The standard average we use to understand everything from stock markets to weather is just a side effect of how we draw coordinate maps.
Standard math defines an average as the sum of values divided by their count, but this assumes we are looking at the data from a flat perspective. A new geometric interpretation treats probability as a curved surface where the average changes depending on which coordinate system you choose to use. This redefines expectations as a geometric operation of averaging across different charts rather than a fixed numerical truth. It implies that the most basic tools of statistics are actually flexible products of geometry. This could change how we calculate risk in complex systems where traditional linear averages fail to capture reality.
Probability Geometry and Kolmogorov Expectations via Coordinate Charts
arXiv · 2605.02104
This paper develops a geometric reinterpretation of probability in which expectation arises from averaging in probability coordinates rather than in value space. By interpreting the cumulative distribution functions as coordinate maps, a real-valued random variable is transported into the unit interval, where averaging becomes a linear operation in probability coordinates and is then pulled back to the value space.Within this representation, the resulting quantities coincide with Kolmogorov mean