# Statistics, averages and deviations.

• Jan 3rd 2009, 07:22 PM
RossBrons
Statistics, averages and deviations.
A population consists of the five numbers 2,3,6,8 and 11. Consider all possible samples of 2 that can be drawn with replacement from this population (i.e. as soon as soon as number has been drawn and its value noted it is replaced and can be drawn again). Find

i) The mean of the population.

ii) The standard deviation of the population.

iii) The mean of the sampling distibution of means.

iv) The standard deviation of the sampling distribution of means.

---------------------------------------

For i) I calculated the mean to be 6.0 however I found the standard deviation for part ii) to be 3.67 to which I am unsure of its reliability.

Regards
• Jan 3rd 2009, 10:27 PM
Aryth
You have found the mean to be:

$\displaystyle \bar{x} = 6.0$

So, the standard deviation is:

$\displaystyle \sigma = \sqrt{\frac{1}{N}\sum_{i=1}^N (x_i - \bar{x})^2}$

Where N is the number of elements in the population, $\displaystyle x_i$ is the i-th element in the set, and $\displaystyle \bar{x}$ is the mean.

$\displaystyle \sigma = \sqrt{\frac{1}{5} \sum_{i = 1}^5 (x_i - 6.0)^2}$

$\displaystyle \sigma = \sqrt{\frac{1}{5}\left[ (2 - 6)^2 + (3 - 6)^2 + (6 - 6)^2 + (8 - 6)^2 + (11 - 6)^2 \right] }$

$\displaystyle \sigma = \sqrt{\frac{1}{5}\left[(-4)^2 + (-3)^2 + (0)^2 + (2)^2 + (5)^2 \right]}$

$\displaystyle \sigma = \sqrt{\frac{1}{5} (16 + 9 + 0 + 4 + 25)}$

$\displaystyle \sigma = \sqrt{\frac{1}{5} (54)}$

$\displaystyle \sigma = \sqrt{10.8}$

$\displaystyle \sigma = 3.29$

And there you go.
• Jan 3rd 2009, 11:24 PM
mr fantastic
Quote:

Originally Posted by RossBrons
A population consists of the five numbers 2,3,6,8 and 11. Consider all possible samples of 2 that can be drawn with replacement from this population (i.e. as soon as soon as number has been drawn and its value noted it is replaced and can be drawn again). Find

i) The mean of the population.

ii) The standard deviation of the population.

iii) The mean of the sampling distibution of means.

iv) The standard deviation of the sampling distribution of means.

---------------------------------------

For i) I calculated the mean to be 6.0 however I found the standard deviation for part ii) to be 3.67 to which I am unsure of its reliability.

Regards

For (iii) and (iv) read this: Sampling distribution - Wikipedia, the free encyclopedia
• Jan 4th 2009, 08:06 AM
RossBrons
Thanks, for the input Aryth, that was very helpful.

Is the standard deviation not...

$\displaystyle \sigma = \sqrt{\frac{1}{N-1}\sum_{i=1}^N (x_i - \bar{x})^2}$

And is the standard deviation of $\displaystyle \bar{X} = \frac{\sigma}{\sqrt{n}}$
Thereby meaning the standard deviation is not 3.29 but 1.47?
I'm lost, can you please untangle this mess?

Thanks

Mr Fantastic, I can't understand how to apply the math on that wiki article. Kind Regards,
Ross
• Jan 4th 2009, 10:23 AM
vg284
the second one in an unbiased estimatior ?
• Jan 4th 2009, 10:41 AM
Aryth
Quote:

Originally Posted by RossBrons
Thanks, for the input Aryth, that was very helpful.

Is the standard deviation not...

$\displaystyle \sigma = \sqrt{\frac{1}{N-1}\sum_{i=1}^N (x_i - \bar{x})^2}$

And is the standard deviation of $\displaystyle \bar{X} = \frac{\sigma}{\sqrt{n}}$
Thereby meaning the standard deviation is not 3.29 but 1.47?
I'm lost, can you please untangle this mess?

Thanks

Mr Fantastic, I can't understand how to apply the math on that wiki article. Kind Regards,
Ross

Both can be used, but the one I provided has a smaller error in estimation, but using the N-1 formula, the standard deviation is the 3.67 that you had initially, and not 1.47.

Your sampling distribution formula is the correct one.
• Jan 4th 2009, 10:44 AM
RossBrons
Quote:

Originally Posted by Aryth
Both can be used, but the one I provided has a smaller error in estimation, but using the N-1 formula, the standard deviation is the 3.67 that you had initially, and not 1.47.

Your sampling distribution formula is the correct one.

Yeah, thanks for that. Greatly appreciated.

Has anyone got any ideas on how to tackle part iii and iv?

Cheers,
Ross
• Jan 4th 2009, 12:46 PM
mr fantastic
Quote:

Originally Posted by RossBrons
Thanks, for the input Aryth, that was very helpful.

Is the standard deviation not...

$\displaystyle \sigma = \sqrt{\frac{1}{N-1}\sum_{i=1}^N (x_i - \bar{x})^2}$

[snip]

This is an unbiased estimator and is used to estimate the sd of a population from a sample taken from the population. You have only a sample (no mention of where it's come from) and you want the sd of only the sample. So the original formula you were given is the one to use.
• Jan 4th 2009, 01:00 PM
mr fantastic
Quote:

Originally Posted by RossBrons
And is the standard deviation of $\displaystyle \bar{X} = \frac{\sigma}{\sqrt{n}}$ Mr F says: Yes.

Mr Fantastic, I can't understand how to apply the math on that wiki article. Kind Regards,
Ross

$\displaystyle \mu_{\overline{X}} = \mu$.

$\displaystyle \sigma_{\overline{X}} = \frac{\sigma}{\sqrt{n}}$.

Substitute $\displaystyle \mu = 6.0$ and $\displaystyle \sigma = 3.29$ and $\displaystyle n = 2$.

It's that simple.