The Cowboy Comedian and the Statistician

In the 1930’s, in the midst of a world wide economic depression, windstorms swept across the American Midwest and the two forces in tandem caused thousands of farm families to lose their homes. Many of them packed everything they could into farm trucks and migrated west to California where there were jobs, almost slavery in honesty, to be had in the rich farmlands. Although they came from all over, they were commonly referred to in the west as Okies, a nickname that has been common to the people from Oklahoma since the beginning of the 20th century.


It is said that the famous Oklahoma cowboy comedian, Will Rogers, commented on this migration by saying, “When the Okies packed up and moved to California, they raised the average IQ of both places.” Now the occurrence of just such an effect in statistics is a concern, and it has often been labeled the Will Rogers effect.


So how does that happen, here is a very simple example using easy numbers. Suppose a school has nine students, whose scores on a state exam last year were 60, 70, 80, 90, 100, 110, 120, 130, and 140. The school has identified students with scores below 100 as “under performing.” They report last year’s results with an average of 75 for the underperforming group, and 120 for the regular students. This year, they are introducing a new teaching approach for both groups, but along the way, someone decides that students scoring 100 will be moved to the underperforming group. This year the test results are 60, 70, 80, 90, 100, 110, 120, 130, and 140. The numbers look familiar, right? But now the school reports the average for the five underperforming students as 80 points, and for the regular students as 125.


Wow, Obviously the new strategy works, both groups improved by five points! But we know the truth the averages don’t tell, the scores are identical. Moving Joe average with the 100 point score from one group to another managed to raise the score of both groups, as Will Rogers had suggested.


It really does happen in real life, and statisticians have to be on the look out. In medicine for instance, we discover new methods of detecting illnesses, such as cancers, earlier and earlier. Now imagine that we have two groups, an “at-risk” group because they have been identified with a disease or disorder, and the “healthy” group. Usually they will have a lower life expectancy than the normal, not at-risk group. Now introduce a new identification measure into the mix. We begin to identify more people who have milder forms of the illness, and move them to the “at-risk” group. They probably have a shorter life span than the average of the “healthy” group, but they probably also have a longer life span than the “at-risk” population. This “stage migration” as it is called, will lead to an improvement in the life expectancy of BOTH groups, even without any treatment improvements. The person researching a new intervention will compare the hospitals results to those maybe five years ago (before the stage migration) and conclude that the new treatment is effective… NOPE, just the Will Roger’s effect.


Here is a footnote to the story about the Dust Bowl migration. The movie “Grapes of Wrath” was a harrowing depiction of the treatment of the Okies, and in general of Man’s inhumanity to man. I was told by a history professor in my youth that the Russian government thought this would be a great opportunity to expose to their people the impending collapse of capitalism by showing the movie in Russia. They offered free screenings all over the country, and millions of Russians saw it. And what was the result? It seems that they were desperate to get to America. Instead of the lessons their leaders had hoped they would see, most of them seemed to pick up on two facts. The poorest people in the US owned cars (almost no Russian below party level had a car) and they were free to pack up and move across country if they wanted. The law of unexpected consequences strikes again. I really don’t know how factual that story is (Gasp!!! Teachers may lie???) but it is a great story, true or false.

Missed Opportunities

Thinking back over my recent blog on moving west, I realize I have missed an opportunity to make a point that tends to be missed when folks talk about statistics. I have just sat through a number of repeated meetings about education and school improvement in which people tried to compare complex distributions of students and subjects by comparing them with a single number. That single number was almost always an average of some kind, although the distinction between mean and median tends to be lost in the drive to have "a number" to use to compare distributions. It seems that people who would question anything you say will suddenly fall silent at statements like, "15 is more than 12, thus God exists." In the nation of the mathematically insecure, the person who quotes numbers has great power. Even if the numbers are purely "Potemkin" numbers created on the spur of the moment... "I have here in my hand a list of 205—a list of names that were made known to the Secretary of State as being members of the Communist Party and who nevertheless are still working and shaping policy in the State Department".
We are a people in love with, and easily duped by statements of quantity. Averages are our favorite pieces of mis-information. But the center, the average location of population, for the 300,000,000 plus US citizens is in an empty field with the nearest residence over 100 yards away (my estimate, not an exact measure). The center of mass of the US population is near a city with a population of about 65 people in an area with a population density of about ten persons per square mile.

It is in some ways very typical of the wide planes areas to the west, which with exceptions in a few big cities have population densities from 0-10 persons per square mile. Only a few miles to the north-east, south-east or north-west the population density rises to over 1000 per square mile in St. Louis, Memphis, and Kansas City.

About 1/3 of the total population of the United States lives in one of ten urban centers, and almost one in ten live in one of the two population centers of New York/Newark in the East and Los Angeles/Long Beach in the west. Both are a long way in both physical and metaphorical distance from the quiet little field south-west of Plato, Missouri.

The US population that is nestled mostly in areas of high density is centered in a grassy field beside a tiny stream near a small quiet village in a very sparsely populated rural area.

Centers are generally a weak measure to describe the great variability of a population in any measure. It's not that averages are bad... it's just that...well, averages ARE bad.

Once More with the Cart Before the Horse

If you read the headlines you would think that Viagra causes Sexually Transmitted Diseases (STD). "Sex Diseases Tripled in Men 40 or Older Taking Viagra, Cialis, Study Says". The Sun Times in Chicago headline read, "Older Viagra Users More Likely to Get STDs".

For teachers of statistics, this is one more case of correlation caused not by causation, but a common cause.

I found the article at a George Mason web site, which points out that the media headlines missed the point.

"Media coverage reveals a classic confusion between causation and correlation, as they implied that Viagra results in greater risk for older men. But if anything, the study suggested just the opposite: Men who are interested in Viagra have riskier sex lives.."

The beauty of data visualization

"David McCandless turns complex data sets (like worldwide military spending, media buzz, Facebook status updates) into beautiful, simple diagrams that tease out unseen patterns and connections. Good design, he suggests, is the best way to navigate information glut -- and it may just change the way we see the world."

A must see for AP stats students.... but be careful..the mind sees patterns...we WANT it to see patterns, and it will... even if????

Standard Deviations of Sums of Distributions

A week or so ago I was at a textbook selection conference with a couple of really good teachers, and one of them (thanks, Dru) pulled out a copy of Robert Hayden's, "Advice to Mathematics Teachers on Evaluating Statistics Textbooks." I mention it now because it has two good pieces of advice. (Ok, it has way more pieces of good advice than that, but I'm mentioning these two in particular)

The first, I hope I follow, "...make sure the textbook mentions assumptions and teaches students to check them rather than make them." One of the ways I try to get students to check assumptions is to make them understand, as much as possible in the limited time of a AP course, the WHY. In order to do that, I frequently violate one of Professor Hayden's other pieces of wisdom; "Be wary of an author who is not familiar with enough real data sets to illustrate a textbook."

Ok, I'm gonna claim some "weasel" room here. First, I'm thinking more along the lines of an exercise to help the students understand why checking independence is so important, and not writing a textbook. Second, even Professor Bob himself says "While there may be places (such as Anscombe’s regression examples , in which a skillfully fabricated batch of numbers illustrates a pedagogical point,.." Ok, so the "skillfully" may not apply to what follows, but I hope the fabricated data at least help drive home a "pedagogical point".

I begin with two simple data populations, X= {1,1,1,2,2,2,3,3,3} and Y= {1,1,1,3,3,3,5,5,5}. Students who have learned the "Standard Deviation as Distance" approach can quickly check and find the standard deviation of the X population (or using a calculator) is sqrt(2/3)or appx .8165. For Y the std. dev. is 1.633. Perhaps for what we will be doing, we remind them that the variance of each is the square of the standard deviation, so Var(X)=2/3 and Var(Y)= 8/3.

So what happens if we add or subtract the populations? It all depends! If the populations are independent, then any X and any Y may (must?) be associated with equal probability. I illustrate this by pairing one of each X value with one of each Y.. (is it possible to have two distributions be independent without this type of each x with each y association?)

X___1___1___1___2___2___2___3___3___3

Y___1___3___5___1___3___5___1___3___5.

and the sum and differences are then

X+Y =2___4___6___3___5___7___4___6___8 and

X-Y =0__-2__-4___1__-1__-3___2___0__-2

I think it is worth drawing the two resulting distributions because many students will NOT see that these are distributions are reflections of each other. So they should have exactly the same standard deviations (this takes a moments reflection for some students).




Wow, that's good news. If the populations items are independent of each other in the way they are combined, it doesn't matter if you add them or subtract them, the spread is the same since the two distributions are symmetric, which means the standard deviations should (and are) the same, about 1.8257. Even better, we can point out that the variance, 10/3, is simply the sum of the original variances, 2/3 + 8/3. For me it is worth pointing out this "Pythagorean" relationship, [StDev(X+Y)]²=[StDev(X)]²+[StDev(Y)]², IFF X and Y are independently associated.....(oops, I have been called out on this mistake... The statement is true IF x and y are independent, but also in any situation in which the correlation coefficient is zero... which does not necessarily require independence...see comment from "gasstationwithoutpumps" below... "mia culpa" and thanks to "gas..."

BUT... what if the original populations were NOT independent. (quick, think of two data sets that you would really combine in real life that are totally independent...better yet, send your ideas in the comments)

Well they might have a positive or a negative correlation, so we slightly rearrange our data sets and group lower numbers somewhat together (no Ones with the fives) like this..

X___1___1___1___2___2___2___3___3___3

Y___1___3___1___1___3___5___5___3___5.

Now our sums and differences are

X+Y =2___3___2___3___4___5___6___5___6 and

X-Y =0__-1___0___1___0__-1___0___1___0

We recognize quickly that the sets no longer have the same shapes. The distribution of sums is almost uniform with the peaks at the ends, while the difference distribution has two peaks closer to the center .

So what are the spread measures now. The standard deviation of the summation distribution is 2.26 or the square root of the variance of 46/9. The differences have a standard deviation of 1.247, the square root of a variance of 14/9, a really big difference. In fact, we help the students notice that the variances are the same distance from the equal variance of 10/3 = 30/9 when the populations were combined independently. The distribution of sums variance is 16/9 higher, the differences are 16/9 lower. Is this just a curious coincidence...(by now my students know that almost NOTHING I bring up is a "curious coincidence" ).

So how can we explain this difference. Slowly you lead their thinking...."If the distributions are NOT independent, they must be dependent,.... and there must be some relationship,..... some measure of how UN-independent they are." Eventually they will think of the correlation coefficient, r. In this association between X and Y they have a positive correlation of 2/3 ... can that help. If the relationship when the association was independent is "Pythagorean", maybe we can look for some extension of the Pythagorean theorem to help... Can we find something like the Law of Cosines that would tie the package together? After all, we need something that will add 16/9 to the sum distribution, and subtract the same amount for the differences... I can't imagine that I would have kids who would see this, and will probably lead them to observe that StDev(X+Y)=[StDev(X)]²+[StDev(Y)]²+2 r [StDev(X)][StDev(Y)]. They can quickly test that the change of sign leads to
StDev(X-Y)=[StDev(X)]²+[StDev(Y)]² - 2 r [StDev(X)][StDev(Y)].

I hope before I get to this point I have laid a foundation for this by giving a short presentation based on a blog from John D Cook at "The Endeavor" that shows this geometrical relation between the correlation coefficient and the cosine of an angle. I hope to write a blog about this relationship in a more vector sense later.

All of this follows in the wake of a warning about non-real data from Professor Hayden, so it is important to follow up with real data that should bare this out. I'm thinking something simple like their own age in months and height. If it is true for all data sets, it should be true with the measures we have about them; but I am very willing to consider suggestions about a more appropriate data base.

Scatter Plot, at 95 MPH

The image above is another gem from the "Statpics" blog of Robert W. Jernigan. It is a plot of 1300 pitches from Yankees pitcher Mariano "Mo" Rivers. See the original here

Pretty great control, guess that's why they call it "painting the corners".

Dear Fox News

Joshua Zucker sent this to the AP Stats EDG, and I thought it worth sharing....



It is from the PhD comic site.

It reminded me of this post I made in December.

They did have another nice comic about the recent financial debacle... http://www.phdcomics.com/comics/archive.php?comicid=1077

A Change of Focus

Had a moment to cruise some other blogs this morning and came across the video below at Casting Out Nines. Just a personal note.... I totally concur with Professor Benjamin's remarks....

BI-MODAL???

Several years ago at the AP Stats reading I remember Ann Watkins (Cal State, Northridge) talking about how rare bimodal distributions are, at least that lots of things we think might be bimodal, really aren’t. The classic example we always talked about in class (before enlightenment) was distribution of heights when both men and women were included.

Turns out it just isn’t so… according to Ann’s speech . So I took some statistics from the National Health Service in the US that said the average height of white males over 20 was 70.2 inches, and for females it was 64.6 inches. Then I pulled out my trusty TI-84. It seems incredible that almost none of the pages which give average height pay any attention to the standard deviation; but since the distributions shown led me to believe it was around 2.5 inches for both groups, I used a generous 3.0 inches for the std. dev. of both groups. In the first image shown the two distributions are shown separately with a window ranging from 60 to 75 inches, with a cursor on the male curve at 67 inches

. If we combine the two, (I simply added the functions and divided by two) as if we measured the nations adults without regard to gender, the distribution looks like this,

So how much would we have to separate them to get a double mode? Well, a lot it seems. Remember that the two means, 70.2 and 64.6, are already almost two standard deviations apart. I decided to see what would happen if we made the males even taller, so I moved them up to THREE standard deviations above the female height, 73.6 inches (ever girls dream, all the guys are over six feet). The resulting combined distribution looked like this.

OK, so why bring this up now? Well, I recently read a report about the entry level salaries of newly minted lawyers (and minted is right for most of them). Most lawyers start out in private practice and make a pretty good salary, (way more than teachers), but some start out in public law and often (almost always) make considerably less. The image posted on the distribution is shown below, and is one of the most striking bimodal distributions I have ever seen for real data.

Yeah, you want to know.. the lower mode is a bridge between the $40,000 and $50,000 (the data seems to have been reported in $5,000 increments). This was 11% of the total data set. And the fat-cats???, there modal hump was at $135,000 to $145,000 (fresh out of the box... wow!) with about17% of the data set.

OK….. TRUTH in Statistics time… data like this is biased by selective reporting… Some of the folks who got low salaries just didn’t send it in…..”Yes, I have the lowest starting salary of anyone in my class.” Others may have lied ( Yes, people do that) and made them a little higher than the truth; but even with very accurate reporting, this looks like one of those true bimodal distributions. OK, keep your eye out for others like this, and when you find them, send me a link… (wow only a month till school starts… )

Picturing statistics

Statisitcs is about data in context, but great statistics, convincing statistics, requires more. In a world where nearly half the people are mathematically illiterate, and way more than half believe the old saw about, "Liars, damned liars, and statistics" you have to be able to present information in a dramatic way. A way that shakes people out of the cold dull stares we reserve for passive entertainment. I think much of what Chris Jordan does falls into that "great" statistics category.

The Ted Talk description says, "Artist Chris Jordan shows us an arresting view of what Western culture looks like. His supersized images picture some almost unimaginable statistics -- like the astonishing number of paper cups we use every single day." How about this for a shocker...we use 4,000,000 plastic cups a day on airlines alone... A Day! Other things that may surprise you...one out of every four people in prison in the world, are in the USA...2,300,000... "Land of the free?"

Remember 9-11? Of course you do... but on that day 3000 (or so) people died from a terrorist attack. Chris points out that on that day, and every day since, 1100 people die from cigarettes, over 400,000 people every year, (rule of thumb, ALWAYS check statistics.. people lie, so I checked with the Center for Disease Control,
.."Cigarette smoking is the single most preventable cause of premature death in the United States. Each year, more than 400,000 Americans die from cigarette smoking. In fact, one in every five deaths in the United States is smoking related. Every year, smoking kills more than 276,000 men and 142,000 women."... …
and 157,000 are JUST from lung cancer). When people think of cancers caused by smoking, the first one that comes to mind is always lung cancer. Most cases of lung cancer death, close to 90% in men, and 80% in women are caused by cigarette smoking. There are several other forms of cancer attributed to smoking as well, and they include cancer of the oral cavity, pharynx, larynx, esophagus, bladder, stomach, cervix, kidney and pancreas, and acute myeloid leukemia. The list of additives allowed in the manufacture of cigarettes consists of 599 possible ingredients. When burned, cigarette smoke contains over 4000 chemicals, with over 40 of them being known carcinogens. Kind of like the lottery of death, except there is a “winner” every time…I am reminded that I teach my students that in statistics we calculate binomial probability using the term p for the probability of success, even if the “success” in question is death.

I'm a teacher. I work with teenagers every day, and it scares me that 65,000 teenagers will begin smoking this month. Maybe a visual impact on their senses will stop one... or two?

A good conclusion, "How do we change?" WATCH !!

Education, Crime, and CCTV

Imagine three towns, all within a few miles of each other, all with approximately the same population and the same distribution of wealth. In a certain year, there were 39 burglaries in the town center of A, only 25 in town B, and 36 in town C. The town council of town A, concerned about their growing reputation as an unsafe place to live, try an experiment with closed circuit TV on the streets of downtown. The two other cities choose not to act.

In the following year, crime drops in town A to only 28 crimes, a reduction of 28% over the previous year. Not only has crime gone down since the cameras were introduced, but in the two neighboring towns crime has increased. Town B has seen a 20% increase to 30 burglaries, and town C is now the crime capital of the tri-cities area. Their burglary total has climbed to 42 for the year, an increase of almost 17%.
Now imagine the town meetings in towns B and C as shopkeepers clamor for the use of CCTV in their downtown areas, and what will be the result? Surely the proof of their effectiveness is clear to everyone. Crime went down in the city that used them, even in the face of a rising crime wave everywhere else in the area.

But then, there is that one nagging doubt; what if the changes from year to year are just the random fluctuations of chance. What if the probability of burglary in all the cities is totally unchanged? And the truth is.... the numbers were randomly created. I had computer software pick 100 numbers randomly with equal probability of being 1, 2, or 3 to represent the burglaries in the three towns. Then, picking the one that was highest (they would be the most likely to adapt a change) I simply repeated the randomization a second time and got the numbers for a second year.

In the statistician's lingo, this is called regression to the mean. The same idea keeps test prep book producers and tutors in the big money. When you go to take courses like the SAT or ACT, there is a certain amount of chance involved, the questions on each form vary slightly, and the topic, or just the wording may favor one student and handicap another. Then most students do a certain amount of guessing on the exams. The probability of a raw guess being right is only 1/5, but if you can eliminate one crazy disclaimer, then you stand to profit. In the end, some kids do better than they expected, and some kids do much worse.

If they all took the test again, the kids who did very much better than they thought, would probably drop back down toward whatever their true ability level was, and the kids who did poorly on the first try would probably go up... but, the kids who did well are probably NOT going to take the test again. Only the ones who did very low compared to their expectations will pay to test again. They will probably buy a review book and may even go so far as to sign up for a tutoring program. Then what happens. BOOM... their score on the second test goes up... some percentage will score MORE than they had reason to expect due to a little luck being on their side now... and off they go to extol the virtues of ACME Study Course...

It isn't exactly fraud, and some may actually teach kids some stuff, but it sort of reminds me of the guy who sends free stock advice to 10,000 people. Half are told that stock A will go UP, the others that it will go down. After a few weeks he is right with half those people, so he sends the "SEE, I told you so." letter to the 5000 he got right. This time it is stock B, and 2500 people are told it will go up, and the other 2500 are told it will drop... and sure enough, he is right on half those. NOW he has them ready, and he offers to send them his weekly tip sheet for ONLY $xx.. for the one year subscription. Well, you think, he has been right twice in a row... better jump on board.

Just a footnote on the CCTV and Crime connection; heard a news interview with a member of one of the British Police force, (don't recall which one) and he asked an interesting question. If you install CCTV and the crime rate increases, does that mean they don't work.... or that they DO? He suggested that many small crimes go unreported because people don't feel the police will be able to do anything, rocks through windows or vandelism in general is an example. But suppose with the presence of CCTV, they think, "Maybe the idiot was caught on camera." The crime gets reported, is still unlikely to be solved, and it looks like both the crime rate and the conviction rate went down. To answer questions about CCTV, he suggests you have to look at crime across the spectrum and learn what kinds of crime CCTV will deter or help bring a conviction, and what it won't. Then he said a most amazing thing... NO ONE knows how many cctv cameras are in Britain... There is a number (in the millions) that floats around that came from a study on three streets in a suburb of London... count the cameras, divide by the population in that area, multiply by the population of Britain... sounds close enough... Ok, there is a lesson in sampling bias in that sentence, find it.

Average, Percent,

Average, Percent,

and

Other Misunderstood Math Terms

“Misunderstood? Surely, you jest!” you reply…. But hear me out.. Okay, I agree that almost every seventh grader knows how to compute averages, but the question is, do they understand them… Consider the following:
A school is asked to report the average student/teacher ratio. Now we could do that two ways; the school could count how many students there are, then count how many teachers there are, and in the end, divide the number of students by the number of teachers to get the average… and that is how it is normally done. But you could also have each student count the number of students in each class they go to (assuming one teacher per class) and then average the class sizes reported by all the students. That takes a little more work, but it should give us the average number of students per teacher also… except, in most cases the two numbers will be different.

Let’s illustrate with a simple (no,, I mean REALLY simple) example. Suppose there is a school where there are only two teachers and two classes. The total enrollment is 6 kids in one class and 4 in the other. From the school point of view, there are 10 students and two teachers, so the average is 5 students per teacher. But when we survey the students, the class sizes reported by the ten students is {6,6,6,6,6,6,4,4,4,4} for an average of 5.2 students per teacher. Go ahead, make up your own numbers, but the only way to get it to agree is if EVERY class has the same number of students….and now the big question…. Which is the CORRECT average??? (“pssst… say ‘both’ ”)… But if they are not the same number, then they can’t both be the average of the same thing… Now you start thinking…. Tick… tick… tick… So what is is that each is the average of?

Ok, let’s go to something even easier; fifth grade percentages. Here is the problem:
Three weeks ago 87% of the students were in favor of the new football coach. Then the team lost and now only 67% of the people support the football coach. Ok, So which is correct Newspaper headline:
…..Support for Coach drops 20%
……Support for Coach drops 23%
…….Support for Coach drops 30%

All of them are a valid percent.. but the question with percentages, like the one with averages, and like most of the questions in applied math, resolve down to the object of a preposition…..percent of WHAT?
See, Math really is Hard.

The Rules of Three

In my youth, back when dinosaurs roamed the earth, there was “the rule of three”… singular, one, and even then the name was often described as “archaic”. More modern books tended to develop “properties of proportions” or similar terms for the problems of proportionalities. Now there seem to be an abundance of them; including one for witches, and one about businesses. There is not space enough to talk about all of them so I will mention three, of course.
The first rule of three is as old as math, and shows up at least as early as the Hindu mathematician Brahmagupta, and in Fibonacci’s famous Liber Abaci(1202). It was once so common that it was introduced into common language. Abraham Lincoln is quoted in his biography as stating that he learned to "read, write, and cipher to the rule of 3."
The most common and longest living form was the direct rule (although there was an inverse rule as well), in which case three numbers would be given and a fourth sought so that the ratio between the third and fourth would match the ratio between the first and second; a:b = c:d. Today students use the ideas in elementary school to complete fraction equivalences, “2/3 is the same as 10/?” Some of the ancient examples grew incredibly complicated.

I suppose the reason I chose to address three of the many “rules of three” is because of the rule of three from language and literature. Three just seems to be the right number for lots of things, there were Three Musketeers, Three Stooges, and Three Coins in the Fountain. It was Goldilocks and the Three Bears, and “bah bah black sheep” had “three bags full.” Comics in the newspaper usually have three panels and many jokes involve a three part ritual where the punch line is the third element, such as the t-shirt with “Great Cities of the World” on the top, and below, one after another, “Paris, Rome, Fargo”. The first two make the last funnier. In language the examples range from “Blood, sweat, and tears, to vidi, vidi, vici. If you don’t think there really is a mental tendency to have three terms, consider that in Churchill’s speech, he actually used four; “I say to the House as I said to ministers who have joined this government, I have nothing to offer but blood, toil, tears, and sweat. “
The final rule of three I would mention is from statistics, and is of more recent origin. It is also, I think, a really clever solution to what is a really difficult problem. Suppose something never happens; how can you assign a probability to it? It is not that it might not happen some day, just not so far. It is just such a problem the statistical rule of there was created to handle. Suppose you stopped at the same gum ball machine every day, but unlike the normal gumball machine, this one did not have a glass you could see into the gumballs inside. You buy a gum ball every day and get red ones, and green ones, but never a blue one. After a while you begin to wonder if they even put a blue one in the machine. So one day, after 20 days of getting all the other colors, over lunch you ask your local statistician (doesn’t everyone have lunch with a statistician?) how to figure out if there really is a blue one in there. He pauses, fork poised in mid-air, and informs you that you can be 95% sure (a common statistical benchmark) that the proportion of blue gum balls is no greater than 14.3%. He had mentally taken three, and divided by one more than the number of failed efforts, to get 3/21 or 1/7 as the upper limit of the possible fraction.
The idea is base on a simple extension of the binomial probability. If you knew that P % of the gum balls were blue, then you could calculate the probability that None showed up in 20 days. The probability would be (1-p)²⁰. Working back through this calculation many times you might notice that the number followed a pattern, a rule of thumb to calculate without tables and calculators, and that turns out to be 3/(n+1), the statistical rule of three. If you wanted greater certainty, you can use the rule of seven, which says that 7/(n+1) will give the 99% interval boundary. So in the case of your gumballs, you can be 99% sure the percentage of gumballs is less than 1/3.

The Pain is All in Your Head (no brain=no pain?)

I’ve been reading several interesting articles lately about the brain, and the strange way it seems to work (and NOT work at times). Some of it ties in with the incredible effect of placebos that I have mentioned in recent posts.

Jack Tsao, A US Navy officer/neurologist at Bethesda Maryland has found some interesting treatments for the phantom pain that often tortures folks with amputated limbs. He recruited twenty-two amputees with one missing leg and randomly assigned three treatments. One group were instructed to sit in front of a mirror and move the non-amputated leg as they watched the movement in the mirror and imagined moving the missing leg. A second group did the same, but with the mirror covered, and a third group was told just to imagine moving the missing leg. And the results? After four weeks of the therapy the second group had over half the patients report increased pain over the four weeks of treatment. For the third group, who only imagined moving the missing limb, two-thirds reported increased pain. For the mirror group, 100% (that’s not a typo).. All of them reported a reduction in pain.

A suggested explanation is that after the amputation the mind notices the lack of sensation from the missing leg, and sort of turns up the sensory volume in the nerve chain for that region of the body, producing increased pain sensation. The therapy seems to fool the mind in regions of so-called “mirror neurons” in the brain which respond when we move or watch someone else move.

As a teacher I get to watch kids behavior during tests, and one of the things you notice is the “looking for the answer on the ceiling” gazes that have nothing to do with cheating. Students will gaze to the left or right or at the ceiling as they ponder the solution to a math question. In the 1970’s there was lots of research about this tendency to gaze off to one side or the other when we are trying to think of the answer to a question. For a while it seemed that there might be an association between the direction of the gaze, left or right, and the type of question but that explanation seems to be too simplistic. There is still definitely an effect, and it appears that if you are prevented from this “lateral eye movement” which actually has an acronym, LEM, that you are less able to answer the question. (See Glenberg, Arthur M., Jennifer L. Schroeder, and David A. Robertson (1998). Averting the gaze disengages the environment and facilitates remembering. Memory & Cognition 26/4: 651-8.)

You can sort of test the idea that harder questions are more likely to induce the LEM by asking a series of questions on this link, that progressively get harder. According to the researchers, only a third of the sample of college students could answer 20 or more of the 33 questions. Ask them of a friend, and watch the eyes. Record the questions that require LEM; according to the research they should occur more on the later, harder questions. Here are a couple of them to whet your appetite, with the answers below.

Six easiest ones are
1. What’s the name of the comic strip character who eats spinach to increase his strength?
2. What’s the last name of the brothers who flew the first airplane at Kitty Hawk, North Carolina?
3. What’s the name of the crime of purposely setting a building or property on fire?
4. What’s the name of Dorothy’s dog in "The Wizard of Oz"?
5. What’s the name of the man who rode horseback in 1775 to warn that the British were coming?
6. What’s the last name of the famous magician and escape artist who died of appendicitis?

And the final six (supposedly very hard) are
28. What’s the name of the mountain range separating Asia from Europe?
29. What’s the name of the first person to run the mile in under four minutes?
30. What was the name of the Cuban leader overthrown by Castro?
31. What was the last name of the artist who painted "American Gothic"?
32. What was the name of the town through which Lady Godiva supposedly made her famous naked ride?
33. What’s the name of the highest mountain in South America?

If you need the answers, here they are for these questions, with the percentage of the college sample who answered them correctly…

1. Popeye (94%)
2. Wright (92%)
3. Arson (88%)
4. Toto (84%)
5. Paul Revere (82%)
6. Houdini (80%)

28. Ural (9%)
29. Bannister (7%)
30. (4%)
31. Wood (3%)
32. Coventry (1%)
33. Aconcagua (0%)

So how did you do… Do you notice yourself looking right or left more on the hard ones??? (HEY, over here!!!)