collective mass education-english-gustav theodor fechner

8/22/2019 Collective Mass Education-English-gustav Theodor Fechner.

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COLLECTIVE MASSEDUCATION

BY

Gustav Theodor Fechner

ON BEHALF

THE

SAXON ROYAL SOCIETY OF SCIENCES

PUBLISHED

BY

GOTTL.FIEDR.LIPPS

LEIPZIG

PUBLISHER OF WILLIAM Engelmann in 1897.

Content

Part One

Preliminary statements

Foreword

I. Introduction. 1, 2

II Preliminary Overview of the key points, which are used in theinvestigation of a collective object into account, and it related names. 3-11

III. Preliminary Overview of the study material and general observations. 12

IV props; abnormalities. 13-23

V. Gaussian law of the random deviations (errors of observation) and itsgeneralizations. 24-37
http://kmasslvw.html/http://kmasslvw.html/


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VI. Characteristics of the collective objects by their determination pieces orso-called elements. 38-46

The theoretical treatment of collective objects

VII Primary distribution panels. 47-52

VIII Reduced distribution panels. 33 - 67

IX. Determination ofA a , , a ', m , m ', , , '. 68-75.

X. Compilation and context of the main characteristics of the three principal valuesofA, C, D: alsoR, T, F 76-86

XI. The densest valueD. 87-92

The asymmetry of collective objects.

XII. Reasons that significant asymmetry of the deviations with respect to thearithmetic mean and validity of the asymmetric distribution law with respect tothe closest value ofDin the sense of generalized Gaussian law (Cape, V) be thegeneral case. 93-95

XIII . Mathematical relations of the connection of major and minor

asymmetry. 96XIV formulas for the mean and probable value of the dependent purely randomasymmetry difference and 97-101

XV. Probability provisions for the dependent of a purely random asymmetrydifference u the outputs from the true center. 102-111

XVI. Probability provisions for the dependent of a purely random asymmetrydifference v the outputs from the wrong means. 112-117

The distribution laws of arithmetic principle of collective objects.

XVII. The simple, two-sided Gaussian law. 118 to 122

XVIII. The sum of the law and the Supplementarverfahren. 123-128

XIX. The asymmetry laws. 129-136

XX. The extreme laws. 137-142

The logarithmic distribution law.


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XXI. The logarithmic treatment of collective objects. 143 to 146

XXII. Collective treatment of relationships between dimensions. Mean ratios. 147-151

Appendix.XXIII. Dependency relationships. 152-155

Second part.Special investigations.

XXIV over the spatial and temporal relationship of the size variations of therecruits. 136-163

XXV. Structure and asymmetry of rye. 164-169

XXVI. The dimensions of the gallery paintings. 170-175

XXVII. Collective objects from the fields of meteorology. 176-179

XXVIII. The asymmetry of the error rows. 180-182

Appendix. The t-table. 183

Preface.

Vorliegendes work has for many years created by me, collected material and toproceed in the same preparation, but this often interrupted by other tasks, all set asidesome time and has thus far delayed the completion of the work. To delay it longer,

like at my age, not be advisable if the work is to appear at all, nor do I dare say it is itcan finally dare after repeatedly coming back to appear, although for not as a perfectwork, but as a backing a further expansion of the doctrine discussed here. Particularthe following introductory chapter speaks about the task of teaching from, and so likehere only the following general remarks can find additional room.

With the new name under which the teaching occurs here, I give it not as a newdoctrine, only that the current level of development need not put the close, they evenset up under a special name for themselves. Everywhere the science so specialized inthe way of their growing development and thus requires separating names of their

various areas. Well probably the most general, most interesting, Verdienstlichste whatexisted of our teaching so far, in Quetelet's "Lettres sur la thorie of probabilits"(1846) and his "physique sociale." (1869) to be found, and if you want, you can at


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him as the father of collectives, such as weaver to see EH psychophysics, but you willbe able to convince from the pursuit of this work, how much occasion was however,not only significantly expanding, but also adjustment transcend it.

In this respect I make a page as a main crop of other than Hauptwur-zel throughoutthe following investigation the opposite hand controlling mathematical reasoning and

empirical validity of a generalization of the GAUSS law random variation claimswhich the restriction of the same on symmetric probability and proportionatesmallness of mutual differences is lifted from the arithmetic mean, and previouslyunknown legal relationships occur, the most important one is compiled 33rd In fact,the general regulator of all coming in the collectives of language relationships in thisgeneralization is also given, as in the simple laws of GAUSS regulator of all physicaland astronomical accuracy requirements, and should still ask themselves if not in

principle also to the recourse to general law would be what one should not, beignored in the comments with 8.

In this respect, the collectives is based on a combination of monitoring andaccounting in mutual relationship, they may be expected to exact teachings. Thelessons that are entitled to such a description, but leave at all to a very different levelof security of their results. At the head are mechanics, astronomy, physics,

physiology is because of the difficulties which confront the complication andvariability of objects far behind; further, because even greater difficulties in thisregard, the psychophysics. The collectives shared difficulties, without being subjectto the same fundamental problems as psychophysics, this offers additional practicalinterest, however, they are far inferior to their philosophical interest. But it is not

missing all the collectives in such a, if the incoming into it subordination of chancecomes under general laws here in an area and in a manner to advantage, which havenot previously been subject to the consideration.

In regard to the shape and width so many versions will be taken into account thatthe work is not intended for both professional mathematicians, which come here inrespect fundamental points are familiar with already, as for those who are to doknowledge acquisition and application of the doctrine is not that they are already in

possession of such knowledge.

Here Next I would like to promote our doctrine still address a request to thecomputer compartment. In the known tables which usually the Gaussian probabilityintegral of the random deviations from the mean (observation error) as

expressed represent the argument is tjust run up to two decimal places, which havelimited use for the physicists and astronomers to make of it, is sufficient tointerpolation by consultation with first and second differences, but for the far moreextensive use of the collectives is to make it out to the same thing as if you reduce themany bills that are to lead by means of logarithms, the number argument, to whichthe logarithms are merely two or three digits and interim provisions would prey only


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give the interpolation. So it would be desirable if the interests of our teaching, whichis incidentally shared by the psychophysical method of right and wrong cases were

available, tables, where tleast four decimals executed is 1) to avoid interpolationsmostly to facilitate part, and in any case I myself have missed such tables inexecution of this work painful. Of course, the expansion of the tables would thus

grow, but the advantage seemed to grow stronger relationships with them. Andshould there be no astronomical or statistical institute that has to have mechanicalcalculating forces, which were to assume the thing! Also could probably make it up a

prize.

1) A version of this table to three decimal places oft, with restriction of the integralvalue to four, respectively. five decimal places, can be found in Appendix 183

I. Introduction. 1 Under a collective object (short K.-G.) I understand an article, consists ofcopies indefinitely many, varying randomly, the type or generic are held together byone.

Thus man is a collective subject in a broader sense, the man of a certain race, acertain age and a certain breed such in the strict sense, as ever, what the extent of itK.-G. may call upon the extent of the genus or Artbegriffs, under which he enters,changes.

The copies of a K.-G. may be spatially or temporally different and thereafter aspatial or temporal K.-G. form. Thus, the recruits of a country or ears of corn field ascopies of a spatial K.-G. . apply So are the (average) temperature of the first January,followed at a given place by a number of years, as many copies of a temporal K.-G.. Instead of 1 January can be any other anniversary, instead of a specific day acertain month, instead of the temperature and the barometric pressure setting etc isthat just as many copies temporal K.-G. receive.

Anthropology, zoology, botany have it much at all with K.-G. to do, because it cannot be a characteristic of individual specimens, but only to the fact that a population

of the same plays that. aspects of this or that is summarized as genus or species ingreater or lesser length The Meteorology has just referred to examples in their non-periodic weather phenomena which represents numerous examples, and can evenperformance art to speak of such, provided books, business cards are among them.

The copies of a K.-G. Now on the one hand high, on the other hand, quantitatively,ie by size and number, determined, and only the latter determination is in thecollectives. A K.-G.does, in fact, in terms of its quantitative determination, the sameclaims as a single object, except that in some (though only some) respect theindividual parts of the object through the copies of the K.-G. be represented. It

applies, for example, recruits a given country, then the question is: how big therecruits are in the middle, how widely different dimensions to their means, how bigare the largest and smallest, the behavior of the recruits dimensions to these


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provisions in the individual cohorts, as in different countries with each other. Theseand related, later to be considered questions can each K.-G. pose, and provided aspatial object has to distinguish different parts and dimensions, they can be in any ofthese parts and dimensions particularly raise and these constitute a special K.-G. treatas skull, brain, hands, feet of a man, height, weight, or volume of the whole man

given parts of people, but also quantitative relationships will come into question, suchas when comparing the people of different races, the ratios of average height, width,length of the skull take a special interest in the claim.

2 All these individual questions but raises a more general, the most important towhich it can act at all in this doctrine and act accordingly to below, is the question ofthe law, how the copies of a K.-G. distributed by size and number. By the expression,

but the distribution is the determination to understand how the number of copies of agiven K.-G. changes with their size. At each present in a greater number of copies K.-G. come from the smallest and largest specimens, short extremes, the rarest, most

often those of a certain medium size. But there is not a common to all or at least mostK.-G. applicable law function of the number of copies of the size? In fact, such aleave is up, and go a main object of the following to its conclusion.

From the outset, however, one can doubt that at the extraordinary difference in theK.-G. legal distribution ratios are sure to find a certain universality at all. Meanwhile,since according to the terms of the K.-G. such varying randomly from specimens is,in any case find the general probability laws of chance - and every mathematicianknows that there are those - use it. In fact, the distribution ratios are the K.-G. generally dominated by those, however, only incidental to the provision of

security means measures obtained are the same laws of probability in physical andastronomical Mabestimmungen contemplated hereby play a very different and muchmore significant status as a measurement gauge in the K.-G..Respect, but thecoincidence in certain of the various K.-G. various plays external and internalconditions, can, by all contingencies through the various K.-G. distinguished bycharacteristic, can be derived from their distribution ratios constants. These are theones where the definiteness of the same resting against each other, and this is it withregard to the general laws of probability to visit. Now it has been taken in this regardhas always been the arithmetic mean of the specimens in the eye and diligence on its

determination at the different K.-G. turned, besides also probably still rarely takeninto account the extremes of the average deviation from the mean. But as importantas these determining factors and will always remain, but so far they have been takeninto account to one side, while others, no less important principle, this usuallyignored.

In so far as the treatment of K.-G. after all the previous relations is subject at allother points and other modes of determination carries than in physical andastronomical measures in consideration come, the measurement gauge of K.-G., orshall we say short collectives, as a doctrine of its kind specially prepared and treated

be, and this will be folgends the task. Since our notion of K.-G. the notion of a random variation of copies received, you


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can anticipate want a definition of chance and explanation of his being. The attemptto give such a philosophical viewpoints from, but would bear fruit for the next littleinvestigation. It must suffice to indicate the, for the following reason to set, factual

point of view of more negative than positive for this character here. Under a randomvariation of the copies I understand one which as a ratio between the size is also

independent of a setting on the size of arbitrary determination, regulatory laws ofnature. Like one or the other of the provisions of the articles have a share, yet onlythe changes of independent random. It can therefore be determined by law nocoincidence how great this or that single copy, although, where size limits of a givennumber will keep the same with this or that degree of probability.

This will not be denied that there is no chance of most general aspects, by the sizeof each copy can be viewed as necessarily determined by the existing laws of natureunder the existing conditions. But as long as we speak of chance, as we ascend to aderivation of the individual provisions of these general regularities neither, nor to

conclude from the facts before it in are unable. Insofar as it is the case, listen to theaccident, and listen to the applicability of the laws vorzufhrenden here or isdisturbed.


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II Preliminary Overview of the most important points, which in the investigation of aK.-G. be considered, and it related terms.

3 The following compilation is to serve, the extent and the nature of the studieswith which we have to deal folgends to make certain overlooked, and to orientthemselves over most of the to need generic terms anticipated in connection, a more

detailed discussion of these points but remains subject to the following chapters. In the random order in which the copies of a K.-G. Maintain perform, neither anoverview of the conditions can win them by size and number, nor a methodical

processing of the same would be possible if their, generally with a degree to bedesignated in the same random order in which they were received and in a wanted so-called original list has listed, leave, so they have to be sorted primarily according totheir size and so arranged in a table, so-called distribution panel perform. If one nowthere are no large number of copies of an article, it is each a or but most a appearonly once in the panel, and the size of distance between the consecutive be a changevery irregular, with numerous developmental objects but that many of which presentspecimens, as they are for the following mainly presuppose, if not all but many ormost ofa, which will bear the scale and the estimation, more or less often occurrepeatedly, and then aimed at the distribution panel in such a way that in a columnofa each a while once only performs, but in a beige discontinued column ofz, thenumberzindicates how often it occurs. , the total number ofA , which enter into adistribution panel, of course, correspond to the sum z, which is by adding togetherallzcontains, match the table and is of me with m respectively.

The establishment of such a panel is to say that the first step you while processingnumerous developmental K.-G. has to do out of the original list.

A second step is this: that the, withA determined to be designated, arithmeticaverage of the individual measurements and the positive and negative deviations, thenumberzof course with the deviating a match.

But to do this, as starting point of the deviations instead ofA also some othervalues which are derived with mathematical certainty from the distribution panel,serve, and by any other choice in this regard are new relationships to the fore, will beto speak of them later. Generally now I call values that are used to develop such

relationships as the starting values of the deviations, home values, and denote itbyH, which thereforeA is only a special case, on the one account so far in thetreatment of K.-G. has only limited, but this is an arbitrary restriction of collectivescarries, as will be readily apparent from the following remarks later. General I calldeviations from which core values they may also be made subject to collectivedeviations.

4 Easy now convinced you are of the following circumstances. An everlargermeters in the distribution panel of a K.-G. arrives, so will the regular course ofthe a correspondingz,and so are certain to put out the legalities of which we

speak. The ideal case would be that you an infinite m would, where you have a veryregular course ofzwould have to be expected and a very exact fulfillment of the


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relevant regularities, which is also ideal conditions and legalities, as they would givean ideal panel and empirical, which consist in a more or less large approximations ithas to be distinguished.

All probability laws of chance at all, and the distribution laws of K.-G. are thosethat have in common that their compliance is more certain to be expected, depending

on a large number of cases they refer to, but as it were a perfect possess valid only forthe case of an infinite number of cases, which does not exclude that already with anempirically well-being procured number of cases, confirming the law shall be held inclose approximation. Respect one now anyway, in reality only K.-G. has to do from afinite number of copies, which represent just as many cases, I refer to the deviationsthat occur from the ideal laws because of finiteness of the number of copies, asinsignificant, and, so far as they go indifferent to one and the other side when caused

by unbalanced contingencies, however, that I, for the assumption of an infinitenumber of cases, our case of copies, current regulations refer to as essential or

normal. The general feature of the immateriality of a provision is that it disappearsthe more so the more one the number of cases, respectively. Copies, subject to theconditions that the concept of K.-G. determine enlarged so that you can assume theywould disappear completely at infinite number of cases, which only ever vielzahligeitems are suitable for investigation of the laws in our case.

Even with a small m but the insignificance of a provision proves the fact that inrepeating the assay with the same small m size and direction of the provision goesundetermined from getting new copies of the same object, whereas in materialitythereof in the Middle of a majority of times a particular size result and a specific

direction out of the same so as to provide a fixed, the larger the number of repetitionsand m each is different.

We speak of a symmetrical distribution of values for a given principal value ofH, ifany deviation ofa positive-ofHequally large negative deviation ofanothera ofHcorresponds to, so that equally strong on both sidesofHdiffers a great equalzbelong . In a K.-G. of a finite number of copies can be dueto the unmatched contingencies did not expect, with respect to any major value tofind a completely symmetric distribution, and of course, a symmetrical distributiondoes not respect more home values also exist, but it is an important object of study, if

can not find the main value in respect of which the distribution approaches the moresymmetric, the more one the m of K.-G. increased, in the way that at infinite m couldassume a symmetrical distribution as really reached in which case you, as aninfinite m is not to have, but can speak of a symmetric probability of deviations.

5 But, there is another than the previous considerations, one can distinguish anideal distribution panel from an empirical and dependent ideal and empiricalresults. For measurements, the specimens can not go beyond certain limits ofaccuracy, as they will bear the classification and the estimate of the scale in

between. One can, for example, even millimeters, even tenths of a millimeter, orhundredth millimeter but not differ beyond. For that differs only millimeters flow allthe individual dimensions that keep within the limits of a millimeter,


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indistinguishably together, so he refers the wholezcopies that are actually distributedon a whole interval of 1 mm to a single value a that forms the middle of thisinterval. Is generally i the still recognizable difference in extent, so part ofthezeach a table of empirical fact throughout the interval of the size i between a

- 1 / 2 i , and a +1 / 2 i on, whereas it is to the empirical table and thus excludes them

from exploitation is usually so composed, as if falling into it made a self-zvorkmetimes. In an ideal, ie walking to the limit of accuracy of measurement and estimation

but would i come upon an infinitely small value 1) , the distinct a of the tablereproduce herewith theirzbut shrink accordingly, this is the perfect panel of theempirical representation.

1) An infinitely small value, here presided in the sense of calculus is not to beconfused with zero, but, although decreased continuously under any anfhrbare size

and its absolute size of indefinable but statement as yet to its relations to otherinfinitely small values determined.

Where now the empirical i is very small, the results of the empirical panel differ, asthey relate to the size and proportions of the derivable principal values and principaldeviation values not significantly different from those of the ideal, but is thedifference generally speaking be considered and will be later this consideration asfind out where he comes into substantial consideration. Empirical rules andconditions in which he is not considered necessary, but it shall be considered as if

really thezeach a this a very zukme, I call raw, those where it is taken into accountas far as possible, sharp.

6 In any case, now you have to be sharp on the results of empirical panel to theideal of the ideal panel, hereby from insignificant to significant, rise of raw search,including a demgeme processing part of the distribution boards.

In this regard, a primary difference between reduced and sheets can be done. Iunderstand from primary plates such as are obtained immediately by order of thedimensions from the original list and the same hereby User Data such as these, but

just ordered, offer. Reduced signs my name, those in which thezfor largerMaintervalle, are distinguished as to the primary tables, and although the total forthe same size throughout the panel, thezbut these larger intervals the centers thereof,as reduced a, be given written , with the advantages, thus a more regular courseofzto get into the panel and a more suitable base for calculations, if not withoutdrawbacks because of conflict with a magnification ofi . whereupon come backlater Incoming is ever traded on the installation methods and the ratios of the primaryand reduced tables in Chapters VII and VIII, with the possibility of different levels ofreduction and reduction principles for language comes.

7 In every non-regular or irregular primary reduction panel is made by followingyou.

The smallestzcan be found by the two limits to the board, which, as already


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touched earlier, the smallest and largest a least likely to occur, the largestzbutgenerally in the middle part of the panel. The maximumzdrops to a certain a part inthe middle, on either side of where thezcontinuously towards the extreme, thoughwith insufficient reduction here and there interrupted by irregularities decrease. Thevalue ofa one not too irregular primary or reduced distribution panel to the

maximumzfalls, I call the densest of tabular or empirically densely value of theobject, which can be certainly regarded as an approximation to the ideal denselyvalue, the one with an infinitely large m and infinitely small i would get, but no lessofA is the board, but even as such, deserves special attention and approach providesthe base to a more accurate approximation by invoice later contemplative manner. Beit empirically or ideal approach taken in this or that, I call it common withD.

One might think that the densest value significantly, so strictly speaking, from avery large, infinite m and with a very small, strictly speaking, infinitelysmall i, determined, would coincide with the arithmetic mean, and indeed soft in the

majority of K. -G. after determination of both large m and small i little enough fromeach other that you may be inclined and has previously held, in fact, ensure that thestill remaining difference is merely a matter of unbalanced randomness. But it will beone of the most important results of the following investigation, that a significantdifference between arithmetic mean values and the thickest is rather the general case,such that the size and direction of this deviation itself characteristic of different K.-G. are. Insofar now also comply with the variations in the two values different ratios,the empirical densest valueD as an arithmetic mean ofA recognize the same table to

be distinguished, important main di baseline value of collective deviations.

To the previous two main valuesA, D but still occurs a previous of the two to bedistinguished, third, I as a central value or value center with Cwill denote di thevalue ofa, of as many more a than smaller has over among themselves and in thisrecognition, the number ofA shares through the middle. At the same, it comes outwhen it is said to be the value for which the number of positive deviations withrespect to equal the number of negative. From the arithmetic mean he differs in thetwo provisions that, while regardingA the sum of the mutual deviations is the same,however with respect to C, the number of mutual differences is the same, and that,while rel.A the sum of the squares of the deviations of a minimum , ie less than

dist. any other initial value is here to mar. C. the sum of simple deviations (negativeexpected it to absolute values) in the same sense a minimum is 2) . With the thirdmain entrees this value to the previous two is now open again new characteristicrelationships for the KG will be talking about what.

2) This, not previously noticed, the median property I have in a special treatise onthe same evidence [over the original value of the smallest deviation sum;Abhandl. the math.-phys. Class of the Royal. Saxon. Society of Sciences, Volume II,1878].

Besides these three main values are other, from the distribution panel


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mathematically derivable as initial values of deviations and hereby serve as the mainvalues and considered largely independent of the previous one, some with the samemay be related, but are certainly the last major, and I remain First there are. In a laterchapter (Chapter X), but I'll irrelevant three other main values as vagina valueR ,heaviest value tand heavy deviation valueFinto account, which in any case offer a

mathematical interest. 8 An animal is characterized by its inner Build to brain, heart, stomach, liver, etc.,the size and location of these bodies against each other, the afferent and efferent

pathways to do so. So a K.-G. its internal quantitative determination of characterizesthem by arithmetic mean, median, densely value and otherwise about zuzuziehendemain values, the size and location of these main values against each other and thedifferences, and these values are no less in mathematics than those organs in anorganic context. A K.-G. forms so to speak a mathematical organism which is capableof decomposition, will be going into the hereafter. And if that is not to say that every

object has to make the implementation of such a claim dissection, as in any case ageneral Kollektionsmalehre has with the general aspects to deal with the same.

To advance can be noted here that although under a certain condition, the twoprincipal values ofD and CwithA and therefore all three coincide each other would,under the condition, namely, that the mutual deviations rel.A possessed asymmetrical probability, ie, with increasing m in the form of a symmetricaldistribution (in the above sense) approached that one at infinite m could consider suchas reached. But it will be seen that for K.-G. rather an asymmetric probability ofdeviations rel.A has to presuppose, according to which with increasing m not a

symmetric distribution, but one to be brought to a certain law, significantlyasymmetric distribution approaches. Yes it can be apart from the exception to beregarded only as a significant coincidence ofD and CwithA absolutely no value forK.-G. find rel. the probability of a symmetric deviations would take place on bothsides.

Now if you have been in the treatment of K. - G. only onA, taking the deviationsfrom it and about the extremes of consideration, one sees not only already from

previous volume, that very important characteristic ratios and differences of theobjects covered thereby ignored, but It will also show that a general law of

distribution of copies of K.-G. is not to be gained by this limited mode of treatment.

But she has not disputed the fact their reason that they transferred the senior aspectsof the physical and astronomical measurement gauge on the collectives, withouttaking into account two important differences that exist between the two, wherebythose limited treatment way for former teaching just as motivated as for latter isdenied. For the former, the arithmetic mean isA the observed values of its dimensionsto be determined each object with the deviations ofA, ie errors of observation, thedominant, so basically alone counting, meaning, as you are known by reason, the

professional mathematicians and physicists are in the values with respect to which thesum of the squares of the deviations, ie, errors which is the smallest possible, thearithmetic mean, also sees the value, which, to the true values to its determination to


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do it in all probability the next , but it is the differences in a means to determine theamount by which the true value but still with a given probability of one or the otherside will be missed. So why not take care of this doctrine to other home values, andtheir differences will help to fulfill the task of teaching nothing! So neither of adensely values, yet central values in the astronomical and physical measurement

gauge the speech, regardless of the various observed values of the same object in it,as a taken at equally well to derive aD and Ccould give rise, as the different copiesof a K.-G. But it would be idle to a special viewing incurring the same, and in anycase does not happen.

For the collectives but has the point, which can favor the arithmetic mean of thedeviations from this principle in the physical and astronomical measurement gauge,no meaning. All copies of a K.-G., they may still so far deviate from the arithmeticaverages or any other principal values, are equally real and true, and preferably aconsideration of one over the other from a void equal for all aspects of course makes

no sense . This counter has any other value to other main characteristic and itsrelationship to the part itself practical significance for a K.-G., thereby contributing todistinguish the same from other objects.

Secondly, however, differ according to the physical and astronomical measurementgauge in the course rather postulated or presupposed as proven beyond doubt,symmetrical probability of observing errors mar. the arithmetic observation agent atgood observation, the three main values not significantly, but only by unbalancedcontingencies of each other, so that in the preferable because of the circumstancesinvoked arithmetic mean of the observed values at the same time mittrifft the most

probable values of the other main values, whereas for the K-G . bemerktermaen anasymmetric probability of deviations rel. of the arithmetic mean to be regarded as thegeneral case is what the different main values significantly fall apart.

Incidentally, it can seem even more questionable whether it really is with thatpostulate in the observation errors in all rights, a question which, although not much

concern us here, but later in a special chapter3) will be considered.

3) [With. consideration of this question is in the second part, Chapter XXVIII,

examines the asymmetry of error rows.]

But we now return to the essential conditions for the collectives.

9 Sub-elements or parts of a provision K.-G. I will understand in the analysis ofsuch at all following values in the following, some of them already used earlier,names.

1) The general with m labeled total number of copies a one contemplateddistribution panel.

2) The generally withHmain levels identified or output values of deviations,which bemerktermaen the arithmetic meanA , the median Cand densest valueD are


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the most important. Since the central value generally betweenA andD to look for ishow to show later, the last three main levels will always be generally in the orderA,C, D are given by me.For this purpose some, irrelevant to consider the main values,which in X. Chapter are discussed.

The arithmetic mean is, from a primary table determines withA 1, from which a

reduced determined with O 2 are known, according to C. InD , no such distinction is

made, because he too because of the irregularities of the standing bids primary panelseverywhere panels has reduced merely out can be derived hereby everywherewithD 2 would be to call. This counter is. to make a difference to the

Herleitungsweise between. After the so-called method of proportion to me, which Iwill give the most confidence, derived, I call himD

p, derived after less secure

interpolation method, withDi. From the differences between the two methods of

procedure, the speech will still be.

All values, which on the positive side of the main value to which they relate, fall, Icall with dashes above, all of which fall on the negative side, with dashes below,however I in those who indiscriminately to mutually related, the dashes completelyomit whereby a 'has a value a known whichHexceeds a , such that byHisexceeded.

Under I understand generally deviations from some main valuesH; under = a '- Hie a positive, under , = a , - H, a negative if the negative characterof , is to be maintained, but since generally the , will be to offset negativedeviations according to their absolute values as positive, is rather put , =H -a , . Hereafter is ' = ( a'-H) the sum of the positive deviationswith , = (H-a , ), the absolute values of the negative deviations,with = ' + , the total sum of the deviations rel.Hrespectively.

3) The main difference numbers ie, the number of deviations of given principalvalues ofH, which, of course, with the number of different values ofa coincident,that is the total number of independent of the nature of the main values are thesame m is, while the number of positive and negative in particular, varies with thenature of the main and as positive values, generally m ', as a negative to m , arereferred to. Ofm ' and m , then the differences are ( m '- m , ) and the ratios ofm': m , and m , : m 'depending which takes m 'and m , can be cited if from them byconsultation ofm , the values ofm 'and m , follow (see below).

4) The principal sum and difference. resulting mean deviations, ie sums of thedeviations divided by the number of them. The total sum of the variances of the twosides together for absolute values, as we believe it always expresses itselfthrough out individually by both sides, particularly by 'and , so

that = '+ , . Depending on this are then the simple average deviationsor mean deviations par4) :


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The totals of variances does not remain as the total number ofmeters , dependingon the principal values equal to, but do not change less than the one-sided sums

depending on the main values.

4) In the physical and astronomical calculation of errors rather maintains a meandeviation simply the root mean square error , rel.A to apply, which I where about

being referred to, by following the specification at the following number 5) as aquadratic mean deviation the above given simple distinction and q will denote.

Regarding the arithmetic meanA particular, the mutual deviationsums 'and , equally necessary, because this is even in terms of this means,however, the mutual deviation numbers m ', m , mar. this means are not equal ingeneral, which carries that even the one-sided mean deviations '= ': m', , = , : m , . bezA generally are not equal.The common force for bothsides = : m is not as simple averages between 'and , = ( '+ , elliott tofind) or to determine how I wrongly in an American treatise on recruits Dimensions

(from 5) ) find specified because one does not thereby to

comes back, but this is only the case when in the middle drawing of 'and , whichtakes into account the weights, which by virtue of them m 'and m , from which theyare to receive, send, hereafter shall:

what on the following simple observation = : m returns. As the product of a

composition of variations in the number of which is equal to the sum of thedifference, that is m ''= 'and m , , = , , that m ''+m , , = '+ , = , on the other hand

m '+ m , = m.

5) [EB elliott, On the military statistics of the United States of America, Berlin,1863. International statistical congress at Berlin.]

The greater the average deviation of a home's value is relative, the more averagein other soft limits each value a of the same from, or the more they vary around thesame average.Besides the absolute size of but is also its relationship to


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theH, whereupon refers, ie :Hinto account what I call the relativefluctuation. The average as relative mean variation for a given m walk Although not

proportional for the different core values, but take it, generally speaking, in so farwith each other and from that a relative of a certain principal value strongly orweakly fluctuating object also concerning the other main values can be assumed to be

strongly or weakly fluctuating, and therefore one can speak without regard to theassistance of a certain principal value of strong and weak in the middle or relativelyvolatile objects.

Thereafter, the following remark. The size of the simple sum of and the simpleaverage error = : m with respect to the arithmetic meanA is not completelyindependent of the numberm of the values ofa, from which the correspondingA isderived, but takes on average with increasing m something, you but may at anyfinite m values obtained and rel.A by multiplying traced back to the normalsituation that they mar. aA of an infinite number ofA obtained what I call the

correction due to the finite m call 6) . While now and = : m are theuncorrected values, so I call with c and c the corrected values:

and .

Only for very small m but the corrected values differ significantly from theuncorrected, and since we in general with large m, have to do while one noticeablydisappears, I am content in general performance of the units, indicating the common,

ie uncorrected values , , resulting in concurrence with the always known m thecorrected values can easily find it when it is doing it. A similar remark isunquestionably for the deviation sum and average deviations rel. Another majorvalues asA apply when the direct examination has been in this respect only to thedeviations ofA is extended. But it is the less reason for citing and recovery of at agiven finite m to prefer the corrected values obtained elements, as not only thedeviation sum and average deviations rel. the different core values, but also thedeviations of the main values themselves from each other under the influence of thesame finite m are the same conditions would therefore not be changed by the joint

correction. When examining the distribution laws but it has to come to us rather tosuch relationships as absolute. Where do you want to go on but those have regardingcorrection of the unilateral values ', , and ', , note the place to find that

they do not respectively by and the, but as of and by

has to be done because otherwise by adding the corrected values ', , thecorrected sum would not find. Here, too, is below the rational standpoint that the

deviation sums each side as members of the sum total deviation of the size oftheirmeters must be influiert together.


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6) It is known that Gauss already long ago for the sum of the squares rel.A and the derived so-called root mean square error of me

the correction due to the finite mdetermined, after which the former by multiplying

by m:( m -l), the latter consistent with our simple correction of the error bymeans happens. The theoretical derivation and empirical validity of ourcorrection ofand but is of me in the reports of the Royal. Saxon Society, Math-Phys. Class, Vol XIII, 1861, pp. 57 f happen, and since the probation is done withdecided success in collective variations, as they may apply for such doubt.

5) The probable deviation w and mean square deviation ofq. Among likelydeviation w bez. a main value is that deviation to understand what has just so much

larger deviations of absolute values themselves, smaller than them, so dist. deviationsof has the same meaning as the central value ofCrel. ofa sub-square. Meanserrorq I understand briefly the root mean square deviations, ie, the value that isobtained when the total variation of a main valuesHparticularly raises the squares,the sum of squares, ie, (to be distinguished well from the square of the amountof di

( ) 2 ,) with the total numberm of the divided quotient and draws the root, just

.

Held jointly for both sides, these values can just as the. simple averagedeviation for both sides specially designed and because of the finite m arecorrected, which I failed to address here, as I verspare what to say about it, even tothe Addendum section on Gauss's law (Chapter XVII), after which these values havecertain relationships with each other, which allow a derivative thereof from eachother, which will save you money, they still perform particularly after performance ofe under the elements.

6) The extreme values ofa the table, ie the smallest and largestA of the table, as

the former, E 'the latter asE, to denote. After the establishment of the panel but ishergebrachten the values for the higher extreme to bottom, the niederere at the top.

10 If two values ofa, to be connected the following way by parentheses, suchas a ( ), this expression is equally valid with a , ie product ofa and , but whenthey are connected by brackets in the following way: a [ beta ], so this does not meanthat a to to be multiplied, but rathera function of is, thus, for example, [A ]represents a difference ofA, [ C] by such a C, etc, m [A ] is the total number ofdeviations rel.A, m [ C] so that the same dist. C, etc.

But at the preferably frequent many other particulars of the principal valuesofA andD , the relevant expressions and formulas by such infliction would it be


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uncomfortable and clumsy, I prefer it in general before, for, m, , depending ontheir function ofA orD equal to several to put a simple name, and indeed this willhappen by the following, which is under the main values of these indications, whichapply without distinction to the mutual deviations without dashes, depending on them

but on the positive or negative side especially belong, yet with a dashes above or

below to be provided are:

A D

m m

E

So that means, for example, a deviation of, such ofD. Since the total numberof deviations is independent of the choice of the home's value, it is generally m==m,whereas is not equal , and is not equal to e is.

The difference'-, (ref.A valid) is short with u , thedifferencem' -m , (ref.D )

with u referred. Ofu follows'and, from u followsm' andm, by the followingequations:

,

.

For the multi considered to be drawn from the upper and lower extreme deviations

from the mean of absolute values of the terms used:U '= E'- A and U, =A -E, .

Rather than the total number of deviations, it was especially moving to either sideor to either side into consideration, we will find occasion it from the main values ofonly up to certain limits, or between given limits, be it their absolute values or theirratios to m , m 'and m , according to consider what is meant by the use ofsigns and is particularly discussed later (in Section V).

As usual, is in the panels of the small dimensions ofa by the larger, ie after the

natural position of the sheet advanced from the upper front of the eyes to the lowerpart of the table, which of course comes in conflict with it, that smaller values thanlower , lower, larger than higher, upper summarizes values. So you have to decide


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according to the context, or explicit indication of whether the terms "higher","lower", "upper", "lower values" are based on the location of the panel or the sizeratio of the values. To avoid this somewhat annoying formal conflict it would be

better in the future, distribution boards with the largest values ofa to have to start,but after I was followed by the previous major part of my studies of the customary

set-up way, I could not change it without my boards rebuilding and running the riskof confusing myself. The dashes above and below the values refer any case to the sizeratio of the values, not their location in relation to the panel.

After this is still the meaning and terminology to discuss the following expressions,which play an essential role in our investigations.

Under Vorzahl, respectively Vorsumme I understand the numberzandsum a ofa, which values a given a go ahead to the board in size, under Nachzahl,

Nachsumme which the a values given a follow blackboard in size. Of course, these

figures and totals change with the values ofa the table, which they precede andfollow, and the prevention of prolixity I introduce here for the cases that it isnecessary to take into account in the applications preferably, a specialdesignations. General like with v , V, n, Nthe Vorzahl, Vorsumme, Nachzahl,

Nachsumme respect to any eligible start a and closing a are referred to a given paneldistribution ofv, V, n , N, the respective values with respect to the a , where thelargestzbelongs, ie the empirical densely valueD , with vi , vi , n i ,N i , with

respect to an a, the radius interval is to interpolate the sharp determination of theelements in later to a decisive way, the way in most cases to the previous, the densest

values coincide, then where can cease the designation by the index. 11 Finally, the following remark. It will be an occasion, an arithmetic and alogarithmic treatment of K.-G. to distinguish which of these items comes in for theformer application, the average deviations with respect to their principal values aresmall, the other for those to where they are relatively large. The former is not only torefer first to this case is far more frequent, and therefore a greater extent than thesecond to be considered, but also easier to be treated case, and all the provisions andterms of this chapter, but would without regard also to the second case of throughoutthe investigation the required generality missing.

The main difference between the two modes of treatment is this:In the arithmetic treatment, the deviations of the individual to be a of their principalvalues in the ordinary sense as arithmetic, ie taken as positive and negativedifferences of its main values and the core values themselves directly to specifiedrules from theA of the distribution panel determined. With the logarithmic treatment,the variations with which it operates have been taken as logarithmic, ie, as differencesof the logarithms ofa so-called logarithmic principal values, ie, the main values to allof the same rules from the log a , as the main arithmetic values simple from theA can

be derived. The transition from arithmetic to logarithmic treatment brings some new

aspects, rules and descriptions, however, on the later to take after will have presentedoccasion to refer to it (see in particular Chapter V ( 36) and XXI) .


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Under is as usual the LUDOLF'sche number = 3.1415927, with e the base ofnatural logarithms, number = 2.7182818, under Mod = log. comm. e understood theso-called logarithmic modulus of the common system = 0.4342945, which is oftendue to the use to be made of it, may be useful to state the common logarithms. Onehas:

log = 0.4971499, log e = 0.4342945, log mod = 0.6377843 - 1

Undert, 't, t, respektiv respektiv are the values:

understood. Undert- table in Appendix A, 183, the following table to which the t-standing respect, to be discussed in Chapter V. values specifies the purposes of theAct GAUSS random variation. since the value exp [- t2 ] 7) is used frequently and

more complicated calculation, as may be specified here, the calculation of itslogarithm, of which he himself is directly derived.

7) [For the sake of simplicity, here and below, the exponential function ex by exp

[x denotes], according to which above exp [- t ] instead ofe t - has been set.]

To log exp [- t] = log 1: exp [ t2 to find], add 2 log tto 0.63778 to 1 (ie log toMod), seeking to logarithms in the number and put it negatively, so you have the

required logarithm is8)

, but in one of the common and divergent for the applicationof logarithms to derive exp [- t] itself from improper form. To get it in to usableform, subtract its absolute value of the higher whole number by 1 and add it to therear differential with the character - too. Thus, when log exp [-t ] = - 0.25 or - 1.25 or- 2.25 would be found, one would have to put it respectively. 0.75 to 1, or 0.75 to 2 or0.75 to 3 usf

8) In fact, the logarithm of exp [ t] is equal to tlogs , hence the Log of l exp [ t]is equal to the negative of logarithm of exp [ t].

UnderEis meant the unit in which the specimen sizes a, the main valuesHanddeviation amounts are expressed thereof.

Instead probability is usually W . ; take collective object, as already noted, K.-G. and instead of Gaussian law for future remark GG set.


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III. Preliminary Overview of the study material and general

observations. 12 A major difficulty for a study such as this lies in the procurement of materialrequired for this purpose. Such may in fact only a plurality of K.-G. be sought fromdifferent areas, each of which is present in such a large number of species thatrandomness of the distribution by size and number nahehin - because it is absolutelynot possible - can be considered as compensated according to the law of largenumbers, and in each of which the to be made following chapter contends that otherlarge props can not be considered fulfilled less than nahehin. Finally, it must containall the information necessary for data processing.

But on some types of K.-G., which should not be ignored to give the necessarygenerality to the investigation, so far nothing was ever before, and if it was not forlack of other information, so for some, like the recruits extent an embarras derichesse exists, but is the same in its present version is not sufficient for all the

purposes of the investigation to be placed on her claims. To your own measurementsbut are only a few items to bid and, since it is to be measured at each very manycopies and bring them into distribution boards, find the time and patience for this, justlangmhigen and lengthy, shops easily their border.

However, it is me but managed to bring on some laborious and cumbersome

processing the folgends recorded material for our study together, which of coursemany of the claims to be made props corresponds only partially so but is also anopportunity to reveal the success of it.

I. Anthropology.

A. recruits dimensions such, ds lengths of even-aged recruits from certain origins,mainly Saxon, which I knew to give me copies of the Urlisten to gain distribution

boards in a form suitable for the investigation of it. Most important for our generalinvestigation in the first part 20 are born in Leipzig student recruits dimensions with a

total m = 2047, soon 17 years crossings so-called Leipzig city size, ie with respect torecruit the rest of the Leipzig population, with a total m = 8402 and also recruitsdimensions of 3 cohorts, respectively. the Borna and Anna BergerAmtshauptmannschaft with m = 2642 and 3067th To be in the second partRekrutenmatafeln mar. other countries, where such existed earlier and are treated byQuetelet, experienced as especially Belgian, French, Italian and American, a partcritical review, QUETELET'schen different part of the treatment, and measurementsof body weight and chest circumference of the recruits are taken into account .

B . Skull dimensions that are put to me by Prof. WELKER Hall up for grabs, a) of

the vertical scope, b) of the horizontal scope of 450 European men's skulls. C. Weight of the internal organs of the human body , according to BODY's


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information 1) .

1) [Dr. Boyd's Tables of the weights of the human body and internal

organs. Philosophical Transactions of the Royal Society of London, 1861].

Botanical II.

Measured by myself rye ( Secale cereale ) from the same locations and yearunderway, 217 six-membered (apart from the Fruchthre) and 138 five-membered;each of the links, particularly measured and partly as a special K.-G. treated, sometaken by its relationship to the other members into consideration.

III. Meteorological.

a) Thermal and barometric daily and monthly values or variations in the detail to bediscussed in 19 and 20 sense. Below are the by Quetelet in hisprob Lettres surla recorded, folgends to be discussed in 21, 10-year so-called ". variations

Diurnes "with a m 282-310, this own compilations thermal and barometric dailyvalues from observations on the Peissenberge by a long series of years, and fromthermal variations month to DOVE'schen treatises.

b) collected daily highs fallen water for Geneva for many years, to the purpose ofthe Bibliothque de Genve (Archives des sciences physiques et naturelles) from me.

IV Artis table.

a) business cards and address cards of merchants and manufacturers, particularlymeasured by myself on the length and width.

b) dimensions, height h and width b , of gallery paintings (specially designed inLichten of the frame) to the catalogs of the collections of reducing to the same unit ofgenre paintings, landscapes, still life of me, taking distinguished the casewhere b > h and where h> b.

This is only for preliminary review; special is going into protruding material underspecific chapters of the second part, where here to's missing further details to find it,and have to point out when to refer already in this first part of this material is .

It may be noted that under the previous objects such exist with which to occupythemselves little or no substantive interest exists. But the point of an objective

interest it is not authoritative been here for their selection and treatment, but ratherbecome important only their usability as a base for our investigation, in which respect


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some insignificant translucent objects than as the dimensions of the gallery paintingsand the daily rainfall heights are.

In that regard, but was present a substantive interest in the goods, one must for thesame reason, do not expect the treatment of the same tired to see this interest here,even if so many results that will fall into the same into contact, by itself as byproducts

of treatment. Each of these objects could give a monographic treatment occasion, buthow large plant would only require the recruits measurements, a comparative

presentation and discussion should be the same for the different countries and in thesame countries for the different vintages or such for cranial dimensions of the various

breeds or to be performed for the structure ratios of the various grasses! Atpenetrations of this kind is not to think here. Other hand, does what explained here onexamples from different areas and it is proved, however, claim to find in any broader

treatment of the same areas of application and consideration. 2)

2 )

[Note: The information in this chapter should be added that a partial newprocurement of test material was necessary, because except for a fraction ofconscripts dimensions and the dimensions of stalks of rye from any of the designatedK.-G. Urlisten or primary distribution boards were present kept. Although the studymaterial was, as far as was practicable, supplemented from the specified sources;

particular were true for gallery painting the catalogs of the Alte Pinakothek in Munichand the picture gallery to Darmstadt, for the daily rainfall heights of Geneva theArchives des sciences physiques et naturelles the Bibliothque universal (see ChapterXXI, and taken XXVI and XXVII). , but instead of the observations of thermal and

barometric daily values Peissenberge served on the appropriate values that arepublished in Utrecht in the Netherlands Year Book of Meteorology (see ChapterXXIII and XXVII). The replacement for the skull dimensions eventually (see ChapterVII and XXII) I am indebted to Prof. WELCKER, who had the kindness to send methe dimensions of around 500 European men's skulls.]

IV props; abnormalities. 13 If a K.-G. allow a successful investigation, it must meet certain conditions,some of which are in his terms, partly to submit to more general considerations.

After the introductory statement is sent ahead one K.-G. one under a certainconcept more tangible, to be its quantitative determination by Random fluctuatingobject of indeterminate number of copies. Now let infinitely many copies does nothave of him, but one must besprochenermaen as many of him looking to get, somany that the strictly taken to be taken only for an infinite number to complete, ideallaws of chance nor a target for the degree accuracy sufficient approximation can beconfirmed. But this condition is sufficiently fulfilled, they must K.-G. still be normalor error from other points of view, as we like to express ourselves short to blend inwith the statutory provisions as to the most common for K.-G. be set up, which arenot subject to these errors.

Among them is that the specimens from any other considerations to K.-G. taken


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together, yet such are excluded, as justified in terms of the subject matter is, that sothe object vielzahlig not only from previous volume aspects, but also in thatvollzahlig was, as all within the limits of its concept that present themselves from himcopies are also really counted with , not from this or that side regardless'm one or theother part of the measuring scale elimination, with this, the object will be mutilated

so to speak, as it would for example be the case if the so-called sub-moderatingshould be excluded in Rekrutenmatafeln, however, to hand the object must beobtained as pure and unmixed, ie specimens that come out of his terms in a randomdirection, must be excluded from it, eg, where the collective term goes to healthyindividuals, specimens with pathologically altered dimensions must come inelimination; therefore to be treated by me WELCKER'schen skull dimensions neither

barrel shaped swollen Hydrocephale yet decided microcephalic skull withenter. Remarks to make it but are of general scope.

14 It is certain that the line between healthy and diseased skulls is not known

with certainty, and a corresponding uncertainty about the definition of the objectreturns in many other cases again, but if only the uncertainty remains within such anarrow speed limits that the boundaries the uncertainty that one has to put up with

because of unbalanced contingencies are not exceeded, then no significantdisadvantage throughout accrue, and you may find yourself satisfied by the successwhen, accrued at our discretion subject to the normal distribution laws inserts, or youcan cut as many copies that is the case.

However, this raises the following important question: It is of course logically self-evident that when healthy individuals or parts are to be examined by such as skulls, in

the allocation ratios of their copies, not those which are identified as sick or acceptedit, with may be mixed, and no less self-evident that the determination of theconditions for healthy specimens has a greater interest than for a mixture of healthyand sick, only it seems to contradict the generality of the objective of the collectivesto run, to determine the general distribution laws of the K.-G. from only healthyspecimens preferable to the subject of a mixture of healthy with sick.

In fact, if the diseased skull from the concept come out healthy, they still fall underthe definition of skull at all, and what justifies us in exploring the general laws for K.-G. eliminate the diseased skull, as we rather only cause the broader term that includes

all skull would apply instead of the narrower the healthy, yes there is, strictlyspeaking, and there are countless other cases where an equal possibility of narrowerand wider version is such anywhere since last all K.-G. can unite under the terms ofan existing system, which can be narrowed only to different directions. But we wouldwith the attempts of our laws generally issued in the K.-G. very wide versions to

prove, bad driving by itself would not prove or imperfect because, while they yet at asufficiently narrow versions of the most diverse K.-G. remain the same and thus

prove their universality. Now the question is, what aspect is decisive for the limitationof the width to be maintained.

This seemingly difficult question is to be answered with regard to the followingactual conditions.


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If we items that correspond with sufficiently narrow version for themselves thecommon to the various objects distribution laws mix, the following condition must besatisfied if the mixture the same laws yet to meet: by which determines the constantsor essential elements, the distribution ratios are, so at least the arithmetic mean andmean deviation of, which the other elements related more or less, may be used for the

composing objects not differ from each other than by unbalanced randomness isexplained, after which we can distinguish unanimous and disparate objects as such,which satisfy this condition, and what they do not meet, on the other hand consistentand ambiguous than those which made unanimous, and which are composed ofdisparate objects. Any extension of the term of a K.-G. However, a compositionthereof with one or more other, possibly disparate objects with it.

From this point of view is now immediately obvious to many items that they cannot be mixed. In fact, there is no one come, men and women, or children and adults inthe same K.-G. to unite, if the distribution of its copies to be considered in terms of

body length, notwithstanding they fall together under the broader term of humanbeings, but you know first of all that there are significantly different mean values forturning them into disparate objects. And it must also have a healthy composition skullwith diseased skulls to a K.-G. be found inadmissible in so far as both disparate

behave towards each other.

15 From this point of view it seems to me very instructive results from studies onthe extent recruits who, after their above (Section III under I. A) is mentionedfleetingly, in the second part of this work (Chapter XXIV) to be notified of incoming.

Recruits dimensions can ever summarized for different countries, times, ages,under the broadest terms of such dimensions, but are also very specialized, and fromthe outset it is, for example, 18 year old recruits a country will not treat mixed with20 years of another country, since both are different dimensions through variousmeans, but also the same age recruits from the same country have specializations indifferent senses. So I have the dimensions of recruits (2ojhrigen), on the other handtreated specially Leipzig students on the one hand and those of the other people inLeipzig, Leipzig city called dimensions. For the first has a very satisfying, for theother one after a certain respect imperfect confirm drawn up general distributionlaws, which I call fundamental, result, by has been shown in comparison between

calculation and observation that occur relatively frequently in the latter, the smalldimensions than it should be the case according to calculation based on thefundamental laws without unbalanced contingencies were sufficient to explain it. Thesame was found for the recruits dimensions of the mixed population of variousdistricts of Saxony greater. What is the difference of the first of the other cases? Therecruits dimensions of the students relate to the limited range of relatively wealthy, anormal Wachstume of individuals means not failing items, and the other toindividuals from a mixture of such objects with booths in which it at the conceptionand birth of more or less lack such means, and abnormally stunted individuals are not

infrequent, whose dimensions are included in the Rekrutenmaliste with, though theindividuals do not set themselves at the service with respect werden.Indieser likelyinterested in the following data.


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In my capacity to ensure 20 years corridors of Leipzig student recruits dimensions

with a total m = 2047 only one individual falls (60 inches) below the level 64 inch 1) ,in 17 vintages of measurements of the other people in Leipzig (short Leipzig citydimensions) with a total m = 8402 drop 197 individuals under 64 inches (the smallestat 48 inches), and we reduce 197 on the ratio of total -m, so fall against one individual

of the Leipzig students dimensions for 48 of the Leipzig city size under 64inches. The Leipzig mixed population includes but how any large city, a large

percentage miserable proletariat. But further 3 vintages recruits dimensions of BornaAmtshauptmannschaft except Leipzig (preferably small towns and farming villagesnestled inclusively) with m = 2642 gave absolutely 50 or, as previously reduced 39degree below 64 inches (with the minimum dimensions 51 inches), and 3 vintagesrecruits Anna Berger Amtshauptmannschaft (much mountainous and poor populationincluding factory) with m = 3067 absolute 62, 41 reduced dimensions under 64 inches(49 inches with the minimum dimensions). So by proportion ofm we have ever

beziehentlich for the specified 4 departments:1 48 39 41

Measurements below 64 2) , and we go to the arithmetic means (according to theprimary panels) over, then the following values are in inches Saxon:

Student Lpzg. M. Borna St. Annaberg

71,76 69,61 69,34 69,00.

So the arithmetic mean of the Leipzig students is more than 2 inches larger than thatof the mixed-Saxon population, and the same is true for median and denselyvalue. On the other hand, the average deviation of the arithmetic mean with respect toa uniform manner for all departments in Saxon inches determination for:

Student Lpzg. M. Borna St. Annaberg

2.01 2.26 2.14 2.33.

And of course, the difference of two relations would be even more if the mixedpopulation of the last three sections divided into those with normal and those withabnormal Wachstume and both could be placed opposite each other.

1) [1 Saxon inch = 23.6 mm.]

2) Less noticeable than with respect to the smallest extent is the difference betweenthe student measures and dimensions of the other three departments regarding thelargest, and also agrees the distribution account for the latter up better than down, butlacks a difference in the largest extent not quite. The students completed measures upwith the three dimensions 80, 80,75, 82,5, and the Leipzig city with dimensions 79.5(4 times) and 79.75, with the Borna 77.25, 77.75, 78.25 and the Annaberg'schen with76.75, 77.25, 78.5.

This is not to say that if we have the recruits of the proletariat really well for


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himself would have before us as that of the wealthy classes to the students, ourfundamental distribution laws are just as good at those as these confirm, because the

proletariat itself another further term, which is capable of specialization in differentdirections, and not a priori is to insure that his specialties are unanimous in the abovesense. Yes, from the outset would be the same to say as little of the represented by the

students affluent classes, but since the experience itself teaches that specialization isdriven far enough into the students measure to allow a confirmation of the laws inquestion in so far as it ever unbalanced due to contingencies is possible, we must alsohelp soothe, while we would like there to drive here the specialization even further ifit was not enough.

Also may very well be conceded that, if we only m increased student recruitsdimensions right, and then there separated according to various criteria, for example,depending on the origin of villages or towns or from different years or different levelsin departments that have a , sufficient m had to discover subtle differences of the

essential elements with certainty, there would be no lack of such, which wouldconflict with a perfect unanimity, and there is nothing to prevent, to make it an objectof investigation.

But if these differences are small, and the various departments that you can do forthe wide range of aspects, herewith the differences between the elements themselves,vary with the nature of randomness, it can be not only reasonably require, but teachesthe fact itself that those differences of elements in the inevitable rise indistinguishableunbalanced contingencies and of probation with the fundamental laws oppose asignificant obstacle.

16 Less so but we must in the variations that the distribution ratios too far calmerand thus ambiguous K.-G. show of the fundamental laws, see a contradiction of theselaws, as it suffices in principle to know the ratios and major elements of composingobjects of an ambiguous object, to calculate the distribution ratios of the compositearticle according to the fundamental laws themselves, so that they thus in this respect,their general validity claim.

Generally follows from the above, in finding and examining the fundamentaldistribution laws that we must not only take care that after various directions apart

retreating distribution results too far calmer, untriftig mixed objects against thegenerality of the for sufficiently narrowly defined, uniform items undrawn law claimsmaking, but also in the choice between the results of a broader and narrower version,under the same circumstances that the preferable closer to the Konstatierung thefundamental laws. The previous considerations in the following order significantly.

The origin of the copies of a K.-G. from different areas or times or both at the sametime easily leads not only qualitative but also quantitative differences of the samewith what a particular attention in this respect deserves as one to a sufficientlylarge m is to achieve a successful examination, usually caused or forced the K.-

G. assemble these specimens, which belong to different rooms or times, quite thesame space and the same time they can not belong. In this regard, a conflict will nowtake place. The specimens from very together increase of one another remote or very


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wide spaces and times, is in danger of uniting disparate objects and hereby to missthe fundamental relations of distribution, and the specimens from too narrow a space-and time limits together increase is the unbalanced coincidences too much room, toderive essential elements at all with any degree of certainty. The mandatory limits inthis respect, but they do not a priori pull, and ultimately, the success has to decide

whether you can take the adopted temporal or spatial distance of the subject to asatisfactory fulfillment of the fundamental distribution laws and if not, thecontraction continued driving, and if you order in too small values ofm comes into itin order to obtain results of sufficient security, to give up the investigation in order toobtain a larger number of copies. In general, this should in any case be the most

practical.

17 A special attention should be paid to the question of whether an object iscomposed of disparate components, following some of them already touchedrelations of distribution boards.

In our fundamental law is justified that thezcontinuously with a up to a certainsize ofa rise, with further increasing a also continuously descend but so that there isa maximum ofzin a central part of the distribution panel (the so-called densely values) and two minima respectively at the beginning and end of the table (at theextreme a ) are. If the a as abscissae, thezas the ordinates takes, you can represent ina known manner by the legal distribution graphically, giving it a curve taken atsmall i rises smoothly to a peak and from there descends again. But in the so-called

by me primary, ie directly derived from the Urlisten the dimensions tables you willinsgemein from entering the beginning of the whole panel, an irregular mounting and

dismounting ofzwith continuous growth ofa find herewith a bumpy texture of thedistribution curve, including the primary distribution tables of Chapter VII providereasonable examples. The most common, never missing cause of such irregularities isnow at least in unbalanced contingencies, and dependent on this cusp of the curvedisappear by a sufficiently broad-driven reduction of the table, ie specified by earlier( 6) a statement Together acquisition ofzheld for equal intervals ofa run throughthe whole panel as described in Chapter VIII and supported by examples of reducedtables. But sometimes the cause can also be that K.-G. have mixed disparate nature oftheir main values is.

In fact, di can already overlooked from general considerations that if we wanted theextent of the same amount of men and women, very different in the arithmetic meanvalue of each test as you mix, for example, so significantly, apart from unbalancedaccidents, a cause for the emergence of two maximum-zthus two closest valueswould arise, yes it could by mixing more disparate objects distribution panels with amuch more maximumzarise. Anyway, now suitable for testing the fundamental lawsof the distribution boards distribution only with a maximumzin the maintenance of astock of the main panel, while small irregularities to the ends of the table are withoutmajor disruption. Therefore are distribution panels, which do not comply with this

condition, they are to check the laws useful only after such reduction, that they byreasonable adjustment of the contingencies of the same match, according to the lawsin question or may contact the reduced panel very well confirm if the majority of the


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maximumzin the main panel really depended on the maintenance of a stock of onlyunbalanced contingencies.

However, it is not to disregard that because their intervals are determined by thereduction of a distribution panel increases, at the same time, related to the unbalancedcontingencies of disparate nature of the components of the board, a majority of the

maximumzcan vanish if this fact on each other neara drop which come together inthe enlarged by reducing the interval are hereby indistinguishable, so we need only tothe reduction and enlargement of the hereby intervals go as far as desired to achievethe safe. Thus, while the rule is that by reducing it to just a maximum in theallocation under test panelzand from there to both sides descending passage ofzbeto reduce, maintain, but any deviation from the fundamental laws will still be of adisparate nature of the components of the panel, which has become blurred by thereduction may depend, therefore, in this respect only the study of the distributionitself can be crucial.

18 However, we are having our props does not end there. Objects, which aredesigned by people with respect to certain purposes or ideas, in short we call themartistic subject, despite the intention that has obgewaltet when they arise, but in termsof size regulations, which still leave blank to chance, the Kollektivmagesetzen;when but secondary considerations or secondary purpose substantially limit thefreedom of chance by preference or exclusion of individual dimensions, so the lawscan also happen much demolition, which is explained by the following examples.

Business cards, as well as the so-called address cards of merchants andmanufacturers can be seen on most manifold on the length as well as width varies,and I thought at first to have an excellent object for examination of our laws is

because they are in large numbers, either from the daily traffic, whether from thepattern books of their maker of which can be found glued sample copies (which Imany have used various Verfertigern to measurements) can be obtained, and thereby

provide the advantage that the accuracy of measurement and estimation more thanmany other objects has in hand. But even though they are, be it on the length, whethermeasured by width, not quite evade our laws, they offer, but only a very imperfect

probation represents the same, which you can find the reason in the followingcircumstances.

With all the variation of their dimensions but the freedom of chance is limited bythe fact that the maker of insgemein such dimensions prefer that permit the sheet ofcardboard from which the cards are cut, possible exploit, ie as completely as possibleto consume, it also may be some particularly popular relationships between latitudeand longitude, in particular 2 : 3 or 3: observed 5 (approximations to the goldenratio), and in fact I was in the measurements of these cards that I made to the pattern

books of a majority of manufacturers, convinced that occur more often in certaindimensions, each of them, as that one could see it as random. The dimensions of the

paintings in the gallery lights of the frame but not subject to the same disadvantageand, after doing a large amount of the same dimensions from the catalogs of thevarious galleries brought together (see Chapter XXVI), an excellent material for the


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logarithmic Magesetze provide probation.

19 In the natural objects on the other hand is one of the requisites imposed by theterm itself that the copies are not in a legal nature are depending on each other, whichemerges from the laws of chance. This point is especially in meteorological K.-G. into consideration. Thermometer and barometer readings and other meteorological

values indicate every place a while in detail by contingencies disturbed, but out-torydecided in mean values be legal on and off already at the Track by the hours in a day,no less by the day or month of year. These so-called periodic meteorological valuesdo not fall within the definition of a K.-G., but only the non-periodic, insofar as theyare considered random changing. In this regard, we will soon be able meteorologicaldaily data, monthly values and annual values, insofar as they deviate from their multi-year funding, and these differences themselves as daily deviations; differ Monthdeviations and annual variations, whereupon something specific will respond as isoften be the cause. return to such. We make the explanation to the thermal values and

deviations, which results in the transfer to other types of meteorological values anddeviations of itself.

Thermal daily values may be determined by any particular date of its annual day,say for example, the first January. Let us take the temperature of the day at a given

place in a given year, just as the thermal value of the first day It was determined fromthe average its 24 hours or the temperature of a, then consistently be retained, certainhour of the day or the average of the maximum and minimum temperature of the dayJanuary. This daily value of 1 January be observed by a number of years in arow. The day after the years randomly changing values represent the copies ofa one

time K. - G. We draw from the arithmetic mean by dividing the sum of the dailyvalues to the same number, which, with the number of years by which it has beenobserved coincident. This means the overall thermal hot daily average of 1 January,and the deviations obtained in different years daily values a of the general dailymeanA then form the individual daily variations, which according to the givennotation with must be described. Provisions may for the 2nd January and everyother anniversary will be available at any particular place of observation.

Rather than for each day of the year but can be obtained even from long-termobservations such provisions for any particular week of the year, for each month ofthe year and for the whole year, then as a weekly values, week variations, monthlydata, monthly deviations, annual values, annual deviations are described. Of these,the thermal month and monthly variations deserve special attention, especially

because many of the provisions in many places is any indication. The thermalmonthly values as a thus obtained, for example, for January (and similarly for everyother month) in particular through a number of years mean temperatures of January,which are the same in the 31 days of winning, and the thermal monthly deviationsfrom the January than in the deviations ofa from the general funds ofa instead ofarithmetic means and deviations, can be, however, other main values and derive

deviations from such values.

Meteorological K.-G. this type are estimated for the study of their general laws at


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all from several points of view, once because of the abundant material, which it is inthe sources of meteorology or can be collected from it, secondly because of thedetailed provisions of the accessible with the meteorological observation means andmethods is, thirdly, because these items have been delivered the only material whichto assess whether temporal K.-G. subject to the same laws as spatial. But they suffer

from the very important disadvantage that, since the m same with the number of yearsby which rich observations, coincides, not easily large m the same, yes nowhere yetsuch is present, as it is for the safety of the resulting would be desirable to be drawn

from results. 3)

3) Among the 70 locations for which dove lists the thermal Month deviations in oneof his essays, it is merely Berlin, where 100 and m is exceeded by the pursuit isthrough 138 years hap-pened, and only Prague and London show m about 90, 94 and92 respectively

20 Now you can, however, a much largerm , obtained from a given number ofyears, as the number of years is in the following way, which is important not todiscard scruples at par.

To start from the specific idea of a QUETELET'schen example (see quete-let'sLettres, last vertical column of the table p. 78), we assume that the temperature of alldays in January as a way between minimum and maximum temperature every day ata certain places (Brussels) was observed by 10 years, we are in the prescribed manner

determining what is believed to be correct, for each of the 31 days of January as K.-G., the first, second, third, etc., a m = 10 is obtained, which is much too little to studythe distribution laws because, here we are with a m = 310 for the whole month ofJanuary as K.-G. obtained if we proceed according to Quetelet's processes in theexamples in question so that we put the 31-day temperatures of January as copies ofthe January daily temperature for the 10 years are 310 copies thereof pull thearithmetic mean by dividing by 310, of which the 310 deviations take and when wewant, the other main values of the deviations from it determine.

Now, however, a light at the outset that, since apart from the random changes in the

temperature of from January 1 to 31 Days grows by law, we hereby obtain acomplica

collective mass education-english-gustav theodor fechner

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