Class CS805 Doctor Howard Carol Project 3 Sampling Distributions

Sampling Distributions
Professor Doctor Carol Howard
Class CS 805
By Tai Cleveland
Project # 3
Due on Monday November 12th, 2007

As defined by David W. Stockburger, “the sampling distribution is the distribution of a sample statistic” (Stockburger, 1998). While it is also a distribution model, the values are statistics and not raw data. For example, if we have 10 samples, and we computed for the mean of those 10 samples several times, the results or the means we derived will be the values in our sampling distribution. The sampling distribution is represented by µ. There is always a subscript to the µ, which tells us what kind of statistics the sampling distribution refers to.
As mentioned, there are also sampling distributions for each statistic. If we will compute for the sampling distribution of medians, then we shall use medians, not means. However, it was found that the mean has a smaller standard of error than the median, even the mode, since the mean takes into consideration all the value or scores included in the sample. The median simply is the middle number, while the mode is the value that most often comes up in a sample. The standard of error (σ), on the other hand, is “the degree by which the computed statistics will differ from one another when calculated from sample of similar size and selected from similar population models” (Stockburger, 1998).
The sampling distribution of means is simply made of means as computed from a sample of scores or values. It was also found that the sampling distribution of means is closely related to the population distribution (Stockburger, 1998), which is called the Central Limit Theorem. This means that the mean of the sampling distribution of means and the mean of the population are equal. There are 2 rules under this theorem. To illustrate the first rule, below is an example of five 5 values, from which scores were derived.
Population of values
Samples from the Population
Means from the samples
1
1, 2, 4, 5
3
2
1, 2, 3, 4
2.5
3
2, 3, 4, 5
3.5
4
1, 3, 4, 5
3.25
5
1 ,2, 3, 5
2.75
Population distribution = 3
(All possible samples) n = 4
Total = 15 / 5 = 3

In computing for the Central Limit Theorem, follow the steps outlined below:
1. Start with the population values. In this case, we had 5 values in our population.
2. Obtain all possible samples. In each population, we were able to derive a maximum of 4 samples.
3. Add up all samples in each population, then divide it by the n or the sample size. This will give us the mean for that particular population.
4. Add all the means, then divide by the N or the sample population. This will yield the same value as that of the population distribution.
The second rule under the Central Limit Theorem states that the sampling distribution of means will have a normal curve regardless of the shape of the population distribution (Sampling Distribution Demo). The reason behind this rule is that even though the samples are taken from different samples, the means of the samples will always be near to the center of the population distribution.
The Central Limit Theorem works well in small sample sizes as shown above, though it works even greater with a larger sample size, as it is closer to the true population. As such, the Central Limit Theorem is often the basis of most hypothesis testing and sampling theory (Stockburger, 1998). Additionally, the Central Limit also serves as a powerful tool for most researchers, as this always has a normal curve, providing scientists and researchers the basis or justification for several studies, even naturally occurring phenomena (Stockburger, 1998).
References
Sampling Distributions. Sampling Distributions Demo. Accessed October 31, 2007, from
http://faculty.uncfsu.edu/dwallace/ssample.html
Stockburger, David W. (1998). The Sampling Distribution. Introductory Statistics:
Concepts, Models, and Applications, 1.0. Accessed October 29, 2007, from http://www.psychstat.missouristate.edu/introbook/SBK19.htm