Hypothesis testing for std. deviation (SD)

[musicgold]

Hi,

What I know: In a hypothesis test for the mean, we compare a sample mean with a hypothetical sampling distribution of means. And depending on how far it is away from the mean of the sampling distribution, we attribute it the probability of getting that value purely by chance.

What I don't understand - In a hypothesis test for the SD, why don't we compare the sample SD with a sampling distribution of SDs? (instead, I have seen people using the the Chi sq. distribution)

As per the applet on the following web page, even the sampling distribution of SDs appears normally distributed around the population SD value. So why is it not used?
[/PLAIN]
http://www.stat.tamu.edu/~west/ph/sampledist.html[/URL]

Thanks.

 

[Stephen Tashi]

Hi,

What I know: In a hypothesis test for the mean, we compare a sample mean with a hypothetical sampling distribution of means. And depending on how far it is away from the mean of the sampling distribution, we attribute it the probability of getting that value purely by chance.

If you did that, what hypothesis would you be testing?

What I don't understand - In a hypothesis test for the SD, why don't we compare the sample SD with a sampling distribution of SDs? (instead, I have seen people using the the Chi sq. distribution)

It isn't clear what hypothesis you are testing. What specific problem are you talking about that was solved using the Chi square distribution?

As per the applet on the following web page, even the sampling distribution of SDs appears normally distributed around the population SD value. So why is it not used?
[/PLAIN]
http://www.stat.tamu.edu/~west/ph/sampledist.html[/URL]

How did you use that applet to create a histogram of sample standard deviations?

 

[musicgold]

Thanks Stephen,

It isn't clear what hypothesis you are testing. What specific problem are you talking about that was solved using the Chi square distribution?

I am talking about the hypothesis test used to verify the standard deviation of a population.
Here is one example.

My question is why don't we use a sampling distribution of standard deviations (SD) to compare the sample SD, as we do for other statistics like mean or median. Why do we need to use the Chi Sq. statistic?

 

[Stephen Tashi]

If f(x) is a 1-to-1 function and X is a random variable then the probability of the event X < v is the same as the probability of the event f(X) < f(v). If you are doing a hypothesis test about X and the distribution of f(X) is easiest to use then its simpler to formulate the test as a hypothesis about f(X). The sample standard deviation and the sample variance are related by a 1-to-1 function, so , when it's simpler, we can use the sample variance to test hypotheses about the sample standard deviation.

This PDF suggests that when sampling from normal populations we might also use the distribution of the standard deviation directly:
http://www.google.com/url?sa=t&rct=...rNp_Tb9hssxhbRQ&bvm=bv.58187178,d.aWc&cad=rja

Different families of population distributions can have different families of sampling distributions for their sample variances and sample standard deviations. Normally distributed populations have chi-squared distributions for their sample variances. Other families of distributions may have sample variances that are not chi-squared. (So one cannot speak of "the" distribution of the sample standard deviation as if there was some type of distribution for it that applies to all situations.)

 

[blue_raver22]

I would like to provide some clarification on hypothesis testing for standard deviation (SD). In a hypothesis test for the mean, we compare a sample mean with a hypothetical sampling distribution of means. This is because the mean is a measure of central tendency and we are interested in determining if our sample mean is significantly different from the population mean.

On the other hand, in a hypothesis test for the SD, we are not comparing a sample SD with a sampling distribution of SDs. This is because the SD is a measure of variability and not central tendency. Instead, we use the Chi-square distribution to determine the probability of getting our sample SD purely by chance. This is because the Chi-square distribution is commonly used for hypothesis testing of variances and standard deviations.

I understand your confusion regarding the normal distribution of the sampling distribution of SDs shown in the applet. This is because the applet is demonstrating the central limit theorem, which states that as sample size increases, the sampling distribution of the SD approaches a normal distribution. However, this does not mean that we can use the normal distribution for hypothesis testing of SDs.

In summary, hypothesis testing for SDs uses the Chi-square distribution because it is specifically designed for testing variances and standard deviations. I hope this helps clarify any confusion.