Mean angle is not a usual average. Means on circle - Intro to directional statistics (3B1B SoME1)

How to indicate the mean direction (or average direction) of the wind? How to calculate the mean position (or average position) on the circle? [Timestamps below] This video shows that such a simple thing like mean or average changes its meaning for the points belonging to the circle or when dealing with an angular variable. We present a generalised understanding of the mean based on the minimisation of the Fréchet function. This approach distinguishes between the intrinsic mean and the extrinsic mean. At the same time, it unifies the way of calculating the mean and median, not necessarily on the circle. It turns out that the circular median can also be intrinsic or extrinsic. This is an introduction to circular statistics (for circular data) or, more generally, to directional statistics (for directional data). These are the simplest and most useful examples of statistics on topologically non-trivial manifolds (on manifold-valued data). The lecture assumes the knowledge of mathematics at the level of a good high school graduate but includes a brief revision of the key issues. (The video is submitted to the Summer of Math Exposition 1 carried out by 3blue1brown.) By Karol Ławniczak Timetable 00:00 Preliminary examples 01:27 Intro 01:50 Revision of the concept of mean 04:00 Revision and clarifications concerning directions, angles, arcs and positions on the circle 05:50 Failure of the usual mean 07:27 Circle as a figure on a plane vs as an autonomous space esp. labelling directions 08:34 Possible confusion with a usual mean over a circle 09:23 Extrinsic mean 11:08 New point of view on ordinary mean, Fréchet function 15:03 Intrinsic mean - optimisation approach 19:00 Intrinsic mean - analytic approach 21:50 Further visualizations and some properties 24:26 Median 29:54 Concluding comment 30:36 Neat physical example 32:06 Further reading Links: 3blue1brown: Euler's formula with introductory group theory    • Euler's formula with introductory group th...   T. Hotz and S. Huckemann: Intrinsic Means on the Circle - Uniqueness, Locus and Asymptotics (2011) https://arxiv.org/abs/1108.2141 A. Brun et al.: Intrinsic and Extrinsic Means on the Circle - A Maximum Likelihood Interpretation (2007) https://ieeexplore.ieee.org/document/... (limited access) Topic on StackExchange: https://math.stackexchange.com/questi...