Percentile Study of χ Distribution. Application to Response Time Data
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 <i>k</i> = 2 to <i>k</i> = 10. The most commo...
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| Language: | English |
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MDPI AG
2020-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/4/514 |
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| author | Juan Carlos Castro-Palacio Pedro Fernández-de-Córdoba J. M. Isidro Esperanza Navarro-Pardo Romeo Selvas Aguilar |
| author_facet | Juan Carlos Castro-Palacio Pedro Fernández-de-Córdoba J. M. Isidro Esperanza Navarro-Pardo Romeo Selvas Aguilar |
| author_sort | Juan Carlos Castro-Palacio |
| collection | DOAJ |
| description | 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 <i>k</i> = 2 to <i>k</i> = 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 <i>k</i> generating Gaussians. In general, the χ probability density, generalized to take any value of the variance, represents an ideal gas in a <i>k</i>-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 <i>k</i> = 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). |
| first_indexed | 2024-03-10T20:42:35Z |
| format | Article |
| id | doaj.art-13ed4341659a4b2fa75948f5122e3685 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T20:42:35Z |
| publishDate | 2020-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-13ed4341659a4b2fa75948f5122e36852023-11-19T20:32:57ZengMDPI AGMathematics2227-73902020-04-018451410.3390/math8040514Percentile Study of χ Distribution. Application to Response Time DataJuan Carlos Castro-Palacio0Pedro Fernández-de-Córdoba1J. M. Isidro2Esperanza Navarro-Pardo3Romeo Selvas Aguilar4Grupo de Modelización Interdisciplinar, InterTech, Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, E-46022 Valencia, SpainGrupo de Modelización Interdisciplinar, InterTech, Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, E-46022 Valencia, SpainGrupo de Modelización Interdisciplinar, InterTech, Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, E-46022 Valencia, SpainGrupo de Modelización Interdisciplinar, InterTech, Departamento de Psicología Evolutiva y de la Educación, Universitat de València, E-46010 Valencia, SpainFacultad de Ciencias Físico-Matemáticas, Universidad Autónoma de Nuevo León, San Nicolás de los Garza 66455, Nuevo León, MexicoAs 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 <i>k</i> = 2 to <i>k</i> = 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 <i>k</i> generating Gaussians. In general, the χ probability density, generalized to take any value of the variance, represents an ideal gas in a <i>k</i>-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 <i>k</i> = 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).https://www.mdpi.com/2227-7390/8/4/514χ distributionideal gas modelMaxwell–Boltzmann distributionresponse times |
| spellingShingle | Juan Carlos Castro-Palacio Pedro Fernández-de-Córdoba J. M. Isidro Esperanza Navarro-Pardo Romeo Selvas Aguilar Percentile Study of χ Distribution. Application to Response Time Data Mathematics χ distribution ideal gas model Maxwell–Boltzmann distribution response times |
| title | Percentile Study of χ Distribution. Application to Response Time Data |
| title_full | Percentile Study of χ Distribution. Application to Response Time Data |
| title_fullStr | Percentile Study of χ Distribution. Application to Response Time Data |
| title_full_unstemmed | Percentile Study of χ Distribution. Application to Response Time Data |
| title_short | Percentile Study of χ Distribution. Application to Response Time Data |
| title_sort | percentile study of χ distribution application to response time data |
| topic | χ distribution ideal gas model Maxwell–Boltzmann distribution response times |
| url | https://www.mdpi.com/2227-7390/8/4/514 |
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