Early Bird Webin 09 - Sample distribution

[frm_daniel]

Hi David,

I refer to the example of demonstration of Sample distribution from Gujarati’s sample of 28 NYSE companies, the sample mean (23.25) is a point estimate of the population mean.

The sample mean 23.25 is simply and average of the 28 NYSE companies. It confuses me as my understanding of sample means is to take all possible samples size of 28 companies PE ratios and prepared a sampling distribution of the sample means. Random sample of 28 companies is selected from a population, and then the mean return of the 28 companies sample is calculated. Repeating this process many times will result in many different estimates of the population mean return (i.e. one for each sample).

However, the example is just simply averaging the28 PE ratios to obtain the sample mean. Is there any I have missed ?

Thanks,
Daniel

 

[David Harper CFA FRM]

Daniel,

I agree with your basic description. In other words, if we take multiple samples, each of size 28, for example:

sample 1 (n=28): sample mean = 23.35
sample 2 (n=28 but different set): sample mean = 24.xx (for example, no reason to expect same unless n = population size)
sample 3 (n=28 but different set again): sample mean = 22.xx (for example)
...and so on...

In each case, just as you say, the sample produces a different estimate of the population mean (the sample mean formula--i.e., sum divided by n--is the random variable ESTIMATOR that produces the point ESTIMATE).

So, in the case of any one give sample Y, the sample mean is a point estimate of the populuation mean, but we do expect the sample mean to vary from sample to sample.

As we repeat multiple samples, each of n=28, our set of sample means (point estimates) ITSELF is a distribution; i.e., the sampling distribution of sample means. Per CLT, we expect this sampling distribution of sample means to be approximately (asymptotically) normal with mean of population mean.

....and so, as we take additional samples (of n=28) and for each sample, we compute the sample mean (a point estiamte), we are (one at a time) building an empirical distribution (of sample means, each a point estimate) that we expect to look very much like the parametric distribution: sample mean ~ N(population mean, population variance/28); i.e., we'd expect it to roughly center at the population mean and its dispersion is less than the population variance (such that, to exaggerate, if you sample grows to approximate the population--i.e., n approaches population--then all the samples will have the same mean and this distribution will have no dispersion, no sampling variation)...hope that helps! David

 

[frm_daniel]

Hi David,
Thanks for your prompt reply. I got your point. However, in TABLE 5-1 P/E RATIOS OF 28 NYSECOMPANIES, it is only one sample showing 28 companies PE ratio. The sample mean 23.3 calculated is only for one sample only. The example in the table takes only one sample, not as to take many samples of n =28 to build an empirical distribution, which is the sample mean that I refer.

Daniel

 

[David Harper CFA FRM]

... totally agree that table 5-1 has no sampling mean (CLT) concept ... did i imply that i webinar, sorry? Thanks, David

 

[frm_daniel]

David,

The example has not made the assumption very clear. It just take an average of the 28 companies (also shown in the excel working )and said it is a simple mean which confused me with the definition of sample mean. If the example give me the sample mean of those 28 companies as 23.35 without showing the table, it would be fine. However, it shows with the calucation of averaging the 28 companies as sample mean, which is confusion. I think in exam the simple mean will be given ?
Thanks
Daniel

 

[David Harper CFA FRM]

Daniel,

Okay, I think we agree if the unstated assumption is "these 28 companies are drawn from a larger population" (e.g., 10,000+ public companies). If n =28 < population = P?, then the 28 are a sample (i.e., subset of population) and taking the simple mean of the 28 is to compute a sample mean; i.e., to compute the sample mean, we do indeed take the average of the 28 observations. This use of "sample mean" (IMO) does not require additional samples (or a sampling distribution). It is simply (no pun intended!) the sample mean of the single sample. David

 

[frm_daniel]

thanks David for your patience.