The most important Master's thesis of all time

Has been defended in 1936 by Claude Shannon, and is titled A Symbolic Analysis of Relay and Switching Circuits. In it, Shannon shows that certain electric circuits are isomorphic to Boolean algebra.

On the necessary existence of the elephant in the room

There is a philosophical view which says that mathematics is a way of evolutionary signaling, i.e. an elaborate way of demonstrating fitness (in this case, intelligence) to one's potential mates. This may very well be true but there is an elephant in the room that it does not address. Which is: while the purpose of doing mathematics may be an accident of evolution, its content cannot. It is possible to conceive of a world in which this function of doing mathematics is not plausible, or even a world in which natural selection does not exist at all; however, it is not possible to conceive of a world in which theorems of mathematics are false. Provable propositions are true in every possible world. Inasmuch as mathematics contains theorems about certain modes of reasoning, it follows that there is a realm of human thinking which cannot be an accident of evolution.

Alan Turing centennial

It could be a weird contraption made of scrap metal and wood:



It could be a "2-state 3-symbol" whatchamacallit:

Or it could be a few lines of Python code.

A Universal Turing Machine, one of the most powerful concepts ever.

Quite mesmerizing

Another short video by Cristobal Vila, this one about the ubiquity of the Fibonacci Series and The Golden Ratio in nature:

Wouldn't you wish your work space looked like this?

So would M. C. Escher.

(HT: God Plays Dice.)

LaTeX fail

From the mathematician Jeff Shalllit:

One problem with the proliferation of "open access" journals is the decrease in quality. A good example is this "proof" of Fermat's Last Theorem by a guy who seems to specialize in rather eccentric papers. This paper was passed around to great laughter at the van der Poorten memorial conference in Australia.

Now I am completely incompetent to judge the content of this paper, but I think I'll take Shallit's word for it. The atrocious quality of its LaTeX formatting is, I think, strong Bayesian evidence that Shalllit is right. The LaTeX code in the paper is hilariously bad. It's just fireworks of insanity. To any of you who have ever compiled a paper in LaTeX, taking a look at it should be good entertainment.

Things I wish would die: The phrase "statistical dead heat"

You hear this phrase a lot any time there's an important election approaching. It refers to a situation when the difference between percentages of respondents declaring they'll vote for candidate A and candidate B is smaller than the margin of sampling error. For example, suppose there will soon be a Republican presidential primary in Florida where voters will be choosing between two candidates, Mitt Romney and Ann Coulter. A public opinion poll comes out, showing that 51% of respondents say they'll vote for Romney, and 49% of them say they'll vote for Coulter. The polling company says that the margin of sampling error in this poll is three percentage points. The media declare that Romney and Coulter are locked in a "statistical dead heat" or "statistical tie" because, given the margin of error, Romney's true vote share could be as low as 48%, and Coulter's could be as high as 52%.

But to represent this situation as a tie is highly misleading, for several reasons. I'll concentrate here on two of those reasons, to show just how misleading it can be. In what follows, I'm making a simplifying assumption that sampling error is the only source of uncertainty in my example poll. This is of course unrealistic, but completely justified, since sampling error is the only type of uncertainty that is reported by the media.

First, the size of the margin of error depends on the significance level chosen for the particular poll. Most polls choose to report a 95% confidence interval. Suppose that's the case with our fictional Romney v. Coulter situation. What this means is that, if this poll were to be redone a large number of times, with the same sample size, then 95% of the time Romney's vote share would fall somewhere between 48% and 54%. Put another way, it means that the difference between Romney's and Coulter's vote shares is not statistically significant at a 5% level (but if we chose, say, a 68% confidence interval, then the margin of error would be approximately 1.53--smaller than the spread). So the fact that the difference between Coulter and Romney is smaller than the margin of error doesn't mean Romney and Coulter are tied; it means that Romney is most  likely ahead but, if we hold ourselves to a 5% significance level, we can't say exactly by how much. Even when your point estimate is not significant at the level you chose, it is still the best guess you have.

Second, when we look at polls, we don't really care about spread; we care about who's more likely to win. The reason we pay attention to spread at all is because we treat it as a proxy for the probability of winning. So let's think about this in those terms (again, assuming sampling error is the only source of uncertainty). At 5% significance level, the margin of sampling error of a statistic is 1.96 times the standard error of that statistic (if you want to know how the standard error of a proportion is calculated, look below the fold). Thus, the sampling distribution of Romney's vote share is normal with mean 51 and standard deviation 1.53 (= 3/1.96). Below is a plot of that distribution. The ratio of the area shaded in red to total area underneath the curve is the probability that it's actually Coulter who's ahead (i.e. it's the probability that the true percentage of voters who intend to support Romney is less than 50). That probability is about 25%. The odds of Romney being ahead of Coulter are 3 to 1; doesn't sound like dead heat to me.

An even better one

Here's the new and improved weasel program. It removes a major inefficiency of the version I posted yesterday, in that in any given generation only one string was reproducing. Since in any generation (except of course the last two of them), the content of the fittest string need not be unique, it could speed up evolution a lot if we allow all maximum fitness strings to reproduce. The program below does just that. Now with the number of offspring = 100 and a mutation rate of 5%, it never takes more than 30 generations to achieve convergence. The output of R code below displays all maximum fitness strings from each generation.

# A Version of Dawkins' Weasel in R
# (c) 2011 Przemyslaw Nowaczyk
 
Weasel <- function(phrase,num.copies,mutation.rate) {
  Score  <- function(x,y) {sum(x==y)}
  Output <- function(x,y,z) {cat(x,y,z,"\n")}
  alphabet <- toupper(c(letters," "))
  split.phrase <- unlist(strsplit(phrase,""))
  new.phrase   <- as.matrix(sample(alphabet,size=nchar(phrase),replace=TRUE))
  max.fitness  <- Score(new.phrase,split.phrase)
  generation   <- 0
  while (max.fitness < length(split.phrase)) {
    offspring   <- new.phrase[,rep(1:ncol(new.phrase),num.copies)]
    mutant.flag <- mat.or.vec(nrow(offspring),ncol(offspring))
    mutant.flag <- sample(c(0,1),prob=c((1-mutation.rate),mutation.rate),
                          replace=TRUE,size=(nrow(offspring)*ncol(offspring)))
    offspring[mutant.flag==1] <- sample(alphabet,size=sum(mutant.flag),
                                        replace=TRUE)
    fitness <- apply(offspring,2,Score,y=split.phrase)
    fit.id  <- which(fitness==max(fitness))
    new.phrase  <- as.matrix(offspring[,fit.id[1:length(fit.id)]])
    max.fitness <- max(fitness)
    generation  <- generation + 1
    apply(new.phrase,2,Output,y=generation,
     z=round((max.fitness/nchar(phrase))*100,digits=2))
  }
}
 
# sample run with timing
system.time(Weasel(phrase="METHINKS IT IS LIKE A WEASEL",num.copies=10,
                   mutation.rate=0.01))

Created by Pretty R at inside-R.org

Weasel

Dawkins' weasel is a toy model of evolution in which the goal is to write an algorithm that evolves the phrase "METHINKS IT IS LIKE A WEASEL" out of a random string of 28 characters via random mutation and non-random selection. The one constraint on the algorithm is that you're not allowed to "lock in" characters; i.e. if one of your strings happens to contain the right character in the right position, you can't exclude that character from the possibility of mutating.

I wrote a weasel program in R, in which the "evolution" proceeds as follows: you draw a random string of 28 characters; the string "breeds" n "offspring," but each character in each of the offspring strings has a probability m of mutating into any character of the alphabet; each mutated string gets a fitness score which is simply the number of same characters in the same position as in the target string; (one of the) strings with the highest fitness survives and breeds in the next generation while the rest are erased.

Below is a sample run of an R-weasel with the number of offspring = 100 an mutation rate = 0.05 (the number to the right of each string is its fitness score):

1 L Z F Z N Y O F N O Z E B I X N P X H B O P U M L A 1

2 L Z F Z N Y O F N O Z E B I N P X H K O P U I L A 2

3 L Z F Z N Y O F N O D E B I N P X H K O P U I L L 3

4 L Z F Z N Y O F N O D E B I L P X H K O P U I L L 4

5 L Z T Z N Y O F S O D E B I L P X H K O P U I L L 5

6 L Z T H N Y O F S O D E B I L P X H K O P U I L L 6

7 L Z T H N Y O F S O D E B I L P X H K O P U I L L 6

8 L Z T H N Y O F S O D E I I L P X H K O P U I L L 7

9 L Z T H N Y O F S O D E I S L P X H K O P U I L L 8

10 L E T H N Y O F S O D E I S L P X H K O P U I H L 9

11 L E T H N N O F S O Q E I S L P X H K O P U I H L 10

12 L E T H N N O S S O Q E I S L P X H K O P U I H L 11

13 L E T H N N O S S O Q E I S L P K I K O P U I H L 12

14 L E T H N N O S S O Q E I S L P K I A P U I H L 13

15 L E T H N N O S S A T E I S L P K I A P U I H L 14

16 L E T H N N O S S A T E I S L P K I A P U M E L 15

17 L E T H N N O S S A T X I S L P K I W P U M E L 16

18 L E T H N N O S S A T X I S L P K I W P U M E L 16

19 L E T H N N O S U A T X I S L P K I W P U M E L 16

20 L E T H N N O S U A T X I S L P K A W P U M E L 17

21 L E T H N N O S U B T X I S L P K A W P U M E L 17

22 L E T H I N O S U B T X I S L P K A W P U M E L 18

23 L E T H I N O S U N T X I S L P K A W P U M E L 18

24 L E T H I N O S U N T G I S L P K A W P U M E L 18

25 M E T H I N O S U N T G I S L P K A W P U M E L 19

26 M E T H I N K S U N T G I S L P K A W P U M E L 20

27 M E T H I N K S U I T G I S L P K A W P U M E L 21

28 M E T H I N K S I T G I S L P K A W P U M E L 22

29 M E T H I N K S I T U I S L I K A W P U G E L 23

30 M E T H I N K S I T U I S L I K E A W P U G E L 24

31 M E T H I N K S I T U I S L I K E A W P U G E L 24

32 M E T H I N K S I T U I S L I K E A W P U G E L 24

33 M E T H I N K S I T U I S L I K E A W P U G E L 24

34 M E T H I N K S I T U I S L I K E A W P U G E L 24

35 M E T H I N K S I T U I S L I K E A W P U G E L 24

36 M E T H I N K S I T U I S L I K E A W P U G E L 24

37 M E T H I N K S I T I S L I K E A W P U G E L 25

38 M E T H I N K S I T I S L I K E A W I U G E L 25

39 M E T H I N K S I T I S L I K E A W M A G E L 26

40 M E T H I N K S I T I S L I K E A W E A G E L 27

41 M E T H I N K S I T I S L I K E A W E A G E L 27

42 M E T H I N K S I T I S L I K E A W E A G E L 27

43 M E T H I N K S I T I S L I K E A W E A G E L 27

44 M E T H I N K S I T I S L I K E A W E A G E L 27

45 M E T H I N K S I T I S L I K E A W E A G E L 27

46 M E T H I N K S I T I S L I K E A W E A G E L 27

47 M E T H I N K S I T I S L I K E A W E A G E L 27

48 M E T H I N K S I T I S L I K E A W E A G E L 27

49 M E T H I N K S I T I S L I K E A W E A G E L 27

50 M E T H I N K S I T I S L I K E A W E A G E L 27

51 M E T H I N K S I T I S L I K E A W E A G E L 27

52 M E T H I N K S I T I S L I K E A W E A G E L 27

53 M E T H I N K S I T I S L I K E A W E A G E L 27

54 M E T H I N K S I T I S L I K E A W E A G E L 27

55 M E T H I N K S I T I S L I K E A W E A S E L 28

user system elapsed

0.09 0.00 0.10

This is not a typical run; it's on the shorter side (for these parameters the median length is something like 70). Note how fast it converges though; R can be quite fast f you do things in vectors and matrices rather than loops.

The next step is to allow the strings to mate and swap their genes.

R code for the weasel is below the fold.

# A Version of Dawkins' Weasel in R
# (c) Przemyslaw Nowaczyk 2011

score <- function(x,y) {sum(x==y)}

weasel <- function(phrase,no.kids,mutation.rate) {
  alphabet <- c("A","B","C","D","E","F","G","H","I","J","K","L","M","N",
                "O","P","Q","R","S","T","U","V","W","X","Y","Z"," ")
  split.phrase <- unlist(strsplit(phrase,""))
  new.phrase   <- sample(alphabet,size=nchar(phrase),replace=TRUE)
  distance     <- score(new.phrase,split.phrase)
  generation   <- 0

  while (distance < length(split.phrase)) {
    m.newph     <- as.matrix(new.phrase)
    offspring   <- m.newph[,rep(1,no.kids)]
    mutant.flag <- mat.or.vec(nrow(offspring),ncol(offspring))
    mutant.flag <- sample(c(0,1),prob=c((1-mutation.rate),mutation.rate),
                          replace=TRUE,size=(nrow(offspring)*ncol(offspring)))
    offspring[mutant.flag==1] <- 
      sample(alphabet,size=length(mutant.flag[mutant.flag==1]),replace=TRUE)
    scores <- apply(offspring,2,score,y=split.phrase)
    new.phrase <- offspring[,which(scores==max(scores))[1]]
    distance   <- score(new.phrase,split.phrase)
    generation <- generation + 1
    cat(generation,new.phrase,distance,"\n")
  }
}

# sample run with timing
system.time(weasel(phrase="METHINKS IT IS LIKE A WEASEL",no.kids=100,
                   mutation.rate=0.05))

Created by Pretty R at inside-R.org

Innumeracy, big time

A New York City-based organization called the Coalition for the Homeless is currently running a TV public service announcement in which they say:

Which is more disturbing: That each night in New York City, more than 40,000 people are homeless, or that the average age of a homeless person is 9?

(Or something to that effect. I'm paraphrasing, but the figures are quoted accurately.)

If the avarage age of 9 strikes you as implausible, you're right. The very same Coalition for the Homeless lists the following as one of the basic facts about homelessness:

In New York City... Each night more than 40,000 people--including more than 16,000 children--experience homelessness.

This basic fact makes the average age of 9 an arithmetic impossibility. 16,000 is 40% of 40,000; so even if each age category (children and adults) is assumed to have the lowest average age possible (1 and 18, respectively), the average age of a homeless person would be 11. But of course these assumptions are empirically completely implausible, which means that the average age of a homeless person is not only certainly greater than 9, but most likely much more so. Assuming group averages of 3 and 30, for example, gives an overall average of 19. The only nationwide data that I have been able to find is here, from which the average turns out to be about 32 (see Exhibit 5-3 on page 43).

How's that for innumeracy?

Binomial nitpicking

Kids Prefer Cheese gives a link to a really funny Sheen-Gadhaffi Quiz, and then says:

I got a 4 out of 10, worse than random.

This is a little bit, what's the best way to put it, wrong.

There's ten questions with two possible answers each, so probability of getting any one of them right by chance is one-half. Ergo, probability that choosing answers via coin flip will get you four or less correct picks is 0.38 (with n = 10 and p = 0.5, cumulative binomial(4) = 0.38). No way can you reject the null.

Saying what you don't mean to say, in extremely precise terms

Observational Epidemiology features a post talking about the "scalar fallacy," i.e. treating scalars as if they were the same as vectors. In their own words:

(...) neither vectors, random variables nor vectors of random variables are scalars. This statement is obvious to anyone familiar with the basic terms. Equally obvious is the fact that when you try to represent one of these complex, multidimensional creatures as a point on a line, you will invariably lose some information. (...) This isn't pessimism; it's mathematics. You lose information when you go from a vector to a scalar.

As immediately pointed out by commenters, this is incorrect. Since sets R and Rⁿ (where R is the real line and n is a natural number) have the same cardinality, for any subset of Rⁿ you can always find a one-to-one (i.e. reversible) mapping onto some subset of R. So what does the author mean when he says "you will inevitably lose information when...?" Let's look at some examples he gives of representing vectors with scalars:

A "rate your experience" question might do a good job comparing the impact of bad beverage service versus that of short delay in take-off but it will probably not do a satisfactory job comparing a forced landing and a seven hour stay on the tarmac on a hot summer day.

A weighted average of nutrients might provide a good way of ranking most of the foods you find in the produce aisle. (...) If, however, you move to the context of the dietary supplement aisle, making that linear assumption about certain nutrients can be dangerous, even deadly.

(...) Take the example of health. There's no meaningful way to boil this complex, multidimensional concept down to one number, but we can come up with scalars that are useful when answering certain questions. Let's say we have formulas for deriving two metrics, L and Q. L correlates very well with longevity; Q correlates very well with quality of life. For most questions about health policy, you will get similar answers with either metric, but there are cases where the two diverge sharply. Both L and Q are good measures of health, but their usefulness depends on the question you need answered.

What is clear is that the author thinks of vectors and scalars not just as of "objects," but as of "objects with some sort of internal structure." Perhaps, then, when he says "you will lose information when trying to represent A as B" he means "there does not exist an isomorphism from A onto B" as opposed to "there does not exist a one-to-one mapping between A and B" (in other words, he's interested in structure-preserving mappings rather than identity-preserving ones). If that's the case, he could be right quite literally, depending on what he means by "an object with structure." If this means "a set S and a relation L on S x S," the loss of information claim is still wrong: In the category of sets, the isomorphism class of a set is determined by its cardinality, and an n-ary relation on set S is simply a set of cardinality |Sⁿ|. But if it means "a vector space V over the field of real numbers with operators p and q," the claim is correct. In the category of vector spaces, the isomorphism class of a vector space is determined by its dimensionality, so you cannot represent a vector space with a lower-dimensional vector space without loss of information.

But is this really what the author meant to say? Is he really saying "health is an n-dimensional vector space and therefore it's impossible to construct a single metric of health that's also a vector space?" It's hard to tell, really, but it seems that this is what he's saying. Which means that the loss of information claim is right, but that his expectations as to the usefulness of linear algebra in applied social science are a bit unrealistic. The concepts "set" and "relation" are much more general and applicable than concepts "vector space" and "scalar multiplication," and insisting on representing social phenomena with vector spaces means there won't be a whole lot out there for you to represent.

Strange beliefs involving numbers: When is a group overrepresented?

One of my hobbies is collecting bizarre and unusual examples of confused quantitative reasoning. Some innumeracies are very common (such as thinking that if an item's price has been discounted twice, first 50% and then 20%, it means that the overall discount is 70%), other ones not so much. Here's an example of the latter kind.

Poland has had presidential elections a few months ago. There were two candidates in the second (and final) round (Poland has a two-candidate runoff system). On election night, when polls were closed but votes were not fully counted, the media were of course talking about election-day polls. Those polls showed a stark contrast between the candidates' relative support among rural and urban voters. About 25% of one candidate's (Komorowski) electorate was rural, whereas the other candidate's electorate (Kaczynski) was reported to be 48% rural. The media concluded that Komorowski was overrepresented among urban voters while Kaczynski was overrepresented among rural voters. One blogger took issue with this interpretation, offering a quite creative argument against it:

The fact is that Kaczynski is represented equally by the whole country. Exactly equally. Because those 2 percent are just statistical error. Rural and urban electorates support Kaczynski equally strongly.

What's bizarre (and, to be honest, quite stupid) about this argument is the implicit assumption that a candidate's representation among different groups is equal if those group's shares in his electorate are equal. Which is absurd, of course; I mean, if some candidate's support was split 50%-50% between people under and over the age of 80, would you say that the candidate is equally supported by young and old voters? In a one-dimensional case, to claim that support is equal it has to be roughly proportional to the base rate. Since about 62% of Poles live in cities, Kaczynski is indeed overrepresented in rural areas.

Interestingly (or perhaps not), in the short passage I quote, the blogger makes two additional quantitative mistakes. First, he writes "2 percent" where he means "two percentage points." Second, he assumes that because the sampling error in the poll is at least 2 points it mean that the true rate is 50% rather than 48%. Sure, it could be 50%. But it's equally likely that it's 46%.

Three staggering mistakes in three sentences! (Yes, I do mean three. The second period mark is artificial.)

When the mean is meaningless

The media often report averages of random variables without saying anything about the shape of the distribution. When the distribution is skewed, this is completely uninformative (or even highly misleading, as most people probably think of "average" as "the most typical value").

For example, the mean monthly salary in Poland is 2,600 PLN (about $830). The mean probability of survival of an airplane incident involving casualties is 38%. What does the mean actually mean here? Not much, as both distributions are positively skewed. About 2/3 of the labor force in Poland earn less than the average salary, and 40% of plane crashes have an extremely low survival rate of 0%.

A perfect world

Mathematicians have axiomatized the real line as a one-dimensional continuum, as a complete ordered Archimedean field, as a real closed field, or as a system of binary decimals on which arithmetical operations are performed in a certain way. Each of these axiomatizations is tacitly understood (...) as an axiomatization of the same real line.

***

(...) there is no way of dealing with mathematical items rigorously except through axiom systems. But this is like saying that there is no way of communicating ideas except through words. Although any idea has to be represented in sentences, the same idea may be expressed by completely different sentences. An idea is "independent" (...) of the words used to express the idea. When we assert that a mathematical item is "independent" of any particular axiom system, we mean this "independence" in much the same way as independence of ideas from language.

Gian-Carlo Rota, Indiscrete Thoughts.

Also online, also free

Is Silvanus P. Thompson's hundred year old calculus textbook, Calculus Made Easy. One of the few, if not the only math textbook that is actually a pleasure to read. It was a supplementary reading in one of the game theory classes I've taken in grad school. In New York, you're exempt from sales tax when purchasing textbooks, as long as you show a valid student ID and a syllabus that has the book you want to buy on it. Whenever I'd buy textbooks, bookstore cashiers would always ask for a student ID but were never very much interested in seeing a syllabus. And for good reason, I suppose; you'd have to be at least slightly insane to be buying, say, "Econometric Analysis" unless you needed it for school. The only time I was asked for a syllabus was while buying "Calculus Made Easy" (it wasn't online yet). "You see this one," the cashier smiled "Some people actually read for fun."

Tail-heavy aircraft, tail-heavy normal distributions

Air Midwest Flight 5481 from Charlotte/Douglas International Airport to Greenville-Spartanburg International Airport is a short (30 minutes) commuter flight, operated by a very small turbo-prop Beechcraft 1900D aircraft. On the day of the accident in January 2003, the plane was filled to capacity (19 passengers and 2 crew members). Few seconds after takeoff, just after landing gear was up, the plane's nose pitched extremely steeply upwards and the pilots were not able to counter it as they found they didn't have enough elevator movement to force the nose down. The plane stalled, lost lift, and dropped to the ground like a rock, killing everyone on board.

NTSB investigation concluded that the crash was caused by a conjunction of two unrelated problems. By itself, neither would have led to the accident, but unfortunately on that day they paired up. First, a flawed maintenance repair of the plane's elevator left it with compromised down movement, which meant the pilots were not able to stop the upwards pitching of the aircraft. Second, the plane was too heavy, which is what caused it to pitch up in the first place. The maximum load of the plane was determined by the airline based on averages: 175 pounds per passenger and 20 pounds per piece of luggage. Based on those directives, the crew's calculations showed the plane to be within limits to take off. The problem was that those averages were determined based on surveys conducted in 1936. When NTSB investigators painstakingly calculated the actual weight of passengers and bags aboard Midwest 5481, it turned out that unbeknownst to the crew the plane was overloaded by 580 pounds.

Of course, using weight averages from 1936 when everyone and their mother knows people have consistently been getting heavier is outrageous negligence on part of the airlines. Due to the finding of the investigation into Midwest 5481 crash, NTSB started lobbying FAA to force airlines to use actual weights for determining whether or not planes can take off. FAA ordered the airlines to do a new survey and update their averages. The results were quite shocking: the average weight of an airline passenger A.D. 2004 turned out to be 195 pounds. (As a side note: this is interesting in itself. The average weight of a male adult in the U.S. is 191 pounds; of a female adult--164 pounds. Why is the flying public so much heavier than the general public?) At any rate, Air Midwest updated their average passenger weight to 200 pounds, which meant that the capacity of a Beechcraft 1900D had to be adjusted downwards to 17 passengers. NTSB investigators felt this was not enough, and are still advocating for regulation requiring to use actual weights.

Which finally brings me to the main point of this post: does it make sense to use acutal weights instead of averages? In other words, is the increase in safety worth the increase in costs? The answer of course depends on two things:

1) How much would it cost to use actual weights; and,

2) What is the probability of overloading a plane if only averages are used?

Knowing precisely nothing about how the airline industry is run, i can't even begin to answer (1). Would using actual weights require installing scales and actually weighing each passenger or not? If so, how much would that cost? Would it mean ticket prices would have to go up and if so, by how much? Etc-these are all questions I can't answer. So I'll just answer (2), as that one is just a bit of trivial probability. But in order to do this, we need to know one more thing: standard deviation of weight. I take it to be 40 pounds. (I couldn't find this statistic, so I'm just making it up as I go along at this point.)

So the official passenger weight capacity of a Beech 1900D is 17 x 200 lbs = 3400 lbs. By central limit theorem, total weight of a group of 17 people is a random variable that follows a normal distribution with mean 3400 lbs and standard deviation of 40 x sqrt(17) = 165 lbs. Now we need to make an additional assumption: how much over 3400 pounds do we have to go to consider the plane to be potentially dangerously overloaded? Surely 10, or even 50 pounds over the limit will not make a whole lot of difference. Let's assume then that exceeding the prescribed limit by 5% or more is unacceptable. This means that we can't allow total passenger weight to go over 3570 pounds. So the question now is: given that we're not weighing anyone and just using averages, what's the probability that, if 17 pax board a Beech 1900D, their total weight will exceed 3570 pounds?

Calculating the z-score and plugging it into the normal distribution we've just found, we can see that this probability is over 15%. That is way too high; completely and utterly unacceptably high. NTSB investigators are absolutely right: with aircraft so small that they board under 20 pax, averages simply won't do; variance is too high. Something else needs to be worked out.

However, as planes get bigger, central limit theorem does start taking care of us. Suppose we're trying to find out the probability of overloading a 60-seater. Sticking with the assumption that the weight limit needs to be exceeded by 5% to consider the plane too heavy, we get the overload probability of just under 3%. Still too high for my taste. In a 100-seater, however, it's 0.6%, and in a 200-seater it's 0.02%. That's probably acceptable. And the bigger they get, the more acceptable it'll be; plus, the bigger the plane, the higher the costs of the alternative (i.e. of actually weighing passengers).

Too complex for its own good

This is a bit late for The Pi Day, but still interesting: an intricate crop circle in England encodes the value of pi! How? Look at the image below; it consists of ten concentric arcs joined by short line segments. Reading from inside out, the angular lengths of those arcs equal 3, 1, 4, 1, 5, 9, 2, 6, 5, and 4 tenths of a circle (the value of pi rounded to the ninth decimal digit is 3.141592654).

I'd like to point out to the designers of this image, whoever they are, that there's a method of graphically encrypting the value of pi that is much simpler as well as infinitely more precise than what they've done. This method consists of drawing a damn circle.

The intellectual most frequently forced to turn in his grave is...

Kurt Gödel, by far. His incompleteness theorem is definitely the most abused piece of formal reasoning ever written. (By "most abused" I mean "invoked as implying the most non-mathematical consequences absurdly far-removed from the domain in which it has any applicability.") I've read and/or heard in conversations, arguments which quite seriously purported Gödel's theorem to imply (in no particular order): postmodernism, creationism, existence of God, non-existence of God, "inevitability of human condition" (I'm not joking), computability of human intelligence, non-computability of human intelligence, impossibility of complete knowledge of mathematics, impossibility of complete knowledge of anything, existence of immaterial soul, non-existence of pretty much anything... etc., etc.

I don't know what it is about this particular bit of knowledge that elicits so much crackpottery.

Nerd test

If you don't think this is funny, you clearly aren't one.

(HT: Alexandre Borovik.)