We randomly selected samples of a given size an " infinite" number of times from a uniform population, calculated the mean of each sample, and plotted the values of the means of each of these samples. We did this three different times -- the first time we took samples with only 5 subjects each ( N=5 ), the second time we took samples with
25 subjects each ( N=25 ), and the third time we took samples with 100 subjects each
( N=100 ). The sampling distributions are presented below.

What can we see from these comparisons?

• With " infinite" numbers of successive random samples, the sampling distributions all have a normal distribution with a mean of 3.0, which is equal to the population mean (µ= 3.0).
  
  
• 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).
  
  
• Taken together, these distributions suggest that the sample mean provides a good estimate of µ and that errors in our estimates (indicated by the variability of scores in the distribution) decrease as the size of the samples we draw from the population increases.

We have one more component of the central limit theorem to learn. Let's see what happens when we take successive random samples from different types of populations.