# Percentile Study of χ Distribution. Application to Response Time Data

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Grupo de Modelización Interdisciplinar, InterTech, Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, E-46022 Valencia, Spain

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Grupo de Modelización Interdisciplinar, InterTech, Departamento de Psicología Evolutiva y de la Educación, Universitat de València, E-46010 Valencia, Spain

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Facultad de Ciencias Físico-Matemáticas, Universidad Autónoma de Nuevo León, San Nicolás de los Garza 66455, Nuevo León, Mexico

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Author to whom correspondence should be addressed.

Received: 19 January 2020 / Revised: 27 March 2020 / Accepted: 29 March 2020 / Published: 2 April 2020

(This article belongs to the Special Issue On Interdisciplinary Modelling and Numerical Simulation in the Realm of Physics & Engineering)

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).