Sample distribution of sample variance

**shellshock** · November 14th 2009, 08:57 PM

I don't quite understand how this question is asking for, and how to do it.

Consider the (small) population: 1,3,4,7

Find the sampling distribution of sample variance(S^2), the sample variance when you take samples of size 2 with replacement.

thank you for your help in advance

**CaptainBlack** · November 15th 2009, 12:55 AM

Originally Posted by shellshock

I don't quite understand how this question is asking for, and how to do it.

Consider the (small) population: 1,3,4,7

Find the sampling distribution of sample variance(S^2), the sample variance when you take samples of size 2 with replacement.

thank you for your help in advance

Well there are six samples of size 2, you can calculate the sample variance of all six and these each are equally likely so have probability 1/6.

CB

**shellshock** · November 15th 2009, 09:46 AM

Originally Posted by CaptainBlack

Well there are six samples of size 2, you can calculate the sample variance of all six and these each are equally likely so have probability 1/6.

CB

Can you show me an example of how to calculate sample variance of one of them?? Thanks!

**CaptainBlack** · November 15th 2009, 09:50 AM

Originally Posted by shellshock

Can you show me an example of how to calculate sample variance of one of them?? Thanks!

Take the sample $\{1,3\}$ its mean is $2$ , and its variance is:

$V(\{1,3\})=\frac{1}{2}\left( (1-2)^2+(3-2)^2\right)=1$

CB

**shellshock** · November 15th 2009, 10:20 AM

Originally Posted by CaptainBlack

Take the sample $\{1,3\}$ its mean is $2$ , and its variance is:

$V(\{1,3\})=\frac{1}{2}\left( (1-2)^2+(3-2)^2\right)=1$

CB

Is it possible for the equation to be $V(\{1,3\})=\frac{1}{2-1}\left( (1-2)^2+(3-2)^2\right)=2$ ???? Because in my notes, the variance of a sample you have to minus 1 in the denominator. Also after I find all of the sample variance how do I find the sampling distribution of S^2? Do I just use the same equation, but use the mean of the results from all the sample variance instead?

**CaptainBlack** · November 15th 2009, 11:05 AM

Originally Posted by shellshock

Is it possible for the equation to be $V(\{1,3\})=\frac{1}{2-1}\left( (1-2)^2+(3-2)^2\right)=2$ ???? Because in my notes, the variance of a sample you have to minus 1 in the denominator. Also after I find all of the sample variance how do I find the sampling distribution of S^2? Do I just use the same equation, but use the mean of the results from all the sample variance instead?

It asks for sample variance not the estimate of the population variance, so I would use 1/2 not 1, but what you use is up to you so long as you say what you have done.

CB

**shellshock** · November 15th 2009, 07:31 PM

Originally Posted by CaptainBlack

It asks for sample variance not the estimate of the population variance, so I would use 1/2 not 1, but what you use is up to you so long as you say what you have done.

CB

O okay thank you, but how do i find the sample distribution of S^2?

**CaptainBlack** · November 15th 2009, 07:49 PM

Originally Posted by shellshock

O okay thank you, but how do i find the sample distribution of S^2?

You calculate s^2 for each of the six possible samples, and these each have a probability of 1/6.

CB

**shellshock** · November 15th 2009, 08:29 PM

Originally Posted by CaptainBlack

You calculate s^2 for each of the six possible samples, and these each have a probability of 1/6.

CB

Is there an equation for it? I'm sorry, this is the first time I'm doing sampling distribution.

**CaptainBlack** · November 15th 2009, 10:41 PM

Originally Posted by shellshock

Is there an equation for it? I'm sorry, this is the first time I'm doing sampling distribution.

See the attachment for the calculation of the variance for each of the possible sample of size 2:

Sample distribution of sample variance-gash.png

So the sampling distribution for variance is v=0.25 with p=1/6, v=1 with p=1/6, v=2.25 with p=1/3, v=4 with p=1/6, v=9 with p=1/6.

CB

**shellshock** · November 16th 2009, 12:12 PM

Originally Posted by CaptainBlack

See the attachment for the calculation of the variance for each of the possible sample of size 2:

Click image for larger version.

Name: gash.png
Views: 34
Size: 3.4 KB
ID: 13871

So the sampling distribution for variance is v=0.25 with p=1/6, v=1 with p=1/6, v=2.25 with p=1/3, v=4 with p=1/6, v=9 with p=1/6.

CB

Thank you very much!

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