Geo Analysis - Demo 4 

 




Demonstration -- Sampling






A  population consists of four values -- 1, 2, 3, and 4.  The mean of the distribution is 2.5.  The population has a uniform distribution.



Two samples are drawn and their mean is taken and recorded.  What is the distribution of the sample means?

The sample space is given below.  Think of the columns as recording the first draw and the rows as the second draw.  After making a draw, the ticket is returned to the bin -- sampling with replacement.







			1						2      34



11,11,21,31,4


22,12,22,32,4


33,13,23,33,4


44,14,24,34,4







The means for each cell in the sample space are given below.







			1						2      34



11.01.52.02.5


21.52.02.53.0


32.02.53.03.5


42.53.03.54.0







The distribution of sample means is given below.








	Mean						Frequency     



11/16


1.52/16


2.03/16


2.54/16


3.03/16


3.52/16


4.01/16












The Central Limit Theorem

If a variable x has a distribution with mean u and standard deviation s, then the sampling distribution of the sample means based on random samples of size N will have a mean equal to u, and a standard deviation of :  s = sx/N 1/2 ... the standard error of the mean and will tend to be normal in form as the sample size becomes large.  In other words, even though the parent distribution was uniform, the sample means will tend to be normally distributed.


There is nothing to be handed in but you should spend about 30 minutes with the following exercise.


Try the  sampling distribution applet.  
    

  1. Start with the normal distribution.  The second window displays the samples selected randomly - I would not use this one as it slows down the process.  Set the third window to show the means of the drawn sample and the fourth window to show the standard deviation of the drawn sample.  Draw 50 sets of 5 samples each.  Make sure that N=5 appears under the third and fourth windows.  For example, imagine that you have chemical analyses of the MgO content of all the basalts in the universe.  You are going to repeatedly sample this population.  The objective is to see how the mean of the sample means approaches the known (target population) mean.  What is the expected standard error of the mean for this experiment?
    Remember that the fourth window is plotting the sample standard deviations and not the standard error of the means.  Compare the mean of these values with the known standard deviation. 
    

  2. Try the skewed distribution and repeat your sampling experiments.

    

  3. Try the uniform distribution and repeat your sampling experiments.
    

  4. Try a "custom" - odd shaped - distribution and repeat.
    
There is nothing to hand in.  Spend time experimenting and thinking about the implications for sampling a target population.
    









Return to Geo Analysis Home Page