Why do the critical values change with degrees of freedom fo

Why do the critical values change with degrees of freedom for the t-tests?

Solution

t-distribution actually dependend on sample size because t-statistic uses sample standard deviation instead of population standard deviation. If the sample size is small, we know that sample standard deviation is not a good approximation of population standard deviation, and this is compensated by inflated t-critical values for small sizes.

When sample size is very large, t-distribution converges to normal distribution, with t-critical being equal to z-critical values for given alpha.