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Re: Simulation for the standard deviation
Posted:
Feb 25, 2013 7:56 PM
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"Ray Koopman" wrote in message
news:9d530ad9-8073-4870-8bf4-c66ea5cf18ec@or10g2000pbc.googlegroups.com...
On Feb 25, 2:26 am, Cristiano <cristi...@NSgmail.com> wrote:
> On 25/02/2013 6:13, Ray Koopman wrote:
>> On Feb 24, 5:08 pm, Cristiano <cristi...@NSgmail.com> wrote:
>>
>>> I randomly pick 3 numbers in U(0,1) and I get, for example,
>>> sd = .1234, but we know that the expected sd is .288675.
>>> How good .1234 is? Is there any way to calculate a p-value
>>> which says how good is .1234?
>>
>> What other information is available about the sample besides
>> its n and sd? Its mean, range, min, max, ... ?
>
> I know all the n numbers in the sample and hence I can calculate
> anything.
If you look at bivariate scatterplots of (range,sd) for a large
number of samples of the same size, it is immediately apparent
that both E(sd|range) and SD(sd|range) are proportional to range.
However, both E(range|sd) and SD(range|sd) are nonlinear.
Since the true sd is proportional to the true range, and the best
available estimate of the true range is the sample range, your
question seems purely academic. You need an expression for the
sampling distribution of the sample sd for samples from a uniform
distribution. I don't know what it is. Problems like that were
popular in the first part of the last century. Maybe someone else
can suggest a reference.
============================================
The standard work, Kendall & Stuart's "Theoretical Statistics" will provide
a formula for variance of the sample variance. This could be used to provide
a sampling interval for the sample variance, and hence for the sample
standard deviation. Indeed, a formula for the variance of the sample
variance exists on the Wikipedia page for "Variance".
In the present context, an alternative is just to do multiple simulations of
the sample standard deviation and use this to get the sampling distribution
empirically. Of course this would assume that the random number generator
has adequate properties.
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