Unlocking Degrees of Freedom: A Clear Geometric Explanation 📊

The most confusing concept in statistics must be degrees of freedom. Students everywhere leave their introductory stats courses totally bewildered about what degrees of freedom means, and why it seems to show up all over the place, such as in the t, chi-square, and F distributions, and also dividing by n minus 1 instead of n in the sample variance. The answer turns out to be complex but delightful, related to the dimensions of random vectors. Along the way we'll learn a powerful visual way to understand data analysis: the geometry of statistics. If you follow along closely, not only will degrees of freedom make way more sense to you, but lots of other statistical concepts will click too.

#SoME4


Chapters:

0:00 Introduction
0:37 Basic Definition
1:34 The Tweet That Started It All
1:57 The Geometry of Statistics
2:20 Prerequisites
2:59 Review of Vectors
4:54 Data as a Random Vector
6:23 Degrees of Freedom as Dimensions
6:53 Decomposing Into the Sample Mean and Residuals
11:03 Sample Mean and Residuals vs. Population Mean and Errors
13:08 To the Third Dimension
16:47 Errors and Mu in Three Dimensions
17:29 Generalizing to n dimensions
18:38 Conclusion and Preview


Further Reading/Viewing:

The Essence of Linear Algebra, by 3Blue1Brown: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

Degrees of Freedom on Wikipedia: https://en.wikipedia.org/wiki/Degrees_of_freedom_(statistics)

Walker, Helen M. “Degrees of Freedom.” Journal of Educational Psychology 31, no. 4 (1940): 253.

Saville, David J., and Graham R. Wood. Statistical Methods: A Geometric Primer. New York, NY: Springer New York, 1996. https://doi.org/10.1007/978-1-4612-0747-4.

Wickens, Thomas D. The Geometry of Multivariate Statistics. Hillsdale, N.J: L. Erlbaum Associates, 1995.

Saville, David J., and Graham R. Wood. Statistical Methods: The Geometric Approach. Corr. 3rd print. Springer Texts in Statistics. New York: Springer, 1997.


Attributions:

Music is from the YouTube Audio Library, by artists Alex Hamlin, E's Jammy Jams, Asher Fulero, Chris Haugen, and Silent Partner.

Guitar slide: E-Gitarre - Slides.wav by BockelSound -- https://freesound.org/s/488623/ -- License: Attribution 4.0


Made with Manim: https://www.manim.community/. The source code will be posted at the conclusion of the series.


Tips are appreciated! Tip me at: https://ko-fi.com/slevey