making it easier for students to take the SAT repeatedly without reporting that to colleges, lack of oversight of the College Board. (While the NACAC commission’s recommendations were put forth to apply equally to the SAT and ACT, most of the anti-testing rhetoric at the meeting was directed at the SAT and the College Board, not the ACT.)

The following passage in Jaschik's article caught my eye:
To applause, [NACAC member Susan Tree] cited "fundamental flaws in a test that has become one that continues to correlate more highly with family income and educational background than academic promise."
The association between poverty and test scores has been much-studied and is not in doubt. What's more, this is not simply a case of correlation without causation. Believe me: it's causation. Poverty causes low test scores. But what people forget, or maybe don't notice, or just skip past, is that this is not a case of direct causation. Think like a physicist with me here for a moment. How are test scores generated? By scanning bubble sheets. So what, physically, are the direct causes of very low test scores? I can think of two.

(1) A very bad scanner malfunction.

(2) A lot of wrong answers.

Now, I really don't think the problem is that poor school districts are ending up with all the crappy scanners. Although we maybe should look into that. After all, they get most of the crappy other stuff. But pending the outcome of that investigation, I think we must believe that what's leading to the low test scores is that poor children are offering a lot of wrong answers.

And why is it the case that poor children give so many wrong answers?

The testing literature details many factors that lead to wrong answers. Test anxiety would be one example. On an individual level, such an explanation makes a lot of sense. We all remember times when we flubbed a test because of anxiety or a bad night's sleep. But when we're talking at the aggregate level, these explanations suffer from an order-of-magnitude problem. In our poorest districts, vast numbers of students score very near rock bottom. It would take a perfect storm of high anxiety, low motivation, and misaligned bubble sheets, not to mention a citywide spate of fire alarms and barking dogs on the day of the test, to explain a phenomenon of this magnitude. No. My inclination is to stick to an obvious and simpleminded explanation. Poor children are giving lots of wrong answers because they don't know very many of the right ones.

And why don't poor children know very many of the right answers?

Well. That's the question, isn't it.

The academic underperformance of poor children is one of the most persistent and serious failures of American public education. I don't know all of the mechanisms that contribute to the problem, any more than anyone else does. But I do know that if the tests go away, then we are flying blind. Last year, 12% of African Americans taking the ACT scored high enough to indicate likely success in college math; whites were four times as likely to score at college-ready levels. (See here, Figure 3.) If the ACT goes away, then so does this damning statistic; and so also disappears our ability to chart growth in response to interventions.

When progressives argue that the tests have to go away because of the way they correlate with poverty, they are cruelly wrong. The time to drop the tests is when they no longer correlate with student poverty.

Friday, October 3, 2008

Listen to the Martingale

Just a short post to recommend Jordan Ellenberg's article in Slate about the financial meltdown. Ellenberg uses a simple explanation of martingales to illustrate the tradeoff between risk and reward.
The [martingale] is a kind of "upside-down lottery." If you play the Powerball, you'll probably lose the cost of a ticket, but you might win big. In the martingale, you'll probably win a little, but if all six numbered balls match your ticket, then the bank comes around and takes away everything you've got.

In the article, Ellenberg refers to a little-known but interesting tidbit, which I also noted in my pamphlet Randomness and Reason - namely, that the ordinary process of coin-flipping slightly biases the coin to land the same way it started (51% vs. 49% or so). The link in Ellenberg's article is to Stanford's press release publicizing the research. Here is the actual study in case anyone wants to see:

Diaconis, P., S. Holmes, and R. Montgomery, "Dynamical Bias in the Coin Toss," http://www-stat.stanford.edu/~susan/papers/headswithJ.pdf

Saturday, September 27, 2008

Gen Xers and Academia, Revisited

Jeffrey Mervis, Deputy News Editor for Science magazine, recently interviewed 26 graduates of Yale's doctoral program in biophysics and biomolecular engineering. The 13 men and 13 women in the interviews entered Yale's program in 1991, the year I myself started graduate school. Today, only one member of this cohort has tenure, and of the other 25 members, only one is even on the tenure track. When I read the interviews, I thought back to an earlier post in which I reviewed Stefanie Sanford's book, Civic Life in the Information Age. Sanford's book describes the way Generation X has redefined the responsibilities of citizenship. At the time, I wrote:
"We" GenXers emerge ... as a prickly group with an intense work ethic, a mania for effectiveness and efficiency, a hatred of talk and meetings, a pragmatic wish to find out what works, a corresponding impatience with ideology, and a risk-taking and entrepreneurial spirit. ...

...I have noticed that while many of my friends in Generation X have gone for advanced degrees, they have finally eschewed academia in favor of writing and consulting; and many have passed on advanced degrees altogether so that they could become entrepreneurs or start new organizations. In my own case, I chose a college without a tenure system so that I, and not a committee of my elders, could direct my scholarship and teaching. The days of climbing the ladder at work and paying one's dues seem to be over. Will my generation's risk-takers, who value effectiveness and efficiency, and who want to be rewarded for their performance, ever find a home in the seniority-based teachers' unions or the gasifying tenured faculties of the colleges? Or will my generation remake those institutions as soon as we get the chance?

Not all of the respondents in Mervis's interviews voiced Gen X sentiments, but a few did. Some quotes along these lines from the Science article:
Eager to apply her knowledge to helping treat and cure diseases, Nagi was chagrined to learn that her Ph.D. wasn't seen as sufficient training for a career in the field. "I was talking with someone from Merck who seemed very interested in my work," she recalls about one job fair she attended as a graduate student. "But then she realized I wasn't a postdoc, and she said, 'I don't see any reason to continue talking with you.' That's when I realized there is a prescribed path that people were supposed to follow, for no good reason."
"Even as an undergraduate in college, I never bought into the concept of being a professor," says Deborah Kinch, associate director for regulatory affairs at Biogen Idec in Cambridge. "Being a grad student is the last bastion of indentured servitude, and being a faculty member is pretty much the same thing, at least until you get tenure. Earning the same low salary and fighting for every grant--that was the last thing I wanted to do."
To Kinch, the idea of moving lockstep into an academic postdoc "started to look like a trap."
Committed to a career in academia, DeDecker has struggled to move up the academic ladder. But NIH's budget has been the least of his obstacles. "The way to get a job is to have a famous person say [about you] that 'this is the best person I've seen in the last 40 years.'
Kosa's days as an academic scientist were numbered once, as a postdoc at Harvard, he got a taste of the burgeoning biotech industry in and around Cambridge. "I wanted to understand the nonscientific side of the biotech industry, and my work was too basic to be applied," he says. So he left Harvard and enrolled in an MBA program at the MIT Sloan School of Management, turning a student project into a biotech start-up that performed liver toxicity testing with a technology billed as a "liver on a chip." Upon graduation, he counseled venture capitalists looking to invest in the biotech industry, and in 2006, he joined Bayer HealthCare before moving last month to XOMA.
That experience pushed her into the arms of industry. "If I'm going to work my butt off, I want to be in control," she recalls thinking. She posted her resume online, where it was picked up by a recruiter for a south Florida company that was setting up a molecular biology unit to develop military sensors that would detect biological contaminants in the environment. The resulting negotiations were a refreshing change from academia, she says. "I asked for double my salary [$66,000], and they gave it to me," she says.
He also prefers what he regards as the greater transparency of commercial research. "In academia, some of the papers are not reliable and the findings are not reproducible," he says. "In industry, if you don't make a very good antibody, there are tests that can prove it doesn't work. So you have to be honest."
In reading the article I registered a skepticism about academia and its ethics...a desire for competitive compensation for hard work and results...an unwillingness to please one's elders and jump through hoops to advance one's career...and a dissatisfaction with a training program that only fits you for one career path.

That said of course, the one member of the cohort who does have tenure is obviously also from "Gen X." And while I have plenty of friends who've stepped off the path, I also have plenty of friends in tenured professorships. Unlike many of the people Mervis interviewed, they were all certain that tenure was something they'd want if they got it. I suspect my friends were also different in that they were highly confident they would be awarded tenure if they made the attempt. The people in Mervis's article don't emphasize this particular calculation, but it doubtless factored into their decisionmaking.

Friday, September 19, 2008

From The Onion

Obama Feels Like He's Taking Crazy Pills

Obama accused Palin of misleading voters by portraying herself as a politician who has disdained the special spending projects sought by some lawmakers, known as earmarks. "When you've been taking all these earmarks when it's convenient and then suddenly you are the champion anti-earmark person, that's not change," Obama said. "Come on. I mean words mean something. You can't just make something up."

Obama was then heard to say under his breath, "Jesus Christ, I feel like I'm taking crazy pills."

OK, I admit it, this isn't actually from The Onion. The first paragraph is actual reportage from Yahoo! News. I made up the second paragraph (and the title) myself.

Friday, September 12, 2008

The Physics of Democracy

Austrian physicist Wolfgang Pauli once ridiculed a colleague's theories by calling them "not even wrong." But trust me, being wrong is worse. I was horrified two or three years ago to find an error in one of my own published papers, more than a year after it had appeared in American Journal of Physics. A cooling effect I had estimated as being roughly 110 millikelvin in magnitude actually appeared in the published paper as 11 millikelvin, because of a typographical error. (The error was mine; I went back to my files and saw that the mistake had already been present in the manuscript submission.)

I quickly published an erratum. This was embarrassing, but of course I had no choice. How could I live with myself if I knowingly allowed such a bald error to remain in print under my name? This is not to mention that the only thing more embarassing than pointing out the error myself would have been for somebody else to beat me to it!

Being right about the math is important, and we physicists pride ourselves on always getting the order of magnitude right. But being right about the phenomena is finally what matters. And that was why I was actually eager to publish the erratum: The additional digit makes the effect ten times easier to measure, and thus it became more likely that an experimentalist might someday try to verify (or disprove) my predictions. I think I finally have the math right, but if nobody ever performs the experiment, then I'll never know whether my calculation passes the most important test: that of correspondence to the facts.


Political campaigns resemble the enterprise of physics in some ways. Campaign strategists are like theorists trying to understand the electorate. They build models and crunch all kinds of numbers. On the day of the election, the big experiment is performed. Once the data are collected, we see whose numbers (if anyone's) are borne out.

Another similarity is that according to the standard interpretation of quantum mechanics, atomic-scale measurements don't straightforwardly reveal pre-existing qualities of the atom; and likewise, I would doubt that a question on a poll always reveals firmly pre-existing thought patterns in the minds of people being surveyed. In both cases, the act of questioning creates some of the reality. Maybe there are even pollsters who adopt an instrumentalist interpretation of the electorate, just as physicists sometimes did back in the 1950s. On the instrumentalist view of campaigning, you don't even try to figure out what the beast is really thinking; you just try to learn which inputs lead to which outputs. "Pinch and poke" the electorate, and take careful notes on how it responds.


The McCain/Palin campaign's study of the electoral beast appears to have led them to the conclusion that victory depends on shoveling garbage at us as fast as they can. Paul Krugman had a good piece on this on September 11, 2008. A reader made the following comment on Krugman's article:
A democracy cannot function in an atmosphere of dishonesty. The point of popular consent is that the people make a reasoned judgment about their leaders and select those that seem most likely to provide what they are looking for. Political dishonesty is inherently undemocratic. It is an attempt to cheat the voters out of their right to make an intelligent choice. Liars should be rejected for that reason alone. — S MacDougall, N. California
This comment brings us full circle back to physics, although at first sight it's not obvious why. To begin with, here is psychologist Emily Cahan discussing the influential American philosopher John Dewey:
The problem of education for Dewey is to discover the means for making scientific thought more widespread in society and giving it a deepening hold among people—for extending his ideals for individual growth in the classroom into society.
. . . if scientific thought is not something esoteric but is a realization of the most effective operation of intelligence, it should be axiomatic that the development of scientific attitudes of thought, observation, and inquiry is the chief business of study and learning.
In schools,
we want that type of education which will discover and form the kind of individual who is the intelligent carrier of a social democracy.

For it will have come about that education and politics are one and the same thing because politics will have to be in fact what it now pretends to be, the intelligent management of social affairs.

It is interesting to contrast this vision of education with that contained in the education chapter of the Republican platform:
Education is a parental right, a state and local responsibility, and a national strategic interest.
(This is the first paragraph in its entirety.)


Dewey wrote on education during the early 1900's. More than a century earlier, Laplace was led to similar thoughts by his investigations into probability theory. As we now understand much better thanks to R.T. Cox and E.T. Jaynes, Laplace extended probability theory far beyond its roots in the mathematics of gambling, building it into nothing less than a set of principles for any sort of reasoning in the face of incomplete information - that is, any sort of of scientific reasoning - that is, any sort of clear thinking in situations with consequences. Laplace described for the first time how evidence should be used to modify our beliefs, sometimes strengthening them when the evidence lines up, but sometimes weakening them when the evidence turns the other way. But this process can only work if we develop certain habits: the habit of seeking evidence in the first place; the skill and inclination to weigh it critically; the flexibility and integrity to change our minds and admit when we are wrong. These habits are not natural, but must be developed through education. Laplace:
It is remarkable that a science, which commenced with the consideration of games of chance, should be elevated to the rank of the most important subjects of human knowledge.

If we consider that [this science] teaches us to avoid the illusions which ofttimes confuse us, then we shall see that there is no science more worthy of our meditations, and that no more useful one could be incorporated in the system of public education.
The world is harsh, and those who subscribe to illusions are eventually crushed by their failure to respect the facts. That's why we so desperately need more honesty from our leaders. That's why McCain's lying campaign not only insults our intelligence but threatens our country.


Here I am, piling blame on the Republicans. Using some high-falutin names as weapons. (And one of them a—a Frenchman.) I know that many of my friends (and not only the Republicans among them) will say that both parties lie to get elected; that all politicians lie all the time. The trouble with this position is that it puts all lies on the same footing. Take for example the lies that got us into Iraq. Those lies, and all of the weak evidence I'm embarrassed to admit I fell for too, have cost us hundreds of billions of dollars and thousands of American lives, while bringing Al Qaeda to Iraq and keeping our eye off the ball in Afghanistan. My nephew and his wife, and doubtless all of America's soldiers, have made their sacrifices willingly. Bravely. Heroically. But the tragedy is that their bravery and their heroism finally do not make their sacrifice worthwhile. We should be furious at the party that led us down this painful road.

To my knowledge, only Barack Obama among the major presidential candidates has said flat out in print that invading Iraq was a mistake. And he said has said it recently, in the heat of this campaign, and he has said it as loud as one can say such a thing: on the editorial page of the New York Times. Here are the first two sentences of "My Plan for Iraq":

Unlike Senator John McCain, I opposed the war in Iraq before it began, and would end it as president. I believed it was a grave mistake to allow ourselves to be distracted from the fight against Al Qaeda and the Taliban by invading a country that posed no imminent threat and had nothing to do with the 9/11 attacks.

This is some of the straightest talk I've ever heard from a presidential candidate. Obama may bob and weave like any politician, but in comparison to John McCain he is a philosopher-king. If it should happen again any time soon that our President leads us to war in Iran, North Korea, or anywhere else, then I want that President to be Barack Obama. He may make mistakes. He may get some things wrong. But he won't lead us to war by manufacturing evidence and lying to us.

Thursday, September 11, 2008

Monday, September 1, 2008

Back-of-the-Envelope Statistics

I recently gave a half-hour talk on statistics for non-experts, which sounds like a half-hour visit to hell for everybody concerned, but it was actually a lot of fun. The audience was sharp and engaged, and I think everybody got something out of the presentation.

In the talk, I was trying to achieve three things. The first goal was to give my audience a gut-level feel for random fluctuations and signal-to-noise ratios. I wanted to convey the same kind of rough, theorem-free intuition that physicists commonly use when approaching noisy data. The second and most important goal was to help my audience differentiate clearly between the "statistical significance" of a measured effect vs. the size of that effect. This is a distinction that the press almost never makes when reporting about scientific research. For that matter, even scholarly authors tend to equivocate on the different senses of the English word "significant" when it helps them further their agenda. Finally, I wanted to show people a toy model of a multiple regression, so that they could at least know what the experts are talking about when they present model coefficients or discuss "the problem of missing variables" in correlational studies.

In case anybody's interested, the Powerpoint document is here.

N.B., the presentation includes a map from American Scientist with no written explanation alongside it; that's because I spoke about the map off-the-cuff. The map is from an article by Howard Wainer that appears here; I also discuss Wainer's map in my pamphlet "Randomness and Reason", available here.


Danimal's Game

Just in case statistics isn't everybody's idea of fun, I thought I'd also take a moment here to develop a comment that Danimal made on an earlier post. Danimal suggested an interesting variation of the standard change-a-word game:
Find a route from one four-letter word to another, but making changes only between neighboring letters in the alphabet, i.e. "b" can only become "a" or "c".

For example, we are allowed to change BELL to CELL, because B and C are neighboring letters in the alphabet. But we can't change CELL to TELL, because C and T are not neighboring letters.

As Danimal pointed out, the new constraint makes it difficult to design instances of this game by hand. A well-designed game, by the way, should satisfy the demand that the starting word and the ending word share no letters in the same position. The BELL example now shows the major difficulty: how are we ever going to get away from that E? Vowels aren't adjacent to one another!

This suggests the strategy of "pivoting" on a double-vowel combination, as in the following three-letter example:

SOY [note, we now have two vowels to work with]
SPY [thus we can pivot from the O to the second vowel]

Admittedly SOX is not a very appealing word, but this was the best I could do by hand.

Using the computer, I did manage to find a (single) viable four-letter instance of Danimal's game:


(A hint is here and the solution is here.)

Graphically speaking

A fair move in Danimal's game is a fair move in the standard game, but the converse is not true. This means that Danimal's rule deletes edges from the standard connectivity diagram (over 90% of them). In other words, the adjacent-letter constraint shatters the standard game's supercontinent of words into hundreds of tiny atolls. The two largest atolls have twelve words each, as shown in the graphs below (spoiler alert!):

The movie below gives a fast review of all of the sizable atolls in Danimal's game. (A clearer version of the movie is here.)


People were kind enough not to send me any congratulatory emails last June 7th, which was roughly when I passed the halfway point of my table of American male life expectancy (see my earlier post on "The Prime of Life"). Today I'm 39 and officially celebrating the beginning of Life, Part II. Now: will it be the motorcycle or the convertible?

Sunday, July 27, 2008

The gender gap in math has disappeared.

A major paper published this week in Science is worth a look.

During the years from 2000 to 2004, I helped to start an education technology company called the Grow Network. In the early days, we sat at makeshift desks in a converted cabbage warehouse, blowing on our fingers to keep them warm. Eventually we became a large and successful company, serving millions of students across the country. McGraw-Hill Education acquired us around the time I returned to academia, and my understanding is that they heat the building in winter now.

As the company's Vice President for Education and Product Development, I led psychometrics for Grow, which meant that I routinely worked with large-scale assessment data for hundreds of thousands of students in grades 3-8. From time to time, I wondered about the gender gap I had heard about between boys' and girls' mathematics achievement. But when I analyzed the raw data myself, I found no obvious effect. Naturally, I wasn't conducting anything like a careful study.

Now in a very significant paper, Hyde et al. have shown convincingly based on multi-state data for millions of students that - whatever used to be the case in decades past - there is no gender gap today. (Gender Similarities Characterize Math Performance, by Janet S. Hyde, Sara M. Lindberg, Marcia C. Linn, Amy B. Ellis, Caroline C. Williams, Science 25 July 2008: Vol. 321. no. 5888, pp. 494 - 495, DOI: 10.1126/science.1160364)

I hope that policy and advocacy groups will run with this message and work it into the culture of K-12 education - and American culture as a whole. I'll certainly look for more opportunities to spread the news.


Here begin some more detailed responses to the paper. Take these for what they are: the first responses of a reasonably intelligent non-expert.

The Hyde et al. paper is about two different things. The first concern of the paper is with math achievement in the mean. Here the findings are clear and straightforward: there is no math gap today, in the mean. This result forms the paper's headline: "Gender Similarities Characterize Math Performance."

The second concern of the paper is much more complex, having to do with gender disparities in the science and engineering pipeline that leads from college majors, to Ph.D. recipients, to tenured positions in academia. The gender gap tends to get more pronounced the further down the pipeline you look. Most notable is the scarcity of women in tenured positions in top-25 research universities.

The gender gap downstream is a very complicated problem, or set of problems, and frankly I do not think that children's math scores are the best place to begin thinking about them. Leaving aside my own instincts, however, it has certainly been suggested in the past that gender differences in math ability at the high end may go some way towards explaining the "tenure gap." The idea is that the means for males and females could be exactly the same, and yet if the variability were higher for males, then males would predominate at the high end of the achievement distribution (as well as at the low end, presumably!). Exactly how this disparity in achievement at the high end is supposed to contribute to the tenure gap is never spelled out, but it is at least an empirical question whether children's scores do or do not exhibit a gender discrepancy in variability (in either direction).

Hyde et al. examined the high end of the score distribution for their data set. They did indeed find greater variability in male students' scores (see graphic further below). This augurs for overrepresentation of males at high percentiles.

Actual gender ratios at high percentiles were not published for the full data set, though the researchers noted that male 11th-grade students in Minnesota did tend to be overrepresented at the 99th percentile and above. (Minnesota 11th-grade was the only state/grade combination for which actual gender ratios were quoted above the 99th percentile, although no reason was given to suspect that Minnesota grade 11 was atypical; the greater variability in male students' scores was significant across the full range of states and grades analyzed, as shown in the graphic further below).

I don't really understand the data on page 494 as they relate to the variance ratios quoted for Asian/Pacific Islanders in Table S2, but for the white students at least, I would call the tail effect substantial. In Minnesota at 11th grade, there was a 2-to-1 ratio of males to females at the 99th percentile and above. The authors note (p. 495) that,
If a particular specialty required mathematical skills at the 99th percentile, and the gender ratio is 2.0, we would expect 67% men in the occupation and 33% women. Yet today, for example, Ph.D. programs in engineering average only about 15% women.

In the conclusion of the paper, this observation is rendered as follows (p. 495):
There is evidence of slightly greater male variability in scores, although the causes remain unexplained. Gender differences in math performance, even among high scorers, are insufficient to explain lopsided gender patterns in participation in some STEM fields.

(One person's "substantial" is another person's "slight.") I think anybody would agree with the authors that a 2-to-1 ratio at the 99th percentile is insufficient to explain the lopsided gender patterns we observe today. 67-33 is simply not the same as 85-15. But the authors are clearly doing their rhetorical best to portray this glass as half-full; 67-33 is not the same as 50-50 either.


The primary result of the paper - the lack of a gender gap in math achievement - looks ironclad. The secondary finding, that of greater variability among males, is also obviously incontrovertible - whatever we may think about its relevance to the pipeline problem. The variability effect is probably not new, but this paper, based as it is on the scores of millions of students ranging in age from 7 through 18, provides a convenient point of reference and a handy estimate (variability ratio male:female = 1.16).

(Click to enlarge. I created this histogram in Excel by hand-keying the data in Table S1 in the Supplemental Online Material. Two outliers have been dropped to keep the scale compact, specifically 1.76 and 2.39. The green shading indicates ratios less than unity. N=64 observations are shown.)


Hyde et al. also looked at performance on difficult questions as another lens on the problem. They found (p. 495) that
At grade 12, effect sizes for [hard] items ranged between 0 and 0.15 (average d= 0.07). At grade 8, effect sizes for these items ranged between 0 and 0.08 (average d= 0.05). Thus, even for difficult items requiring substantial depth of knowledge, gender differences were still quite small.

This shows that gender differences were small in the mean even for hard problems - a significant finding. But now the tail question arises once more: What about gender disparities at the high end of performance on the "subtest" of hard questions? Is the variability ratio for the hard subtest similar to the 1.16 quoted elsewhere in the paper? What male:female ratio do we find among those achieving perfect scores on the difficult subtest? I think this is an important question - possibly much more important than the overall variance ratio given in the paper - because, having looked at many standardized tests during my time at Grow, I have long known that many of the questions on accountability exams are "gimmes" that are not exactly selecting our future star researchers. Less anecdotally, Ginsburg et al. (p. xiii) reported in 2005 that (emphasis added)
The questions on Singapore’s high-stakes grade 6 Primary School Leaving Examination (PSLE) are more challenging than the released items on the U.S. grade 8 National Assessment of Education Progress (NAEP) and the items on the grade 8 state assessments. ... Singapore’s most challenging questions are designed to help Singapore identify the best students. These are more difficult than the most challenging questions on the state grade 8 assessments as well as on NAEP.

For this reason I was surprised by the tone of surprise in the paper's final sentence (p. 495):
An unexpected finding was that state assessments designed to meet NCLB requirements fail to test complex problem-solving of the kind needed for success in STEM careers, a lacuna that should be fixed.

The net result of all this discussion might be that PISA or TIMSS could be a better way to investigate the STEM pipeline than U.S. exams (whether state or NAEP).


The facts reported by Hyde et al. are facts, of course, but even so, all of this research seems backwards to me. Instead of sifting through the tea leaves of seven-year-olds' math scores, shouldn't we be looking directly at the traits of successful scientists? Though we'd probably find fairly high math ability in this group, I think we might find even greater deviations in certain relevant dimensions of personality. Think for example about the way curiosity factors into a scientific career. Or creativity. Then there's ambition...the ability to concentrate for extended periods of time...relative freedom from family responsibility during the 20's and 30's...and a level of perseverance and attention to detail bordering on the obsessive-compulsive. Or who knows - maybe scientists would turn out to be just like the rest of us.

Saturday, July 5, 2008

A Sound of Thunder

Most of the windows in our house look out across the little valley that separates New York's Taconic Range from the Green Mountains of Vermont. The elevation of the house, 1250 feet, gives us an eye-level view of approaching thunderstorms. On the hottest summer days, thunderheads gather on the far western horizon all afternoon, bunching higher and higher, until grey misty clouds spill over into the valley. The grey raft floats towards us, dropping a curtain of rain at its leading edge. Standing in the living room and looking out to westward, I can almost imagine I'm sailing into a squall.

Lightning occasionally strikes the marsh down below the house; the picture below shows a lightning bolt from a spectacular electrical storm just a few days ago.

This strike was about 300 yards from the house. We've had closer. A few years ago, our well pump motor was apparently fried by lightning that must have struck one of the trees on the south side of our yard. The current flowed into the ground and jumped to the wiring that leads into our mechanical room.


A couple of weeks ago, I was awakened during the night by a storm. The thunderclaps were loud, but it was actually the brightness of the lightning that disturbed my sleep. I lay for a time in the intermittent dark, calculating distances and velocities from the delay information, when it occurred to me that there was something peculiar about the sound of the thunder. Mixed in with the sounds of the storm, I heard a regular beat of deep booming sounds spaced a few seconds apart. The uniformity of the spacing in time marked the sounds as unnatural. It was then that I remembered an article I'd read a few days earlier in the local newspaper: Hail Cannon Stirs Complaints.

The regular booms I had heard were those of a "hail cannon." You can google around to learn more about these devices, but the point of them is to emit a loud sonic boom every few seconds, the goal being to suppress the formation of hailstones. Hail cannons were popular in Europe during the 1890's and 1900's, but they were gradually abandoned after they were perceived to be ineffective. Hail cannons have made a comeback in the last decade or two, and this year a large orchard in Bennington has purchased one and put it to use, causing dozens of complaints in Bennington and neighboring towns.

It is unlikely that hail cannons actually work. According to a 2006 paper by Dutch meteorologists Jon Wieringa and Iwan Holleman in Meteorologische Zeitschrift, the few publishable experiments that have been carried out yielded results ranging from mostly negative to inconclusive at best. And there appears to be no known meteorological mechanism that would lead one to believe in their effectiveness a priori. Wieringa and Holleman conclude that "the use of cannons or explosive rockets is waste of money and effort." The Commission for Atmospheric Sciences Management Group of the World Meteorological Association also declared in 2007 that "...hail cannons...have no physical basis and are not approved."

Nevertheless, hail cannons are growing in popularity. In one sense, this is understandable. Hail is a very destructive force, costing billions of dollars of damage in the U.S. each year. And premiums for hail insurance are apparently extremely expensive. Caught between a hailstone and a hard place, a farmer (especially a gullible one) might view a $30,000 hail cannon as a lifesaver.

Of course, even a correct calculation about the risk of hail, the cost of insurance, and the cannon's own cost and effectiveness ignores another relevant factor, the property rights of the neighbors who are being rattled every time the cannon fires. In Bennington, a noise ordinance prohibits the cannon from being fired at night, but the orchard has violated the ordinance twice. The Town may be considering litigation against the orchard, and the neighbors may be doing so as well. In any event, I think it will take a judge to balance the perceived and real interests of the orchard against those of the surrounding property owners.

A healthy agricultural sector is in everybody's best interest. For this reason, some residents of the Town support the hail cannon, which the farmer has said he believes in "100 percent." But how can it be good for agriculture when a farmer spends $30,000 on a machine that in all likelihood fails to protect his crops? Would we applaud the farmer if he had spent $30,000 to have the Rite of the White Tiger performed on his property? Tibetan farmers used to believe in this anti-hail ritual 100 percent too. But belief alone didn't make it effective at preventing hail.

It can't be pleasant for farmers and car dealerships to have no good options for dealing with hail. But a noisome gimmick isn't a good option either. Perhaps Vermont and other states could consider setting up a taxpayer-subsidized hail insurance program. After all, insurance is a proven method for dealing with Acts of God. And it's also very quiet.

Sunday, May 18, 2008

Randomness and Reason

This past term, it fell to me to teach the perennial service course in statistics at Bennington College. In addition to covering various standard topics (descriptive statistics, significance testing, correlations, etc.), I spent a great deal of time with my beginning students on the theme of probabilistic reasoning based on evidence, as codified by Bayes's Theorem. I needed to do this if we were going to enter the richly confusing space between randomness and reason appropriately armed. I have assembled some of the most informal of the course lectures here. By way of apology for the overall state of things, I confess that during a busy term I struck off some of these mini-essays during the wee hours of the morning before class. (I thank my students for giving me a lot of room to experiment and improvise during the semester.) The lectures are collected with the vague idea that I might fix them up at some point in the future. Any comments are therefore very welcome.

Saturday, May 17, 2008

Four-Letter Words

The Change-a-Word is a staple of supermarket puzzle books. In Change-a-Words, you have to change a word like GULF into a word like YAWN by changing one letter at a time. (So you might say: GULF, GULL, BULL, BURL, BURN, BARN, YARN, YAWN.)

These puzzles are sort of fun, but they are mostly amusing for the variations they inspire (see e.g. my 75th Anniversary Lecture), and for the logical and linguistic questions they raise. For example,

1. Can most words be reached from most words?

2. Which words are "furthest" from each other? Specifically, if the shortest chain of words connecting A and B has length N, then which words A and B maximize N, and what is this maximal value of N?

I consulted my handy electronic dictionary to answer these questions.

As to question 1: For four-letter words, it appears to be the case that about 95% of the words are connected in a massive "supercontinent." This means most four-letter words can indeed be reached from most four-letter words - hence the steady stream of supermarket Change-a-Word books.

The 5% of four-letter-words lying "offshore" are either isolated "island words" (can you think of any?), or isolated pairs of words (such as IDOL-IDYL), or isolated triples such as OGEE-OGRE-OGLE or the distinct case IDLE-IDLY-ISLE, etc. The largest of the offshore islands is the eight-word "atoll" {QUAD, QUAI, QUAY, QUID, QUIP, QUIT, QUIZ, QUOD}. (See figure.)

As to question 2: In my dictionary at least, the Change-a-Word puzzle that requires the greatest number of steps to solve is this one:


These two words lie at opposite ends of the dictionary, so to speak. I'll leave it to the reader to find a chain of words connecting them! A hint is available here, and the solution is available here.

P.S. One assumes that 15-letter words are not concentrated in a massive continent...but I haven't looked at those yet!

Saturday, April 26, 2008

Some puzzles for fun

1) Homophones are words such as main and mane that are spelled differently but pronounced the same. Recently I decided that the coolest homophones are surely those that contain their own homophones. (Maybe call them "isophones"?) An example would be buss, which contains its homophone bus. How many examples like this can you think of?

2) A quickie. Starting with the ten-letter word STREAMBEDS, remove a letter to yield a nine-letter word. Then remove another letter to yield an eight-letter word. Continue in this way until you have a one-letter word.

3) The same as above, starting with INSOLATING.

4) Say that a letter is "special" if it can be used together with the letters A, F, and W to form a four-letter word. (You are allowed to rearrange the letters.) Find all of the letters that are "special" in this way. Then rearrange the special letters to form a new word.

5) The same as above, using the letters I, C, and L in place of the letters A, F, and W.

6) (Some math knowledge is required for this one.) I'm thinking of a game with a curious feature. The probability of winning the game once is exactly the same as the probability of losing the game a million times in a row. Are the chances of winning this game one in a million, greater than one in a million, or less than one in a million?

(Math wizards may wish to show that if "a million" is replaced by a large number N, then the probability of winning is given approximately by W(N)/N, where W is the Lambert W-function.)

7) This one is based on the Paradox of the Liar (background here). The classic paradox is the sentence "This sentence is false." The sentence is paradoxical because we cannot assign it a truth value that respects its meaning. Also well-known is the paradoxical pair of sentences, ("The next sentence is true", "The previous sentence is false.") Again, there is no way to assign truth values to the sentences in a way that respects their meaning.

One day a couple of years ago, it occurred to me that these two famous examples can be seen as instances of a general pattern. We consider an ordered n-tuple of sentences (S1, S2, ..., Sn) in which each sentence asserts the truth or falsehood of the next. The last sentence closes the loop by asserting the truth or falsehood of the first. Under what circumstances is such a collection of sentences paradoxical?

As a warmup exercise, you might want to verify that the following pair of sentences is not paradoxical: ("The next sentence is false", "The previous sentence is false.") There is no paradox here because we can read the first sentence as true and the second sentence as false; this reading respects the meaning of the two sentences.

(For another take on this, see my vanity-published puzzle book Word Puzzles for the Seriously Smart.)

Enjoy! And feel free to put answers or partial answers in the comments (if anyone's reading this, that is.)