Statistical ensemble

“ If an ensemble, whether grand or petit, is identical so far as generic phases are concerned with one canonically distributed, we shall say that its distribution is canonical with respect to generic phases. Such an ensemble is evidently in statistical equilibrium with respect to generic phases, although it may not be so with respect to specific phases. ”

“ It follows that to human experience and observation with respect to such an ensemble as we are considering, or with respect to systems which may be regarded as taken at random from such an ensemble, when the number of degrees of freedom is of such order of magnitude as the number of molecules in the bodies subject to our observation and experiment, , , would be in general vanishing quantities, since such experience would not be wide enough to embrace the more considerable divergencies from the mean values, and such observation not nice enough to distinguish the ordinary divergencies. ”

“ We must, however, draw a very important distinction. There is one method by which uniformities of a certain kind can be detected in the behaviour of purposive intelligent beings, without insight into the nature of their individual purposes—the method of statistical averages. ”

“ X. I took the liberty of telling you last night that statistics are a mathematical form of telling a lie, and that, to use the expression of a clever Frenchman, a statistical table is like a loose woman who is at the service of anyone who wants her. ”

“ When the word "significantly" is used in comparisons, it is meant to show that there is a significant statistical difference between two or more samples. Whenever eight or more specimens from one locality were available, the mean, range, standard deviation, standard error of the mean, and coefficient of variability were calculated. ”

“ Statistical facts are usually numerical or quantitative in character, so that, if many causal agencies are in operation at the same time, their effects are simply added together algebraically, and are inextricably merged into a general total. ”

“ The statistician is primarily the man of facts, not merely a recorder of facts, but one upon whom is laid the duty of their orderly arrangement, their exposition and scientific coordination. Our author inferentially recognizes these functions by mapping out a general work to bear the title of the science of the statistics. Herein will be arranged the principal contents of statistical knowledge, in two separate volumes, the one having particular reference to the needs of sociology, the other of economics. ”

“ Not only is this true, but the further interesting fact may be discovered that social statistics of every category employed or known are based upon a frank recognition of that coefficient of resemblance, physical or mental, which I have contended is a mark of social phenomena. The first step in statistical tabulation is classification, and classification invariably starts from an assumption of real or supposed resemblance. ”

“ Yet no less Herculean a task confronts us. To find out about their motions, we are therefore driven to what is called the statistical method of inquiry—which is simply a branch of the doctrine of probabilities. ”

“ He uses the results of statistical inquiry because the method of obtaining and presenting them is scientific, and because he recognizes with the German Schlosser, that "statistics is history ever advancing," and that if he wishes to grow with advancing history and keep himself fully and thoroughly informed of progress in every direction, he must use the statistical or historical method ”

“ The method, I believe, is the surest for ascertaining conditions, and the truest on which to base conclusions; but the method must be supplemented by full and frank analysis. A statistical table, independent of such analysis, is to me what a red flag is to a bull. It immediately excites antagonism and invites attack. The value of any statistical presentation must depend upon the basis upon which it is made, the integrity of the collection of the various elements of it, and the analysis which accompanies it. ”

“ The experiments to which allusions have been frequently made during the present chapter, favourably as they were received, might have shared the fate of many other discoveries which did not admit of an immediate and obvious application to the purposes of common life; for statistical value is a necessary passport to popular favour. ”

“ One must not infer from the title that statistics in the scientific meaning of the term is an integral part of sociology, but that sociology, the general science of society needs to make use of statistical laws and conclusions just as it would the generalizations of political economy, of politics or of social ethics, in studying the organic development of society and in formulating a social synthesis. Population is the most important subject matter of statistical science, it is likewise a central fact in sociology. ”

“ Now it happens that a great many investigations in descriptive sociology do not as yet admit of the introduction of exact statistical—that is, arithmetical—inquiries which, nevertheless, do admit the use of algebraically quantitative methods. ”

“ As becomes the orderly mind of a statistician, the society has been very regular in its publications, having for seventy years issued a yearly volume in quarterly parts, which form a veritable mine of statistical information. ”

“ Let us consider three attributes, say A, B and C, of the individuals of a population and suppose that the attribute C is positively correlated to the attributes A and B, so that to great or small values of A or B correspond respectively great or small values of C. Now if the individuals in the population are statistically selected in such a way that we choose out individuals having great values of the attributes A and small values of the attribute B, then we get a statistical series regarding the attribute C, which consists of two seemingly distinct normal frequency distributions. ”

“ In the same way, to measure the healthiness of a population and ascertain whether it is safely under the level of its food, tending to pass 124beyond it, or simply rising up to it, and to ascertain by what ways and means the process is going on, we need instead of one single general criterion a whole array of particular tests. It is in the infancy of statistical science that men yield to appearances and “suppose a greater uniformity in things than is actually found there.” ”

“ Provide Assessments of Variability Nearly all biological measurements, when repeated, exhibit substantial variation, and this creates uncertainty in the result of every calculation based on the data. A basic purpose of statistical analysis is to help assess uncertainty, often in the form of a standard error or confidence interval, and one of the great successes of statistical modeling and inference is that it can provide estimates of standard errors from the same data that produce estimates of the quantity of interest. ”

“ To analyze statistically significant differences between variables, we selected appropriate statistical tests based on the distribution of data including the Kruskal Wallis test, chi square tests, Wilcoxon Rank sum test, and Spearman’s correlations for non-parametric data and one-way analysis of variance and Pearson’s correlations for continuous data. ”

“ Every article on statistics requires at least one caveat. Here is ours: we refer in this article to “science” as a convenient shorthand for investigations using data to study questions of interest. This includes social science, engineering, digital humanities, finance, and so on. Statisticians are not shy about reminding administrators that statistical science has an impact on nearly every part of almost all organizations. ”

“ The development of statistics is causing history to be rewritten. Till recently the historian studied nations in the aggregate, and gave us only the story of princes, dynasties, sieges, and battles. Of the people themselves—the great social body with life, growth, sources, elements, and laws of its own—he told us nothing. Now statistical inquiry leads him into the hovels, homes, workshops, mines, fields, prisons, hospitals, and all places where human nature displays its weakness and its strength. ”

“ On account, then, of the difficulties of the subject and of the profound importance of results which can be reached by no other known method, it is of the utmost consequence that the principles and processes of statistical mechanics should be put upon a firm and certain foundation. ”

“ Any experiment may be regarded as forming an individual of a “population” of experiments which might be performed under the same conditions. A series of experiments is a sample drawn from this population.
Now any series of experiments is only of value in so far as it enables us to form a judgment as to the statistical constants of the population to which the experiments belong. ”

“ Such is the great complexity of our mental life that it is not possible to deny that constant mean values, when obtained, are of the nature of such statistical constants. To test that, I examine the distribution of the separate numbers represented in an average value. If it corresponds to the distribution found everywhere in natural science, where repeated observation of the same occurrence furnishes different separate values, I suppose—tentatively again—that the repeatedly examined psychical process in question occurred each time under conditions sufficiently similar for our purposes. ”

“ A central and common task for us as research investigators is to decipher what our data are able to say about the problems we are trying to solve. Statistics is a language constructed to assist this process, with probability as its grammar. While rudimentary conversations are possible without good command of the language (and are conducted routinely) , principled statistical analysis is critical in grappling with many subtle phenomena to ensure that nothing serious will be lost in translation and to increase the likelihood that your research findings will stand the test of time. ”

“ Data have now been gathered in regard to the families of the men of science previously selected. In our present state of ignorance a statistical study of any homogeneous and objectively chosen group should be of value, both as a contribution to psychological and vital statistics in general and for comparison with other groups which may be similarly studied. ”

“ Statics, like sociology, studies the structure of society, but there is a wide difference between the methods employed respectively. Statistics investigates in the first instance statistical conditions, and phenomena of uniformity, whereas sociology is concerned particularly with the general direction in which human evolution is progressing. ”

“ In these works he shows how the numbers representing the individual qualities of man are grouped round the numbers referring to the “average man” in a manner exactly corresponding to that in which single results of observation are grouped round the mean result, so that the principles of the theory of probabilities may be applied to statistical researches on the subjects. ”

“ The attempt to make comparison between the percentage of growth of population and the percentage of growth in the expenditures of the Federal Government, using no facts relative to the great increase in mechanical production and of wealth, is a vicious use of statistical data. Such showings are the results of the work of the statistical mechanic, the man who constructs statistical tables to order.
Statistics really take the place of observation. ”

“ All that is necessary to make the examples appropriate is that we should admit that in their case statistical data are our ultimate appeal in the present state of knowledge. Of course if the empirical laws can be resolved into their component causes we may appeal to direct deduction, and in this case the employment of statistics, and consequently the use of the theory of Probability, may be superseded. ”

“ He defines the statistical science as “die systematische Darlegung and Erdrterung der thatsächlichen Vorgänge and der aus diesen sich ergebenden Gesetze des gesellschaftlichen menschlichen Lebens auf Grundlage quantitativer Massenbeobachtungen” (the systematic statement and explanation of actual events, and of the laws of man's social life that may be deduced from these, on the basis of the quantitative observation of aggregates) . ”

“ Statistical averages, and the like, are nonsensical productions unless they apply to objects that cluster toward a common center; and composite pictures are equally monstrous or meaningless unless they are compounded of objects that have a common similarity to a central ideal type. ”

“ Some students of verse are doubtless offended by the use of statistics in connection with a subject of this kind; and it is easy to ridicule the obvious incongruity of mathematical methods and poetry. A recent magazine critic makes merry over certain statistical studies in rhythm, carried on in a [Pg 440] laboratory by recording the beats in nursery-rimes on the one hand and in hymns on the other. "There is a certain class of problems," he observes most justly, "whose external aspects may possibly yield to statistical tabulation, but which in the last resort must be spiritually discerned." ”

“ It is of this form of probability that Mill advises to know, before applying a calculation of probability, the necessary facts, i.e., the relative frequency with which the various events occur, and to understand clearly the causes of these events. If statistical tables show that five of every hundred men reach, on an average, seventy years, the inference is valid because it expresses the existent relation between the causes which prolong or shorten life. ”