Question

There are about 4200 college presidents in the United States, and they have annual incomes with a distribution that is skewed instead of being normal. Many different samples of 40 college presidents are randomly selected, and the mean annual income is computed for each sample. What is the approximate shape of the sample means?

Answer 1

Because a mean is an average, the means of each of the samples of 40 college presidents' incomes will have less of a skew than the actual income distribution in dollars of all of them. That is because outliers, like a president who is paid $2 million a year, will be averaged in with many others who are paid less. The shape of the curve of the means will depend upon how many different samples of 40 presidents are included, with the shape becoming more bell-like (normal distribution) the more random samples are included. Another consideration is how often the outliers are included and how big the income skew is that you start with. For example, if 2 college presidents are earning $2 million a year and the other 4,198 are earning $200,000 or less then the shape of your curve will depend greatly upon the number of times those two high earners are randomly selected and factored into the mean.