Linearity of Data and Linear Probability Space

TL;DR

This paper explores the linearity properties of different data types, emphasizing transformations like log odds for bounded data to achieve linearity, and discusses methods for analyzing untidy data.

Contribution

It introduces a framework for understanding data linearity based on boundedness and proposes transforming probabilities into a linear probability space using log odds.

Findings

Log odds transformation improves mean and standard deviation calculations.

Two-sided bounded data are inherently non-linear without transformation.

Methods for analyzing untidy data are discussed.

Abstract

Some data is linearly additive, other data is not. In this paper, I discuss types of data based on the boundedness of the data and their linearity. 1) Unbounded data can be linear. 2) One-side bounded data is usually log transformed to be linear. 3) Two-side bounded data is not linear. 4) Untidy data do not fit in these categories. An example of two-sided bounded data is probabilities which should be transformed into a linear probability space by taking the logarithm of the odds ratio (log10 odds) which is termed Weight (W). Calculations of means and standard deviation is more accurate when calculated as W values than when calculated as probabilities. A methods to analyze untidy data is discussed.