Quantitative analysis

“ It is natural, therefore, to conclude that they not only lack precision but are unnecessary. Obviously our quantitative analysis must be expressed without using any quantitatively vague expressions. And, indeed, as soon as one makes the attempt, it becomes clear, as I hope to show, that one can get on much better without them. ”

“ The fact that two incommensurable collections of miscellaneous objects cannot in themselves provide the material for a quantitative analysis need not, of course, prevent us from making approximate statistical comparisons, depending on some broad element of judgment rather than of strict calculation, which may possess significance and validity within certain limits. ”

“ So far as it is true that the last sovereign of my expenditure in bread equals in utility the last sovereign of my expenditure in books, that fact proceeds not from a comparison, conscious, or unconscious, of these separate items at this margin, but from the parts assigned respectively to bread and books in the organic plan of my life. Quantitative analysis, inherently incapable of comprehending qualitative unity or qualitative differences, can only pretend to reduce the latter to quantitative differences. ”

“ A preliminary qualitative examination is generally indispensable, if intelligent and proper provisions are to be made for the separation of the various constituents under such conditions as will insure accurate quantitative estimations. ”

“ It is assumed that the operations of qualitative analysis are familiar to the student, who will find that the reactions made use of in quantitative processes are frequently the same as those employed in qualitative analyses with respect to both precipitation and systematic separation from interfering substances ”

“ Therefore, one has much to say for the thesis that the whole logical value of the quantitative conceptions in science is due, not to any peculiar advantage of the concept of quantity, as such, but to the exactness of the forms of serial order which are discoverable in case of any quantitative realm of facts. ”

“ 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. ”

“ 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 prevailing unfamiliarity with statistical methods may also favor a skepticism as to their proper application to education, since it is not an exact science. But the fact remains established that it is always possible to measure qualitative differences if stated in terms of their quantitative amounts. ”

“ The necessity of qualitative measurement of statements not only per se, but with reference to the sources from which they are drawn, is generally recognized: but historians as yet seldom heed the twin requirement of quantitative measurement of our data in comparison with the sources whence they come. ”

“ Moreover it is to be remarked that while, on the one hand, we can discover the laws of the greater proportion of phenomena only by investigating them quantitatively; on the other hand we can extend the range of our quantitative previsions only as fast as we detect the laws of the results we predict. For clearly the ability to specify the magnitude of a result inaccessible to direct measurement, implies knowledge of its mode of dependence on something which can be measured—implies that we know the particular fact dealt with to be an instance of some more general fact. ”

“ 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) . ”

“ Thus the extent to which our quantitative previsions have been carried in any direction, indicates the depth to which our knowledge reaches in that direction. And here, as another aspect of the same fact, it may be observed that as we pass from qualitative to quantitative prevision, we pass from inductive science to deductive science. Science while purely inductive is purely qualitative; when inaccurately quantitative it usually consists of part induction, part deduction ”

“ So soon, however, as a result is known to be discoverable, the scientific man ought to proceed to the quantitative inquiry, how great a result follows from a certain amount of the conditions which are supposed to constitute the cause? The possible numbers of experiments are now infinitely great, for every variation in a quantitative condition will usually produce a variation in the amount of the effect. The method of variation which thus arises is no narrow or special method, but it is the general application of experiment to phenomena capable of continuous variation. ”

“ The higher laws alone can assume a numerical form, and it is a truism to say that the perfection of any science is measured by the quantity of mathematics which it involves. Not that mathematical formulæ imply or confer any magical virtue, but they are the sign of reduction to clear and simple relations, and are frequently an instrument of further progress. It is true that in the domain of empirical law, there are many processes which attempt to imitate quantitative determination: graphic records, curves, statistics, percentages, etc. ”

“ There is an interesting class of experiments which enable us to observe a number of quantitative results in one act. Generally speaking, each experiment yields us but one number, and before we can approach the real processes of reasoning we must laboriously repeat measurement after measurement, until we can lay out a curve of the variation of one quantity as depending on another. We can sometimes abbreviate this labour, by making a quantity vary in different parts of the same apparatus through every required amount. ”

“ The second inquiry through which we may learn somewhat of the present position of sociology relates to the development of method. Exact method in social research is statistical. Wherever we can obtain numerical data within the domain of social phenomena, there we arrive at exact or quantitative knowledge. The development and application of statistical methods to social problems has been one of the most striking scientific achievements of the present century. ”

“ Size, on the other hand, and its alteration regarded merely as size, is for the qualitative determination, when it is unable to assert itself in terms of measure, an indifferent determination. That is to say, measure is quantity, precisely in so far as it can give to itself an aspect which qualifies it, and thereby a qualitative determination is united to the purely quantitative one. ”

“ Hence, if several different quantitative effects are known to enter into any investigation, we must have at least as many distinct observations as there are quantities to be determined. Every complete experiment will therefore consist in general of several operations. ”

“ Suppose that one man has collected statistics including 1000 instances, and another has collected 4000 similar instances. Common sense can recognize that the latter are better than the former; but it has no idea how much better they are. Here, as elsewhere, quantitative precision is the privilege of science. The answer we receive from this quarter is that, in the long run, the modulus,—and with this the probable error, the mean error, and the error of mean square, which all vary in proportion,—diminishes 486 inversely as the square root of the number of measurements or observations. ”

“ Whence it appears not only that in proportion as knowledge becomes quantitative do its previsions become complete as well as certain, but that until its assumption of a quantitative character it is necessarily confined to the most elementary relations. ”

“ As it is impossible to comprehend aright the value of quantitative laws without constantly bearing in mind the degree of quantitative approximation to the truth probably attained, I have devoted a special chapter to the theory of approximation, and, however imperfectly I may have treated this subject, I must look upon it as a very essential part of a work on scientific method.
"It then remains to illustrate the sound use of hypothesis, to distinguish between the portions of knowledge which we owe to empirical observation, to accidental discovery, or to scientific prediction. ”

“ And yet, were we to take members of the Mathematical Society, who, having severally devoted themselves to the laws of quantitative relations, know that, simple as these are intrinsically, a life's study is required for the full comprehension of them—were we to ask each of these his opinion on some point of social policy, the readiness with which he answered would seem to imply that in these cases, where the factors of the phenomenon are so numerous and so much involved, a general survey of men and things gives data for trustworthy judgment. ”

“ In some rudimentary experiments we wish merely to establish the existence of a quantitative effect without precisely measuring its amount; if there exist causes of error of which we can neither render the amount known or inappreciable, the best way is to make them all negative so that the quantitative effects will be less than the truth rather than greater. ”

“ 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. ”

“ The application of quantitative methods to the study of mental functions as thus defined, quickly revealed the fact that human beings, sampled at random, in large numbers, do not fall into distinct types. ”

“ But it must constantly be remembered that quantitative thinking does not necessarily or even generally mean thinking in terms of numerical statistics. Number, which obliterates all distinction between the units numbered, is not the only, nor always even the most exact means of representing quantitative facts. ”

“ Where expert skill is required, as at state and national headquarters, it can be found. Every layman can train himself to use skillfully the seven ingredients of the statistical method which it is his duty to employ, and to know when to pay for expert analysis and advice. ”

“ We cannot, it is true, arrive at the same certainty and precision of result in these inquiries as in those branches of physical research where exact quantitative appreciation is possible, and we must content ourselves with probabilities and approximations. ”

“ Such statistics while convenient and fashionable, and assuredly not without interest, have a purely quantitative basis—such is the way of statistics—and of themselves do not reveal a vital cultural development. ”

“ The quantitative basis of scientific procedure. Science is science, some scientists insist, in so far as it is mathematical. That is, in the precise determination of facts, and in their repetition with a view to their exact determination, quantities must be known. ”

“ Neither shall we make it a part of our present task to study the equally interesting and more important problem of measurement, which is concerned with quantitative determinations of the variation of the ordinal characteristic; it will here suffice if we say that the possibility of measurement of a given continuous aggregate is dependent upon the existence of a certain relation between every two elements of the continuum, which relation it is possible to express numerically, and to compare as to equality or inequality between various pairs of elements ”

“ Somewhat generalized statistics are used as a means of gaining perspective. No effort has been made for detailed accuracy or for completeness. So far as possible the quantitative features are expressed in general proportions, and where specific figures are given they are meant to indicate only such general proportions. ”

“ Stem and Leaf Displays Histograms Frequency Polygons Box Plots Bar Charts Line Graphs Dot Plots As discussed in the section on variables in Chapter 1, quantitative variables are variables measured on a numeric scale. Height, weight, response time, subjective rating of pain, temperature, and score on an exam are all examples of quantitative variables. Quantitative variables are distinguished from categorical (sometimes called qualitative) variables such as favorite color, religion, city of birth, favorite sport in which there is no ordering or measuring involved. ”

“ Strictly speaking, only the things which I have mentioned belong to the category of quantity: everything else that is called quantitative is a quantity in a secondary sense. It is because we have in mind some one of these quantities, properly so called, that we apply quantitative terms to other things. We speak of what is white as large, because the surface over which the white extends is large; we speak of an action or a process as lengthy, because the time covered is long ”