Percentile Study of χ Distribution. Application to Response Time Data

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).

Development of Perceptual Inhibition in Adolescents—A Critical Period?

“…In Figure 2 the histograms of the variables that represent the RTs through the different conditions in relation to the number of distractors, TR4, TR8, TR16 and TR32, referring to 4, 8, 16 and 32 distractors, respectively ( [76] for similar analysis on RTs) are shown. By dividing histograms by the area under the curve, we obtained a probability distribution for each of these RT variables (BVTR4, BVTR8, BVTR16 and BVTR32).…”

“…The ex-Gaussian distribution is commonly used to represent the reaction time data [76]. This positive-skewed distribution function comes from the convolution of a Gaussian function and an exponential decay function.…”

Development of Perceptual Inhibition in Adolescents, a Critical Period?

“…In Figure 2 the histograms of the variables that represent the RTs through the different conditions in relation to the number of distractors, TR4, TR8, TR16 and TR32, referring to 4, 8, 16 and 32 distractors respectively ( [76] for similar analysis on RTs) are shown. By dividing histograms by the area under the curve, we obtained a probability distribution for each of these RTs variables (BVTR4, BVTR8, BVTR16 and BVTR32).…”

Monte Carlo Simulation of a Modified Chi Distribution with Unequal Variances in the Generating Gaussians. A Discrete Methodology to Study Collective Response Times

“…In previous works [14,15], we have developed a methodology to analyse the response times to visual stimuli of a group of individuals which is based on the use of the MB distribution [2] instead of the ex-Gaussian distribution [3]. The MB distribution emerges when the reaction time distributions of single individuals are combined at a higher level of hierarchy, which allows us to capture the collective behaviour in a single distribution [14,15]. The details of the computer-based experiments and for the design of the visual stimuli can be found in our previous works [4,14,15].…”

“…In our previous work [14,15], we integrated the response times of a group of individuals of similar age in a Maxwell-Boltzmann (MB) distribution in order to model the behaviour of a collective of individuals. Relevant correlation has been found between the RTs series to visual stimuli of individuals who perform experiments in a short time scale (hundreds of milliseconds) when cultural elements are not expected to have influence and without exchanging information.…”