Why n-1? Least Squares and Bessel’s Correction | Degrees of Freedom Ch. 2

What's the deal with the n-1 in the sample variance in statistics? To make sense of it, we'll turn to... right triangles and the Pythagorean theorem! We'll recast the problem into vectors, and explain the n-1 in terms of the lengths and dimensions of these vectors. This is Chapter 2 in a series on Degrees of Freedom, or, The Geometry of Statistics, which is trying to rigorously but intuitively explain what is easily the most confusing concept in statistics, Degrees of Freedom. Check out the other videos here:    • Degrees of Freedom   This series assumes you are already comfortable with introductory statistics concepts like probability distributions and expected value. If you are less comfortable on the linear algebra concepts like vectors and components, check out the series The Essence of Linear Algebra, by 3Blue1Brown. Chapters: 0:00 Introduction - Why n-1? 0:38 Title Sequence 0:54 Look ahead 1:08 The Problem: Estimating the mean and variance of the distribution 2:17 Estimating the mean geometrically 3:54 A right angle gives the closest estimate 4:51 Vector length 6:06 The Least Squares estimate 8:33 Higher dimensions 9:05 Turning to the variance 10:11 Variance vs. the error and residual vectors 12:00 Why the variance isn't just the same as the length 12:40 Greater degrees of freedom tends to mean a longer vector 14:46 Averaging over degrees of freedom corrects for this 15:19 Review of the geometry 15:54 Previewing the rest of the argument 17:25 The residual vector is shorter than the error vector 18:42 The sample variance comes from the residual vector 18:58 Finding the expected squared lengths 21:03 Putting it together to prove Bessel's Correction 21:49 Recap 23:02 Conclusion Further Reading/Viewing: Bessel's Correction, from Wikipedia: https://en.wikipedia.org/wiki/Bessel%... So, S. (2008). Why is the sample variance a biased estimator. Griffith University, Tech. Rep., 9. https://www.marcovicentini.it/wp-cont... Saville, David J., and Graham R. Wood. Statistical Methods: A Geometric Primer. New York, NY: Springer New York, 1996. https://doi.org/10.1007/978-1-4612-07.... 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: 'Sunday Smooth' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au Other music is from the YouTube Audio Library, by artists Alex Hamlin, E's Jammy Jams, Chris Haugen, and Silent Partner. 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