Degrees of Freedom - Introduction

The degrees of freedom which i discuss is purely based on Statistics and it might sound simple but understanding them is very important. There are plethora of application of degrees of freedom like hypothesis tests,regression modelling, t-test, F test, ANOVA, Chi-Squared test. Even though we do often use it at different perspective, many fails to understand them. I do not directly hit in to the usual practice of definition but kickstart with a simple example.

Suppose in a class of 5 students, professor provides 5 topics for a seminar, where students needs to chose the topic randomly and each topic must be different among the students. The first student has a option to pick from 5, 2nd student has a chance to pick only from 4| topics since one topic has been already chosen, 3rd student can pick with option of 3, 4th student has a chance to pick from 2 and the 5th person does not have any choice and is forced to pick the leftover topic .This implies that n-1 i.e 5-1=4 degrees of freedom.in which the topic which student chose could vary. They are usually defined as number of values or observations which can be varied while estimating the parameters.

This is why several others mention in their books and manuscripts that they are usually number of observations minus the parameters which needs to be estimated. The other important feature is that 5th student does not have independence since he has no freedom to vary in the topic. Further, degrees of freedom has been primarily used to study the significance of the tests and they imply a different meaning for each of the tests. The idea varies in regression, chi squared and so on which will be discussed in further posts.