Remember that the central limit theorem states that when an infinite number of successive random samples are taken from a population, the sampling distribution of the means of those samples will become approximately normally distributed with mean μ and standard deviation σ/√ N (∼N(μ, σ/√ N)) as the sample size (N) becomes larger, irrespective of the shape of the population distribution.

We randomly selected the following samples of five subjects (N=5) 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 25 samples, the second time we took 100 samples, and the third time we took an "infinite" number of samples.

What can we see from these comparisons?

Frequency distributions of sample means quickly approach the shape of a normal distribution, even if we are taking relatively few, small samples from a population that is not normally distributed (such as the uniform distribution).

As we randomly select more and more samples from the population, the distribution of sample means becomes more normally distributed and looks smoother.

Let's go on to our next component of the central limit theorem and see what happens when the sample size increases.