Abstract
As a continuation of our previous work, where a Maxwell–Boltzmann distribution was found to model a collective’s reaction times, in this work we will carry out a percentile study of the χ distribution for some freedom ranging from k = 2 to k = 10. The most commonly used percentiles in the biomedical and behavioral sciences have been included in the analysis. We seek to provide a look-up table with percentile ratios, taken symmetrically about the median, such that this distribution can be identified in practice in an easy way. We have proven that these ratios do not depend upon the variance chosen for the k generating Gaussians. In general, the χ probability density, generalized to take any value of the variance, represents an ideal gas in a k-dimensional space. We also derive an approximate expression for the median of the generalized χ distribution. In the second part of the results, we will focus on the practical case of k = 3, which represents the ideal gas in physics, and models quite well the reaction times of a human collective. Accurately, we will perform a more detailed scrutiny of the percentiles for the reaction time distribution of a sample of 50 school-aged children (7200 reaction times).
Highlights
The χ distribution is a well-known probability density function (PDF) for its use in applied statistics [1], and for its appearance in several contexts, such as the Rayleigh distribution (k = 2) and the Maxwell–Boltzmann (MB) distribution in the ideal gas model [2]
We have proven in our previous work [3] that the collective behavior of individuals at short times is ruled by an MB distribution which depends on a single parameter B, namely, a measure of the dispersion of the distribution
It is important to remark that we have numerically proven that the ratio r is independent of the value of the parameter B in Equation (1), which is easy to demonstrate analytically: Let xp (B) be the point where the cumulative distribution function (CDF) takes the value of the percentile p for a given value of the parameter B, that is
Summary
The χ distribution is a well-known probability density function (PDF) for its use in applied statistics [1], and for its appearance in several contexts, such as the Rayleigh distribution (k = 2) and the Maxwell–Boltzmann (MB) distribution in the ideal gas model [2] This PDF is interesting because it is built from Gaussian-distributed random variables combined through a Euclidean distance operation. We are interested in applications to biomedical and collective behavior contexts In this respect, and in line with our previous works [3,4], we will carry out here a percentile study of the χ distribution.
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