The t distribution and the normal distribution are both bell-shaped and symmetric around the mean, but they differ in some key ways.
Firstly, the t distribution has heavier tails compared to the normal distribution. This means that it is more prone to producing values that fall far from the mean. The reason for this is that the t distribution is used when estimating the mean of a population when the sample size is small and/or when the population standard deviation is unknown. The heavier tails account for the extra uncertainty that arises from estimating the population standard deviation from a small sample.
As for how the shape of the t distribution depends on the sample size (n), the t distribution becomes closer to the normal distribution as the sample size increases. With larger sample sizes, the estimate of the population standard deviation becomes more accurate, and thus, the t distribution’s tails become less heavy. This means that with very large sample sizes, the t distribution approximates the normal distribution very closely, and is virtually indistinguishable from it.
In summary, while both distributions are similar, the t distribution is broader and has heavier tails, especially when sample sizes are small. As n increases, the t distribution approaches the shape of the normal distribution.