On Wed, 21 Jul 2010 11:50:45 -0700, Kevin Wampler wrote:

> Jim Henderson wrote:
>> On Wed, 21 Jul 2010 13:58:06 -0400, Warp wrote:
>> 
>> 
>>>   If the curve of people's intelligence is not linear, it can mean
>>>   that
>>> the majority of people are actually below average than above it (or
>>> the other way around, of course, depending on the sign of the
>>> derivative of the curve).
>> 
>> For a large statistical sample, IIRC, it's basically a bell curve.
> 
> It depends on what it's a statistical sample of.  It only comes out a
> bell curve under certain assumptions about how what you're sampling
> behaves, and these will not hold in many circumstances.
> 
> That said, I think IQ tests do tend to come out more or less a symmetric
> distribution (and close to a bell curve) so pretty close to half of the
> population will indeed be below average, so the theoretical point Warp's
> making doesn't really apply in this case.

They do tend to come out to be pretty symmetric - though it seems recent 
study suggests that this may not be accurate; but it has been the 
'standard' for some time.

>>>   As a simple example, if we have 4 people with IQs 90, 90, 90 and
>>>   130,
>>> the average will be 100, but the majority of these people is below
>>> average. In other words, three quarters of the people are "stupid" and
>>> one quarter is "smart".
>> 
>> Except that when measuring "average" intelligence, your sample size is
>> a population, not a small discrete number.
>> 
>> 
> It turns out that Warp's overall point is correct in this case, and you
> can also generate examples for continuous probability distributions.
> This occurs in cases where the distribution is asymmetric (see
> http://en.wikipedia.org/wiki/Skewness).  I'm not sure what he meant by
> "the sign of the derivative", but maybe this was what he was thinking
> about.

Well, yes, his point is correct in this case because he's crafted a small 
sample size that actually does bear out his assertion.  That's kind like 
a friend of mine saying that the red telephone boxes in England are 
extremely rare, only to have one show up pretty much everywhere we went - 
his sample size was his local neighbourhood, where there weren't any - 
ie, his assertion was based on a sample size that wasn't representative 
of anything other than what he was asserting.

Warp's done the same thing here as well - and in a small sample, sure, I 
can prove that over 50% of people are of above average intelligence as 
well by picking numbers that prove that.  That doesn't prove anything 
with regards to a large population distribution, though.

> I don't totally see how it applies to the conversation since "stupid"
> and "smart" in this context haven't been given rigorous definitions, but
> if you are to define them by which side of the IQ median you lie on,
> then he would be correct if the IQ distribution were asymmetric (which
> it mostly doesn't seem to be).

Well, true, neither of those terms really has a rigorous definition; it 
seems that when talking about IQ, those terms tend not to be applied 
along the curve.

Jim

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