Re: Stat problem

anewbie <chayanidh@xxxxxxxxx> wrote in
news:1177854773.579259.224810@xxxxxxxxxxxxxxxxxxxxxxxxxxxx:

On Apr 29, 1:45 am, David Winsemius <doe_s...@xxxxxxxxxxx> wrote:
anewbie <chayan...@xxxxxxxxx> wrote in news:1177749690.286300.58130
@n76g2000hsh.googlegroups.com:

I was just wondering if there was any method that I could use to
find the minimum standard deviation of a group of 21 numbers. Given
that min=10, max=90, median=45, Q1=30, Q2=60, mean=45 and mode =35.
I have actually found a number, which I believe is the lowest
possible. However, I have no idea of how to prove that it really is
the lowest one. Any ideas?

Since the count is odd, you know one of the numbers is 45. You have
mode (at least two points), max, and min and maybe the quartile
bounds as well depending on how those are defined.

Because one of your constraints is the mode and all of the values are
integer, the answer probably depends on whether your domain is
integer or real. Since Q2 is the median, also suspect you meant to
type Q3=60. Once you provide further details and corrections of the
problem statement, progress might be possible.

--
David Winsemius

Thank you for your reply.

And, sorry for the mistake I made, I actually meant to type Q3=60.
Also, all the numbers have to be whole numbers. And, unfortunately,
that's all the information I have.

That's quite a bit of information. You have 7 points already. It not
really a statistics problem, anyway, since there is no "randomness" in
the problem, but rather an integer programming problem. See
<http://mat.gsia.cmu.edu/orclass/integer/integer.html>

You want to minimize the SD (or equivalently, the variance) subject to
one constraint that the count of 35's needs to be greater than the count
of any other number, and another constraint that they sum to 21*mean.

ConstraintA would be the ordering:
10 ? ? ? ? 30 ;these need to be <= 30
35 35 ? ? 45 ;these need to be >=30 ; and <=45
? ? ? ? 60 ;these need to be >=45 and <=60
? ? ? ? 90 ;these need to be >= 60 and <=90
ConstraintB: Mean=45 ; equivalently sum=945;
ConstraintC: count("35")>count(x_i) for all i

goal: minimize sum((x_i-45)^2)

My esperiments on a spread*** suggest to me that answer might have
count of "35" as 3. When I set up a trial solution with 4 35's I was way
off on the mean constraint, and my current try on a 3 35's answer is
better than my 2 35's trial. What is you current solution and how did
you construct it?

--
David Winsemius


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