Degrees Of Freedom

Defination

Degrees of freedom are the number of independent values that a statistical analysis can estimate. You can also think of it as the number of values that are free to vary as you estimate parameters.

Scenario

Your hockey team has 20 players registered for a weekend tournament. During each match, only 11 players can be on the pitch, while the rest remain as substitutes. A day before the tournament, your best defender and the only goalkeeper inform the coach that they cannot make it. Since the team has no backup goalkeeper, the coach insists that the goalkeeper must attend. Fortunately, there are three defenders available, so the coach replaces the absent defender with another.

In your team, 11 players must be on the pitch. That’s a fixed requirement.
If the goalkeeper has no substitute, the coach has no freedom in choosing that position, it’s fixed (0 degrees of freedom for the goalkeeper).
For defenders, there are 3 options to fill 1 spot (since one defender is absent but you still have choices). That means the coach has freedom to decide which defender to bring in (1 degree of freedom for that position).
Similarly, for the other positions, depending on how many substitutes are available, the coach has more freedom of choice.

How to calculate Degrees Of Freedom in Statistics

The total degrees of freedom is one less than the total sample size

DoF=n-1
where (n) is the sample size. 

That's the general formula .

Importance of Degrees of Freedom

Degrees of freedom matter because they affect the shape of probability distributions used in tests. For instance:

1. The t-distribution changes depending on the degree of freedom, with larger degree of freedom, it approaches the normal distribution.
2. In chi-square tests, they determine the critical values needed to decide whether to reject a hypothesis
3. In regression analysis, they help calculate residual variance and test model fit.

In short, degrees of freedom ensure that statistical tests adjust for the amount of information and constraints in your data.

Conclusion

Understanding degrees of freedom helps us interpret statistical tests more accurately, ensuring that our conclusions are based on the right amount of free information.