Orton's Arm

He's claiming that, since 1980, education spending has risen by only 30% in nominal terms, which (at a 3% inflation rate) implies that, in real terms, education spending is only 60% of what it was back then. Even if the annual inflation was only 2%, real spending would only be 78% of what it was back then. Do you honestly think there's any truth to this claim?

I find your education spending number exceptionally difficult to believe. The U.S. spends more per child on education than any other major nation in the world. However, the U.S. education produces the worst results of any industrialized nation. The problem with education isn't that we're failing to dump enough money down the rathole of our existing system. The problem with the U.S. education system is that education dollars are allocated by federal, state, and local bureaucrats, instead of by parents choosing the best schools for their children. Now consider the influence of the National Educators Association--the most powerful teachers' union in the U.S. For many years, its head was an outright socialist, who wanted to produce equality of educational results for all children. To achieve this goal, he successfully influenced public schools to dumb down their textbooks. Sentence structures were simplified, complex ideas removed, difficult words eliminated. While this change harmed gifted children the most, it also harmed children of average intelligence. The least intelligent students did receive some benefit however; because the material was no longer beyond their ability to understand. To address the other part of your post, I also favor a decrease in prison spending. This decrease should be achieved by a dramatic increase in the number of executions, coupled with a speedier and less arduous pre-execution process. Edit: the other crime fighting measure I advocate is a far broader interpretation of self-defense. The fact that someone is breaking into your house while you're in it ought to be a legal justification for lethal force against the intruder.

Both.

I don't like it on several levels. First, once he gets out in 15 - 25 years, there's the chance he might rape again. Then, there's the chance that some liberal might get elected governor, mismanage the budget, and decide to cut prison spending to free up money for waste elsewhere in the system. I've seen liberals cut prison spending by releasing criminals early. Dead criminals can't be released from jail.

Back on page 12, I wrote Bungee Jumper responded with this: Wraith is right to say that my choice of words hasn't always been ideal. But it's been very frustrating for me to continuously try to explain a statistical phenomenon, and have people just not get it. And not only did they not get it, they repeatedly ridiculed the explanations I provided. You'll notice, by the way, that my explanation on page 12 says the same thing as my explanations on page 23. I've been saying the exact same thing for the last twelve pages! I'm glad that finally someone's joined this thread who actually gets it.

If you look at those quotes in the context of the examples I was providing, you'll get a better idea as to what I was getting at.

You make a very good point. In fact, you make an outstanding point--especially with regards to rape. On the other hand, there are those who feel life begins at conception. Are they wrong? When do you feel life begins? At what point do you say, "it's wrong to kill this baby, no matter how the conception might have taken place"? I don't have easy answers to these questions. But whatever else we do, we should be increasing the punishment for rape. There is only one punishment that's appropriate; and that punishment doesn't involve jail.

No you don't, trust me. Think of all the people whom you've called "idiots" over the past month or two. Imagine each of those people challenging you to a duel. Maybe you survive the first, and the second, and even the tenth. But it's only a matter of time before a duel doesn't go your way. Bringing back dueling would, for you, be a death sentence.

It's possible I should have gone into more detail when explaining how I created my simulation. What I did was as follows: 1. created a population of 1000 members, and assigned each a true I.Q. The true I.Q.s were based on a normal distribution. 2. Gave each population member an I.Q. test based on their true I.Q. plus a random error term. 3. Assigned those members with the highest test scores to the Threshold group. 4. Gave Threshold members a second I.Q. test based on their true I.Q.s plus a random error term. On the second I.Q. test, Threshold members obtained slightly lower scores than on the first I.Q. test. This is because Threshold members were selected in part based on their luck on the first I.Q. test. The second time Threshold members were given the test, good luck presumably balanced bad; resulting in slightly lower scores.

An excellent post, and I find the rubber band analogy very apt. I feel you truly understand the phenomenon I've been trying so hard to describe.

To the best of my understanding, the conflict is about this: I feel that someone who scores a 190 on an I.Q. test will, upon being retested, generally obtain a lower score the second time around. I feel that this is because there is an element of luck (or measurement error) in determining someone's true I.Q. Thus, someone with a true I.Q. of 180 might be mislabeled as a 190, or vice versa. The population of people who score a 190 on an I.Q. test consists of those with true I.Q.s of 180 who got lucky, 190s who scored correctly, and 200s who got unlucky. There will be more lucky 180s in this group, than unlucky 200s. When someone who scored a 190 gets retested, that person's expected second score reflects the fact that he might well be a 180 who got lucky the first time around. His expected second score is lower than 190, because the odds that he's a lucky 180 are higher than the odds he's an unlucky 200. Suppose that two people who scored a 190 on the I.Q. test get married and have kids. Suppose further that their kids have I.Q.s in the low 180s. On the surface, it appears as though the children's I.Q.s are closer to the mean than those of their parents. But that's not necessarily the case, because the parents would likely have gotten a score in the low 180s themselves had they retaken the I.Q. test.

It was a hypothetical example, intended strictly to show what happens in systems without measurement error versus systems with such error. The fractional inch problem you described isn't relevant to that hypothetical example. Mostly, the point I'm illustrating is that the presence of measurement error on the first test means that the results from second test will tend to regress toward the mean. The larger the measurement error, the greater the expected regression toward the mean. For instance, say that your measurement system had the potential to be off by a foot. Someone who measured 7'5" the first time around is likely to regress toward the mean quite considerably upon being retested. This is because there are more people who are 6'5" available for getting lucky than there are people who are 8'5" available for getting unlucky.

You have someone's true height. Then you have measured height, which is a function of their true height plus a measurement error term. For example, error could be normally distributed, with a mean of zero and a SD of 1". Or it could be uniformly distributed, with 20% chance of being underestimated by 2"; a 20% chance of being underestimated by 1", a 20% chance of getting correctly measured, etc. I'm not that fussy, because regression toward the mean will happen regardless. The people with very low measured heights are, on average, less lucky than average. They will tend to regress toward the mean upon being remeasured.

An excellent question, Bill. It's too bad that what started off as a good thread managed to turn into a Losman debate. On the other hand, we need to debate Losman somewhere, and it's not like the topic's come up elsewhere on TSW. To answer your question, Green Bay came into the game with a losing record. They moved the ball up and down the field seemingly at will; whereas the Bills' offense looked ineffective and boring. On the one hand, the way we won the game gives almost no hope for future wins. If we only pass for 100 yards next week, if we allow nearly 300 passing yards to the other team's QB; do you honestly think we'll win that way? Maybe, but not likely. Add to that the fact that Green Bay had more rushing yards than us, and it's a truly uninspiring win. So this win created a feeling of pessimism about how we'll do the rest of the year; while at the same time hurting us in the draft. Whether your focus is on doing well this year, or doing well next year; it's a pretty depressing win.

It doesn't sound to me like you've learned a whole lot in those stats courses.

Consider a test with no measurement error--height for example. You stand on the scale at the doctor's office, they take out that height measure thing, and measure you. Someone with a height of 6'2" isn't going to get lucky and have a measurement of 6'4"; nor unlucky with a measurement of 6'0". It's the same every time. If your height is measured at 6'2" the first time, it will be 6'2" the second time, and the third time, etc. No regression toward the mean. Now suppose that you introduce measurement error into this test. Someone who measured out at 6'2" may actually be a 6'0" person who got lucky on the first test. Because there is now measurement error in the system, those who obtained exceptionally high measurements the first time are likely to regress toward the mean upon being remeasured. This is because there are more 6'0"s available for getting lucky, than there are 6'4"s available for getting unlucky. The logic is the same for the math section of the SAT. You want to know what someone's average score would be if they were to take the test 1000 times. You give them the test one time to estimate this score. This system involves measurement error--someone with a true score of 725 could get lucky and score a 750; or unlucky and score a 700. Therefore, someone who scores very well the first time will tend to regress toward the mean a little upon retaking the test. Take away the measurement error on that first test, and you take away the regression toward the mean.

You're wrong on two fronts: first, you are the one who apparently needs a basic education in statistics. Second, the correct answer (in this case) is 725; though it will vary based on the value of Pearson's correlation. In your pair of dice example, the Pearson's correlation is zero: a pair of dice that both roll sixes are neither more nor less likely to get a high roll the second time around than a pair of dice that both roll ones. With the SAT score (or an I.Q. test score) Pearson's correlation is positive, but less than one. Someone who gets a 750 on the math section of the SAT will, on average, get a 725 upon retaking the test.

If you want to see what they think, maybe you should read the article.

A dozen? Where did you come up with that number? Ramius's contributions to this discussion have shown me only that it would be a mistake for any employer to hire him to do serious statistical work.

The fact that you're afraid to answer my question suggests a (slightly) growing awareness that you don't fully understand the regression toward the mean article.

It makes sense to anyone who understands the article to which I linked. Apparently, that category doesn't include you.

Do you remember that big guy from last night--the one who slipped something in your drink when you weren't looking? Do you remember how everything sort of went all blurry after that? And you're wondering why you feel sore today? Um, yeah. About that . . .

For the die roll example to be consistent with the I.Q. test example, the goal of a single die roll has to be to predict the average value of a die roll over the course of 1000 rolls. The stuff about discrete faces, and all that other junk, is just an excuse for you to a) hit me over the head by pointing out perfectly obvious facts about dice as though I was unaware of these things, and b) hide from the question I asked earlier which you didn't answer. Suppose someone scored a 750 on the math section of the SAT the first time they took it. They're going in to retake the test. Is this person's expected score on the retest 750, or is it less? That question is at the heart of the regression toward the mean debate, and not all this junk about dice.

Obviously you didn't understand a single word of my post. Thanks for being consistent.

A very good question. The answer is that in the real world, you'd never do something so foolish as that. Suppose you wanted to know what someone's average score would be if given 1000 I.Q. tests. Your way of estimating this average score would be to give the person just one I.Q. test. Unfortunately, the person might get lucky or unlucky on that one I.Q. test, so your estimate of their true I.Q. might contain some error. Bungee Jumper introduced dice into this discussion. To make dice analogous to the I.Q. test, you have to roll a single die a single time to estimate what that die's average roll would be over the course of 1000 rolls. It's actually a more confusing concept to understand with dice than it is with test scores, but Bungee Jumper challenged me to present the topic by means of dice. As long as error is symmetrically distributed, regression toward the mean will take place. By "symmetrically distributed error" I mean that your chances of getting lucky on the test are equal to your chances of getting unlucky. I chose normally distributed error for my simulation because I had to choose something, and normally distributed error is as good as anything. The average was zero, meaning that on average someone got neither lucky nor unlucky. You ask about different error distributions. Suppose a uniform error distribution. Someone taking an I.Q. test has a 20% chance of getting really unlucky and scoring 20 points too low, a 20% chance of getting mildly unlucky and scoring 10 points too low, a 20% chance of getting scored correctly, etc. Now suppose you're testing a population with 10 people who have true I.Q.s of 190, 100 people with true I.Q.s of 180, etc. Of your 10 190s, 2 will get really lucky on that I.Q. test, and score a 210. If, therefore, someone who scored a 210 came up to you and said, "hey, buddy, I'm retaking the test, what do you think I'll get," the answer would be 190. You know that you're dealing with a person with a true I.Q. of 190. This person might once again get really lucky and score a 210, or he might get really unlucky and score a 170. If you average out all the scores (weighted by probability) this person's expected score is 190 for the second test taking. The same logic applies to those who scored a 200 on the test the first time, those who scored a 190, those who scored a 180, or anyone else who scored above the mean. All of them are expected to get a score somewhat closer to the mean upon retaking the test. When two exceptionally smart people have children, their children's measured I.Q.s tend to be somewhat lower than the measured I.Q.s of their parents. I'm arguing that, in general, people with exceptionally high I.Q. scores were not only disproprortionately smart, but also disproportionately lucky on I.Q. tests. As the article I linked to stated, someone who gets a 750 on the math section of the SAT is more likely to be someone who should have gotten a 725 but got lucky, than someone who should have gotten a 775 but got unlucky. This is because there are more true 725s than true 775s. Suppose that two people who got a 750 on the math section of the SAT got married and had kids. Suppose those kids scored 725s on the math section of the SAT. Do the lower scores mean the kids are dumber than their parents? Not necessarily. The average person who scored a 750 on the math section the first time will, on average, score a 725 the second time. In this example, the children's scores reflect the 725s their parents would have gotten had they neither been lucky nor unlucky. The fact that children's scores slightly regress toward the mean when compared with those of their parents is largely, perhaps entirely, due to the fact that their parents' scores were measured incorrectly in the first place. This is why regression toward the mean isn't the dynamite objection to a eugenics program that some have claimed.