Science of Statistics

The work on gas theory had many extensions. In 1865 Johann Josef Loschmidt used estimates of the mean free path to make the first generally accepted estimate of atomic diameters. In later papers Maxwell, Ludwig Boltzmann, and Josiah Willard Gibbs extended the rrratherrratics beyorrd gas theory to a new gerreralized science of statistical mechanics. Whenjoined to quantum mechanics, this became the foundation of much of modern theoretical con-derrsed matter physics. 

To describe particle distributions, we will use our own nomenclature, but will soon find that it is related to the science of statistics as well. We begin by the use of the Probability Law. 

In the history of the development of mathematics, one important branch was the study of the behavior of randomness. Initially, there were no highfalutin ideas of making science out of what appeared to be disorder rather, the investigations of random phenomena that lead to what we now know as the science of Statistics began as studies of the behavior of the random phenomena that existed in the somewhat more prosaic context of gambling. It was not until much later that the recognition came that the same random phenomena that affected, say, dice, also affected the values obtained when physical measurements were made. 

So it is the changing nature of the world and the types of analyses we do that dictate how we go about organizing the calculations we use to do them. This comes from fundamental considerations of the behavior of the modeling process, which the science of Statistics can tell us about. 

We have been writing about statistics and chemometrics for a long time. Long-time readers of the column series published in Spectroscopy magazine will recall that the series name changed since its inception. The original name was Statistics in Spectroscopy (which was a multiple pun, since it referred to Statistics in Spectroscopy and Statistics in Spectroscopy as well as statistics (the subject of Statistics) in Spectroscopy (see our third column ever [1] for a discussion of the double meaning of the word Statistics . The same discussion is found in the book based on those first 38 columns in the earlier Statistics series [2])). 

In Chapter 69, we worked out the relationship between the calculus-based approach to least squares calculations and the matrix algebra approach to least-squares calculations, using a chemometrics-based approach [1], Now we need to discuss a topic squarely based in the science of Statistics. 

The early work of Gibbs (22) in the field of statistical mechanics, which has been supplemented and extended by many workers (20, 72), has assisted in the interpretation and coordination of experimental results. The science of statistical mechanics has been of particular value in establishing the heat capacity of hydrocarbons at infinite attenuation. However, much progress must be made before it will be possible to predict from statistical mechanics the characteristics of a phase from a knowledge of its state. The period between 1925 and 1950 has been characterized by substantial progress in the accumulation of experimental data from which concordant theories or generalizations may be developed. 

Molecular equilibrium, by contrast, is complicated by entropy. Entropy, being a measure of randomness, reflects the tendency of molecules to scatter, to diffuse, to assume different energy states, to occupy different phases and positions. It becomes impossible to follow individual molecules through all these conditions, so we resort to describing statistical distributions of molecules, which for our purposes simply become concentration profiles. The molecular statistics are described in detail by the science of statistical mechanics. However, if we need only to describe the concentration profiles at equilibrium, we can invoke the science of thermodynamics. 

For the rest of this book we will be concerned solely with the problem of random error. Bias just should not be there - good experimental design should see it off. However, random error is ever with us and the science of statistics is there to allow us to draw conclusions despite its presence. Statistics will not make random error disappear, it just allows us to live with it. 

This last point is important. In practice we do not consider that only results studied for a given subgroup are relevant to that subgroup we accept that results carry over to some degree. This is, in fact, a necessary condition for any science of statistics. At the very least we have to be prepared to act as if results from one individual had some relevance to another. In fact, both frequentist and Bayesian approaches to statistics allow ways in which this can be done. The Bayesian, for example, can place an informative prior on possible differences in treatment effects between subgroups. Such a prior would state that she believes with high probability that any possible difference between sexes as regards treatment effects will be fairly small. If the treatment effect... 

Furthermore, it is also plain from the definition of validation above that documentary evidence is the lodestar of validation and the tool by which the assurance is derived. In respect of software, the assurance needed will be in the form of documentary evidence, not of the product directly (though this may be included) but, as we have shown, from an examination of the development process that produced it. This strongly implies that the product is inseparable from the process, and the fitness for use of the product can only be secured through the rigorous control of the process that produced it. This ought to come as no surprise, since it is precisely this principle, the essence of the science of statistical process control, that determines the quality of any manufactured or fabricated product. In other words, the quality (of fitness for use) of an item is entirely determined by the process that produced it, where quality is defined according to the ISO 8402 [3] ... 

The science of statistics is purely mathematical with probabdity theory as the cornerstone. Because all statisticed methods ace based on probability concepts, it is necessary for one to understand the... 

Another method is to analyze S(test) as a percentage of the total variation, S(total). Since S(total) represents all sources of variation — raw materials, process, test method, etc. — this comparison makes good intuitive sense. Again, the smaller the percentage, the more sensitive the test is to real differences in the product. On the surface this seems like the best approach however, it is not statistically sound The science of statistics teaches one to compare variances (the square of the standard deviation) rather than standard deviations. One needs to think of comparing the area under the test distribution curve to the area under the total distribution curve rather than the lineal distance of the two standard deviations. 

A more candid view of how politicians view the science of statistics is attributed to Winston Churchill, another British Prime Minister ... 

Medical Imaging with MRI The NIH has funded the Biomedical Informatics Research Network (BIRN), a federated repository of three-dimensional brain images to support the quantitative science of statistical comparison of the brain subsystems. The task includes integration of computing, networks, data, and software as well as training people. 

The science of statistics involves the collection, organization, and interpretation of nnmerical data. Engineers use statistics for many purposes, including ... 

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