Re: Contradictory tests of normality
- From: RichUlrich <rich.ulrich@xxxxxxxxxxx>
- Date: Fri, 02 Jan 2009 12:37:45 -0500
- I'm leaving in the previous posts, for the sake of context.
On Fri, 2 Jan 2009 14:58:12 -0000, "Dan" <tesfes@xxxxxxxxx> wrote:
"RichUlrich" <rich.ulrich@xxxxxxxxxxx> wrote in message
news:bl7ll4d9ifvardrvo3qupjbvief263p2vb@xxxxxxxxxx
On Tue, 30 Dec 2008 12:12:27 -0000, "Dan" <tesfes@xxxxxxxxx> wrote:
"Ray Koopman" <koopman@xxxxxx> wrote in message
news:c13f52fa-9b5c-4693-a101-2077416a4d1a@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Dec 29, 2:53 pm, "Dan" <tes...@xxxxxxxxx> wrote:
Here's the distribution, in more detail.
x p(x) histogram
-2 1/12 x
-1 2/12 xx
0 6/12 xxxxxx
1 2/12 xx
2 1/12 x
sum p(x) = 1
sum p(x)*x = 0 = mean
sum p(x)*x^2 = 1 = variance
sum p(x)*x^3 = 0 = skewness
sum p(x)*x^4 = 3 = kurtosis
But it's still not normal.
A ha I see what you mean about the proportional probabilies now. I also
didn't realise that to calculate the variance, skew and kurtosis you raise
the power each time in that equation you put there, so thanks for that.
What I am still confused about though, is that, to my understanding, the
distribution of those values is as normal as can be. Even the shape looks
normal in a histogram/the text one that you have produced above. How
could
the shape be any more normal? Granted it is not 'smooth' but that's just
because the sample size is so low.
No. The reason that it is not "smooth" is because it is
*defined* to be integers. The moments will remain the
same if you multiple each count by 100. Or a million.
"Approximately normal, but discrete". And truncated, since
there will never be any extremes beyond the observed 2 SD
range (-2 to +2), regardless of sample size. For practical
applications, like generating Normal deviate for a Monte Carlo
experiment, both facts could matter a lot - the truncation and
the discreteness.
But it is also possible to match a normal distribution with
a continuous distribution on the first 4 moments, while
higher moments differ - as Herman was emphasizing.
--
Rich Ulrich
I've tried looking around the net for what the higher moments are. I'm no
mathematician so I'm probably getting out of my depth here. As far as I
understand it, the 1st is the mean, the 2nd is variance, 3rd skew and 4th
kurtosis.
- You imply, but you don't ask the implied questions. I don't know
any names for the higher moments. I've never seen them
taken as a direct concern, except for knowing that they can
be computed. The "central moments" would be computed
following the simple model -- definition, actually -- that Ray showed.
The Nth moment is the average value at the Nth power.
Why is the distribution Ray gave abnormal according to what these
higher moments are?
"Why?" That's more complicated that assuring you that the
higher moments *will be* different. If all the powers matched,
the distributions would be the same (I think). Ray's example
can't be "normal" -- because, for one thing, it is discrete. For
another, it is truncated.
The Normal distribution has various special and interesting
properties. Those are not all contained in the first four moments.
Also, could we not say that if a distribution does meet the the first four
moments it is probably normally distributed although not necessarily,
- it is sometimes "safe" to assume normality" -
and if
this is the case, could we work out a probability statistically?
- No.
"If a distribution does match the first four moments..."
Where does this distribution exist, if not in your head?
Anything that is created as a simple average of similar units
is going to tend to be normal, as the N increases, under the
Central Limit Theorem. For some distribution that you want
to describe which is created this way, matching the first
four moments is pretty robust as indicating 'normality'.
But, so what? If, for instance, the "scores" are integers
and not on a continuum, they can't be "normal" in the
sense of being continuous. Does that matter? Well,
sometimes....
Ray handed you a set of numbers that gives you a match
for 4 moments. What is the probability that a set of numbers
that you fall across will be generated as a simple average,
as oppsed to being "devised" by someone or some process
that you haven't contemplated?
.
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