Average IQ scores have gone up a lot over the years, although they seem to be plateauing.
Do I think that people really got smarter over that period? Based on real-world accomplishments? Not one bit. Probably they’ve gotten a bit dumber, from selection, and more than a bit dumber, from demographic change.
Most importantly, math subscores haven’t changed much. “There is a subtest of the Wechsler called Arithmetic. A typical question: If a widget costs 18 cents, and if you buy 3 widgets and give the clerk 1 dollar, how much change should you get back? This is one of the subtests showing the smallest Flynn effect. Over 50 years it showed 0.23-SD increase in adults, whereas Raven’s Matrices showed a 2.39-SD increase. ”
West Hunter 
True.
IIRC, vocabulary has barely budged too. Most likely a change in thinking styles rather than ability.
By the time you reach the actual question you already know the answer. At least if you were taught your three times table.
Doc T, a question. The UK underwent a “natural experiment” when the God-given pounds, shillings, and pence were replaced by Napoleon-given metric money. Did anyone bother trying to see whether there was a subsequent decline in people’s ability to do mental arithmetic?
At today’s prices, not sure anybody would still be minding how many denarii went into a solidus. How much longer will they even go on burdening schoolchildren with those tedious decimal points?
Many observers decided that a lot of what IQ tests measured is pattern-recognition, and decided that was a central part of g. It sounds plausible. Sure, pattern recognition…that’s the Holy
Grail… Ravens goes straight for that ability, and it probably is the best test of pattern-recognition. Yet even if that single ability is the core of g, and thus IQ, it doesn’t seem to be everything.
The Wechsler doesn’t have a high enough ceiling for the upper IQ societies, so it has fallen in disfavor among the hard-chargers. But I prefer it, as do most clinicians doing a full evaluation. I thought I had saved the link but I haven’t. I recall reading that the WAIS was not showing the Flynn Effect other tests were. Perhaps I only wanted that to be true and got that wrong.
Maybe pattern recognition is important, but then maybe other factors are as well.
One of the things I do is write code that writes code (in the same language, thankfully, so I don’t have to deal with two different syntaxes.)
To a certain extent this simply requires bloody-mindedness since you have:
C1 writes C2.
So, C1 has to compile correctly. But then C2 has to compile correctly as well. But then C2 has to produce the correct result.
Easy, n’est-ce pas?
Yes I was recently thinking of the following question: Would Jefferson or Madison or Adams score high on Ravens? I’ll bet not. Meaning the problem is with Ravens (or comparing Ravens scores across time).
There may be a genuine Flynn effect visible in mathematics, at the very end of the bell curve. The highest level math competitions (IMO/Putnam) have gotten steadily harder since the 70s, in order to maintain the same distribution of scores. This could be due to a new influx of chinese competitors, or higher participation in general. In research, the low hanging fruit was exhausted in the 20th century, and now the bleeding edge is growing steadily more arcane and complex, yet progress has not stalled.
I feel there must be some genetic component to this, since the most talented undergrads I’ve encountered were all obvious products of assortative mating.
Voevodsky’s father was a top Soviet physicist who was the director of a high-energy physics center. His mother had a PhD in Chemistry or Physical Chemistry or something like that.
Essentially every big name in math has similarly impressive parents.
I wonder if Terence Tao is as smart as von Neumann. I think his mental arithmetic is about as good, but can he memorize A Tale of Two Cities verbatim?
“[…] I was never very good at school with … humanities … anything which was more a matter of opinion.”
The article mentions that he translated A Hitchhiker’s Guide to the Galaxy into Latin while in high school, so maybe he’s not selling himself properly here.
Tao took the SAT-V/SAT-M quite a few times as a young man. First scored 760 on the SAT-M at age 8 (a couple months shy of 9).
This paper goes into his early verbal and math performance.
http://www.davidsongifted.org/Search-Database/entry/A10116
Can’t find it atm, but I seem to recall reading he hit the culmulative SAT ceiling at some point. So I’d take any self-deprecation of his verbal abilities with a grain of salt.
Just have to search out the low hanging fruit from other plants. our host eg discovered that plenty of biologists can’t do math.
“UESTION: Do you think genetic barriers to further progress are becoming obvious in some areas of art and science?
CHOMSKY: You could give an argument that something like this has happened in quite a few fields. It was possible in the late nineteenth century for an intelligent person of much leisure and wealth to be about as much at home as he wanted to be in the arts and sciences. But forty years later that goal had become hopeless. Much of the new work in art and science since then is meaningless to the ordinary person. Take modern music — post-Schšnbergian music. Many artists say that if you don’t understand modern music it’s because you just haven’t listened enough. But modern music wouldn’t be accessible to me if I listened to it forever. Modern music is accessible to professionals, and maybe to people with a special bent, but it’s not accessible to the ordinary person who doesn’t have a particular quirk of mind that enables him to grasp modern music, let alone make him want to deal with it.
QUESTION: And you think that something similar has happened in some scientific fields?
CHOMSKY: I think it has happened in physics and mathematics, for example. There’s this idea, which goes back to the French mathematicians known collectively as Bourbaki, that the development of mathematics was originally the exploration of everyday intuitions of space and number. That is probably somewhat true through the end of the nineteenth century. But I don’t think it’s true now. As for physics, in talking to students at MIT, I notice that many of the very brightest ones, who would have gone into physics twenty years ago, are now going into biology. I think part of the reason for this shift is that there are discoveries to be made in biology that are within the range of an intelligent human being. This may not be true in other areas.
QUESTION: You seem to be saying two things. First, that whatever defines our common human nature will turn out to be a shared set of intuitions that owe much of their strength and character to our common genetic heritage — our species genotype. Second, that the exhaustion of these intuitions in many areas is producing a peculiar kind of artistic and scientific specialization. Further progress in music or mathematics, for example, requires a scientist or artist with an unusual heredity.
CHOMSKY: Well, it’s a different mental constitution — something like being a chess freak or a runner who can do a three-and-one-half minute mile. It’s almost a matter of logic that this change is going to occur sooner or later. Has it happened already? That’s a matter of judgment. It’s a matter of looking at, say, the twentieth century and seeing whether there are signs of this change. Is it the case, for example, that contemporary work in the arts and sciences is no longer part of our common aesthetic and intellectual experience? Well, there are signs. But whether the signs are realistic or whether we are just going through a sort of sea change and something will develop, who knows? Maybe a thousand years from now we’ll know.”
I’ve heard Chomsky use the same wheeze about post-structuralism and critical studies that he uses to defend Schoenberg’s music. Perhaps he should consider the possibility that rather than modern music being so advanced that intelligent humans can’t appreciate it without advanced study (provided by the faculty at MIT no doubt) the truth may be that it is simply decadent and a descent into nonsense merely for the sake of being “new” and esoteric.
At least vis a vis modern music, it could be both haha. To appreciate the esoteric harmonic innovations (in the absence of any hedonic enjoyment) of someone like Schoenberg does is impossible below certain cognitive threshold.
It is, of course, I agree also pure dreck and mental masturbation.
I prefer Buxtehude. One meets a better class of intellects at Baroque chamber concerts or performances by medieval choral groups. I don’t think anybody really likes modern classical music. The folks attending those performances always have painful smiles on their faces as they try to pretend that they are oh so avant-garde.
Search for “Noam Chomsky postmodernism” or “Chomsky post-structuralism” and you will find critical remarks of these fields similar to what you are expressing about Schoenberg’s music. Elsewhere he does use the same technique as above when describing those fields, talking about how they sound like total nonsense, but maybe they’re just too sophisticated for him to understand(sarcasm?). Considering he describes Schoenberg’s music the same way, I expect he has considered the idea that it is decadent nonsense. In any case, certainly I see no defense in that Chomsky passage.
Once I have incognito attended a class lecture (on Calculus ?) by a Professor at Cornell U. While the English pronunciation of that Professor was definitely much better than my own, a slight French accent could be observed. In the process he used term “mapping”,
and it took some time for me to understand that he meant just “function” (be it of one variable, or of several variables.) The stench of “Bourbaki style” making high words for simple things made me nauseous, but I kept myself tight.
This is to be contrasted with attitude of great mathematician V. I. Arnold, known for KAM diffusion, for solving Hilbert’s thirteenth problem, and for many other things,see
https://en.wikipedia.org/wiki/Vladimir_Arnold From there:
“Arnold was an outspoken critic of the trend towards high levels of abstraction in mathematics during the middle of the last century. He had very strong opinions on how this approach—which was most popularly implemented by the Bourbaki school in France—initially had a negative impact on French mathematical education, and then later on that of other countries as well.”
How is a “mapping” from one space to another more complicated terminology than a “function”?
Have you ever kept in your hands a textbook with the title “Mapping which define Bessel mappings”, or a treatise “Zeros of Riemann mappings”, or “First course of analytic mappings of a complex plane to itself” ?
I’ve never heard the Riemann Mapping Theorem referred to as the Riemann Function Theorem.
the term “mapping” is widely used in mathematics. I’m not even sure that Bourbaki uses this term more than others. In the index to the Bourbaki calculus book “Functions of a Real Variable” there are 39 references listed under the term “function”, none under the term “mapping”. The term “function” appears in the Table of Contents in the title of 25 sections. The term “mapping” does not appear anywhere in the Table of Contents.
Henri Cartan was for many years the leader of the Bourbaki group. Of the thirteen individuals who were his doctrinal students, two – Serre and Thom, received Fields Medals. Among his other doctrinal students were Cartier, Cerf, Douady, Dolbeault, Godement, Karoubi and Koszul. If Cartan was a poor teacher it sure doesn’t show up in his students.
A friend of Pierre Deligne gave him some Bourbaki books to read when Deligne was about 12 or so. The early exposure to Bourbaki does not seem to have done much harm to Deligne’s mathematical abilities.
The English translations of Arnold’s treatises on ODE’s make extensive use of the term “mappings”.
I’m sure I heard the term “mapping” used for “function” often during my university education. It was a bit more informal of a term, but not uncommon. But I did take both Math and Physics and I noted they had slightly different vocabularies. Even slightly different notation.
I can’t remember where I heard “mapping”.
I think the SMPY study found a modest Flynn effect.
But the effective population size from which these competitions draw has also drastically increased, esp in China, India, etc.
No matter how smart you are, top performance in Putnam and IMO requires a good deal of training. So many more children are being allowed to follow their natural abilities and being exposed to this sort of advanced mathematics than were in the 1970s. They start in elementary school with MathCounts and move to AIML, AIME, etc.
It seems obvious that the difficulty would have to increase substantially from the comparatively uncompetitive days of Feynman. Even though there’s been probably little to no meaningful change in raw smarts.
School achievement test scores have been decent for math over the decades. Verbal has tended to be more troublesome. I think American schools have worked harder on math since, say, 1983 and paid tutoring in math is a lot more common today than when I was in school in the 1960s and 1970s.
The widget question probably doesn’t change because everyone gets it right. How in God’s name would anyone not?
Your life must be an example of restriction of range.
What, those who have attended second grade?
I’m not familiar with the WAIS, but aren’t there dozens or hundreds of questions? I can imagine a fatigued person making a mistake.
The long one is the full battery of WISC sub-tests given to young people whom one assumes have plenty of stamina. As I recall, the WAIS for adults is only about 90 minutes.
I know people who would struggle with the widget question. And I live in a white middle-class town with about half the national poverty rate — and about half the national college grad rate. The mental abilities of the average person are mildly horrifying. When I was in high school, I worked for a family of rich Indian immigrants, tutoring their daughter. I was paid $10/hour for this. The girl was probably average college material, while her older brother was a doctor and her father a cardiologist. Nevertheless, whenever it came time for the girls mother to pay me, she would bring a calculator. “How many hours have you worked since the last time we paid you?” she asks.
“Sixteen hours.”
At this point she pulls out the calculator, types in “16 x 10”, looks at the answer, and hands me $160 dollars. And she used a calculator every damn time. Her daughter always looked a little embarassed by this. (The mother was by far the shortest one of family, so consider this an n=4 correlation study on height vs. IQ.)
And these were upper-class people. Probably 1-percenters.
The restriction of range most middle class and above people encounter in IQ makes the full range of IQ hard to understand till you encounter the full swath of humanity…I think the quasi segregation of people by income has contributed to the commonly held idea that people don’t differ all that much in IQ.
I went to a Jewish school for elementary/middle then a gifted program in high school and while I didn’t think too many people were /brilliant/ at those schools, everyone i knew was capable of reading a book or doing some arithmetic or understanding some basic health information. I was /so/ naive. I remember an argument with a friend where I was trying to convince her the only reason anyone seemed like a genius as math or reading was because “they had a random early interest in it which led to lots of practice early on in life” . I was smoking some deliberate practice BS.
But when you work at a hospital…I shadowed some docs a couple times and I was continually amazed at the ineptitude of some patients. At first I thought they had some mental issues– but no, some.people just don’t understand why HIV meds need to be taken regularly, or what calories are, or why you should clean a wound after surgery if instructed to, or how to properly calculate insulin dosages, etc.
You’re really exposed to the full range of human stupidity and competence. I think it’s part of what makes docs think they’re hot shit. Not too many middle class and above jobs give you so much exposure to the rest of the population like the healthcare system.
I attended a public high school in a small town. So I’ve seen a cross-section. But so have many people: I think a lot of them genuinely don’t remember.
I attended high school in a large city. I saw some pretty dim bulbs, but at graduation there were hundreds more I’d never seen at all. None of they were quite knuckle walkers, but I bet many couldn’t spell their own names.
I wonder about that. In Canada you have a GST (goods and service tax) plus most provinces have a sales tax (amount varies). Some goods are exempt, from one or both.
So the widget question isn’t quite so simple. Especially now that Canada has done with physical pennies. If you pay by debit card, it’s the same as usual. But with cash, the clerk must round up or down to the nearest nickel.
This is an example of differing culture, maybe?
NZ abolished the cent coin and adopted a rounding system. When we lived there the results In the university refectory were funny. it didn’t matter to The Young who paid for everything by card anyway while looking over their shoulders and chatting to their friends, or staring at their mobile phones. Their transactions took longer than they did for the codgers who would simply do mental arithmetic and hand over the cash.
You never bump into clerks who struggle to give you correct change?
I bought a light bulb. It was marked 2/$1. The clerk took out a piece of paper and a pencil and laboriously divided it out. (She did get it right.)
Haha ! Good example. I often have clerks give me incorrect change.
I also have clerks stare at me in bewilderment when I give them more money than necessary – say $21.25 for an $11.24 purchase – because I don’t want a bunch of change and dollar bills in my pocket.
Modern cash registers tell the clerk how much change is due.
I don’t think they did when I was a child. But in that era everyone was drilled extensively on arithmetic.
It’s all different now.
(I’ve stopped carrying cash and pay everything by debit card.)
Some places, like gas stations, do not treat debit cards as cash for price level. They are treated as credit cards. The bank may charge a processing fee. I don’t know.
I know there isn’t a fee for the customer (but I’m in Canada). I have no idea about the US.
Then either every cash register is not “modern” or the problem with low IQ is bigger than I imagined.
And the percentage of Americans who use cash for at least half their purchases is still near 50 percent.
An advertisement for a cash register in 1957 shows it displaying change due.
http://clickamericana.com/eras/1950s/national-cash-registers-that-figure-your-change-1957
Frau Katze,
I had no idea you considered that a “modern” cash register. But that ubiquitous machine still doesn’t help a young clerk figure out how many quarters, dimes, nickels and pennies are in the change to be given back to the customer. The clerks still have to work it out in their heads, and I’ve seen kids struggle with that.
When you mentioned “modern cash registers”, I assumed you were talking about machines which told the clerks exactly how many of each type of metallic change to give the customer. In your ad’s example of $3.98, that change would be three quarters, two dimes, and three pennies.
I’d never heard of such a thing. But they do have some modern cash registers with elaborate functions now, so I couldn’t be sure they didn’t exist.
Supermarket checkout machines here show the amount of cash received and the change due. The machines are honest, but prices shown on the shelf and the amounts debited at checkout are different. Who can remember and compare? So some feel justified to consume sweets on the spot to make it up. I heard that in Denmark this is not done.
OK. I didn’t realize you meant that the clerk couldn’t assemble the bills and coins, given the amount. I would say: such a person is not intellectually up to being a clerk. You’d have to be borderline retarded to be unable to do that.
I can’t imagine fast food servers using an actual cash register from forty years ago. It would be constant chaos if they had to ring up 1.39 for every cheeseburger, add it all up, then add the tax. Today they push a button that has a cartoon of a dancing cheeseburger and a computer does the rest. OK, that is a slight exaggeration, but just slight.
When I worked at a supermarket in the 70’s, the cash register had 4 columns of buttons – so if an item coast 23.45 (most items in a grocery store at the time were under a buck), you’d hit the 2nd button from the bottom in the left-most column, the third down in the next column, the 4th in the next, and 5th in the last, then hammer the “enter” key, for each item. It did not have a change calculation feature. But registers that calculated the change were coming into fairly common use at the time.
As I suspected. I’m positive I’ve seen registers display the change due. But I don’t have any recent experience since I use debit cards (and shop online a lot).
I can confirm that this is pretty much the exact process today. Just be thankful that we still have to be able to read because I feel like graphic representations for the orders are just around the corner.
Ever met folks like those in this video: https://www.youtube.com/watch?v=OBXWZKeWvGo&list=UU5maSolHQX9er0BOxrzjMwA ?
Every small town (of a few thousand) will have a family group or two like it. The fact that they don’t catch on on to offering their last names is what really strikes me.
Apathy, perhaps?
I know a teacher who asked high school students the factors of 56 (in relation to factorizing a quadratic equation.)
After no one answered she said 7 and 8 are factors of 56.
One female student challenged her and said: “How do you know?”
This is for a high school in the San Francisco Bay Area, although …
As Cochran says, I think your experience is limited.
Well, if kids are not forced to memorize the times tables what do you expect?
We also were given extensive arithmetic problems throughout grade school to help pound it in. I bet that schools don’t teach things like long division anymore. How many exercises are given multiplying, say, two 3-digit numbers? Including decimal points! No calculator allowed.
So for my age group (born 1951) you bet we know our times tables.
We did the times tables up to 14 and 16. Not much call for those in the metric era I suppose.
If “nurture” has any effect, it’s by way of practicing multiplication tables (and the rest of the three R’s)
I would be willing to bet that the girl in question did know her times tables. She just did not realize that she could apply that knowledge to the question “what are the factors of 56?”
I think it was a failure of creativity, not knowledge.
If the widget question is typical of the Weschler arithmetic questions, then this result is consistent with increasing g combined with a decrease in practice-based skill at mental arithmetic. It doesn’t necessarily contradict the increase in the Raven’s results.
Are there results that aren’t confounded by changes in teaching methods and emphasis? (Other than, er, Raven’s… 😉
But you’re not taking into account the rise of machines that count for us. So maybe people aren’t as practised in mental math as they once were because they have a machine calculate for them.
anon’s hypothesis makes sense, and to test it I asked my class to solve equations without the pocket calculator. I was always fast in mental calculations and exercise myself all the time but my students were very good. The machine did not cripple them.
Sometimes when I go shopping I get a cashier who can’t add up or do change. I feel really embarrassed for them. Indian immigrants who work at convenience stores do not tend to have this problem.
The example of 50 years of the Math subtest versus the Matrices subtest is dramatic and it must have something to do with the driver of the Flynn effect. Could it be that math is something so fundamental in terms of cognitive processing – so “core” to g – that it is extremely difficult to improve – but that pattern recognition is susceptible to relatively simple problem solving strategies that require a much lower level of processing? It would be interesting to compare the two subtests using Item Response Theory, and I wonder if this has been done. How would the level of the latent trait required to pass compare for each of the subtests? Doing this kind of study with an Amish population – which has been exposed to simple (8th grade?) Math but for which very little pattern recognition demand exists – and contrasting this with a “modern citizen” population – might show some contrasting results. Maybe.
How do Victorian reaction times factor into these trends?
The Popeye forearm was probably more common back when horses were more common than cars. You still see a cute version on cute horse-crazy girls.
This sort of thing bothers me. It sounds like this: Sure, IQ has gone up over time. But what does IQ measure anyway? Surely not real intelligence, right? Just the ability to perform on an IQ test, right? Surely that’s not meaningful.
You’ll recognize that kind of argument.
On the other hand, I’m only 30. I’ve never met anyone from the past. Maybe they really were cleverer than us.
Could you source your quote?
Personal communication. From an expert.
You could say that IQ measures an ability to discern what a question on an IQ test is really asking. Yet many people take year after year of IQ-similar tests at school starting in early grades, yet when they take their SAT’s in 12th grade still haven’t figured out the simple strategies. The smarter people have figured those out, sometimes on the first try in 4th grade.
… pattern recognition again.
I have a theory about the Flynn Effect. I just thought of it, so I’m not deeply committed to its validity.
I’ve always done very well on multiple choice question based tests. I liked to think it’s because I’m just so damn smart but at least part of the reason is simply because I’m only smarter that most of the people who devise the tests – much less of an achievement.
I’ve made up a lot of multiple choice tests in my day. Most teachers who have no background in test theory or construction make up lousy MC tests. One of the simplest errors is to ask the same quest more than once in two slightly different ways, and give away the answer by so doing. Smart test takers will go back and revise their early answers based on clues they pick up from later questions. Many test constructors pad their list of answers with obviously wrong answers. This means guessing becomes a better bet – your odds go from one in five down to one in four or three or lower.
My hypothesis is that test takers are getting better at exploiting the weaknesses in multiple choice tests made improperly. The test constructors are in a race against the test takers and so far the takers have been winning, but eventually multiple choice tests will get as good as they possibly can and the Flynn Effect will die out.
Yeah, this is a good point.
Also, “math scores have changed less than matrix scores, so let’s declare that math is closer to g”…what reason is there to assume that? Am I missing something?
I just that think that math is more important than whatever.
This is true for stem fields and the technological and productivity advancements that flow from those fields into the economy and the physical superstructure. The problem that I see is that the people who have gained control of the political process come from the verbal side. At some point, if that verbal faction decides that everyone who knows how to use a slide rule needs elimination or re-education, we might not be as lucky as Red China in making a recovery. In which event, one could say that it just as “important” to have “smart” people working on some things other than math.
Do any high school students still take a class in the slide rule?
Probably not.
Here’s a meta-analysis of dysgenics studies: https://www.demographic-research.org/volumes/vol18/5/18-5.pdf
the trend: http://imgur.com/a/xjwO1 (Table 1 is also useful)
the dataset: http://www.demographic-research.org/volumes/vol18/5/files/StatusFertilityDataset.xlsa
It’s not too apparent from the graph but it does look like things got noticeably worse for women born in the latter half of the twentieth century.
Some recent data from Sweden is also available: https://www.demographic-research.org/volumes/vol14/16/14-16.pdf
correlation by occupation: http://imgur.com/a/ZeuOU
“The study reveals that as fertility declines, there is a general shift from a positive to a negative or neutral status-fertility relation.”
In English: lower fertility is almost entirely driven by high status women not breeding anymore. Dr Cochran has talked about this https://westhunt.wordpress.com/2014/03/23/burning-seed-corn/
People talk about welfare as a dysgenics program. I wonder if women in the workforce is even worse.
But more women worked in the past than you might think. The 1950s were an anomaly. The economy in North America was great because so much of Europe was in ruins. But our factories were fine.
My grandmother (born in England, 1895) was from a family of mostly girls. They were all taught trades.
Same with my husband’s grandmother (also England, born 1901). After being widowed with two small children in the 1930s she went back to her trade (seamstress) so her kids would not have to go an orphanage.
On the farms, women certainly worked as much as men (although the men did the work that required strength).
In this day and age surviving on one income is difficult. The price of housing in particular is unreal (I live in Victoria, B.C.)
I worked myself and brought up two children, as did my mother.
Women in those jobs still have kids. Not so for the high powered stuff.
“The study reveals that as fertility declines, there is a general shift from a positive to a negative or neutral status-fertility relation.” I take English like that as a pretty good indicator of declining intelligence among the academic population. Indeed, how else could universities cope with the increase in the numbers of students?
In real life the answer would be dependent on some preconditions.
3 x 18 is 54c. From a dollar, that would be 46c. Unless you are in Canada where the penny was abolished for cash transactions, and the results are rounded. In which case you would get back 45c. But if you paid electronically you don’t get change and you pay 54c. Plus perhaps transaction fees.
It’s like the old joke about a teacher asking little Johnnie “you have 20 sheep. One finds a hole in the fence. How many are left?” Johnnie answers, “zero”. Teacher says “Wrong. You don’t know subtraction?, 20 take away 1 is 19.”
Johnnie retorts “You don’t know sheep! If one gets out, the whole flock follows!”
You’re forgetting Canadian GST, 5%. Most provinces also have a sales tax. Some things are exempt (like food). It’s more complicated than it appears.
Hardest level PISA MATH question and percentage of students who got it right:
Is this a joke?
For cyclists in Copenhagen, the average cycling speed is 15.5 km/h. Helen did 28 km/hour. It must be a joke.
Perhaps her ride was all down hill.
How I missed that? To the river and back, downhill all the way!
Jokes aren’t as funny when you explain them..
A story.
For several years in a row teacher of mathematics gave her students the following exam problem.
“Given on a plane a right-angle triangle with hypotenuse of length L=10 (e.g. inch).
The height drawn from right angle to the said hypotenuse has length h=6.
All those years students made the teacher happy by writing
Area = L * h /2 =30 (square inch.)
At some year a student came, with poor knowledge of English, who was unable to solve the problem above.
Quiz: what was the matter with that student ?
I forgot to say that it was the Area of triangle was to be calculated.
Um, he didn’t associate the length of the hypotenuse with the base of the triangle, so answered Area = bh/2 = 86/2 = 24?
Didn’t like the asterisks.
Area = bxh/2 = 8×6/2 = 24?
It is the teacher who had the poor English. All right triangles have hypotenuses. She was probably from South Asia.
“It is the teacher who had the poor English. All right triangles have hypotenuses. She was probably from South Asia.”
You need to know the 10 is the hypotenuse to solve the problem. The issue is the math, not the English.
Dude, the question is not whether or not it is a 6-8-10 right triangle. The question is what didn’t the new foreign student understand about the teacher’s approval of 30 square inches as being the right answer to a very poorly stated problem by Florida resident (if it was even a properly phrased question by the hypothetical teacher in the first place.)
Answer to the quiz:
Draw the hypotenuse of length 10 and find midpoint, which is at the distance 5 from each end. Take a compass with one leg at the said midpoint, and draw the circle of radius 5.
Potential right-angle vertices of right-angle triangle are all located on that circle: it is a well-known property in geometry on the plane. It means the maximum value of h ( height towards hypotenuse 10) is only 5. it means that
{a right-angle triangle with hypotenuse L=10 and height h=6 towards it}
can not exist.
Student could not solve the problem, since the question was about the area of the figure, which can not exist.
30 square inches?
You need to right this so-called quiz down in your native language first, and then get a competent translator to render it in English.
Write.
And perpendicular. And side.
I now realize that the answer depends on how you picture the triangle on the plane, and what you consider to be the height. That might be clearer to someone who had taken the class than to someone reading the problem on the internet. As written I wouldn’t have answered 30. And I know some geometry. (Euclidian and non-Euclidian.)
I hear Denmark is pretty flat. Where I biked as a kid there were plenty of hills.
Hey, maybe she was on a motorbike.
I remember many bikers overtaking me at about that speed, and it wasn’t during competition.
These 28 km/h twice more than “average” but twice less than the Olympics record for such distance.
If Helen didn’t stop waiting for traffic or examining the scenery, then figure is completely OK.
Oops of course I meant ‘cyclists’ not bikers.
“For cyclists in Copenhagen, the average cycling speed is 15.5 km/h. Helen did 28 km/hour.”
If a velocipede were to coney a maiden at such a tempo, she would be in jeopardy of having her uterus expelled from her body.
17mph sustained for 15 minutes? Impossible.
BS. Get outside and look at some real cyclists.
Note that most modern bikes have gearboxes so frequency of pedaling
Also, one could argue that it’s not contiguous 15 minutes but 9 min + rest + 6 min.
Anyway, this speed is completely reasonable for amateur sportswoman.
Helen pedals pretty fast.
I looked up PISA math test questions on the net. The test writers not only made poor Helen pedal like a maniac, but their questions are dumb and misleading. Vide:
“Nick wants to pave the rectangular patio of his new house. The patio has length 5.25
metres and width 3.00 metres. He needs 81 bricks per square metre. Calculate how many bricks Nick needs for the whole patio. ”
Quoting Marx: “A five years old could answer this. Bring me a five years old!”
The wise landscaper figures in an overage.
The landscaper would fail PISA. To succeed in PISA and similar tests, one has to figure what the test-writer – who probably is unfamiliar with bicycles, landscaping, sheep, etc. – thinks that the correct answer should be. That ability is the IQ that these tests measure.
P.S.: Mr Florida Resident above and his kind, who would not answer “30 sq.inch” to that elementary problem, would risk being classified as feeble-minded. Rightly, if you allow me to say.
The point is that the maximum height of a right-angle triangle with an hypotenuse of length 10 is 5. The calculation is easy, but the triangle doesn’t exist in Euclidian space.
The maximum height is 10 – epsilon.
Esteemed j:
I definitely allow you to say that that I will go for the risk of being classified as feeble-minded. I do not mind if you personally consider me as feeble-minded.
*
I am really glad that “Bert” understood the joke correctly.
One has to add to Bert’s formula that not each height (out of 3) is subject to inequality
h< L/2, but only the height drawn from the right angle towards hypotenuse L.
I am sure that “Bert” meant it, but it is rater longish to write.
That is why the original formulation of the “story” looked so clumsy, and not only due to mediocre quality of my English writing.
*
To explain this to statistically mature participants, here is an analog of the “story”;
Suppose an experimenter has measured the following 2-by-2 correlation matrix:
<xx> = 1, <yy> = 1,
<xy>===<yx>=1.3.
Professional statistician should explain to the experimenter, that there was some error in measurements.: Probability of eigenvector (a.k.a. Principal Component {x-y} ): that probability is negative (!?!). To visually catch the falsity of that type in a 5-by-5 matrix is much more difficult. But for 2-by-2 matrix it is easy: both trace and determinant must be non-negative. For me the “story” is a very simple example of “non-negativity condition.
*
My sincere apologies to all who felt offended. I did not mean do do that.
Dear Florida Resident: Thanks for enlightening me. Richard Feinstein, asked by the Army psychologist if he felt that people was staring at his back, answered “sure”, correctly assuming that those behind him waiting to be interviewed were watching him. He was classified 4-F. Please do not be annoyed by the analogy: as for this feeble-minded me, my answer is an emphatic “30 sq.inch” while pointing to eppur si muove, like in Bartolomé Murillo’s painting.
Reply by esteemed “j” of
May 20, 2017 at 10:37 pm
is read and understood,
with pleasure. F. r.
“Richard Feinstein” is a good one.
Spell-checker detects an error,
and directs to “Einstein”,
but mentions
neither Feynman, nor Feinman.
Feinman is also detected as error,
and suggested change is “Feynman”.
Florida resident – The values x.x = y.y = 1 and x.y = 1.3 clearly do not satisfy the Cauchy-Schwarz Inequality. No need to consider the eigenvalues of the matrix.
Comment by “Jim”
of May 21, 2017 at 1:24 pm
reminds me bed conversation:
-John, do you love me ?
-And what do you think I am doing right now ?
*
Sure I have designed the example
with the Cauchy-Schwartz inequality
in mind, or something equivalent to it.
Difficult part is not C-S inequality itself,
but to remember, to what class
of math. objects this C-S is applicable.
“Hey wise landscaper, how many bricks will you need?”
The wise landscaper stares into space for a few moments, then says: “I’ll need 81 bricks per square metre.”
I’ll bite. 28 km/h, right?
I think it’s meant to catch out people who average the two speeds, like 60*(4/9 + 3/6)/2 = 28.3, i.e. the wrong answer. Harmonic mean vs arithmetic mean.
Yeah. You just take the total distance and divide by the total time to get the average speed.
When I taught statistics to business students I gave the class a similar little problem as a way to introduce the harmonic mean.
I suspect people today really are smarter at dealing with the kinds of electronic logic devices that are omnipresent today due to Moore’s Law:
http://www.unz.com/isteve/the-flynn-effect-across-time-and-space/
The early IQ folks had a correct sense of which way the world was moving and IQ tests measure styles of thinking that became more common over the course of the 20th Century. Perhaps it’s not a coincidence that the father of IQ testing in the U.S., Louis Terman, was the father of the man with perhaps the best claim of being the father of Silicon Valley, Fred Terman.
By the way, both got their name taken off of Terman Middle School in Palo Alto, the highest test scoring public middle school in California, this year for the father’s crimethinking, because what did the Terman family ever do for test scores in Palo Alto?
I dunno Steve.
I have said this before, but there are really smart people who write those UIs so that they are ‘intuitive’ for dumb people but still they manage to get some things wrong.
I don’t always catch all the little song allusions Greg throws in here, I’ve seen Simon & Garfunkle and Shakira (SNP’s Don’t Lie) of course and recently Led Zep (Ramble On) and the Who/TRex – but is this the first Metallica sub-reference we’ve seen?
Did you miss Blue Öyster Cult? Godzilla?
Dude, you have almost been elevated to the level of the God Emperor Trump!
groan
I did completely miss that one:(
‘History shows again and again how nature points out the folly of men’?
I admit my Blue-Oyster-Cultology is lacking
I’m missing all of them. For me, pop music went downhill after the early 1970s and I quit listening.
I did like Simon and Garfunkel, but I don’t recall a post with a title alluding to them.
I just thought that Greg came up with really stupid post titles, to be honest. Culture again.
they’re not always titles, sometimes phrases in sentences. The recent S&G one was a paraphrase of the line from “The Boxer” – “a man hears what he wants to hear and disregards the rest”
What’s scary are the answers to the question, “Youre watching television. Suddenly you realize theres a wasp crawling on your arm” on the Wechsler.
Replicant Leon had a good answer to a like question on the Voight-Kampff test.
My mother? Let me tell you about my mother.
I’d kill it.
You must be my lost twin.
This is slight OT, but I find it appalling. The American Mathematical Society let a rabid fruitcake write on their blog saying cis white male [mathematicians] should quit their jobs. She ranted on and on. I can scarcely believe it.
http://blogs.ams.org/inclusionexclusion/2017/05/11/get-out-the-way/
You should check out her thesis, what a trainwreck
Click to access PiperThesisPostPrint.pdf
I read the list of AMS members to get an impression of whom she wants to eliminate to make place for her kind. I smell a Feminazi.
Something else matters.
I hope history is accurate in telling the truth in what just transpired. It must have been incredible to be in that room May 18th, 2017. 100 senators flied into it on Thursday and listened to a briefing by Rod Rosenstein, Deputy Attorney General. What he presented to the United States Senate was a thorough overview of what the FBI knows about the Trump Russia connection. Pretty dramatic setting, it is called SCIFF, Sensitive Compartmented Information Facility. Whatever is said in there is supposed to stay in there.
But guess what…
People confide in others closest to them when what they here is incredibly disturbing. Then those closest confidants talk to a few people the next day. “You won’t believe what I just heard from Senator X.”
I am not sharing any specifics. Know this. Trump’s goose is cooked.
Take me with multiple grains of salt. But watch how much defense Trump gets from anybody in the US Senate or US Congress after this. The briefing to the US Congress was on Friday.
What the heck kind of comment is this?
Yeah, Rep. Devin Nunes went to a SCIFF.
The Senate was given a “broad overview” by Rosenstein from all I’ve read. He refused to answer many questions in any detail for “fear of disrupting the new counsel’s investigation.”
What came out of the meeting with 100 senators was that Trump was going to fire Comey before Rosenstein wrote the memo.
Trump as much as said he was pissed at Comey about the “Russia thing.”
If you’ll recall, on the morning of the day Obama political hacks Sally Yates and James Clapper testified, Trump tweeted “Ask them about the leaks.” (Trump believes Yates was a major leaker as she was Acting AG for a time as the Dems purposely took their good ole time confirming Sessions.)
Finally, one Senator did ask Yates if the FBI had interviewed her about leaks. She responded, “No.”
Trump was incensed. He knew for sure then that Comey was doing all he could to NOT determine the leakers while the press skewered Trump several times a day with the Russia stories.
Result: You’re fired.
Sorry for going OT.
Neither of these posts shows much understanding of what’s going on. However …
It’s a “SCIF”, one “F”.
The Rosenstein briefing had no big impact on Trump’s public support from Senators.
Sessions was confirmed in less than three weeks, hardly a long time for a confirmation.
ursiform said on May 20, 2017 at 8:31 pm
“I now realize that the answer depends on how you picture the triangle on the plane, and what you consider to be the height. That might be clearer to someone who had taken the class than to someone reading the problem on the internet. As written I wouldn’t have answered 30. And I know some geometry. (Euclidian and non-Euclidian.)”
*
F. r.: Info for connoisseurs of geometry. (Euclidian and non-Euclidian.)
Height in a triangle is the straight line connecting a vertex of a triangle with opposing side (or with straight line continuation of that side): perpendicular to the said side or continuation. This definition does not depend “on how you picture the triangle on the plane” as a consequence of invariance of planar geometry with respect to rotation group. Since there are 3 (and only 3) vertices in a triangle, there are 3 distinct heights in a triangle, each is uniquely defined by the corresponding side. In the “story” that side was clearly announced: “The height drawn from right angle to the said hypotenuse has length h=6.”
Not being a connoisseur in IQ, nevertheless l have read somewhere that one of components of IQ is the ability of mentally rotate 3-D objects, which could help “to someone reading the problem on the internet.”
My best to esteemed “ursiform”.
By the way, does ursi stand for this: https://en.wikipedia.org/wiki/International_Union_of_Radio_Science ?
F.R.: No matter how I rotate that triangle in my mind. it does not appear to be a 3-D object.
i agree.
However I once played with a 7-year old girl (not a relative of mine: grand-daughter of my boss and friend.) Real story.
We drew two triangles on a paper with A, B, C, such that A > B > C, and the other one with a=A, b=C, c=B. The paper was white on one side, and was red on the other side. We cut (a,b,c) triangle by scissors. It took her some time to see that she could not superimpose (a,b,c) with (A,B,C) by translations in the plane, but going into 3-D space we were able to “flip” (a,b,c) so super-imposition with (A,B,C) was made possible.
*
Here is your rotation of triangle as a 3-D object.
*
Unfortunately, after two continuous years of 1.5 -hour weekly studies with me they had to hire a professional teacher to coach her in arithmetic. She is 13 y.o. now, and she is doing reasonably in school.
My best to you, dear “j” !
Your F. r.
Rather general question on re spatial and visualization component of IQ.
*
Mentally rotating objects in 2-D and 3-D is a sub-set of transformations of 2-D and of 3-D spaces, namely, the transformations which do not change the sign of orientation. In the case of linear transformations those are the transformations x’=A*x with det(A) > 0.
*
Is anything known about mental transformations with det(A) < 0, i.e. the ones which include reflection in a mirror ?
Reflection by a still surface of water in a lake is something pre-historic people and even animals could be exposed to ! This also should cover the mental processes for octopuses.
Translations on the Moebius strip definitely can change the sign of orientation.
It is better to say that the sign of orientation can not be defined on Moebius strip in the first place. Is mental play with Moebius strip a part of IQ ?
I expect Dr. Cochran to know the answer to this question.
*
By the way, the “Area” in my “story” is positive quantity by definition, and can not account for the sign of orientation, and thus does not depend “on how you picture the triangle on the plane (ursiform)”
Orientation on a Moebius strip can only be defined locally not globally. On any manifold there are two orientations as in Euclidean space. Together these orientations form the orientation sheaf. For connected orientable manifolds the orientation sheet is disconnected consisting of two separate sheets the choice of either of which defines a global orientation on the manifold. For non-orientable connected manifolds such as the Moebius strip the orientation sheaf is connected. The two sheets of the orientation sheaf join. One can pass continuously from one local orientation to it’s opposite by a suitable path. There is no global notion of “right” vs. “left”.
I meant to say “On any manifold there are two orientations at any point as in Euclidean space.”
“What the heck kind of comment is this?”
More is known about Trump’s connection to Russia than has yet come out. The senators and congressmen were briefed. The information was enough that further defense by republicans of Trump seems unlikely.
Want to bet?
No bet, but I mail you 100 dollars if after the new evidence comes out republicans still stand behind Trump. About time I gave you some money anyway. I definitely would not believe some random guy on the internet claiming what I just claimed. If you are reasonably smart you would place about a 99% chance I am full of shit, but time will tell. 🙂
Evidence of what? Has Trump secretly pledged allegiance to the worldwide Orthodox conspiracy? Is he a secret Skoptsy?
The Soviet Union was a self-declared enemy of the United States, stirring up anti-American trouble everywhere on earth.
Russia doesn’t do that, or anything close to that. They have supported Assad in Syria, while we have supported groups like the Al Nusra Front, which is far more hostile to he US – baked in hostile, ideologically hostile – than Assad is. And there are areas of agreement: Russia thinks they should be dominant over their near neighbors, and we think we should dominate those same countries.
Looking at the world as the big Risk board that it is, the coming strategic problem for the US is China. If we play our cards with even minimal skill (which includes dropping them all on the floor and barfing on them, as usual) Russia should be on our side in that.
Never blame on a conspiracy what could just as easily be blamed on stupidity. Trump will continue to prove himself as stupid.
It’s a conspiracy.
I think some of trump’s aides were compromised by russians, and that Trump realized and tried to ignore it.
Who? compromised how? Why by Russians? Why not by China? Why not by everyone on the planet?
Americans today are working themselves up on the Yellow (Chinese) menace, just like a generation ago they went hysterical about the Japanese buying up Hollywood and Detroit. Yet the Japanese did not conquer the world but went bankrupt and retreated to their island. The Chinese will spend their savings and energies on the roulette in Macao, what is what they like to do best when left alone.
In my opinion, the most unappeased and energetic country is Germany. German public opinion is virulently anti-American, and their tanks (NATO) are stationed on the Russian border and advancing. Give them a chance and they will do what they love to do: war.
Would you be interested in playing Risk for money?
Germany’s armed forces aren’t much: Israel’s armed forces are much stronger, not even counting nuclear weapons. Germany’s military potential is much larger of course, but they don’t seem interested.
I’m baffled by why Russian control of the Crimea is a vital threat to the US. I believe they originally got that back in the 18th century. Nobody regarded Russian control of the Crimea as a vital threat to the US during the Cold War. How many Americans could find the Crimea on a map or for that matter have even heard of it?
It’s true that today Russia and the US really have more fundamental reasons for cooperation particularly in regards to China than they have reasons for conflict.
But as the old geopolitical maxim goes “He who controls South Ossetia controls South Ossetia (and little else)”. Getting involved in a conflict between Georgia and Russia over South Ossetia must be considered one of the zaniest things ever in US foreign policy.
Fresh news from those unpacified Germans: Merkel just declared that the Anglo-Saxons are unreliable and Germany cannot base its defense on the alliance with the US. Meaning that they need to rebuild their armed forces. Against whom? Against Russia (and the USA under Trump is not a reliable partner against Russia). BTW, “disarmed” Germany today has a trice-larger army than Israel, and qualitatively much better. German army is stronger than the Russian one ohne nuclear weapons. I am hearing the gong calling for the next round.
Israel has more tanks, more armored personnel carriers, more fighter/interceptors, and more strike aircraft than Germany. And nuclear weapons.
You are mistaken.
Duplicate (slightly edited, do drop sexism, and international aspect of) a riddle (story).
For several years in a row teacher was giving students the following exam problem.
“Given on a plane a right-angle triangle with hypotenuse of length L=10 (e.g. inch).
The height drawn from right angle to the said hypotenuse has length h=6.
What is the area of that triangle ?
All those years students made the teacher happy by writing
Area = L * h /2 =30 (square inch.)
At some year a student came, who was unable to solve the problem above.
Question (riddle): what was the matter with that student ?
He used the common meaning of “height” to mean height, the vertical or y-axis dimension of the triangle, rather than your non-standard meaning of “the length of the line segment perpendicular to the hypotenuse which passes through the opposite vertex of the triangle”. The usual term for that is “altitude”, and it isn’t specific to the hypotenuse, there are three altitudes for every triangle, so the question leaves out an important piece of information, that the altitude of 6 is perpendicular to the hypotenuse and not one of the other two sides.
As written, the problem could be visualized with the right angle at the origin, one side rising 6 units on the y-axis, another side running 8 units to the right on the x-axis, and the hypotenuse sloping down. If one knows the lengths of the perpendicular sides of a right triangle then of course using the altitude to calculate the area is a waste of time. The answer is 24 for the question as written. Of course such questions, or at least the lessons leading up to them are virtually always accompanied by a diagram which makes the meaning of “height” clear. So my guess is that the student was blind or for some other reason couldn’t see the diagram.
If there was no diagram, then there was nothing wrong with the student, but rather the teacher. Such badly written questions are unfortunately quite common on tests teachers make themselves, but worse, they also often show up on state exams.
Esteemed “savantissimo”:
Thank you for your attention to my humble joke.
You may want also to look at my comments of
May 27, 2017 at 11:04 am, and around.
Hint.
At L = 10, if the value for h were any smaller than or equal to 5 , the formula
Area = L * h / 2 would work OK.
66 (the h) + 88 (the other side) = 1010 (the hypotenuse). The area of the triangle is 68/2 = 24 square inch.
It seems that the multiplication sign is eaten by the format. I wrote 6 times 6 plus 8 times 8 equals 10 times 10. The area is 6 times 8 divided by 2.
Hello, R49 !
OK with multiplication sign. But where 8 came from ?
h = 6 is not (repeat, not) the side (edge) of the whole large tight-angle triangle; it is its supposed height !!!
It goes back to the problem being unclearly written. If you envision a 6-8-10 triangle with the base being 8 it is a geometrically possible triangle with area 24. That can be viewed as a more reasonable interpretation than the problem describing an impossible triangle.
Dear “ursiform”:
IMHO, the problem as a whole is written clearly enough.
It is quite another thing, that you do not like it, as it is formulated:
what was the matter with that new student, why he could not solve the problem presented by the teacher.
You are welcome to compile another problem and discuss it with whomever you want.
Participants: HELP, HELP, HELP !!!
On the un-related topic: my senior (out of two) late grandfather-in-law attended URSI congress in London in 1959. That is the source of my curiosity about URSI.
I still remember my (late) previous wife in a leather-like jacket he bought for her in London.
Ursiform means bearlike. There is nothing complicated about it.
Just because you think that what you write has only one reasonable interpretation doesn’t make it so. That’s a reality of life.
Hey, dear “Bret”,”j”, and “jim” ! Can you kindly help me with explanations.
I feel that something is missing in my explanations.
*
Dan Quayle joke: D.Q. asks a dermatologist:
— Can you make a birthmark on my forehead, just like Gorbachev has ?
http://www.telegraph.co.uk/travel/travelnews/8418789/Tourists-head-to-Gorbachev-birthmark-archipelago.html
—Yes, I can make such a mark. But why would you want it ?
— Oh, when I met Gorbachev, he touched his forehead and said to me:
“Looks like you are missing something here.’
*
Disclaimer: I (F. r.) personally think Dan Quayle is quite smart person, he is just a traditional object of jokes.
*
In that manner, I feel that I am missing something here.
Help, Help, Help !!!
*
“tight” in my previous comment should be changed to “right”.
As you stated this problem you are completely correct. However if I were grading an exam and there was a question like this that a lot of the exam takers misunderstood I would throw out the question.
The following reference may contain some extra info about triangles:
The Math of Love Triangles – feat. Rachel Bloom – “Crazy Ex-Girlfriend”
May be this reference will work better.
The male singers of the song claim that a triangle has several centers.
This gives me a good excuse for the following comment.
*
Here is a curious mathematical problem for your entertainment.
Most textbooks prove 4 theorems about intersections in single points in a triangle
1) of 3 bisectors,
2) of 3 heights,
3) of 3 perpendiculars to mid-points of the sides,
4) and of 3 medians.
But only # 4) is usually proven with the use of similarity of triangles, unique parallels;
in other words, with the use of Euclidean 5-th postulate.
Can you prove # 4) without 5-th postulate ? In other words, is # 4) valid for triangles made of geodesics on the 2D surface of 3D sphere ? Is it valid for the triangles on the Lobachevsky plane ?
Your truly, F.r.
For “jim”:
Here is an example, when each of 3 Cauchy-Schwarz inequalities is satisfied, but one eigenvalue is negative (about -0.0034):
===1,
=1, =1, =0.99.
By this I meant to show that C-S inequalities are necessary conditions, but may be are not sufficient, to get non-negative probabilities.
Thank you for your C-S comment, which stimulated me to invent the example above.
Which, by the way, has nothing to do with the fictitious triangle from the riddle (story) above.
My best to you, dear “jim”.
Your F. r.
For “jim”: corrected formatting
Here is an example, when each of 3 Cauchy-Schwarz inequalities is satisfied, but one eigenvalue is negative (about -0.0034):
averages xx=yy=zz=1, xy=1, xz=1, but yz=0.99.
By this I meant to show that C-S inequalities are necessary conditions, but may be are not sufficient, to get non-negative probabilities.
Thank you for your C-S comment, which stimulated me to invent the example above.
Which, by the way, has nothing to do with the fictitious triangle from the riddle (story) above.
My best to you, dear “jim
A real symmetric matrix is positive definite if and only if all it’s eigenvalues (which are all real for any real symmetric matrix) are positive. A necessary and sufficient condition which seems to have been first stated by Sylvester is that all the leading principal minors are positive. For a matrix of order 2 or 3 this is a practical test. For matrices of high order it’s totally impractical and it’s easier to numerically calculate the eigenvalues by some one of the numerical algorithms for doing so.
Doing a Cauchy-Schwarz check on every pair of variables is the same as checking that all 2×2 principal minors are positive which is necessary but as you indicate not sufficient for the matrix to be positive definite. In fact all the principal minors must be positive and all of them being positive is a sufficient condition however as Sylvester noticed it suffices that all the leading principal minors are positive.
Dear Jim:
Thank you very much for your moral support in previous comment. I agree that the question was invented for the “story”, it was not a real story.
As for the comment directly above, the name of Sylvester came to my mind also, but I did not remember the exact formulation. My excuse: I really did not work with minors.
Anyhow, in my example
xx * yy = (xy)^2
(11=1)
xx * zz = (xz)^2
(11=1)
yy * zz > (yz)^2
(1*1 )> (99/100)^2
negative eigenvalue of that matrix is -334/100,000, it is definitely present there
Which C.S. inequality (<=) did I miss ?
Your truly, F. r.
Do you (and Sylvester) consider the determinant of the whole matrix as one of the “leading principal minors” ? But for 3 by 3 matrix it is not a C.-S. condition.
Yes the whole determinant must be included. The test can also be done with the trailing principal minors. The inequalities coming from n x n principal minors for n > 2 can be considered as generalizations of the Cauchy-Schwarz Inequality. They also express the fact that an inner product on a vector space V induces canonically an inner product on the exterior algebra of V and |x1^x2..^xk|>=0 gives these “higher Cauchy-Schwarz” inequalities. If one works out exterior algebra and then gets Cauchy-Schwarz from |x1^x2|>=0 one gets the longest and most ridiculous proof of Cauchy-Schwarz.
Anything to do with minors is probably best expressed using exterior algebra.
Thank you for clarification. I never worked with exterior algebras. Dealt a little with Li algebras of generators of Li groups.
Now, C.S.:
https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality
From there.
The Cauchy–Schwarz inequality states that for all vectors u and v of an inner product space it is true that …(inequality itself) … is the inner product. Examples of inner products include the real and complex dot product, see the examples in inner product.
in the definition of inner product, also in Wiki, there is requirement for that bi-linear form
to never be zero, except for one of vectors being zero.
In this sense, correlation matrix is not exactly inner product: it is allowed to produce zero,
even for non-zero vector. That was the source of my vacillations.
My gratitude and best to you, dear Jim.
Your F. r.
Inner product (a,b).
(a,a)>=0.
(a,a)= if and only if a=0.
The covariance matrix is always positive but in degenerate cases it may not be positive definite. Sylvester’s test for being positive is that all principal minors are non-negative. If all principal minors are positive then the matrix is positive-definite. However for positive-definiteness it is enough that all the leading principal minors are positive.
The non-negativity of the 2×2 principal minors is Cauchy-Schwarz. The non-negativity of the principal nxn minors for n>2 gives “higher Cauchy-Schwarz inequalities”. But these “higher Cauchy-Schwarz inequalities” follow from Cauchy-Schwarz in the exterior algebra.
I should have said “follows from the non-negativity of the corresponding inner product on the exterior algebra”. Cauchy-Schwarz is (x1^x2).(x1^x2)>=0 and for k>2
Dear Jim:
See my comment of May 27, 2017 at 11:04 am .
Sophisticated variant of a riddle (story).
For several years in a row Professor was giving to graduate students the following exam problem.
“Given 3 real random variables x, y, and z, with zero average values each.
Averages of their products (elements of correlation matrix) are
xx=yy=zz=1, xy=xz=1, but yz=0.99.
Calculate average value of D=(x+y+z)* (x+y+z).
All those years graduate students made Professor happy by writing
D = xx +yy +zz +2xy +2xz+ +2yz = 9 -20.01 = 8.98.
At some year a graduate student came, who was unable to solve the problem above.
Question (riddle): what was the matter with that student ?
In previous comment should be
D = xx +yy +zz +2xy +2xz+ +2yz = 9 – 2(0.01) = 8.98.
*
Resolution of the riddle. New student made an attempt to calculate also
average of G = (2x-y-z)*(2x-y-z), and following the same procedure, he got G = -2/(100),
i.e. got negative value, instead of expected non-negative !!!
His conclusion was that the input conditions of the problem were internally contradictory,
and therefore refused to deal with the problem any further.
F. r.: calculations of average D and of average G are really elementary.
Sophistication comes in the ability to choose the expression for G as above.
None of 3 Cauchy-Schwarz inequalities was not violated,
but evidently something is wrong with the conditions of the problem.
*
My gratitude to “jim” of pointing to C.-S. inequalities
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