"Gottfried Helms" <[EMAIL PROTECTED]> wrote in message
[EMAIL PROTECTED]">news:[EMAIL PROTECTED]...
> Hi ,
>
> there was a tricky problem, recently, with the chi-square-density
> of higher dgf's.
> I discussed thath in sci.stat.consult and in a german newsgroup,
> got some answers and also think to have understood the real point.
>
> But I would like to have a smoother explanation, as I have to
> deal with it in my seminars. Maybe someone out has an idea or
> a better shortcut, how to describe it.
> To illustrate this I just copy&paste an exchange from s.s.consult;
> hope you forgive my lazyness. On the other hand: maybe the
> true point comes out better this way.
>
> Regards
> Gottfried
>
>
> 3 postings added:
> ---(1/3)---
> [Gottfried]
> > Hi -
> >
> >im stumbling in the dark... eventually only missing any
> >simple hint.
> >I'm trying to explain the concept of significance of the
> >deviation of an empirical sample from a given, expected
> >distribution.
> >If we discuss the chi-square-distribution
> > |
> > |*
> > | *
> > | *
> > | *
> > | *
> > | *
> > | *
> > |*
> > +-
> >
> >then this graph illustrates us very well, that and how a
> >small deviation is more likely to happen than a high deviation -
> >thus backing the concept of the 95%tiles etc. in the beginners
> >literature.
> >Just cutting it in equal slices this curve gives us expected
> >frequencies of occurences of samples with individual chi-squared
> >deviations from the expected occurences.
> >
> >If I have more df's, then the curve changes its shape; in this
> >case a 5 df-curve for samples of thrown dices, where I count
> >the frequencies of occurences of each number and the deviation
> >of these frequencies from the uniformity.
> >
> > |
> > |
> > |
> > |
> > |*
> > | **
> > |**
> > | **
> > | * *
> > +-
> > 0X²(df=5)
> >
> > Now the slices with the highest frequency of occurences
> > are not the ones with the smallest deviation from the
> > expected distribution (X²=0) - and even if I accept, that this
> > is at least so for the cumulative distribution, it is
> > suddenly no more "self-explaining". It is congruent with
> > the reality, but our common language is different:
> > the most likely chisquare-deviation from the uniformity
> > is now an area which is not at the zero-mark.
> > So, now: do we EXPECT a deviation from uniformity?
> > That the count of frequencies of the occurences of the
> > 6 dices numbers is NOT most likely uniform? HÄH?
> > Is this suddenly the Nullhypothesis? And do we calculate
> > the deviation of our empirical sample then from this new
> > Nullhypothesis???
> >
> > I never thought about that in this way, but since I do
> > now, I feel a bit confused, maybe I only have to step
> > aside a bit?
> > Any good hint appreciated -
> >
> > Gottfried.
> >
> --
>
> ---(2/3)---
> Then one participant answered:
>
> > Actually, that corresponds to the notion that if a "random" sequence is
> > *too* uniform, it isn't really random. For example, if you were to toss
a
> > coin 1000 times, you'd be a little surprised if you got *exactly* 500
> > heads and 500 tails. If you think in terms of taking samples from a
> > multinomial population, the non-monotonicity of the chi-square density
> > means that a *small* amount of sampling error is more probable than *no*
> > sampling error, as well as more probable than a *large* sampling error,
> > which I think corresponds pretty well to our intuition.
> >
>
> ---
>
> --(3/3)-
> I was not really satisfied with this and answered, after I had
> got some more insight:
>
> [Gottfried]
> > [] wrote:
> > > Actually, that corresponds to the notion that if a "random" sequence
is
> > > *too* uniform, it isn't really random. For example, if you were to
toss a
> > > coin 1000 times, you'd be a little surprised if you got *exactly* 500
> > > heads and 500 tails. If you think in terms of taking samples from a
> >
> >
> > Yes, this is true. But it is the same with each other combination.
> > No one is more likely to occur (or better: one should say: variation?).
> > But then, a student would ask, how could you still attribute a
near-expected-
> > variation more likely than a far-away-expected variation in generality?
> >
> > The reason is, th