Statistical degrees of freedom - What are they REALLY?

When degrees of freedom are introduced in intro stats classes, students often get confused. However, there is a quite easy way to understand them using the geometry behind statistics. This video builds upon an earlier video about variance estimation, but it is not necessary to have watched it:

Timeline:
0:55 Usual explanation
3:24 Introduction to geometry of statistics
6:02 Data partition into mean and residual vectors
8:48 Degrees of freedom as possible vector movement
13:30 Differences due to sample mean vs. population mean
15:37 Residuals vector and variance
17:38 Why underestimation can be compensated by "sample size reduction"

[statsandscience]

I made a quite bad mistake around 10:46 onwards (written and spoken) - it is of course the dot product that is the relevant concept here, not the cross product. I am sorry if that caused confusion, I added some links to videos explaining dot products together with a correction text and I hope that helps... thanks @a.r.k.2734 to point that out in the comments.

[johnnytremain]

Thank you for answering all those so-called "naive" questions. These are exactly the questions I have been having, that I haven't been able to get answers to anywhere else.

[nugaelatinae]

What an awesome video. I love LA. And having the concept explained like that makes total sense now. I hope you make more videos.

[knmrt]

That was very interesting, I've studied maths and some statistics but I've never quite understood the concept behind degrees of freedom I always took them for granted and used them as advised to make my formulas work etc...

[dan_nad]

This video helped a lot, thank you for providing such intuitive animations❤❤

[a.r.k.]

This is a really cool way to look at degrees of freedom in stats! Thank you for sharing. I believe you misspoke around 11:30 when you said the cross product is zero when two vectors are perpendicular; it's the dot product that's zero when two vectors are perpendicular.

[AlanSadeeq]

thanks for making it harder than it was

[kyledagman]

This is a really great explanation of the geometric interpretation of degrees of freedom with random vectors. You covered basically the same ground as the "Of random vectors" section of the Degrees of Freedom Wikipedia article, but your visualizations really helped me to understand better. Thank you so much for the great presentation!

I will need to do more research on how this interpretation makes sense when looking at things other than the unbiased estimator of population variance, such as for the Chi-Square and Student's t Distributions. Does this interpretation also work in those contexts?

[idlesummer]

hi! this video was greatly insightful. i do get where the divisor n-1 is coming from both geometrically (as the projection of a vector) and how it was derived mathematically. but im still confused as to how the degrees of freedom and the sum of squared deviations of the sample are related from this geometric perspective.

[davecorry]

1:42 "Suppose you have only one data point, let's say 2. Then someone else tells you they added a second value but don't tell you which number exactly ... So the degrees of freedom for those two numbers is 2." But why isn't it 1? Because 2 is *not* free to vary: it's the unvarying number 2.

[benjaminp.vallieres]

I don't understand this, but I feel like I have to if I ever want to understand those degrees of freedom.

[sara-qlxs]

3:08
Me a mathematician watching and asking the same questions this cause I never really understood degrees of freedom: O_O

[elenamascarenas]

what do you mean by "the residuals vector needs to lie on a single line"?

[sashode]

I wasted half a day trying to understand your speech

[johnnytremain]

OMG! What overkill!! I just want a simple definition of degrees of freedom, not the geometry of statistics. Anyway, it doesn't make sense. If you had 10 degrees of freedom, would you then have 10 dimensions of geometry? That is impossible to wrap my head around, or to visualize. It is not helping.