Question

Why is the concept of degrees of freedom so important? How does it relate to t statistics? What is an example of degree of freedom ?       

Answer 1

The number of of independent values or quantities that can be assigned to a statistical distribution is known as degrees of freedom. Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the estimate of parameter are called the degrees of freedom. t statistics follow t distribution and the particular form of t distribution is determined by it's degrees of freedom. When estimating mean from a single sample the number of independent observations is equal to sample size minus one. Say sample size is 8, then degrees of freedom of t distribution will be 7. An example is say we have a multivariate random variable which follows multinomial distribution with k parameters (unknown). We know that sum of probabilities of these k parameters should be 1, hence this constraint reduces the degrees of freedom to k-1 as once we fix k-1 probability values the kth gets fixed.