The Emergence of the Normal Distribution in Deterministic Chaotic Maps.

The central limit theorem states that, in the limits of a large number of terms, an appropriately scaled sum of independent random variables yields another random variable whose probability distribution tends to attain a stable distribution. The condition of independence, however, only holds in real systems as an approximation. To extend the theorem to more general situations, previous studies have derived a version of the central limit theorem that also holds for variables that are not independent. Here, we present numerical results that characterize how convergence is attained when the variables being summed are deterministically related to one another through the recurrent application of an ergodic mapping. In all the explored cases, the convergence to the limit distribution is slower than for random sampling. Yet, the speed at which convergence is attained varies substantially from system to system, and these variations imply differences in the way information about the deterministic nature of the dynamics is progressively lost as the number of summands increases. Some of the identified factors in shaping the convergence process are the strength of mixing induced by the mapping and the shape of the marginal distribution of each variable, most particularly, the presence of divergences or fat tails.

Accuracy and Precision Improvement of Temperature Measurement Using Statistical Analysis/Central Limit Theorem.

This paper describes a method for increasing the accuracy and precision of temperature measurements of a liquid based on the central limit theorem. A thermometer immersed in a liquid exhibits a response with determined accuracy and precision. This measurement is integrated with an instrumentation and control system that imposes the behavioral conditions of the central limit theorem (CLT). The oversampling method exhibited an increasing measurement resolution. Through periodic sampling of large groups, an increase in the accuracy and formula of the increase in precision is developed. A measurement group sequencing algorithm and experimental system were developed to obtain the results of this system. Hundreds of thousands of experimental results are obtained and seem to demonstrate the proposed idea's validity.

Normal y lognormal: dos distribuciones de frecuencias y una Teoría Neutral Unificada para estudiar los bosques tropicales / Normal, and lognormal: two frequency distributions, and one Unified Neutral Theory for studying Tropical forests

Los inventarios de biodiversidad en sitios contrastantes obtienen datos con distribuciones normales y lognormales, útiles para cuantificar cómo el cambio climático afecta a los bosques del mundo. Sin embargo, casi todos los biólogos están familiarizados con la normal, pero menos discuten el por qué la distribución lognormal de frecuencias relativas de especies ocurre en tantas comunidades bióticas. Pretendemos incorporar a más estudiosos a dicha discusión. Tanto la normal como la lognormal tienen medias y valores extremos. Ello es consistente con el teorema del límite central; válido cuando los datos de un muestreo provienen de procesos aleatorios y el muestreo ha sido estocástico y representativo. Según la Teoría Neutral Unificada de la Biodiversidad y la Biogeografía de Steve Hubbell, basta considerar que la natalidad, mortalidad, migraciones y especiación en una comunidad, y desde la metacomunidad circundante, ocurren al azar y simétricamente entre especies, para explicar que las frecuencias relativas de la comunidad sigan una distribución lognormal. Ello es consistente con la Biogeografía de Islas, y se puede aplicar -por tanto a la articulación de abundancias relativas de especies arbóreas en bosques que se regeneran por sucesión secundaria, donde el sitio talado constituye una isla que luego es colonizada. En el sofisticado siglo XXI, conocimientos numéricos tan simples, como la normal y la lognormal, siguen siendo necesarios para mover las fronteras de la ciencia afrontando temas permanentes: por qué en tantos lugares hay especies más abundantes que otras, y cómo se puede contrarrestar la pérdida de las especies en dificultad.

Biodiversity surveys among contrasting sites get normal, and lognormal distributed data used for quantifying how Climate Change affects forests around the world. Yet most biologists are familiarized with the normal distribution, while few discuss why the lognormal distribution of relative frequencies of species is so common in many communities of living beings. We aim to add more researchers into such a discussion. Both normal and lognormal have mean and extreme values -which is consistent with the Central Limit Theorem. Such a theorem is valid when the data come from random processes, and when the sampling excercise of collecting the data has been stocastic and representative. According to Steve Hubbell's Unified Neutral Theory of Biodiversity and Biogeography, random birth, death, migration and speciation in a community -and from the surrounding metacomunity are enough for generating lognormal distributions of relative frequencies of co-existing species. That is consistent with Island Biogography, and is applicable to the assembly of relative abundances of tree species during secondary succession, where the clear-cut site is an island further colonized by tree species. Deep into the sophisticated 21st century, simple numerical knowledge like the normal and lognormal are still needed for moving the borders of science by facing permanent subjects: why in so many places some species are more abundant than others, and how to tackle the loss of endangered species.