Let's randomly select an infinite number of samples of the same size from a population that follows a Poisson distribution and calculate the mean of scores in each sample. We will do this three separate times -- once for small samples (N=5), once for samples of 25 subjects (N=25), and once for samples of 100 subjects (N=100). The sampling distributions are presented below.

What can we see by comparing these distributions?

• With " infinite" numbers of successive random samples, the sampling distributions all have an approximately normal distribution with a mean of .78, which is equal to the population mean (µ= .78). The distribution based on the smallest sample (N=5) only approaches a normal distribution.
  
  
• As the sample sizes increase, the variability of each sampling distribution decreases so that they become increasingly more leptokurtic. The range of the sampling distribution is smaller than the range of the original population. The standard deviation of each sampling distribution is equal to σ/√N (where N is the size of the sample drawn from the population).
  
  
• These principles work for the Poisson distribution as well as the uniform distribution and show that, with large enough samples, the sample mean provides a reasonably accurate estimate of the population mean.

Let's look at this one more time, sampling from a population that follows a normal distribution .