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{"created":"2022-01-31T15:20:37.749661+00:00","id":"lit28768","links":{},"metadata":{"alternative":"Studies from the Yale Psychological Laboratory","contributors":[{"name":"Scripture, Edward W.","role":"author"}],"detailsRefDisplay":"Studies from the Yale Psychological Laboratory 2: 1-39","fulltext":[{"file":"p0001.txt","language":"en","ocr_en":"ON MEAN VALUES FOR DIRECT MEASUREMENTS,\nBY\nE. W. Scripture.\nSources op error.\nIn making measurements we are subject to errors which can be classed as: 1. errors of scale, 2. errors of observation, 3. errors of definition, 4. errors of number, 5. errors of calculation.\n1. Errors of scale. Since the divisions of the standard used in measuring come into consideration, it must be determined on every occasion what aliquot portion of the unit, or what ultimate subunit shall or can be finally considered. Let this sub-unit be r. On the axis of IK (fig. 1) let the value of the quantity measured be\nP\tP\n-* m\tJim*\nPig. 1.\nat P when expressed with infinite accuracy. Let the sub-unit be applied in succession from 0, producing a series of points Pv P\u201e . \u2022\t-, Pm-1, Pm, Pm+i- If P,n 1\u00ae the nearest point to P, the\nnumber mr is noted as the value of P. The measurement thus consists not in giving the absolutely true value of P but in noting the nearest mark of the measuring instrument. The error of scale P\u2014Pm in any particular case cannot be known, as all the points between Q and P will be denoted by the same result. It is usually\nT\tV\nassumed that results between Q=zPm+- and P = Pm\u2014~ are\n2\tz\nequally probable and that in the long run the result Pm = mr will differ but little from P.\nV\tT\nThe assumption of equal probability for an error of + - and \u2014 - is\n2\t2\nnot strictly correct. Particularly wrong is the further assumption\nV\tV\nthat all values between 4- - and \u2014 - are equally probable.\nIf all other sources of error be made negligible, i. e. if the errors of observation and of definition be made sufficiently small, the re-","page":1},{"file":"p0002.txt","language":"en","ocr_en":"2\nE. W Scripture,\nsuiting value P can be determined in each case with an approximation so close that it can be regarded, comparatively, as the true value. The scale-readings for a set of results can then be compared with the \u201c true \u201d values. Thus a series of differences\npm-Pm= F,; P^-Pm= Fs; . . .; P\u00ab-Pm= Vn\nare obtained, which can be called the errors of scale.\nThe real values of the measurements are thus distributed over a region P\u00ab\u2014P\u00ab for which one value Pm is read as a representative. It thus becomes a question how well Pm represents the true values.\nThe influences that affect the error of scale, whereby the actual reading may be too great or too small, may be said to be of two kinds : 1. instrumental, 2. psychological.\nAn example of the former may be seen in all instruments in which the reading depends on a bar dropped into the teeth of a rack, as in the Hipp chronoscope. If the edges of the bar and of the teeth were infinitely sharp, the bar would (other sources of error supposed absent) drop and slide down just as often on one side of a tooth as the other. If, however, the edges are rounded, as they must eventually become by wear, the inertia of the body in motion, whether the bar or the rack, will carry the bar constantly to one side whenever it strikes on the rounded edge. Thus in the Hipp chronoscope the readings are all slightly too large. A reading of mP is supposed to represent all values between + and\twhereas in an 61d\ninstrument rnP may be the reading possibly for {m + \\)a to (to\u2014\u00a3)ff.\nAn example of the latter is found in the familiar case of reading in tenths of the index-unit on a graduated scale, as in getting thousandths of a second from a tuning-fork curve in hundredths. A large portion of the work on the least perceptible difference might be used to determine the law of frequency for the error of scale.\nThe assumption of equal probability for an error of + - and\n2\nr 2\nis not strictly correct, as the law of probability followed by the measurements in general will be followed here also. It has been proven, however, that the error introduced by the assumption is of the second order of smallness as compared with the error of scale.1 2. Errors of observation. In making measurements the error of\n1 Lehmann-Filh\u00e9s, Ueber Ausgleichung abgerundeter Beobachtungen, Astr. Nachr., 1889 CXX 305.\nPizz\u00c8TTi, Sur la th\u00e9orie des observations arrondies, Astr. Nachr, 1890 CXXIV 33.","page":2},{"file":"p0003.txt","language":"en","ocr_en":"On mean values for direct measurements.\n3\nscale can be made so small as to be negligible in comparison with all other sources of error. The measurements xlt xrj, .\t.\t,, xn\ncan be considered as having been recorded with practically perfect accuracy. They will disagree owing to sources of error thus classified by Lambebt:1 inaccuracy of the instruments, lack of care of the observer, dullness of the senses, and circumstances of the observation. The errors of observation resulting herefrom falsify the separate measurements. As these sources of error are made smaller the values of x differ less from one another and we can suppose that they would ultimately tend toward a value Xwhich is usually called the true value of x. If we could know this true value, the separate errors of observation would be given by\nV=x-X; Vr1=x1 \u2014 X; . . .; Vn=xn-X.\nAs we cannot know X we cannot know the values of V. If we take some representative value R, we obtain\nvl=xl\u2014R', v,=xt\u2014R;^. . . .\\vn=xn-R.\nThe best we can do is to determine that the value R shall not differ from X by more than a given amount. This is the problem of adjustment of measurements as it is usually proposed in works on physical measurements.\n3. Errors of definition. When the errors of scale and of observation are made so small as to be entirely negligible, the results xx, .\t.\t., x\u201e will be practically true as far as these errors\nare concerned. They will still not agree owing to the infinite number of factors entering into the definition of the quantity. No matter how carefully we define it, we can never make it absolutely complete.\nLet the quantity measured be the height of the American soldier. The object is already limited to a nation, a sex, a class and a range of ages. Let the unit of scale be 0.001m and the method of measurement be a blunt point descending on an arm supported without shake by a rigid vertical bar. The recorded heights will extend over a range of 0.60m. The height of the American soldier is thus uncertain to that amount.\nLet the quantity be further limited to a given age, say 30 yrs., and a given nativity, say Massachusetts ; this may render the result definite to 0.30m. Let it be further confined to a given company of a given Massachusetts regiment at'a given day. The result might , well be uncertain to 0.151\u201d.\n1 Lambert, Theorie d. Zuverl\u00e4ssigkeit d. Beobachtungen u. Versuche, Beytr\u00e4ge z. Gebr. d. Math., I 424.","page":3},{"file":"p0004.txt","language":"en","ocr_en":"4\nJE. W. Scripture,\nThese limitations are still indefinite. As the age may range over 12 months and the constituency of the company might change, let the measurement be limited to a given individual, and let 10 measurements be made in the manner prescribed. The results will depend on the altitude and inclination of the head. Even if these be defined as the extreme height while standing on the heels, the individual will, owing to practice and to fatigue, never stand twice alike. Let the required height be the maximum attainable under any conditions ; the results will vary unless the - occasion be fixed at a given time. Although the subject has never been investigated, the varying atmospheric conditions during the day undoubtedly affect the activity of the nervous system and thus the tension of the muscles maintaining the upright position. Even when limited, to a definite occasion the results will vary unless the pressure of the point on the skin be defined. Here, however, the limits of accuracy of the proposed method of observation will be reached ; there will probably be no changes comparable to the inaccuracy of the observer\u2019s eye in adjusting the point. A finer method with multiplying levers and air-transmission by Mabey capsules will reveal continual, though minute, fluctuations. The error of definition could in this manner be reduced probably to O.OOOl\"1 by taking the height thus indicated at a given instant. With still finer methods familiar to physicists the instant of time would necessarily be more closely defined. Enough, however, has been said to show that owing to the impossibility of defining with infinite accuracy the quantity measured, the true value of the quantity is to be considered as that which is obtained with an accuracy sufficient for the purpose in hand ; whether it be a general -figure by which to compare the American with the French soldier, by which to compare those from the different states, by which to designate a particular individual, or by which to determine the individual\u2019s change at each instant.\nLet it be required to determine the reaction-time of a given individual on a given occasion to a given stimulus of a given intensity. Let the external conditions be further defined by perfect quietness and perfect darkness ; let the air-supply be of a given quality, the pulse-rate of a given frequency, etc. Let the method of recording be accurate to 1\u00b0, i. e. the error of observation shall be less than that amount. Even under these circumstances the results will vary owing to the continually changing subjective conditions of attention, fatigue, etc., which are still beyond control. It is only in exceptional cases that the disagreement can be reduced below an average","page":4},{"file":"p0005.txt","language":"en","ocr_en":"On mean values for direct measurements.\t5\nof 10ff. As the methods of psychology are perfected we shall be able to define and control each of the subjective conditions influencing the time, until we can define the reaction-time so that it shall not vary to the extent of 1\u00ae under the given conditions.1 * Of course, when this occurs the errors of recording must be proportionately small.\nWhen the error of definition is negligible in comparison with the errors of observation in direct measurements, physicists speak of the \u201c true \u201d value of the quantity which would be attainable with an infinitely accurate method of observation. As physicists are able in many fundamental cases, e. g. length, time, etc., to define more accurately than they can observe, the \u201c true \u201d value in this sense is often sharply distinguished from an average value. There is a \u201c true \u201d value for the height of the barometer for any given minute because we cannot observe finely enough to detect the continuous fluctuation.\nEnough has been said, I think, to indicate that the distinction sometimes made between the average of observations and the mean of a series of quantities3 is really a distinction between the mean of a series of quantities subject to variation in observation and the mean of a series of quantities subject to variation in definition.3\n4.\tErrors of number. The result of a measurement is recorded with a definite number of decimal places, whereas a perfectly true value would require an infinite number. The possible numerical error of any result recorded to a places is 0.5/?, where \u00df is 1 unit in the a place. Owing to the partial cancellation of these errors in combining measurements, the mean numerical error in the a place is 0.25/?,\n5.\tErors of calculation. The errors caused by mistakes in reading scale-numbers, in writing them down and in performing the necessary arithmetical calculations can be treated by the principles of probability.4 For the sake of simplicity they will be left out of consideration here.\n1 Scripture, Accurate work in psychology, Am. Jour. Psych , 1894 VI 421.\n8 Quetelet, Th\u00e9orie des prob., Lettre XI, Bruxelles 1846.\nHerschel, Quetelet on prohabilites, Edinburgh Rev, 1850 CLXXXV 1; Essays, 365 (404), Lond. 1857; also in Quetelet, Physique sociale de l\u2019homme, I 35, Bruxelles 1869.\nJevoxs, Principles of Science, 2. ed , 862, Lond. 1887..\n3\tVenn, Logic of Chance. 2. ed., 96, Load. 1876.\n4\tHofmann, Ermittelung der Tragweite der Neunerprobe bei Kenntniss der subj. Genauigkeit des Rechnenden, Zt. f. Math. u. Phys., 1889 XXXIV 116.\nEmmerich, Zur Neunerprobe, Zt. f. Math. u. Phys., 1889 XXXIV 320.","page":5},{"file":"p0006.txt","language":"en","ocr_en":"6\nE. W. Scripture,\nREPRESENTATIVE VALUES.\nOwing to the various sources of error the measurements as,, xa, .\t. x\u201e will generally disagree. The disagreeing results of\nthe measurements, regarded in themselves, are actual concrete matters of fact. For practical reasons these values are replaced by one value deduced from them on some given principle. The proper selection of the representative value requires a clear conception of the purpose for which it is to be used and of the method of deduction.\nIt is sometimes desirable to choose an extreme value to represent the actual ones, e. g. the highest mast to be allowed for in building a bridge, the smallest difference that might be noticed, the largest variation for a given degree of probability; but for the purpose here under consideration some mean value is desired and to this the discussion will be confined.\nProposed means.\n\u00ab\nMeans have been proposed on four principles: 1. on consideration of the properties of a geometrical or material figure employed in representing the results and their probabilities; 2. on the principles of probability a priori ; 3. on the possible ways of practically calculating means ; 4. on certain relations of the powers of the variations.\n1. General analytical or geometrical deduction. If we denote the number of occurrences of the result xk by mk, where k\u2014\\, 2, . . ., n, then the quotients\nm1\tm\u201e\nm \u2019 m \u2019\t'\t'\t\u2019\u2019 m \u2019\nwhere m = mt + mt + .. . . + m\u201e, will denote the relative frequencies of xn xa, . . ., xn respectively. These frequencies stand in the relation\nm1 :\t: .\t.\t. : mn.\nWhen the individuals are taken at random, as m is made greater, the quotients tend toward definite limits,\n<KX,)> <KX,)> \u2022 \u2022 \u2022> <f>(x\u00bb)>\nknown as the probabilities of xt, x2, . . ., xn respectively.\na. Selection on the basis of y = <j>{x). Let the given values be laid off on the axis of X (fig. 2) and the ordinates y\u201e ya, .\t,\t., ynf","page":6},{"file":"p0007.txt","language":"en","ocr_en":"On mean values for direct measurements.\n1\nbe erected proportionately to <\u00a3(\u00e6j,\t<\u00a3(*\u201e). For con-\ntinuous values,\nV = <KX)\t0)\nwill be the expression for the curve or law of frequency or probability.\nFig. 2.\nThe probability of any value x may be regarded as the probability of a value falling between x and x + dx. This will be represented by the area inclosed by the curve over the base dx with the mean ordinate\tThe total area is\nXn\nX,\nSince the probability of a result outside of the given limits is not 0 but is infinitely small, it is justifiable to write\n_+oo\nW = 14>(x)dx.\t(3)\n\u2014oo\nThe form (2) will, however, be retained here as the extension of the limit\u00bb has occasioned some misunderstanding.1\nSince it is certain that all values are included between \u2014 co and + oo,\nW = 1.\t(4)\nThe value of maximum probability is the abscissa xh of the highest point of the curve. It is found by putting\ngftfo) _ 0\ndx\nand taking that one of the resulting values for which d*cf>(x)/dx'1 is negative.\n1 Cattell, On errors of observation, Am. Jour. Psych., 1893 Y 287.","page":7},{"file":"p0008.txt","language":"en","ocr_en":"8\nJE. W. Scripture,\nThe value of mean area xm is the abscissa whose ordinate divides the area of the curve into two equal parts, and is found from\nXm\tXn\nJ'cj>(x)dx \u2014 J'fi(x)dx = $ W = \u2022\u00a3.\t(6)\nX,\tXm\nThis value of mean area was called by Laplace1 the value of middle probability. For the sake of brevity the name can be shortened into \u201c middle value.\u201d\nThe probabilities <\u00a3(*,), 4>(xJ, .\t.\t.,\tcan be regarded as\nparallel forces acting on the points xlt\tof a straight\nline. The centroid or mean center will be at\n\u201e _ ^4>{x)x f \u201c *Ux)\n(6)\nIf 4>(xt),\t. . ., <(>{xn) be regarded as masses at the points\nof a straight line, the position of the center of gravity will be expressed by the same equation.\nFor continuous values,\nXn\nX?~\n(7)\nor, on account of (2) and (4),\nXn\nX,\nThis equation also represents the abscissa of the center of gravity of an area of uniform density bounded by the axis of JXJ and the curve y =\nThese values are represented in fig. 2. The highest point of the curve is at h ; its ordinate is xh. The ordinate m, xm divides the area of the curve into halves. The mean center or center of gravity is on \u00a3 ; its abscissa is x^.\nb. Selection on the basis of Y=fcj>(x)dx. Starting with x, on the a,xis of X, erect the ordinate\tAt x=xt, erect Y1=<f>(x1) +\n1 Laplace, M\u00e9moire sur la probabilit\u00e9 des causes par les \u00e9v\u00e9nements, Mem. de math, et de phys. par divers sa vans, Acad. Paris, 1774 YI 621 (636).","page":8},{"file":"p0009.txt","language":"en","ocr_en":"9\nOn mean values for direct measurements.\n<\u00a3(*,,) ; at xk, (*=1, 2, . . ., n) erect Yk=<j>(xt) +<\u00df{xj + . . . + <f>(xk). The unit of abscissa is here, as before, dx, and just as in the previous case, these values can be transformed into continuous ones. Thus\nXk\nTk=Jj>(x)dx.\t(9)\nThis is the integral curve for y\u2014<f>(x), and may be plotted or drawn directly from it.1\nThe difference between the ordinates at the beginning and end of the curve will correspond to the total area of the frequency curve. Thus\ny\tXn\nYn- Y=f j>(x)dx= W=l.\nThe value xm whose ordinate Ym is halfway between Y1 and Yn is determined by\nT \u2014 Y = Y \u2014 Y or\nX m\tXn\nJ'$(x)dx=J'<ti(x)dx=)s W =i.\t(10)\nThis value \u2022 xm is evidently the value of middle probability noted above.\n1 Abdank-Abakanowicz, Les integraphes, la courbe int\u00e9grale et ses applications, Paris 1886.","page":9},{"file":"p0010.txt","language":"en","ocr_en":"10\nJE. W. Scripture,\nThe over-quartile x, and the under-quartile are determined by The octiles are determined by\n\n-, 8),\nThese may be used as characteristic values to indicate the form of the curve.1\nThe percentiles\n^=/(C(3^~ Y,)), (\u00ab= 1, 2, . . ., 100)\nhave formed the basis of the method of percentile grades.5 In practice the percentiles generally become vigintiles or deciles,\nGalton, -who first introduced the practical use of this method of considering measurements, treats them in a way which, mathematically stated, is as follows. With any arbitrary unit on the axis of -5T, erect in succession the ordinates\nXl\u2018> -^2--5 *\t*\t\u2022')\t-Xj>\nextending the processes till the value x, has been used as many times as it occurs in the set of results. Likewise let\nY\u2018ml+2\u2014X-i * \u2022\t\u2022\t\u2022>\nwhere xt has occurred m\" times. Repeating this process we have\n1 mt+mit-f-\n\n+\u2122r_l+2---Xr>\n1 ml+mrl-}-\n\n1 Galton, Hereditary Genius, 33, London 1870.\nGalton, Statistics by intercomparison, Phil. Mag., 1875 (4) XLIX 33.\nGaltost, Inquiries into Human Faculty, 51, Hew Tork 1883.\nMcAlister, The law of the geometric mean, Proc. Eoyal Soc. London, 1879 XXIX 367 (374).\nEdgeworth, Problems in probabilities, Phil. Mag., 1886 (5) XXII 371 (374).\n8 Galtoe, Natural Inheritance, London 1889. (I have not seen this work.) Bowditch, Physique of women in Massachusetts, XXI. Ann. Rept. Mass. State Board of Health, 285, Boston 1890.\nBowditch, Growth of children, XXII. Ann. Rept. Mass. State Board of Health, 47 9, Boston 1891.\nSeaver, Manual of Anthropometry, 1. chart, New Haven 1890.\nGeissler, lieber die Vorteile der Berechnung nach perzent\u00fcen Graden, Allg. stat. -Arch., 1891-1892 452;","page":10},{"file":"p0011.txt","language":"en","ocr_en":"On mean values for direct measurements.\n1.1\nT.\nml+mH+\nYmi+m\"+\n\u2014 \u2022Tn j\t\u2022\n\u2022 +\u201d>\u201e_! =X<\n+<\u00bb+\u201e_l+3---X\"\u2019 \u2019\nThus in general Ym is the ordinate for\nmk\n(*=1\u20192\u2019 \u2022 \u2022 - ^\u2022\nm\nBy considering the whole interval covered by X to be m=m' + m\" +\n+ with the sub-intervals\nany point Xk\nm m m \u2019 m\n\u2014* we have for m\n(*= 1, 2, .\n*)\u25a0\nm'\nFor continuous values this evidently becomes of the same form as (9) and would become exactly the same if the 3r-axis had been used in place of the JT-axis. Galton\u2019s elementary illustration1 2 of a group of men placed side by side in order of size with a curve just touching the tops of their heads has led to the use of the axis of Y for the values of xx, xv .\t.\t., xn. As Galton\u2019s method and\nillustration have been widely accepted by statisticians, confusion is introduced by the neglect of mathematical conventions. To the non-mathematical statistician it seems more natural to erect height-ordinates vertically and to imagine a row of men standing on their feet. He should remember, however, that in the simple probability curve the heights were laid off horizontally on the axis of X and that if he wishes to have the height-ordinates vertical they must in both cases be laid off on the axis of Y. Galton\u2019s \u201c ogive \u201d curve is obtained by tracing the integral curve on tissue-paper, looking at it from the back of the paper and turning it through 90\u00b0. But if this is done, the simple probability-curve must be treated in the same way. In respect to the proper choice of axes Shaver\u2019s table, for example, is correct, Bowditch\u2019s tables and curves are not.\n2. The most probable value. If x be taken to represent a set of values xx, . . ., \u00e6\u201e, the differences xk\u2014xP, (&=1, 2, . . ., n), can be considered as errors or detriments.\u2019\nThe antecedent probability of any set of errors is proportional to\n$(*.\u2014*\u25ba) \u00a3(*.\u2014\u00bb(.) \u2022 \u2022 \u2022 Hx\u00ab\u2014xp)-\n1\tG-alton, Statistics by intercomparison, Phil. Mag., 1875 (4) XLIX 33.\n2\tGauss, Theoria combmationis observationum, I, 6.","page":11},{"file":"p0012.txt","language":"en","ocr_en":"12\nE. W. Scripture,\nThat value of xp which renders this product a maximum, is a priori the most probable value. This requires that\n0__J________ d+fa-xj __________1____d<t>(x,-xp)\n<KX* ~xp) d(xp)\td(xp)\n+ _J______ d<t>(xn\u2014xT)\n' <t>{x*\u2014xp) d(xp) \u2019\nwhich is the equation of condition for the determination of xp.1\nThis equation is used in two ways :\t1. to determine the law of\nprobability required for an assumed most probable value ; 2. to determine the most probable value for an assumed law of probability. In either case an arbitrary assumption must be made, as Gauss clearly recognized.2 3 If the average be assumed as the most probable value, then, with the usual additional assumptions,\n,, , h -hV\n<Hv) = -^e\t.\t(11)\nwhere\tvk=xk \u2014 A, (k= 1,2, . . .,*\u00ab)\nIf another mean, e. g. the median or the geometric mean, be assumed as the most probable value, the law of frequency takes a different form.\n. It has been customary to regard the assumption of the arithmetical mean and the exponential law of error therefrom deduced, as practically verified, although theoretically not correct.*\nSince the fundamental supposition of symmetry of the probability-curve is quite unjustifiable for psychological measurements, and since the theory of means has been treated by Gauss independently of any definite law of error, it is justifiable to omit all further consideration of this treatment of the most probable value, although Gauss\u2019s earlier treatment is followed by most of the text-books on the adjustment of measurements.4\n1 Gauss, Theoria motus oorp. coel., II, 3, 171.\n* Gauss, Theoria motus corp, coel., II, 3, 177.\nGauss, Anzeige, Gott. gel. Anz., 1821 Feb. 26; Werke, IV 98.\n3\tBertrand, Calcul des probabilit\u00e9s, 183, Paris 1889.\n4\tEncke, lieber d. Methode d. kleinsten Quadrate, Berliner Astr. Jahrb., 1834 249.\nChauvenet, Manual of Spherical and Practical Astronomy, 469, Phila., 1864.\nHelmert, Die Ausgleichungsrechnung, Leipzig 1872.\n. Meyer, Vorlesungen \u00fc. Wahrscheinlichkeitsrechnung, 243, Leipzig 1879.\nMerriman, Method of Least Squares, New York 1894.\nWeinstein, Handbuch der physikal. Maassbestimmungen, I 54, Berlin 1886.","page":12},{"file":"p0013.txt","language":"en","ocr_en":"On mean values for direct measurements.\n13\n3. Algebraic selection. Means can be classed according to the way in which they are computed. a. Combinatory means.' These are\nFA*)=V\nFfx)-.\nSz\nn\n\n\tXaX\u00df\t\t\nn(n\u2014\t1)\t\t\n/1.2. . .\t\u2022 ^ \\\u00df, \u2022 \u2022\tX X0 .\t, .,p a \u00df\t,\t. X P\n\tn[n\u2014 1) .\t.\t. (n\u2014r)\t\n\t\u2022\t\t\ni?T\u201e(*)=Va!\u2022 \u2022 \u2022 *\u00ab\u2022\nOf these means Only two have come into use, namely, the arith-\nmetic mean\n\u2022 \u2022 \u2022 \u00b1*=A\nand the geometric mean\nFn{n)=yx(x^ .\t. . xn=G.\nb. Power means. These are\n1 Sch EiBNER, Ueber Mittelwerthe, Ber. d. math.-phys. Cl. d. k\u00f6nig. s\u00e4chs. Ges. d. Wisa., 1873 XXV 562.\nFbchnbe, lieber d. Ausgangsw\u00e8rth d. M. Abweichungssumme, Abhandl. d. math.' phys. Cl. d. k\u00f6nig. s\u00e4cha. Ges. d. Wiss., 1878 XI 1(76).","page":13},{"file":"p0014.txt","language":"en","ocr_en":"14\nE. W. Scripture,\nBy analogy we might put\nbut since \u00a3B\u00b0= 1, then p0(x)=\u00b0\\/ 1 = lff=l\u00e6, which represents an indeterminate quantity1 2 between 0 and oo . This can be taken to express the fact that the median M=p0(x) is not dependent on the numerical values of xlf x\u201e .\t. xK.\nThe value *\nP,(x):\nx,+x,+\n+ \u00ab\u201e\nn\nis the arithmetic mean. When x represents the errors from an average, their mean\nP,(x)\n=v-\njg,+s, +\n+ \u00ab\u201e\"\nzm\nis the mean-square-error as used in the method of least squares. The quartic mean pfx) has also been used in the calculation of precision.\u2019\n4. Selection to minimize a function of the variations. Means have been so selected as to make\ndR \u2019\nwhere\nvk=xk-R, (*=1, 2, . . ., n). For /(w)=2(u), we have3\n10adcht, Cours d\u2019analyse algebr., 69, Paris 1821.\n2 Gauss, Theoria eombinationis observationum, I, 11.\n8 Boscovich, De littera expedition\u00a9 ad dimetiendos duos meridiani gradus, Romae 1755. (I have not seen this work. It is described in Todhunter, Hist. Theories Attract., I 305, 332, Lond. 1873.)\nBoscovich, De recentissimus graduum dimensionibus, Philosophia recentior a B. Stay, II 420, Romae 1760. (I have not seen this work. It is described in Todhun-ter, Hist. Theories Attract., 1321, Lond. 1873.)\nLaplace, M\u00e9m. sur la prob., M\u00e9m. de math, et de phys. par divers savans, Acad. Paris, 1774 VI 621 (635).\nBernoulli, Milieu, Encyclop\u00e9die m\u00e9thodique, Math., II 404, Paris 1785: Diet, encyel. d, math., Paris 1789.\nLaplace, Sur les degr\u00e9s mesur\u00e9s des m\u00e9ridiens, Hist.'Acad. Sei. Paris 1789, M\u00e9m. de math., 18.\nLaplace, M\u00e9canique celeste, III 40, Paris 1804; Oeuvres, II 144.\nLaplace, Th\u00e9orie anal, des prob., Suppl. 2, \u00a72.\nFechner, lieber d. Ausgangswerth d. kl. Abweichungssumme, Abhandl. d. math.-phys. Cl. d. k\u00f4nig. saehs. Ges. d. Wiss, 1878 XI 1.\nGlaisher, On the law of facility of errors of observations, Mem. Roy. Astr. Soc. Lond., 1872 XXXIX 75 (123).","page":14},{"file":"p0015.txt","language":"en","ocr_en":"On mean values for direct measurements.\n15\n\u00c6g(t>)\ndB '\n:0,\nand\nFory(\u00ab)=2y3, or, what amounts to the same thing, for\n- , l/W\nf{v)=y \u2014 =m,\nfb\nwe have1 *\n\u00df _!gl+g!a+\t\u2022\t\u2022\t\u2022\t+ Xn_^\nn\nAvbeage.\nAverage and centroid. The average is defined as\n^ \u2022 \u2022 \u2022 +\nn\nIf m\u201e mit .\t.\t., m, denote the number of times the values\nxlt x2, .\t. xr occur respectively, then\nA _ !\u00ab, + \u00ab, + \u2022\t\u2022\t\u25a0+ a* _ mx xt+ m2 xa + . . . + m, x,\n\u00bb\t\u00ab,+ \u00bb!,+ .\t.\t. +mf\nSma:\t/\n\u2019\nThe average thus corresponds to the centroid of a system of parallel forces.\nIf the results are so numerous that the values of x can be treated as continuous and cj>(x)dx can be substituted for rn, the average can be substituted, with a small error e, for the abscissa of the centre of gravity. Thus\nxn\n\nwhich in consideration of (2) and (4) becomes\n1 Legendre, \u00eeTouv. m\u00e9thodes pour la d\u00e9termination des orbites des com\u00e8tes, VIII,\nParis 1805.\nMerriman, List of writings relating to the method of least squares, Trans. Conn. Acad., 1877 IV 161.","page":15},{"file":"p0016.txt","language":"en","ocr_en":"16\nE. W. Scripture,\nIf <f>(x) is unknown and the number of results is large, the average A represents the centroid-abscissa x^ with a degree of certainty and within limits determined by Poisson.1 2 * For a large number of results\nA = Xg dt\n2yyA\n~W'\n(12)\nwith a probability of\n7\n\u00aeM= 7? /\u00ab\u201c'<*\u2022\nV 0\nwhere \u00c4 is a quantity derived from the means of the first and second powers of the errors but is not amenable to practical calculation except for known <f>(x).\nVarious other considerations bearing on the relation of the average to the centroid and to the individual results,5 * * 8 although necessary to a just appreciation of these relations, cannot be touched here. The assertion that the use of the average is the mean supposition of all possible suppositions as to the mode of obtaining value,\u201d in addition to its questionable character,4 rests on the assumption of symmetrical probability which cannot be admitted here.\nPrecision of the average. The calculation of the mean variation, the mean-square-error, the probable error, the constant of precision, etc., are to be found in the numerous works on measurement. They generally start with the assumption of a symmetrical curve of probability and pass over assymmetrical curves as being symmetri-\n1\tPoisson, Recherches sur la probabilit\u00e9 des jugements, ch. IV, Paris 1837.\n2\tLagrange, M\u00e9m. sur l\u2019utilit\u00e9 de la m\u00e9thode de prendre le milieu entre les r\u00e9sultats de plusieurs observations, Miscell. Taurinensia, 1770-1773, V (math.) 167; Oeuvres, II\n171.\nEncke, \u00fceber d. Anwendung d. Wahrscheinlichkeits-Rechnung auf Beobachtungen,\nBerliner Astr. Jahrb. f. 1853, 310.\nPizzetti, Sopra una generalizzazione del principo della media aritmetica, Atti d. R.\nAccad. dei Lincei, Rend., 1889 (4) Vi 186.\n8 De Morgan, On the theory of errors of observation, Trans. Oamb. Phil. \u2022 Soc., X 409 (416).\n.4 Glaisher, On the law of facility of errors of observation, Mem. Roy. Astr. Soc. Lond., 1872 XXXIX 75 (90).","page":16},{"file":"p0017.txt","language":"en","ocr_en":"On mean values for direct measurements.\n17\ncal curves with systematic errors. An elementary treatment without this assumption is given by Bertrand.1 2\nNumerical error of the average. The computation of the average involves a decision on the number of decimal places to be retained in the observed values and in the mean after division.\nThe limitation of the number of decimal places used in writing a result introduces an error into each result. Each result differs from the truth by not more than one half-unit in the last place ; its mean error is J of a unit in the last place, or 0.25/3 where \u00df denotes 1 unit in the last place.\nWhen n results are added to form an average, the mean error of\nthe sum will be 0.25\u00df\\/n: The average itself will have a mean 0.25\nerror of \u2014>\u2014 P-y u\nAlthough this supposition is the usual one for numerical work, it is not strictly true. Such a treatment of the mean error is valid only when the law of frequency is expressed by (11). The error thus introduced is, however, negligible.\nSince it is understood that the last decimal place is subject to a mean error of 0.25/3, the extra decimals obtained in calculating the average of 100 results may be retained to one place beyond the original results.\nEven when the number of results is less than 100, the retention of one place further introduces less error than rounding off to a places. Thus for 25 results the mean error of the a place is 0.05/3, and of the a-f 1 place is 0.50/3'. If we round off to the a place on a supposition of a mean error of 0.25/8 in the usual way, the uncertainty of the a place is increased.\nGiven the set of 9 values 213,215,213, 210, 212, 214, 215,210, 212. The mean numerical error of each value is 0.25/3. The mean error of the average will be\n0-258\t* \u201e\n-----=-=0.08/8, or 0.83/3.\nV 9\nThe average is\n1914\n\u2014 =212.666 .............\nwhich is subject to a mean error of 0.08/8 in the first place before the point or 0.8/3' in the first decimal place. If we round off to 213,\n1 Bertrand, Calcul dea probabilit\u00e9s, ch. X, Paris 1889.\n2","page":17},{"file":"p0018.txt","language":"en","ocr_en":"18\nE. W. Scripture,\nwe introduce an uncertainty corresponding to a mean error of 0.25/? in the whole numbers, whereas by retaining the 6, the uncertainty corresponds to only 0.08/3 in the unit-place or to 0.8/3' in the first decimal place, being a gain corresponding to 0.17/8. Rounding off to 212.7 adds an uncertainty of 0.25/3' in the first decimal place, giving a total of 0.83/8' + 0.25/3'=1.08/8' for that place or 0.108/8 for unit-place, being a loss corresponding to 0.028. Since under any circumstances the loss would have to be 0.025/8, the writing of 212.67 has practically no advantage over 212.7.\nThe case is different when the uncertainty of the values is not due simply to the omission of decimal places. Let the measure of the uncertainty of a value x be denoted by \u00b1 A\u00e6. Then the uncertainty of the average of n results will be given by\n\u00b1 A\u00e6\nVn\nThe mean error is the most convenient measure of uncertainty.\nUnder very favorable conditions the record of the Hipp chrono-scope1 * is liable to a mean error of A*=1.5'7. The average of 9 records is reliable to 0.5^. To obtain a result numerically precise to 1\u00b0, i. e. with a mean error of 0.25^, it would be necessary to have 36 original records.\nThus, if a body, e. g. a control-hammer, were known to fall with perfect constancy, 36 records with the chronoscrope would be required to determine its time of fall to lff.\nDependence on characteristic variations. The result of a set of direct measurements is stated to be A\u00b1d, A\u00b1m or A\u00b1r. The quantity d is the mean variation or mean error, m is the mean-square-error and r is the half probable variation or probable error. These characteristic variations are determined from the formulas\nd=\nm-\nr\u20140.674y/t)'\u20191 + w\n>+(\u00ab,)+ \u2022 \u2022 \u2022\t+ (0,\nVw(w\u2014 i)\t, 1\nV+<+\u2022 \u2022\t. +v\u201e3\nn\u20141\t\n/\u00ab,\u2022 + \u00ab,\u25a0+ \u2022 \u2022\t. +\u00fc\u201es\n0.708\n0.708\nn\u2014 1\nV<\nn-\nIn these formulas the signs \u00b1 have the meanings usually given them in works on adjustment.\n1 K/\u00fclpe and Kirscbmann, Ein neuer Apparat zum Contr\u00f4le zeitmessender Instru-\nmente, Phil. Stud., 1892 VII 145.","page":18},{"file":"p0019.txt","language":"en","ocr_en":"On mean values for direct measurements.\t10\nThese results indicate the range within which any other single measurement under the same circumstances may be expected to differ from A with a given degree of probability. Thus, we can wager 1 to 1 that a repetition of the measurement under the same conditions will give a result differing from A by not more than r.\nIn the final statement of the average it is unnecessary and misleading to use more figures than would be justified by the characteristic variations.\nThus if a set of 25 measurements on reaction-time gives an average of 0.2346 sec. with a probable error of 0.012 sec., the second figure, 3, of the- result is uncertain to the extent of more than \u00b1 1 unit. The third figure, 4, of the result is uncertain to the extent of \u00b112 units, and the last figure, 6, to the extent of \u00b1120 units. As figures when rounded-off are understood to be uncertain to the extent of a mean error of \u00b10.25 unit in the last place, the statement that the result is 0.23 sec. is somewhat less reliable than the figures themselves indicate and the statement that the result is 0.235 or 0.2346 is quite misleading. The usual method, whereby the average is given to the last place justified by the compution, while the amount of d, m or r is independently stated, is justifiable or not according to the purpose in hand. When the purpose is simply the determination of an average, there can be no ground for affixing meaningless decimal places ; the average should not be stated further than the first place rendered insecure by the characteristic variation.\n' <\nMedian.\nMedian and middle Value. The median is that value which is obtained by counting off an equal number from each end of the series of results arranged singly according to size. If pv p2, .\t.\t., pn\nrepresent the relative frequencies of \u00e6\u201e \u00e6a, .\t.\t., \u00e6\u201e, then\nM x\u201e\nP = ^P-M\nThe values xlt xt, .\t.\t., xn have all an equal influence in the\ndetermination of xm. Each quantity is either above or below xm ; how far above is not regarded. Those above xm might be called positive results and those below xm might be called negative. Thus we might put","page":19},{"file":"p0020.txt","language":"en","ocr_en":"20\nJE. W. Scripture,\nx\u201e\txm\n2P==r\t2P=ZS\nXm\t\u00bb,\nand\tr + s=[jt\nor\tr=s=\n&\nThe relation between Mand xm can be determined by Bernoulli\u2019s theorem. When /j. is large,\nM=xm\u00b1yr\\/ -\t(18)\n\u00df\nwith a probability of\n7\nP= j\u2014 f?-'2 (It +\nV 7T\u00ab-'\n0\nThe values of y are to be determined from the usual table for\nr\nComputation of the median. The median is defined as that value which occupies the position given by\nX\u00b0 + X\u00b0+ .\t.\t. +\u00a3Cb0 + 1_W + 1\n2 \u2014 2\nin the series of values x\u201e \u00e6a, .\t.\t., xn taken in order of size from\nthe smallest to the largest and from the largest to the smallest.\nLet the number of occurrences of each value of x be denoted by ma, mh, .\t.\t., m,. The series of differing values x\u201e x\u201e .\t.\t., x\u201e\nfinitely expressed, can be regarded as having arisen from the series x,, x,, .\t.\t., x\u201e expressed each to an infinite number of decimals\nby rounding-off all the decimals to the a place. In the a place the. set \u00e6, x\u201e .\t.\t., xa will all agree and can be expressed by m\u201e xa.\nLikewise we have the sets mb xh, .\t.\t., mr xr.\nWhen these sets are arranged in order of size\n\u00bb1, xa, m,, xb.......... m,_, x,_u m, x\u201e m,+1 xw, .\t.\t., m, x,\nthe set containing the median will be mt xt where\n(ma + mb+ .\t.\t. +m,_,) \u2014 (mH.1+ml+i+ .\t.\t. +mr) <m,.\nThe set containing the median is not necessarily the middle set.\nThe median will be one of the mt values which have been rounded\n'f fx. e r/f\u00ef-K","page":20},{"file":"p0021.txt","language":"en","ocr_en":"On mean values for direct measurements.\n21\noff to the same value xx. Each value a:, represents some value x,\u00b10.5\u00df where \u00df indicates 1 unit of the order a.\nWhen\n(m\u201e + mh+ .\t.\t. + m,_,) \u2014 (m;+1 + mt+i + .\t.\t. +mr) = \u2014 c,\nthe median value occupies a place among the m, values given by c where\n,\t. m,\u2014c\tm,\u2014 c .\n+\t. . . + TO;_]) + c -1--\u2014\u2014\u2014\u2014\u2014h (W;+1 + m,+2+\n.\t.\t. + mr).\nThe median is thus not the middle value of the group mx but of the group m,\u2014c.\nIf the whole interval from which the values of xt were derived be denoted by S, then the position of M within the interval S will be given by\nM=xl+^~ S 2 m,\nIf the extreme values for xx be\nXi'=x,\u20140.\u00f6\u00df x\u00dfz=.Xi + 0.5 \u00df\nwhere \u00df is 1 unit of the last place, then\nM=xl+~\u00df 2 mx\nFor the results given on p. 17, 2,10 is the smallest value, 210 the next, 212 the next, etc. As there are 9 values, the median\nwill be the ^^=5^th value from the smallest ; this is 213. The\nlargest value is 215, the next 215, the next 214, etc. The 5th from the largest is 213.\nThe 4th value from the largest is also 213ff. Thus a;i=213<r, m,=2, c= + l and \u00df=\\ff. Consequently\nilf=213'T + il,T=213.25(7.\nThe following example of calculating the median is given by Fechner.1 As a historical interest is present, the rather na\u00efve method of using decimals is left untouched.\n\u201c Take the case where the result runs as follows :\nresult\t1\t2\t3\t4'\t5\nnumber of times it has occurred\t2\t5\t16\t10\t7\n1 Fechner, Ueber d. Ausgangswerth d. kleinsten Abweichmgssumme, Abhandl. d. math.-phys. Cl. d. k. s\u00e4chs. Ges. d. Wiss., 1878 XI 1 (19).","page":21},{"file":"p0022.txt","language":"en","ocr_en":"22\nE. W. Scripture,\nTb 1 1\n\u201c The total number n is here 40, and the-i. e. the 20J. value\ncounted from either left or right end of the series cuts into the number 16; thus the median is to be sought by interpolation in the series of the 16 results giving the value 3, but as the 20th, not as the 20jth. The limits of this series of 3s are 2.5 and 3.5. Counting from the left side we have 7 values up to the limit 2.5 of the series and 13 more are needed to make the 20th value which falls among the 3s ; thus according to the simplest principle of interpolation we have to take from the limit 2.5 upward still |f=0.8125 of this interval in order to reach M, making i!f=2.5 +0.8125=3.3125. Going from the right hand end, we have 17 values up to the limit 3.5 of the series, and lack still 3 of the 20, which fall in the series. Thus we have to subtract from the limit it\u2014 0- 1875 of the interval in order to reach M, which gives 3.5 \u2014 0.1875=3.3125 as before.\u201d\nNumerical error of the median. The mean numerical error of a single value being 0.25/3, that for m, values will be\n~\u00df\u00df.\nV m,\nIn the foregoing example where w\u00ee,=2, the mean error of the result 213ff is 0.18/3 and of the extended result 213.3CT is 1.8/3', where \u00df=la and \u00df'=0.la.\nAs a slightly different example take the values 44, 51, 46, 50, 47, 49, 47, 45, 48, 50. Tbe median will be the ^th or 5|th value.\nThe fifth from the smallest is 47 ; the fifth from the largest is 48 ; the 5|th will lie between the two. As there is no reason to prefer one extreme of the interval 48 \u2014 47 to the other, the simplest method is to take M\u2014 47.5. The numerical uncertainty of 48 is represented by a mean error of 0.25/3 ; that of 47, derived from two results by 0.25/1\n\u2014=\u2014=0.18/3. The mean error of their sum will be\nV 2\t^\t____________\n-^(0.25)\u201c + (0.18)s/3=0.31/8 ;\nand of tbeir average\t\u00b0'3_-/3=0.16/3.\nThe numerical mean error for the median in this example is thus 0.16/8 for the unit-place.\nThe decimal place, already uncertain by a mean error of 0.25/8', becomes uncertain to the extent of a mean error of 1.85/3'. Although 47.5 is less uncertain than 47 or 48, it has nevertheless quite a numerical uncertainty.","page":22},{"file":"p0023.txt","language":"en","ocr_en":"On mean values for direct measurements.\n23\nThe numerical uncertainty of the average of the ten results would\n0 25\u00df\nhe indicated by a mean error of -i-r==0.08\u00df for the unit-place\nV 10\nor 0.8/3' for the first decimal. The median is thus at a disadvantage numerically.\nAs the results gather more around a middle value in more accurate work, more values will coincide with the median, mt will become larger and the numerical error for a given number of results will become less. The numerical error of the average will remain the same.\nAs mentioned on pages 2 and IV the influence of </>(\u00e6) on the numerical error is negligible. Thus in calculating the numerical error of the median, as long as the unit of number does not exceed the size of the mean error, we can safely suppose the values xt to have arisen by rounding-off equally frequent values throughout \u00df.\nDependence of accuracy on the number of results. Gatjss has deduced an expression for the accuracy of the mean value of any power of a variation as depending on the number of variations.1 With slight changes the results can be stated as follows on the supposition of (11). Let xk be the mean as determined from the k\npowers of the n observations. Then with not too small numbers of results the probable uncertainty, in the same sense as the probable\n0.VS2 ;\nerror, for \u00e60=Af determined from xf xf .\t.\t., x\u201e is \u00b1\t----\u2022\n\\tn\u20141\nfor x,=A determined from x1, x1, .\t.\t., xn' it is \u00b1----:.\n\\/n \u2014 1\nOther things being equal, it is necessary to take 249 observations to gain the same accuracy for the median as is given by 114 observations for the average.\nDependence on characteristic variations. As noted on p. ? the median is that representative value which corresponds to a minimum for the sum of the absolute values of the first powers of the variations. The mean variation from the median will bear to the median a relation similar to that which the mean-square-error bears to the arithmetic mean. The median will thus be stated as\nM\u00b1a, M\u00b1l or M\u00b1s, where\n1 Gauss, Bestimmung d. Genauigkeit d. Beobachtungen, Zt. f. Astr., 1816 I 185; Werke, IY 109.\nLipschitz, Sur la combinaison des observations, C. R. Acad. Paris, 1890 CXI 163.","page":23},{"file":"p0024.txt","language":"en","ocr_en":"24\nJE. W. Scripture,\n0.756 \\\t\nV\u00ab(\u00bb\u2014i)V\t'fn \u2014 1/\n'\tn \u2014 1\t'\tf\t0.708 k 'fn\u20141\ns = 0.674 l.\nThe number of significant figures to be retained in stating the median is regulated in the same way as for the average. For the example given on p. 17, the median is calculated to be 213.3^ ; the mean variation is l.e17. The last figure is uncertain by at least 1.6ff and cannot justifiably be used for final statement. Even the last figure in 213er is slightly uncertain.\nIn Fechner\u2019s example on p. 21 the mean variation is 1.14. This makes even the whole number 3 rather uncertain and for final statement renders utterly valueless the four-place decimal.\nIn the example given on p. 22, the mean variation of the individual results from the median is 2.0. Whence it follows that for the mere statement of a representative result, the decimal place is totally worthless and even the unit-place is unreliable. The mean variation from the average 47.7 is also 2.0. The conclusion is the same. Thus the average in a case of this kind has no advantage whatever.\nAs the mean variation becomes less, owing to better agreement of the results, the numerical mean error of the median will also decrease, whereas that of the average will remain the same. Thus the numerical advantage of the average is valueless, for a final statement, when the mean variation is large, and this advantage itself is lost as the mean variation decreases.\nThe labor of obtaining the extra decimal is thus not justified when the results disagree to such an extent. By a preliminary estimate or computation or by a cursory glance at the values themselves it is generally possible to determine the number of places required and thus to adjust the amount of the labor to the worth of the result.\nIn the consideration of the characteristic variations both for the median and for the average, I have, for the sake of using Gauss\u2019s deductions, made the supposition that the law of error is\n<\u00a3(u)=\u2014e\t.\t(11)\nV 7r\nLabor of computation. There is one important property of the median which can be understood only after practical acquaintance with it, namely, its economy. Suppose the original results of the","page":24},{"file":"p0025.txt","language":"en","ocr_en":"On mean values for direct measurements.\n25\nexample already used to be set down in a line or a column: 213, 215, 213, 210, 212, 214, 215, 210, 212. Let the higher numbers be marked off successively by a small check, thus 2\u00cf5, till 5 have been checked. This is the median. After a little practice this can be done for 10 or 25 values with unexpected rapidity. The result is rapidly verified by checking off the numbers from the smallest upward. The determination of the average requires the addition of 9 figures and the division of the result by 9. Even in this example where the first two figures can be neglected and the whole work can be done mentally, yet the time required is much longer.\nIn all examples mentioned, owing to the size of the mean variation there would have been no gain in computing the average, whereas the additional labor would have been a decided loss. When it is remembered that saving in labor in computation means additional opportunity for obtaining results, it is justifiable to claim that the most economical method of computation should be employed.\nIn the typical examples given the additional uncertainty of the median is negligible in comparison with the characteristic variations. When this is not the case, it is a question whether to obtain double the number of results for the median or to perform Jhe additional computation required by the average, in order to obtain the mean with a given precision.\t.\nMedian error. On the assumption of (11) the probable error should be nearly the same whether determined by Bessel\u2019s formula\nor by Peters\u2019s formula\nft>^ + K)+ \u2022 \u2022' \u2022 + (\u00ab\u00bb)\nVn (\u00ab\u20141)\n\u00bb\u2022\u201e = 0.845\nor by counting off in order of size till the middle error is reached. The probable error is thus in the last case the median error. In the same way that the mean error for the median corresponds to the mean-square-error for the average, so the median error for the median would correspond to the mean error for the average. In a similar manner the median error for the median can be compared with the probable error for the median, in order to test the validity of the assumption mentioned.\nOther discussions on the median. In addition to the productions cited elsewhere, several other articles containing accounts of or refer-","page":25},{"file":"p0026.txt","language":"en","ocr_en":"26\nE. W. Scripture,\nences to the median have been consulted. Those whose titles I have noted down are by Cournot,1 Edgeworth,2 Turner,3 Scripture,4 * Venn.6\nWeight and influence.\nIn forming a direct average each measurement in a given set of n measurements has an influence of\nJl~n \u2019 Jt~n \u2019 \u2022\non the result. The influence of a quantity is thus directly proportional to its numerical value. The numerical values as,, sea, . . ., x, can thus be called the relative influences of the 1, 2, .\t.\t., nth\nmeasurements.\nIn combining results from different sets of measurements it is not always desirable for them to influence the average in direct proportion to their face-values. This has led to the use of a system of multipliers, called weights, by which the influence of a quantity is modified. The various results are each multiplied by a coeflicient p\u201e\tpn such that Sjo=n. The average of the weighted\nresults will be the \u201c weighted mean,\u201d\nA\t/u\\\n?\ti\\+r\\+ \u25a0 \u2022\t\u2022 +Pn\ty \u2019\nThe fact that in concrete cases %p is not equal to n arises from a tacit division of both numerator in (14) by the same number.\nThe weighted mean agrees with the direct mean only when\nP,=Pf= \u25a0\t\u25a0 \u25a0 =Pn- .\t(16)\nIt is a very natural step to apply this concept to individual measurements ; some measurements are naturally better than others. But when all measurements have been made apparently with equal care and there is no reason to prefer one to another, it may be said that there is no a priori reason for weighting one different from\n1 Cournot, Exposition de la th\u00e9orie des chances et des probabilit\u00e9s, 120, Paris 1843.\n5\tEdgeworth, On discordant observations, Phil. Mag., 1887 (5) XXIII 364. Edgeworth, New method of reducing observations, Phil. Mag., 1887 (5) XXIV 222. Edgeworth, Empirical proof of the law of error, Phil Mag, 1887 (5) XXIV 330. Edgeworth, Discussion on Dr. Venn\u2019s paper, Jour. Roy. Statist. Soc. Lond., 1891\nLIV 453.\n8 Turner, On Mr.,Edgeworth\u2019s method of reducing observations, Phil. Mag., 1887 (5) XXIV 466.\n4 Scripture, On the adjustment of simple psychological measurements, Psych. Rev.,\n1894 I 281.\n6\tVenn, On averages, Jour. Roy. Statist. Soc. Lond , 1891 LIV 429.","page":26},{"file":"p0027.txt","language":"en","ocr_en":"On mean values for direct measurements.\n27\nanother. The simplest assumption is the purely arbitrary one that all are of equal weight.\nThis gives a very good result if the values of x run along very regularly and close together. No dissatisfaction is felt as long as the individual variations fall within the limits.\n\u2014I ( Vk < +1\nor1\nvk<(l)\nwhere\nvk=xk-A, (*=1,2,.\t.\t.,}>),\t(16)\nI\tnot being considered a large quantity. But if some very large value xT occurs so that\n(\u00ae0> (l),\nthe natural supposition is that xr is not so reliable as the other values of x. Sometimes it Is rejected, i. e. the weight pt\u20140 is attached to it while pY\u2014pt= . . .\t. . . \u2014Pn= 1. Sometimes it receives a weight\tor pr=$, while the others receive p=\\, in a\npurely arbitrary fashion.\nTo know when to reject values it is necessary to assign some value to l. This has led to various criteria for rejection, the best known of which are those of Peikck,2 3 Chauvenet8 and Stone.4 * * *\n1\t(%) is used for abs x, or x taken without regard to sign.\n2\tPeirce, Criterion for the rejection of doubtful observations, Astr. Jour. (Gould), 1852\nII\t161.\nGould, Report . . . containing directions and tables for the use of Peirce\u2019s criterion, U. S. Coast Surv., Kept. 1854, 131*.\nGould, On Peirce\u2019s criterion for the rejection of doubtful observations, with tables, Astr. Jour. (Gould), 1855 IV 81.\nAiry, Letter from . . . [remarks on Peirce\u2019s criterion], Astr. Jour. (Gould), 1856 IV 137.\nWinlock, On Prof. Airy\u2019s objection to Peirce's criterion, Astr. Jour. (Gould), 7856 IV 145.\nNewcomb, A generalized theory of the combination of observations so as to obtain the best results, Am. Jour. Math , 1886 VIII 343. (The note on p. 344 contains two strik-*ng deductions )\n3\tChauvenet, Manual of Spherical and Practical Astronomy, 564, Phila. 1864.\n4\tStone, On the rejection of discordant observations, Month. Not. Roy. Astr. Soc. Lond, 1868 XXVIII 165.\nGlaisher, On the rejection of discordant observations, Month. Not. Roy. Astr. Soc. Lond., 1873 XXXIII 391.\nStone, On the rejection of discordant observations, Month. Not. Roy. Astr. Soc. Lond., 1873 XXXIV 9.\nGlaisher, Note on a paper by Mr. Stone . . . , Month. Not. Roy. Astr. Soc. Lond., 1874 XXXIV 251.\nStone, Note on a discussion . . . , Month. Not. Roy. Astr. Soc: Lond., 1875 XXXV 107.","page":27},{"file":"p0028.txt","language":"en","ocr_en":"28\nJE. W. Scripture,\nThe criteria for rejection have never proven satisfactory. The general sentiment seems to be that the rejection of an honestly made observation simply because it differs largely from the expected value amounts to an attempt to make the work appear more accurate than it is.1 * 3 4 * 6 * The rejection of observations by calculation of the average first with it and then without it, is said to be like what happens in war when two detachments of the same army meet in the dark and fire into each other, each supposing the other to belong to the common enemy.\u201c\nThe very questionable justification for any rejection of results on account of their divergence has led to various systems of weights.\u00ae The main objections to these systems are 1. the assumption of the validity of Simpson\u2019s law* cj>(\u2014vk) = <l>( + vk), where vk is defined as in (16) and <f> indicates the relative nuihber of times of occurrence of vt; 2. the labor of computation which is often out of all proportion to the gain.8\nThe rejection of observations has been a troublesome question for psychologists. The results did not agree and absolutely refused to group themselves around the arithmetical mean. It was not a question of a single result differing from all the rest but of several results tending toward an extreme value. This led to a process of wholesale rejection0 which reached a crisis in giving double sets of results, once\n1 Hall, Orbit of Iapetus, 40, Astr. and Meteor. Obs. for 1882, U. S. Naval Obs,, App. I., Wash. 1885.\nNewcomb, A generalized theory of the combination of observations so as to obtain the best result, Am. Jour. Math., 1886 VIII 343 (345).\nFate, Sur certains points de la th\u00e9orie des erreurs accidentelles, C. R. Acad. Soi. Paris, 1888 CVI 18.3.\n* Doolittle, The rejection of doubtful observations, Wash. Bull. Philos. Soc., 1884 VI 162, in Smithsonian Mise. Coll., 1888 XXXIII.\n3\tDe Morgan, Theory of probability, Enpyc. Metropol., II 456, Lond. 1847.\nGlaisher, On the law of facility of errors of observation and on the method of least\nsquares, Mem. Roy. Astr. Soc. Lond., 1871 XXXIX Pt. I. 75 (103)-\nNewcomb, A generalized theory of the combination of observations so as to obtain the best result, Am. Jour. Math., 1886 VIII 343.\nSmith, True average of observations, Nature, 1888 XXXVII 464.\n4\tSimpson. An attempt to show the advantage arising by taking the mean, Mise. Tracts,\n64, Lond. 1757.\n6 Edgeworth, Choice of means, Phil. Mag., 1887 (5) XXIV 268 (271).\n6 Exner, Exper. Untersuch, d. einfachsten psych. Processe, Arch. f. d. ges. Physiol. (Pfl\u00fcger), 1873 VII 601 (613).\nv. Kries and Auerbach, Die Zeitdauer einfachster psychischer Vorg\u00e4nge, Arch. f. Physiol. (Du Bois-Reymond), 1877 297 (307).","page":28},{"file":"p0029.txt","language":"en","ocr_en":"On mean values for direct measurements.\t29\nwithout rejection and then with rejection,1 * * and found its reductio ad adsurdum in making experiments in sets of 25 of which 20 were selected for the calculation of the average.\u201c\nAll this has made it evident that, even if the arithmetic mean of the observations is justifiable in the physical sciences, it is not a priori the best representative value for psychological measurements.\nThe trouble lies in the fact that, because the average is the most plausible representative for certain kinds of differing quantities, it has been treated as the best one in all cases.8 In the use of the average each individual quantity influences the result in direct proportion to its numerical value. The value \u00e6; = a contributes to the\nresult ^ times as much as the value xk=b. This is unquestionably\nthe correct method to pursue when the individuality of the quantity is of no account. If r cubic meters of soil must be removed for a railroad-cut, it makes little difference just how much each particular car of a train carries, provided the average is satisfactory. Each car counts not merely as an overloaded or an underloaded car, but as overloaded or underloaded to a definite extent.\nThis is not the case in most measurements. The measurements tend to group themselves around some mean value, and when a widely different value occurs it is looked upon with distrust. To use it in an average is to make it count, not as one single value above or below the mean, but as an individual counting for every unit of divergence. The more it differs, the more it counts.4 * For example, in a set of values 3, 4, 2, 2, 3, 10, it is evident from mere inspection that the grouping is around 3. The average is 4, because the one extreme value 10 contributed to the formation of the average as much as the four values 3, 2, 2, 3 put together. If we had 15 instead of 10, the influence of this extreme value would have been more than that of all the rest. In the particular case of measurements the more a value differs from the rest, the less we think of it ; if it\n1 Beroer, Ueber d. Einfluss der Reizst\u00e4rke auf d. Dauer einfacher psych. Vorg\u00e4nge, Phil. Stud., 1886 III 38 (61).\nCattell, Psychometrische Untersuchungen, Phil. Stud., 1886 III 305 (317).\ns Jastrow, Studies from the Univ. of Wisconsin. Am. Jour. Psych., 1892 IV 382 (413).\n8 De Morgan, On the theory of errors of observation, Trans. Camb. Phil. Soc., X 409 (416).\nVenn, On averages, Jour. Roy. Statist. Soc. Lond., 1891 LIV 429.\n4 Bowditch, Note to Laplace\u2019s M\u00e9canique celeste, translated, vol. II, 434, Boston\n1832.","page":29},{"file":"p0030.txt","language":"en","ocr_en":"30\tE. W. Scripture,\ndiffers too much, we think so little of it that we are tempted to throw it out altogether.\nInstead of allowing the individual result to have an influence proportional to its value, why not let it enter into the formation of the mean as one individual differing from the mean regardless of the amount of the difference ?\nThus, if the graduating class in college happens to contain one very tall man, the average height will be greater than the averages for other years. The tall man contributes to the representative value not simply as one man but with the influence of several men.\nIt would seem more natural that each individual should influence the representative value merely as one individual. If all the men were placed in order of height, the most natural representative would be the man in the middle, or the man who would be determined by counting off an equal number of individuals from both extremes. This is equivalent to determining M by the formula on p. 20; and the height of this middle man is the median height. If there happens to be a very tall man among them, a few millimeters more or less in his height will make a difference in the average, but as long as he remains taller than the middle man the median will remain the same.\nLet the set of results be indicated by xx\u00b0, . . ., xn\u00b0, where these letters are used merely as symbols for single quantities. Thus, if the measurements are the heights of a set of individuals, x will designate a certain individual ; if they are observations, each will designate an observation. In taking the average, we influence each individual quantity with a number indicating its numerical value. Thus\nS\u00e6\u00e6\u00b0 jx,x\u00b0 -\\-xp:^ + .\t.\t. + x\u201ex\u201e\u00b0^\nn ~\t11\nBut let a tall man count as only one tall man regardless of just how tall he is, provided he is above the mean, and let a short man count likewise as one man. Since tall and short are only relative terms, there must be some value above which all the men are to be called tall and below which they are to be called short. This is the value that we have called the middle value. Each individual counts as one unit. In general terms,\nx\u00b0 + x\u00b0+ . . . +\u00e6\u00bb\u00b0 _\t_ M 0>\nn\tn\nEach value thus has the same influence.\nThe influence of an individual measurement can be defined as its relative effect in the formation of the mean; its weight is an arbi-","page":30},{"file":"p0031.txt","language":"en","ocr_en":"On mean values for direct measurements.\n31\ntrary multiplier prefixed to its numerical value. The influence of a measurement in taking an average is thus the product of its numerical value by its weight.\nThe general formula (14) for the weighted mean becomes in the notation just used,\nA Pix>x\u00b0 +Pix*x* + \u2022\t:\t\u2022 +P*X\u00abXn\nP,+Pl + \u2022 \u2022 \u2022 +Pn\nThe supposition of\nPx=Pt= \u2022 \u2022 \u2022 =Pn=1\t\u2018 (17)\nhas, as just noted, not proved satisfactory, extreme values being less trustworthy than moderate ones.\nIt has been proposed, as an assumption more satisfactory than (17), that each quantity be weighted inversely as its numerical difference from the average, whereby a corrected average will be obtained.1 2 Let\nvk=xk\u2014A, {k= 1, 2, . . ., n), then take \u2014 as the weight of xh whereby\nAc-\n1\t1\n-X.+\u2014 x,+ V.\tv\u201e\n\n1 1 ---1-----h\nV, V\u201e\n+ -\nSince the average differs from the centroid on account of n<' co, this corrected average A-, may be treated to a still further correction in the same way. This process, when repeated tends to one of the given measurements as the mean.3\nWhat is here approached in a way so awkward as to preclude any but a theoretical interest, can be very simply stated.\nLet each quantity have a weight inversely proportional to its difference from the middle value. The centroid of a series of observations thus weighted will coincide with the middle value within the limits of error necessitated by n<f co.\n1\t-------, Dissertation sur la recherche du milieu le plus probable, Annales de math.\n(Gergonne), 1821 XII 181.\nDe Morgan, Theory of prob., Encycl. Metropol., II 440, Lond. 1847.\nGlaisher. On the rejection of discordant observations, Month. Hot. Boy. Astr. Soc. lond., 1873 XXXIII 391.\nStone, On the rejection of discordant observations, Month. Not. Roy. Astr. Soc. Lond., 1873 XXXIY 9.\n2\tGergonne, Note, Annales de math. (Gergonne), 1821 XII 204.","page":31},{"file":"p0032.txt","language":"en","ocr_en":"32\nJE. W. Scripture,\nThe case will be represented by a rod without weight having on it a number of particles x\u201e x\u201e . . ., xn with masses inversely proportional to their distance from the point of suspension. The distance of any particle from the point of suspension a is x\u2014a-, its\nweight is ~~c^ its moment is 1. The point on which the rod will\nbalance is thus determined by the condition that there shall be an equal number of particles on each side.1 The center of gravity of this system of particles will thus be its middle particle for an uneven number, or a point between the two middle ones for an even number.\n----*-----\u00ab--------------x-#-+\n7 6\tH\t3 Z\tt 2\nFig. 1\nFor a continuous mass governed by the same law, the centre of gravity x^ and the middle value xm coincide.\nFrom these considerations regarding influence and weight the following conclusions can be drawn:\nThe median represents the series of quantities in such a way that each quantity has an influence of unity.\nThe average represents the series of quantities in such a way that each quantity has an influence directly proportional to its numerical v alue.\nThe median is equal to the weighted mean where the weights are inversely proportional to the differences of the values from the mean.\nThe average is equal to the weighted mean where the weights are equal.\nChoice of means.\nThe selection of representative values may take place under three different conditions: I. the results may be so numerous that the empirical frequency curve can be used as the probability curve;\nII.\tthe results are few but the form of <f>(x) has been determined;\nIII.\tthe results are few and <f>(x) is unknown.\nI. Numerous results. The fundamental difference between statistics and ordinary measurements lies in the number of measurements executed. Although the passage from one class to another is grad-\n1 Wilson, Note on a special case of the most probable result of a number of observations, Month. Not. Roy. Astr. Soc. Lond., 1878 XXXYIII 81.","page":32},{"file":"p0033.txt","language":"en","ocr_en":"On mean values for direct measurements.\n33\nual, we can confine ourselves here to the extreme cases. When the results are so numerous that the 'curve of frequency can be plotted and can be regarded as identical with the curve of probability, every value for x and <f>(x) is assumed as known. The representative values can be calculated from the formulas : for the centroid\n2<\u00a3(x) x\nX^^\\xY\nfor the median\nand for xh\nxm=M\nd<f>(x)\ndx\nxm\u2014M\n=0,\nwhere dt\u2018<f>(x)/dx'\u2018 is \u2014.\nIt is to be noted that there is often more than one value for xh.\nThe allowable difference between the actual frequency \u2014 and the probability <j>(x)dx for each value of x can be determined by Bernoulli\u2019s theorem.\nWhen the object of the statistical measurements is merely the determination of the objects measured under a single set of conditions, the result is generally presented in the form of a curve of frequency; thus, all values and their weights being given, there is no need of a selection of any representative value. In scientific work, however, the purpose is generally to determine the change in the results as dependent on a change of conditions ; and the observed value is treated as a function of one of the conditions. Given x=f(z) to determine x for each value of z where the measurements for each value of x are numerous. Such an example would be furnished by measuring 10 000 persons each year to determine the law of dependence of height on age. Although the curve of frequency of heights at each year could be made out, still, aside from the impracticability of so much labor in most cases, it would be impossible to give any intelligible expression to the law of relation. Some representative value or values must be picked out for each step of the change. Owing to the fact that Xp xm and xh almost never coincide, it is very desirable that all three shall be given. There will thus be three curves,\nxS=A(z),\n*\u25a0\u00bb=/, 0),\n3","page":33},{"file":"p0034.txt","language":"en","ocr_en":"34\njE TF. Scripture,\nThe changes in relative position of these values indicate changes in the character of the quantity measured.1 2,\nII. Results not numerous but <f>(x) known. The law of frequency can be considered to be known :\t1. when numerous results have\nbeen taken on previous occasions under the same circumstances whereby the law of frequency has been determined with the requisite accuracy ; 2. when a knowledge of the circumstances indicates what the law must be.\nIn this connection it may be well to call attention to the fact that the statement of the law of error as\n^=v^e~\"\to1)\nrests (a) on the assumption that the average is the most probable value, (5) on the assumption that experience has shown such a law to be true, (c) on the fact that it is the limiting form for combinations of symmetrical frequency curves, or .(d) on the simplicity of treatment thereby rendered possible. I have already pointed out that Gauss clearly recognized and distinctly stated1 (a) as an assumption, and that long experience has shown (b) not to be strictly justifiable. This law of error has done probably better service in astronomy than any other could have done and long familiarity with both assumptions has made them appear almost as axioms. Weinstein,3 who is careful to call attention to the fact that the assumption (a) is not an axiom, is mistaken in asserting that Gauss regarded it as such. Weinstein is also mistaken in supposing that Schiaparelli4 * attempted to prove analytically that the average is the most probable result. Schiaparelli showed that, under certain assumptions the average is the most, plausible mean, and stated that it becomes the most probable mean only when (11) is the law of error.\nAccording to Fekrero6 the utmost defence of the use of the arithmetic mean for all cases is that, when the observations are closely grouped, no mean will differ much from the arithmetic mean.\nApparently still more axiomatic is the law <#>(\u2014'\u25a0p) = <j>( + v). By\n1\tBowditch, Growth of children, 2XII Annual Rept. Mass. State Board of Health, 479 (496), Boston 1\u00d691.\n2\tGauss, Theoria motus corp. coel., II, 3, 177.\n8 Weinstein, Physikalische Maassbestimmungen, I 46, Berlin 1886.\n4 Schiaparelli, Sur le principe de la moyenne arithm\u00e9tique, Astr. Nachr., 1876\nLX XXVII 65.\n6 Ferrero, Esposizione del metodo dei mininimi quadrati, Firenze 1876. (I take the statement from a review by Peirce, Am. Jour. Math., 1878 I 69.)","page":34},{"file":"p0035.txt","language":"en","ocr_en":"On mean values for direct measurements.\n35\nmost writers it is so regarded. Nevertheless Gauss makes the statement purely as a hypothesis1 and Laplace gives a special paragraph to the consideration of unsymmetrical facility.2\nEven if not assumed as an axiom the law is almost universally supposed to have been verified by experience. It is not in place here to consider whether it has been verified for astronomical measurements or not.3 It has not been verified for psychological measurements. Since the errors of observation in astronomy are in part due to psychological causes, it seems likely that all astronomical records involving an observer would show some assymmetry.\nThe assumption, without proof, that this law always holds good and that all cases of assymmetry are cases of constant or systematic error, is purely arbitrary.\nIt is thus evident that many of the cases, supposed to belong in this section, really belong to the following one where is unknown.\nAccording as <j>(x) is (A.) symmetrical or (B.) assymmetrical the treatment of the results and the selection of the representative value will be different.\nA. Symmetrical results. When the curve of frequency is symmetrical, the ordinate of middle area and the ordinate of the centroid will be the axis of symmetry.4 * 6 Thus xm=x and M\u2014 A within the allowable limits of error corresponding to the required certainty.\nIf, according to usual experience, the extreme values occur less frequently than those between the extremes, xh will in general be the same as xm and Xc, and will be'represented by M and A. The curve of probability may, however, have several maxima, none of which may fall at xm.h\nIn a general fashion the law of frequency for physical, geodetical and astronomical measurements has been found to resemble (11). This law was not, however, originally established on the basis of experience, but was deduced as a necessary result of the arbitrary assumption that A is the most probable value.8\nAlthough it has been approximately verified on many occasions, a closer examination shows considerable disagreement in the assymmet-\n1\tGauss, Theoria combinations observationum, I, 5.\n2\tLaplace, Th\u00e9orie analyt. d. prob., 3. \u00e9d., 329, Paris 1820.\n3\tDe Forest, On an unsymmetrical probability curve, Analyst 1882 IX 135, 142 ; 1883 X 1, 61 (71).\n4\tDeforest, On unsymmetrical adjustments and their limits, Analyst, 1880 YII 1.\n6 Edgeworth, Observations and statistics, Trans. Camb. Phil. Soc., XIV 138 (161).\n6 Gauss, Theoria motus corp. coel., II, 3, 177.","page":35},{"file":"p0036.txt","language":"en","ocr_en":"36\nE. W. Scripture,\nrical position of xh and in the undue number of extreme values. Newcomb even concludes that cases where it is fully valid are exceptional.1 2 3\nIn any case of symmetry, whether (11) is valid or not, the median and the average will be theoretically the same.\nSince the number of results, is small and since according to the principles of probability it is seldom likely that in a small set of measurements the values will be actually symmetrical, the median and the average will frequently differ within the limits consistent with theoretical symmetry. For facility curves of the ordinary exponential form the average is most advantageous\" as giving a .smaller probable and a smaller huge error;8 for curves very high in the center and widely extended at the extremes the median has the advantage for the same reasons.4 5 In neither case is the advantage a great one ; in fact, when the ordinary law of probability is assumed, it is practically indifferent which is used.6\nIt is noteworthy that De Forest apparently proves that, on the supposition of a symmetrical probability-curve, the influence of the smallness of the number of results renders a symmetrical adjustment of the mean less probable than an unsymmetrical one.6\nB. Assymmetrical results. When the law of frequency is not symmetrical, the vaules xm and can correspond only in those cases where the centre of area falls on the centroid by some peculiar formation of the curve. Such cases, if they ever actually occur, are to be treated as cases of symmetry.\nIn nearly all psychological and statistical measurements the curve of probability is assymmetrical. If the abscissa of maximum ordinate be determined, the values above it will be found to be much more frequent than those below it. That is, xmf>xh. For all assymmetrical curves of this general form, as the assymmetry increases, x^ departs more rapidly than xm from the main mass of results,7 and consequently does not represent them so well.\nThe general expression for the usual cases of assymmetry has been\n1\tNewcomb, A generalized theory of the combination of observations, Am Jour. Math., 1886 VIII 343.\n2\tLaplace, Th\u00e9orie analyt. des prob, 2 Suppl., \u00a7 2.\n3\tMerriman, Method of Least Squares, New York 1894. -\n1 Edgeworth, Observations and statistics, Trans. Camb. Phil. Soc., XIV 138 (161).\n5 Edgeworth, Choice of means, Phil. Mag., 1881 (5) XXIV 268 (210).\n\u00ab De Forest, On an unsymmetrical probability curve, Analyst, 1883 X 61 (14).\n\u2019 De Forest, On an unsymmetrical probability curve, Analyst, 1883 X 67.","page":36},{"file":"p0037.txt","language":"en","ocr_en":"On mean values for direct measurements.\n37\ndeduced by De Forest.1 It includes constants determined from the squares and cubes of the errors. Any method, however, that introduces more calculation than the usual average will be at even greater disadvantage than that value.\nIf Fechner\u2019s law of the estimate of differences of sensation could be relied upon in all cases, the best representative value would unquestionably be the geometric mean. If the geometric mean g be assumed as the most probable value, it is easily shown that\n<\u00a3(x)=Bea (log g) which with the usual assumptions becomes2\nThe general result of experience, however, goes to show that is of a form intermediate between (11) and (19).\nFor cases of assymmetry the most natural representative value to take would be xh. As this is not determinable from few results, some other value must be used. Any other value would be justifiable only from a consideration of the purpose for which it is wanted. There is no reason, as far as I can see, for taking any one of the values around the maximum rather than any other one except in so far as it comes nearer the maximum. If we are to take xm or and not xh, the middle value xm would be the better on account of its nearness to xh.\nIII. Few results and unknown\tSince it is impossible from\nthe few results given to make any deductions concerning cf>(x), it is evident that other things being equal, that value will be preferable for which the fewest assumptions need to be made.\nLet it be assumed that the curve of frequency is symmetrical. Then x^=xm. One .value is as good as the other, for both should be the same. The value of maximum probability xh cannot be directly calculated. For any assumed form of $(\u00ab), a comparison of the mean variation, the mean-square-error and the probable error will\n1\tDeforest, On an unsymmetrical probability curve, Analyst, 1882 IX 135, 161; 1883 X 1, 67.\n2\tMcAlister, On the law of the geometric mean, Quart. Jour. Pure a. Appl. Math , 1880 XVII 175.\nMcAlister, The law of the geometric mean, Proc. Roy. Soc. Lond., 1879 XXIX 367.","page":37},{"file":"p0038.txt","language":"en","ocr_en":"38\tE- W. Scripture,\nshow whether they stand in the proportions required by the assumption for (j>(x).\nSince 4>(x) is in general unknown, or only roughly suspected, it becomes desirable to either occasionally or constantly compare x^ and xm in order to judge of the symmetry or assymmetry of <f>(x).\nFor a symmetrical curve x^=xm. If for a supposed symmetrical curve over x^=x'm the values r', s' and p' have meanings as defined on p. ? and if r, s and p be the corresponding values for xm, it can be expected, according to Bayes\u2019s theorem, with a probability of\nr\nV v **\n0\nthat\n4=-\u00b1y\np p\t' p\n\nor since\nr___ 1\nP ~ 2\nInstead of x^ and xm we have M=xm\u00b1rj and A=x^\u00b1e where rj and c are determined by (12) and (13). It would not be difficult to calculate on general principles the limits of difference between M and A within which we could, with a given degree of probability, suppose that the law is symmetrical. For the present I will assume that the desired degree of probability is 50$ = and that the question to be decided is the symmetry or assymmetry of a curve whose equation is given in the case of symmetry by (11).\nIf the curve be symmetrical around the average, the probable error r of the variations\nXi\u2014A,\t(*=1, 2, . . ., n)\nwill be the limit of variation for A. If the curve be symmetrical around the median, M can be considered as representing the average of such a curve; the probable error r' calculated from\nxk\u2014M,\t{k\u20141, 2, . . . n)\nwill give the limit of variation for M. Let the difference between the average and the median be denoted by\nB=A-M.","page":38},{"file":"p0039.txt","language":"en","ocr_en":"On mean values for direct measurements.\n39\nR \"4\u201c R*\nAh long as the limits of e and k' overlap, that is, \u2014-\u2014<fA \u2014 M, the\nR \"j- I?/\npresence of assymmetry cannot he asserted. But when \u2014-\u2014f>A \u2014 M,\nit can be said with a probability of SO $ that the curve is assym-metrical.\nIf M and A indicate symmetry in a large number of cases, either of these values can be used in similar cases for the reason that both are practically the same.\nIf they indicate assymmetry, there are the same reasons for pre-fering Mto A as in the case of known considered above.\nTo Profs. Gibbs, Newton and Elkin of Yale and Prof. Merriman of Lehigh, I am under very great obligations for discussion, criticism and correction. Of course, they are in no wise responsible for my deductions or conclusions, from which each one dissents at some point. Nevertheless, any value this article may have is due to their patient labor with one who is not a mathematician but who is obliged to use mathematical means to solve practical problems.","page":39}],"identifier":"lit28768","issued":"1894","language":"en","pages":"1-39","startpages":"1","title":"On mean values for direct measurements","type":"Journal Article","volume":"2"},"revision":0,"updated":"2022-01-31T15:20:37.749667+00:00"}