degress of freedom aren't explained well

Why do we divide with n-1 instead of n, when population variance is estimated from the sample? Yes, sample will have less variability, so our estimate based on n will underestimate population variance – so if we divide with something smaller, like n-1, it will be bigger, makes perfect sense. But why n-1?? The argument is that we are getting information from only n-1 independent pieces, because the last one is already given by the mean, therefore only n-1 values are free to vary. I understand the flow of argument and clearly one can determine the last value from average, however I don’t understand how exactly it applies here. If one wanted to determine just sample variance (not as population estimate) than you would divide with n, but you only have n-1 values that are free to vary? So why doesn’t the same logic apply. Also, one can potentially estimate last value from the mean but we are not doing that. Maybe I don’t understand enough statistics yet, but please somebody explain this.
also sometimes degrees of freedom depends on how many parameters are involved in a certain distribution or formula... like for example for the population variance, there is one parameter involved (mu - population mean) that's why we subtract one in our n (count)... or in simple terms, my prof says that it assumes or allocates the error that may be involved in computing the problem...

i am glad that you are curious with some 'ignored concepts' , since most of the students only accept what is written or taught in schools... i am very glad that you had some interest with it.. hehehe :D hope i answered your question


New Member
Ahh mysterious degreees of freedom :)

Q: What exactly is degrees of freedom?
A: The rank of a quadratic form.

Try explaingn that to an engineering student. How many of them know about quadratic forms, and that you can represend the variance in thee quadratic form?

Please read the following article about degrees of freedom