Induction and statistical analysis
Induction follows experiment (adduction). Its conduct is associated with many difficulties.
Are there any exact values for the measured parameters? It seems that each attribute (parameter) can have a so-called exact, or point, value. But to justify this opinion, apparently, it is impossible. Even in the absence of measurement errors, the value of the characteristic value must be correlated with a certain interval. The value of the characteristic value is always interval. It is not so that the interval value distorts the exact value. In fact, the point value is a simplification with respect to the interval value. It is the interval value that is intended to fix the corresponding device.
Selective average, probability, mathematical expectation and uncertainty
So far, problems associated with determining the value of a parameter with a narrow interval have been considered. But with respect to the so-called probabilistic magnitudes, this is clearly not enough, for in this case one of the central plans yields the concepts of probability and mathematical expectation. Both concepts seem rather unusual, but their nature can be quite clearly interpreted.
The key value in understanding the concepts of probability and mathematical expectation has a selective mean. Let's assume that the quantity Y is considered. Let's designate the value Y in the s -th test by . The total number of trials included in the corresponding sample is n. In this case, the sample mean A is determined by the formula
The determination of the sample mean requires the experimenter to be highly competent in selecting the appropriate samples and determining their characteristics, in particular stability. Both these values represent some limits of the sample mean. In the case of mathematical expectation ( E ) deal with the value of the measured parameter. E [Y] is the limit of A [ Y ], defined on the basis of not one, but many samples. When taking into account the probability (P), we are talking about the limit of the sample mean with respect to the relative frequency of outcomes:
where т is the number of favorable outcomes from the total number n.
Since t is determined from many samples, it also acts as a certain averaged value. It seems that the experimenter, with all his efforts, is not in a position to determine either the mathematical expectation of the magnitude or the probability of its occurrence, for the limiting transitions considered above assume an infinite number of both tests and samples. But in the conditions of a shortage of time, he is forced to confine himself to a very definite number of tests. The experimenter seems to have the right to state that he should strive to approach as close as possible to the exact (true) value of the quantities corresponding to the mathematical expectation and probability, respectively. But this so-called exact value is introduced a priori, which should alert the experimenter. In the author's opinion, the paradox of unattainability of the exact value of mathematical expectation and probability can be completely overcome if one carefully considers, on the one hand, the status of concepts and, on the other hand, the correlation of certain stages of conceptual transduction in chemistry. Let us consider this imaginary paradox using the example of probability analysis.
There are various understandings of the nature of probability. Especially frequent bewilderment seems to be the complete lack of possibility to reconcile the understanding of probability as the relative frequency determined in the experiment and its mathematical counterpart. In the latter case, the probability is understood either by R. von Mises, namely, as the limit of the relative frequency, or according to AN Kolmogorov-as a measure given on the algebras of sets. The paradox arises insofar as the mathematical realities are accepted for quite real idealized objects and their signs. In the experiment, this kind of realities can not be detected. To avoid paradoxical judgments, there seems to be only one possibility, namely, to assume that the stage of idealization follows the stage of experimentation. The most genuine in science is supposedly idealization.
The way out of the situation is if we recognize mathematical objects not as idealizations but as formalizations. Attentive researchers of mathematics can not be transferred directly and directly to the field of chemistry. The mathematical apparatus is surely checked for its suitability.
So, mathematical expectation and probability, being the most important scientific concepts, are not measured directly, but are determined by means of initial experimental data, i.e. facts. The term mathematical expectation hardly probable. Strictly speaking, in the context of chemistry, the mathematical expectation turns into a chemical expectation. Nevertheless, the literary norm forces us to use the term mathematical expectation & quot ;.
The definition of mathematical expectations and probabilities is associated with numerous difficulties, each of which gives this or that certainty to the experiment as a stage of transduction. We point out some of them, following the main work of Yu. I. Alimov and Yu. A. Kravtsov.
1. Enumeration of factors that are relevant for determining mathematical expectations and probabilities. It turns out to be possible only after a careful study of the peculiarities of the experimental situation. Factors are ranked, but some of them are unaccounted for.
2. Subjective (expert) assessment of probabilities. It turns out to be necessary if there is a need for a new theory. The work of experts needs to be understood.
3. Restoring the statistical ensemble for a limited experimental sample. As a rule, the data is insufficient, so they are thought out. Criteria for preconception are themselves in need of critical analysis.
4. Determination of mathematical expectations and probabilities under conditions of nonstationarity and instability. In these conditions, all forecasting is provoked by new difficulties.
5. Interpretation of rare phenomena. Since rare phenomena, as a rule, are not reproducible, then their study is difficult.
6. Attracting the law of large numbers. Contrary to popular belief, increasing the sample size does not necessarily entail a reduction in the scattering of experimental data. The law of large numbers occurs only in the presence of factors that ensure its existence.
The concept of uncertainty is closely related to the concepts of mathematical expectation and probability. In occasion of this concept there are big ambiguities. Usually the vague is interpreted as the negation of a certain. From this point of view, a quantity that does not have an exact value must be recognized as undefined. In epistemology, uncertainty is often associated with a lack of knowledge that can be overcome. This understanding was questioned by the discoveries made in quantum mechanics. The Heisenberg uncertainty relation: indicates that when the momentum is measured simultaneously along the and the coordinates () their uncertainty is unrecoverable. Understanding the Heisenberg uncertainty relation has shown that uncertainty is real and, therefore, it is not associated with a lack of knowledge. The connection of uncertainty with probabilities also became apparent. At least that's the way things are in physics and chemistry.
Consideration of the ontological status of uncertainty deserves attention. Probability characterizes the possibility of the onset of certain events. But in this case, it should be recognized that there is a certain concentrate of activity ensuring the onset of the events mentioned. According to the author, precisely the characteristic of this activity is precisely uncertainty. It is not enough just to emphasize the uncertainty of the magnitudes of the signs. It is extremely important to single out their origins. And they are such that they overturn our usual ideas. It is very significant in this connection that, because of the uncertainty characteristics of elementary particles, even ... universes arise. The world is saturated with not exact values, but with uncertainty generating a wide range of probability events.
Above, we considered the basic concepts needed to analyze the data extracted from the experiment, namely, the mean sample value, mathematical expectation and probability. To this splendid triplet, it is also necessary to add dispersion (from Latin dispersion - scattering). The variance ( DX) of the quantity X is defined as the square of its deviation from the mathematical expectation. Dispersion is necessary in two respects. First, it allows to keep in the field of researcher's attention the whole set of measurement results, which does not reduce to mathematical expectations. Secondly, with support on it it is possible to characterize various kinds of errors.
Below are various ways to analyze the experimental data. They will be considered only to a degree that allows us to come to certain methodological conclusions.
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