We can compare the relationship of the t-distribution with n âˆ’ 1 degrees of freedom with the normal distribution in several ways. As n gets large, the t-distribution converges to the standard normal. But what happens when n is â€œsmall,â€ and what do we mean by â€œlarge?â€
A few comparative graphs can give us an idea. For n = 10 we can use boxplots of simulated data to examine the tails, or we can compare plots of theoretical quantiles or densities. These plots are created as follows:
Repeat the above for n = 3, 25, 50, and 100. What value of n seems â€œlargeâ€ enough to say that the two distributions are essentially the same?