Posted on 11/21/2001 11:02:28 PM PST by pcl
That's an interesting statement the Central Limit Theorem. Another is, if a random variable, "Y", is the sum of number of other independendent random variables, Xi, {Y = X1 + X2+ ...+Xn} as the number of independent variables becomes large, {n approachs infinity} the distribution of "Y" approaches a normal distribution, with certain restrictions on the higher moments of the distribution of the underlying variables, the Xi's".
The assumption of a normal distribution is a "model" not a fact of nature. The usefulness of model is limited by how well the actual distribution conforms to the normal and the sensitivity of your conclusions to deviations from this ideal. There are lots of examples in engineering and physics where the assumption of a normal distribution is worthless.
A couple simple counterexamples are useful:
1.) If x1 and x2 are indentically distributed random normal variables and Y = sqrt(x1^2 + x2^2) then Y will assume a rayleigh (not a normal) distribution.
2.) It the distribution of a variable, "Y" is normal, then the distribution of a postive number raised to the "Y" most certainly is not normal. Another way of saying this is that if log10(Z) = Y; Z = 10^Y, (Z might be the thing that I want to measure and plot) either Y may normally distributed or Z but not both.
I'd be happy to tell you a cheerful story wherein Lonesome did not pay attention to the "higher moments of the distribution of the underlying variables" and wound up underestimating the cost of production of a run of 25,000 high performance military receivers by about $250 bucks a unit. What's 6.25 Megabucks among friends?
"Insanity is inherited...you get it from your children."
True. The younger the age, the less reliable the tests. Girls, who tend to be more verbal than boys at young ages, will usually have higher scores due to that factor.
One of the problems with "gifted" programs in the public school system is that they tend to test students at young ages when the tests aren't very reliable, and track those kids for the remainder of their years based on those early scores. Things tend to look somewhat different in high school AP/honors classes, when students tend to self select themselves for difficult course work.
And some parents will say that's because "giftedness" is not all academics, but the funny thing "is" that most schools are supposedly testing for "academic" giftedness.....not sculpting or ballet. :)
2. What good is there in getting less intelligent people to waste their time and effort trying to achieve what they cannot? They will only be discouraged at their failures and think they are worthless. Few people could ever succeed as physicists, for example, and it would be a tragedy to spend years studying only to learn that they are no good or at best mediocre. But almost everyone can do other useful and important things that don't require quite as much brainpower.
So you have to have the golden middle: neither underestimate, nor overestimate your abilities.
Disclaimer: Opinions posted on Free Republic are those of the individual posters and do not necessarily represent the opinion of Free Republic or its management. All materials posted herein are protected by copyright law and the exemption for fair use of copyrighted works.