Stories tagged Math
So, I open up my web browser this weekend to check the news, and I see the following three polls, all on the same page:
- Rasmussen: Obama up by 5 points
- Gallup: Obama up by 10 point
- Zogby: McCain up by 1 point
These can’t all be right, can they?
Actually, they can. Or, at least, they can all be properly conducted, and just lead to wildly different results.
The only way to get a perfect result is to interview everyone in the country. (In fact, that’s exactly what we do on Election Day.) But that takes so much time and money that no individual pollster can do it. Instead, they interview several hundred people, maybe a couple thousand, and from there extrapolate what the country as a whole will do.
Now, mathematically, you can do this. You just can’t be sure of your answer. Here are a few of the reasons why.
Margin of error
Most opinion polls will state the margin of error. For example, they may say that that Candidate X is ahead by, say, 5 points, with a margin of error of plus-or-minus 3 points. Meaning, the real answer could be as high as 8 points or as low as 2 points.
(Sometimes, the margin of error is actually larger than the result. The poll shows Candidate X leading by 2 points, but with a margin of error of 4 points. Meaning, he could be ahead by 6, or he could actually be behind by 2! This seems to have happened a lot this year.)
A range of a few percentage points, when applied to a country with over 100 million voters, can lead to some pretty huge differences.
Confidence interval
In addition to reporting a margin of error, polls also report a confidence interval, usually 90% or 95%. This means that, according to the laws of mathematics, there is a 95% probability that the real result is the same as the poll result, within the margin of error.
But what about the other 5% or 10% of the time? Well, the folks reporting the numbers don’t like to tell you this, but, mathematically speaking, the poll can do everything right, and still be completely wrong, as much as 10% of the time.
There have been over 700 polls released this election season, and over 200 just in October. No doubt, many of the polls you have heard about fall into this category.
Weighting
In most elections, more women vote than men. If you conduct a survey and talk to 100 men and 100 women, you are going to have to give the women’s answers more weight to accurately reflect the Election Day results.
How much more weight? That depends. Do you think this election will be pretty much the same as previous years? Is there something happening this year that will make a lot more women come out to vote? Or, perhaps, something that will attract a lot more men?
The fact is, nobody knows. Weighting is just educated guesswork. And this year, it is more complicated than usual:
- Black voters are expected to come out in record numbers to support Barack Obama. How many will actually vote? Nobody knows.
- Young people generally do not vote as much as other groups. But many analysts expect more young people to vote this year. How many more? Nobody knows.
- Democrats, having lost the last two elections, are likely to turn out in larger numbers. How much larger? Nobody knows.
- New voters. There has been a great push to register new voters, many of them poor or minority members. These groups tend to vote Democratic. But, the NY Times reports that as many as 60% of those registrations may be fake. If you are a poll taker, you really have no idea how many new voters there actually are.
- Likely voters. Every pollster ends up talking to some people who will not vote on Election Day. Different polls use different methods of figuring out who is likely to actually vote—based on whether they voted last time, how much interest they have in the election, or just taking the voter’s word for it.
The different weighting factors used by the different polls probably accounts for most of the variability we see in the results.
Human factors
Let’s face it – humans are complicated and sometimes uncooperative beings. There are lots of ways they can foul up a perfectly good poll.
- Lying. People have a tendency to tell a pollster what they think he wants to hear. Maybe they just want to be nice; maybe they want to avoid an argument. This skewing has been found in many, many types of polls.
- Refusal. In any poll, a certain number of people refuse to participate—they don’t want to be bothered, or they don’t want to talk politics with a stranger. If refusals are more likely to come from one party than the other, this can skew results.
- Hard-to-reach folks. For years, pollsters called people on the telephone. But today more and more people have cell phones, or call screening, and are hard to reach. Again, if people with such gadgets are more likely to support one candidate or the other, it will skew the results. (One blogger has noticed that John McCain does better in polls conducted during the week than on the weekend, and speculates that's when McCain supporters are home.)
- Bias. Poll takers are only human. They have their own thoughts and opinions. And while they take great pains to be neutral, those opinions sometimes come through in the questions asked or the way the answers are interpreted. Most of the major polling companies are based in big cities like New York, Los Angeles, Boston, or Washington, DC – all cities that are heavily Democratic. This may explain why, in the last two elections, the Democrats did better in the polls than they did on Election Day.
So, with all these problems, how can we figure out who is going to win the election? Well, never fear – there is one sure-fire way to find out the winner:
Read the newspaper Wednesday morning.
And don’t forget to vote!
Tags : Math is fun, i enjoy it a lot, one day, it will help me to the top, I like to do math, cause it is cool, i like to do math in high school, math is like poetry with a neverending flow, math will lead me to where i want to go.
Garrett Lisi uses math to describe everything
E8: the largest and most complex of the exceptional simple Lie algebras
Courtesy Claudio Rocchini Surfer dude, Garrett Lisi lives in his van on a beach in Maui. Using a type of algebra he calls E8, Garrett has developed an exceptionally simple theory of everything -- a grand unified theory that explains all the elementary particles, as well as gravity. (link to pdf of paper found below)
Lisi describes how gravity, the standard model bosons, and three generations of fermions can be unified as parts of an E8 superconnection. This unified field theory attempts to describe all fundamental interactions that physicists have observed in nature, and stands as a possible theory of everything, unifying Albert Einstein's general relativity with the standard model of particle physics.
"I think the universe is pure geometry - basically, a beautiful shape twisting and dancing over space-time. Since E8 is perhaps the most beautiful structure in mathematics, it is very satisfying that nature appears to have chosen this geometry."
"This is an 'all or nothing' kind of theory -- meaning it's going to end up agreeing with and predicting damn near everything, or it's wrong. At this stage of development, it could go either way." Garrett Lisi
This video explains how mathematics predicts reality
Warning, even though I have a degree in physics education, the material presented was way over my head. I will watch it again though, because it does give me a glimpse of how mathematics can lead to understanding, perhaps even someday making possible something like electrogravity. Click this link if the video below does not work
Learn more
Garrett Lisi forum frequently asked personal questions
Garrett Lisi forum frequently asked questions about E8 and Theory of Everything
31 page paper (pdf) An exceptionally simple theory of everything
Cats and dogs, living together: You need more proof?
Courtesy darkmavisPut a star on your calendars today, Buzzketeers, because science is taking a vacation! And I don’t know about y’all, but I’ll be celebrating by eating my own head and diving into a good book. Literally!
Some of you (the sassboxes) might point out that natural laws apparently still apply, and that scientists the world over continue to fiddle around with their science-tools to learn things about the world. But the day I start paying attention to scientists is the day I give up my lifelong dream of eating my own head.
No, I’m listening to the people who have a more direct influence on my day-to-day life: bookies.
See, it seems that a bunch of bookies are concerned that there’s some chance that an alien spaceship will be landing today in the American desert. And so I’m concerned too.
The idea… Nay, the fact of the impending landing came from the Australian psychic, Blossom Goodchild.
—A little side note: I can’t believe it! My name used to be Blossom Goodchild too! I just changed it to JGordon in junior high. I can’t believe we were both born “Blossom Goodchild”! Amazing!
Anyway, Blossom Goodchild, Aussie supernatural, delivers the secrets of love, light, and laughter by channeling “a native American spirit energy” by the name of White Cloud. I’m not entirely clear on why a native American spirit energy would go to Australia for channeling, but who am I to question the ways of the spirits?
Goodchild, presumably with the help of White Cloud, has started a wave of Internet-enthusiasm by predicting the imminent arrival of a massive space ship full of aliens (or “light beings”), which will supposedly be happening today!
The enterprising residents of Earth, not wanting to be caught with our pants down by light beings, have rushed to prepare… by betting on the arrival! Betting so much, in fact, that bookmakers have had to suspend further all wagers.
There has been no evidence of the coming aliens—no radio transmissions, no detected incoming spaceships, and no precedence—except for the word of an Australian good child, and her wandering spirit. And so it would make sense that the odds are set up against the landing. Yet human gamblers aren’t into odds (who ever won something by betting on a sure thing?) or evidence, and they stand to make a lot of money if (when) the ship arrives. See, we shoot from the hip, and we follow our guts, and the bookies know it, and they’re afraid to take any more bets on the spaceship.
So today’s the day, Buzzers. Try to do something impossible. You won’t be the only one.
Fluid flow separation: The fluid flow becomes detached from the surface of the object, and instead takes the forms of eddies and vortices.
Courtesy jaganath
Fluid flow separation explained with mathematics in 1904
In 1904, Ludwig Prandtl, considered the father of modern aerodynamics, derived the exact mathematical conditions for flow separation to occur, but only in two dimensions for steady flows.
Unsteady fluid flow in three dimensions explained with mathematics in 2008
A century later, George Haller, a visiting professor in the Department of Mechanical Engineering at MIT led a group that explained the mathematics behind unsteady separation in two dimensions. This month, his team reports completing the theory by extending it to three dimensions. Papers on the experiments and theory are being published in the Sept. 25 issue of the Journal of Fluid Mechanics and in the September issue of Physics of Fluids, respectively. Haller's coauthors are Amit Surana, now at United Technologies; MIT student Oliver Grunberg; and Gustaaf Jacobs, now on the faculty at San Diego State University.
Fluid mechanics theorists are excited
The equation will forever change the face of advanced fluid dynamics and will have a profound impact on many industries, including the aerospace and automotive industries. This quote from Daily Tech Review shows that this breakthough has theorists in fluid mechanics excited;
The new work -- if it survives the extensive peer review that is to come -- will likely go down as the greatest scientific advance of the decade. The research has already survived a strenuous initial round of peer review.
Equally important, this month Thomas Peacock, the Atlantic Richfield Career Development Associate Professor and his colleagues report important experimental work verifying the theory.
"This is the tip of the iceberg, but we've shown that this theory works," Peacock said.
Fluid dynamics makes a difference
Understanding how surfaces effect how an object flows through a fluid (including air) can make big differences in maximizing performance. Did the new swimsuits make a difference in breaking world records in Olympic swimming competition? How about the surfaces of baseballs, golf balls, and tennis balls? The effects on miles per gallon for autos and airplanes can save millions (billions?) of dollars.
Source: MIT News
On November 4, America will go to the polls and choose its next president. But we do not vote for the president directly. Rather, we vote for electors to represent our state in the Electoral College, and they ultimately choose the president.
By a strange quirk of math, voting in an indirect, divided election such as this actually gives vote4rs more power than if we voted in a direct election. The best way to explain is through an example:
- In 2004, over 121 million people voted for the top two candidates. If we had held a direct election, the winner would need one more than half of those votes—60,500,001. (Roughly.)
- But in our electoral system, a candidate doesn’t need a majority in the entire nation. Instead, they need to win a majority in the ten largest states. In 2004, those states had a combined voting population of over 66 million. So a candidate would need only 33,000,010 votes to win.
So, your favorite candidate needs only about half as many votes to win a divided election as they would to win a direct election. Which means your vote has the potential to be worth almost twice as much!
But what if you don’t live in one of the ten biggest states? That’s OK—those states almost always split between the major candidates, so that voters in other states also become crucial to winning the election.
It is true that under the Electoral College system, there are years when your vote doesn’t matter at all. But in the years that it does matter, it matters so much that, on average, you still come out ahead.
We recently put together a web exhibit to demonstrate this phenomenon. It includes an interactive calculator that allows you to change the voting populations of states and see how this affects voting power.
The exhibit is a simplified version of the work of Dr. Alan Natapoff, a physicist at MIT who first figured this out. You can find a nice summary of his work here,and an update here.
Three black cats: So that's a triple negative... run! Run!
Courtesy heyjupiterEvolutionary biologists and math wizards have put their minds together to summon a pulsating, glistening packet of truth from the void.
Biting into the fruit of this magnificent spell, the meta-scientists gained the following information: superstitions, it seems, are an evolutionary adaptation.
This isn’t an entirely new train of thought, even here on Science Buzz, but this research takes the notion a little further. It has already been proposed that superstitions—false connections between cause and effect—prepare us for “just in case” scenarios. That is to say, as Gene put it, it won’t actually rain on a particular day just because we forgot our umbrellas, but thinking that that’s true will encourage us to bring our umbrellas just in case. The scientists behind this new study are looking at that idea in a more mathematical way.
They started with a similar premise: that assuming a potentially false connection between cause and effect will sometimes be beneficial. For example, to a prehistoric man, rustling grass might sometimes mean that there’s a lion getting ready to pounce on you. Even though a lion isn’t the only thing that will make grass rustle, treating rustling grass as a sign of danger isn’t a bad idea in the long run; the caveman looses nothing by avoiding grass that is actually being disturbed by the wind, but gains everything by avoiding grass the few times that it actually hides a predator.
The scientists then decided that the theory could be tested mathematically. By weighing the losses of false associations (avoiding wind rustled grass) against the gains from when those associations turn out to be real (hungry lions hiding), we can see if that sort of behavior is beneficial to survival in the long run, and will therefore be selected for evolutionarily. The model gets more complicated when there are multiple potential causes to connect to an effect (is it the rustling grass, the full moon, or the random sneezing that means a lion is on its way?), but it seems that assuming false causes is, in general, a decent survival strategy. Fortune favors the timid, apparently.
In modern times, the scientists say, this behavior can manifest in things like attitudes toward alternative and homeopathic medicines; while most of them may be ineffective, the chance that some work is enough to get people to use them all.
Superstitions like avoiding black cats, paths under ladders, and opening umbrellas indoors, however, may have more to do with evolutionarily superstitious behavior getting mixed up with culture and “modern life.” These days, the researchers point out, superstitions are probably less beneficial than they used to be.
That’s a little bit of a copout, I’d say. Fortune, after all, favors the bold, so why not go out on a limb here?
You don’t want black cats crossing your path, obviously, because a much larger black cat could be chasing them—and you don’t want to mess around with huge black cats (especially if they’re being chased by an even larger cat).
Walking under ladders is an easy one. There’s always the chance that a bucket of paint could fall on your head, and once you’ve got a bucket stuck on your head any number of awful things can and will happen to you. Trust me.
Opening umbrellas indoors—if you’re in a very small house, you could seriously damage your umbrella.
Unlucky number thirteen? Thirteen of anything can’t be divided fairly between friends, leaving you with no other option than to kill one of your friends. That’s how blood feuds start.
I should be a scientist. Or a fortune teller.
Once more into the breach: We few, we happy few, we band of kitties; for he to-day that sheds his blood with me shall be my puss-puss.
Courtesy o205billege
The general holds the binoculars up to his eyes and surveys the battlefield. This will be the first test of the new Coordinated Autonomic Tactical force—C.A.T. for short—an army of robot warriors with electronic brains as complex and powerful as a small mammal’s.
The exercise begins, and all goes exactly to plan. The mechanized warriors sweep across the terrain in formation. Faced with unexpected obstacles, they improvise their own solutions. Soon, they are overwhelming the enemy positions.
Suddenly, a squirrel darts across the field. The entire right flank breaks rank to pursue. Corporal Whiskers beings licking himself. Sergeant Buttercup and Lieutenant Muffy begin hissing at each other. Private Snookums climbs a tree and can’t get down.
The general lowers his binoculars. Staring off into the middle distance, he says to his second-in-command, “We may not have thought this through thoroughly.”
The day we all die: September 10, 2008
in Earth and Space Science, Physical Science, Structure of Matter, Math, Scientific Enterprise, and Forces of Nature
Ask not...: It tolls for thee, bro, it tolls for thee.
Courtesy carl.jonesJust messin’, y’all!
Only some of us will die on September 10th! And that’s only because we were going to die anyway. There will be sudden heart attacks, tragic car accidents, hilarious full-body prolapses, and possibly some mysterious cases of spontaneous combustion, and none of that will have anything to do with the Large Hadron Collider turning on on the tenth of September.
That’s right, everyone, you can stop holding your breath, and start crossing your fingers, because the LHC now has a date for its first proton collision.
Some people have raised concerns that turning on the LHC could lead to the destruction of the earth in one of several very sciencey ways. Other people have shouted down these jokers, however, because they are very, very, very probably wrong.
And if the world doesn’t end, well, we’ll probably learn all sorts of rad things about the nature of the universe. We might even get some visitors from the future. But I might put a larger bet on the destruction of the solar system (but, you know, fingers crossed).
So, Buzzketeers, on September 10, do your best to protect yourself from the everyday dangers of existence. Wrap your head in packing foam, fill your tummy with starch-based peanuts, and keep yourself wet and/or naked to prevent sparks catching in your clothing and hair, because you probably won’t want to miss what’s coming out of the LHC.
Some fingers: Some fingers down, more fingers up.
Courtesy anna_tResearchers from MIT have found that a tribe in remote northwestern Brazil has no words for specific numbers.
The language of the tribe, of which there are only about 300 members, seems to be unique in that it has no numbers. Counting was thought to have been innate in human cognition. Apparently that isn’t totally the case. Specific numbers weren’t useful to this culture, so they never developed them in their language.
Instead of specific numbers, the group, called the Piraha, has a couple of relative terms, translating to something like “some” and “more.” Piraha math classes, I assume, would be awesome.
Some + some = more (obviously)
Nothing + some = some (duh)
Nothing + more = some (interesting!)
More – some = some (probably)
Some – more = your mind blown (Whoa!)
Something very anthropologically and linguistically crazy is going on here. Something about how even though we think our thoughts shape language, language actually ends up shaping our thoughts. So if you come from a culture whose language has no concept of specific numbers, how does that shape your perception of the world?
Oh, if only I had been a better student.
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