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{"created":"2022-01-31T15:26:17.164077+00:00","id":"lit28758","links":{},"metadata":{"alternative":"Studies from the Yale Psychological Laboratory","contributors":[{"name":"Scripture, Edward W.","role":"author"}],"detailsRefDisplay":"Studies from the Yale Psychological Laboratory 8: 109-123","fulltext":[{"file":"p0109.txt","language":"en","ocr_en":"COMPUTATION OF A SET OF SIMPLE DIRECT MEASUREMENTS\nBY\nE. W. Scripture.\nI. Theory of the average and of representative errors.1\nGiven the single measurements xv x2, \u2014, xn where n is a finite number, a good representative value is found by taking the average\nxl + x2 + - + xn\nx = \u2014----------------\nThis average is not the true value but merely a representative of the group of particular values. Thus in measuring the simple reaction time of a given subject at a given time under given external and internal conditions the results will not coincide if the unit of measurement is small enough ; there may be errors in the apparatus, and there are undoubtedly fluctuations in the internal condition (for example, of attention) that cannot be taken into account in the stipulations. Let the apparatus and the external conditions be made constant and correct to less than two-thirds of a thousandth of a second ; the results of the measurements are still irregular and their irregularity can be ascribed entirely to the fluctuation in the internal conditions, that is, to mental sources.\nWhat is the \u201ctrue\u201d reaction time among all the irregular measurements obtained? The problem is like that of determining the \u201ctrue\u201d height of a class of students ; each individual has a particular height which is true for him buta \u201crepresentative\u201d height is wanted for the class.\nThe average is generally assumed as the best representative value. Yet each additional measurement is likely to alter the average ; the average for five measurements will probably differ from the average for ten measurements, etc. The simplest assumption is that the ideal average, or true value, is that for an infinite number of measurements. Thus, if\n1 References, for general works : Weinstein, Handbuch d. physikalischen Maassbe-timmungen, Berlin 1886 (to which the present account owes much); Czuber, Theorie d. Beobachtungsfehler, Leipzig 1891 ; Pizetti, Fondamenti matematici per la critica dei resultati sperimentali, Genova 1892 ; Bertrand, Calcul des Probabilit\u00e9s, Paris 1889; Airy, Theory of Errors of Observation, London 1875 j Merriman, Method of Least Squares, 6 ed., New York 1894.\n*\t109","page":109},{"file":"p0110.txt","language":"en","ocr_en":"I IO\nE. W. Scripture,\nwe could measure the given simple reaction time for the given subject under the given conditions for an infinite number of times, we would have a value that we could well call the \u201c true \u201d value for the given conditions.\nTrue errors.\nLet this true value for the set of measurements xv x2, \u2014, xn be X. The individual measurements would vary from the true value X by the amounts V1 = x1 \u2014 X, V2 \u2014 x2 \u2014 X, \u25a0\u25a0\u25a0, Vn = xn \u2014 X. These are the \u201c true deviations,\u201d or \u201ctrue variations,\u201d or \u201ctrue errors,\u201d of the individual measurements for the true value. The term \u201c error \u201d has in this case no detractory meaning ; it is simply the usual expression for a deviation in such a case and, being almost universally adopted, it is probably best to adopt it in psychology also.\nRepresentative errors.\nIt is nearly always necessary to know the amount of fluctuation of the individual measurements around their representative value. When the true value is used as the representative value, the fluctuations are Vv V2, \u25a0\u25a0\u25a0, V. A good representative of Vv V2, \u2014, Vn would be the average of them all regardless of sign, that is, the average of the amounts by which the single measurements disagree with the true value\u2014not considering kind of disagreement whether positive or negative. Thus\nI?_\\K\\ + \\K\\ + - + \\K\\\nn\nwould be a true average deviation, or true average variation, or true average error, the vertical lines indicating that the Vs are used without the signs + or \u2014. Such a value can be called a representative variation or representative error.\nThe true value of X, however, cannot be known and the average x is usually assumed as the best practicable representative value. The deviations from the average, or the practical errors, are then v, = xl \u2014 x, z> = x2 \u2014 x, \u2014, vn = xu \u2014 x. These practical errors are not the same as the true errors. The average of these practical errors without regard to sign is\nd H + hi + - + N\nn\nThis is what has been improperly called the mean variation, the mean error, the average variation, the average error, or the average deviation. It is, however, merely an approximation to the true average error. If a","page":110},{"file":"p0111.txt","language":"en","ocr_en":"Computation of a set of simple direct measurements.\nhi\ncloser approximation can be found, this rougher one should be rejected in favor of the better. To find this better quantity we must first consider another representative variation.\nMean error.\nAnother representative deviation might be found by taking the square root of the average of the squares of the deviations, or\nThis is known as the \u201cmean error,\u201d or \u201cmean-square error,\u201d or \u201c standard deviation.\u201d The term \u201cmean variation\u201d means this quantity and not the previous one, D, which is termed the \u201caverage variation.\u201d\nSince the true value X cannot be known, the true errors Vv V2, \u2014, Vu cannot be calculated. The practical errors vv v2, \u2014, vn can be used to calculate a value\nh2 --------1- Vn\nm ==\n= \\!\n71\nwhich is only an approximation to the mean error, or mean-square error. This approximation can be improved in the following way :\nThe difference between the average and the true value may be called the \u201c resultant error \u201d ; it will be\nK=x \u2014 X.\nIt is evident that\nVl \u2014 7J1 + X, V2 = v2 + K, \u2022\u2022\u2022, Vn=vn + K.\nSubstitution of these values in the equation\nM2 = V1 + v% + \u2022\u2022\u2022 + vn + 2K(vl + v2 + \u2022\u2022\u2022 + vn) + nK*\nn\nBut the definition of the average gives\nTherefore\nv1 + v2 d--------f- vn = o.","page":111},{"file":"p0112.txt","language":"en","ocr_en":"I 12\nE. W. Scripture,\nThis would give the degree of approximation of \u00ab to M if K could be known.\nWe might have considered the difference between the true and the practical errors as separate quantities ; thus\n= Vx\u2014 vv d2 = V2 \u2014 v2, - , 5n = Vn \u2014 vn.\nThese values of 8 would indicate the precision of the set vv v2,\t, v\u201e\nas compared with the set Vv V2, \u2014 , Vn and their mean error would be a good characteristic error for the purpose. This mean error would be\n-J\n<5,2 + V + - +\nBut K = 8X = 8a = \u2022\u2022\u2022 = 8n and thus \u00ab = K. We now have\nM2 = nil + //,\nwhere \u00df indicates the closeness of approximation of m to M.\nWith an infinite number of measurements x = X ; whereby K = o, and \u00df = o. With only one measurement vx = o and M \u2014 \u00df. Thus \u00df varies between the values o and M. Now it is a well established fact that the measure of uncertainty of a set of measurements varies inversely as the square root of the number of measurements. The precision of M may plausibly be assumed to vary in like manner and the quantity \u00df may well be replaced by a value that depends on n.\nThus we may put\nV n\nwhere C is an arbitrary constant.\nBut for n = i, \u00df = M, therefore C = M and\nFinally\nM\nor\n\nwhence\nV +\t+ \u2022\u2022\u2022 + V,","page":112},{"file":"p0113.txt","language":"en","ocr_en":"Computation of a set of simple direct tneasurements.\n\"3\nThis new value for M gives a true mean error calculated from the practical errors. It is evident that the calculation of m is hardly desirable when Mcan be obtained from the same data by using n \u2014 i instead of n.\nMean error and az>erage error.\nThe relation of M to D in a given system of probabilities is constant ; in the system of probabilities generally assumed for measurements\nM\n\u2014 =1.25331.\nM\nSince the relation \u2014 is a constant one, it can at once be concluded that\nD = H + N + - + H\n\u2022 v n (n \u2014 1)\nThus the average deviation, or average error, calculated in this way is practically a true value. There can hardly be any excuse for using the value d. The extra labor in calculating D is not worth considering. For example, if there are 25 measurements the divisor will be 24.5 instead of 25; with Crelle\u2019s Rechentafel at hand the extra trouble is small. On the other hand, the average deviation thus formed is not a rough approximation but one so close that it is practically true.\nThis same consideration holds good for statistical researches, although it seems not to have always been taken into account.\nWhich of the two characteristic deviations D and Mshould be used? The quantity M calculated for the second powers of the variation is preferable to D calculated from the first powers or to any value calculated from powers higher than the second for the following reasons :\n1.\tA change in the conditions of the measurement will change the values of V If the conditions are more precise, the Fs will be smaller, if less precise, larger. But for any particular method the Fs will be grouped more closely around a certain value than they will be around that value in any other method whether more or less precise ; this value is M. M is in this way an error specially probable in the particular set of measurements.\n2.\tThe probability for the occurrence of a set of errors Vv F2, \u2014, /\u201e is the same as that for the occurrence of a set in which each V = M.\n3.\tThe fact that n is not infinite gives an uncertainty to D and M when they are calculated for the practical errors v. This uncertainty has been found by Gauss 1 to be for D\n1 Gauss, Bestimmung der Genauigkeit der Beobachtungen, Werke, IV 109.","page":113},{"file":"p0114.txt","language":"en","ocr_en":"E. IV. Scripture,\n114\nand for M\n\u00b0-756\n1/n \u2014 i\no. 708 V n \u2014 i\nThus M is more precise than D.\nProbable error.\nStill another representative error is in use. If the errors i\\, vv ..., vu are arranged in order of size, for example the smallest to the left and the largest to the right, then the one in the middle could be chosen as the characteristic variation. For an infinite number of measurements this would be what is called the \u201c probable error.\u201d In an infinite number of measurements there would be just as many variations, or errors, bigger in size than the probable error as there would be those smaller ; or if any one measurement were picked out by chance from the infinite series, the proper wager would be 1 to 1 that its error would be larger (or smaller) than the probable error. In an infinite number of measurements the relation holds\nThus or closely\nIt is also true that\n0.67449-\nP = 0.67449 M\nP = - M.\n3\nP= 0.84533 D.\nIn a finite series of measurements the probable error cannot be picked out, but may be calculated from the mean error.\nThe fact of the almost exclusive use of the probable error by the other sciences makes it obligatory for psychology to use it in preference to the mean error or the average error, although in discussion of the results it is often favorable to use the mean error also.\nII. Example of computation.\nThe character and purpose of these formulas may perhaps be made clearer by an illustrative example.\nLet the quantity to be measured be the simple reaction-time to the click of a sounder by a release movement of the index finger, while a certain subject is in a condition of prepared and undistracted attention.","page":114},{"file":"p0115.txt","language":"en","ocr_en":"Computation of a set of simple direct measurements.\t115\nThe latter stipulation is met by placing the subject in a quiet, comfortable, well-ventilated room, by making the experiments in small groups, separated by short intervals of rest, by giving a warning signal before each experiment and by the other usual precautions. The apparatus is so arranged that the stroke of the sounder makes the stimulus -record by a spark, a marker or otherwise ; the key is so arranged that the slightest movement of the finger makes the reaction-record.\nThe records for Subject A are 251, 257, 265, 240, 219 ; pause ; 231, 240, 242, 237, 245; pause; 241, 235, 243, 237, 239; pause; 243, 249, 239. 25i, 235; pause; 254, 237, 263, 263, 237.\nThe record-blank will show the figures as indicated under the word Record.\nRecord.\n251\t4- 7-3\t53-29\t\t+ 5\t25\t4)1272\n257\t+ 13-3\t176.89\t\t+11\t121\t318\n265\t+ 21.3\t453-69\t\t+19\t361\t3/ = 18\n240\t\u2014 3-7\t13.69\t\t\u2014 6\t36\t%i/ = 12\n219\t\u2014 24.7\t609.09\t246\t\u201427\t729\t\n231\t\t 12.7\t161.29\t\t\u2014 8\t64\t4)102\n240\t\u2014 3-7\t13.69\t\t+ \u00ab\tI\t26\n242\t\u2014\ti-7\t2.89\t\t+ 3\t9\tVO 1!\n237\t\u2014 6.7\t44.89\t\t\u2014 2\t4\tcO II N\n245\t+ i-3\t1.69\t239\t+ 6\t24\t\n241\t\u2014 2.7\t7.29\t\t+ 2\t4\t4)40\n235\t- 8.7\t75.69\t\t\u2014 4\tl6\tIO\n243\t\u2014 o-7\t\u202249\t\t+ 4\tl6\tII go\n237\t\u2014 6.7\t44.89\t\t\u2014 2\t4\tII\n239\t\u2014 4-7\t22.09\t239\t0\t0\t\n243\t\u2014 0.7\t\u25a049\t\tO\tO\t4)197\n249\t+ 5-3\t28.09\t\t+ 6\t36\t49\n239\t\u2014 4-7\t22.09\t\t\u2014 4\tl6\tV= 7\n251\t+ 7-3\t53-29\t\t+ 9\t81\t%V= 5\n235\t- 8-7\t75.69\t243\t\u2014 8\t64\t\n254\t+ 10.3\t106.09\t\t+ 3\t9\t4)689\n237\t\u2014 6.7\t44.89\t\t\u201414\t196\t172\n263\t-j- I9-3\t372.49\t\t+ 12\t144\tt/ = 13\n263\t+ 19-3\t372-49\t\t+ 12\t144\t%V= 8\n237\t\u2014 6.7\t44.89\t251\t\u201414\t196\t\n245\t+104.7\t24)2802.05\t\t\t\t\n\u20141-3\t\t104.2\t116.75\t\t\t\t\n243-7\t+ 0.5\tf/= xo.81\t\t\t\t\n\t\t%V= 7-21\t*\t\t\t","page":115},{"file":"p0116.txt","language":"en","ocr_en":"116\tE. IV. Scripture,\nComputation of the average.\nAs all measurements were taken under the same conditions, they should be averaged together directly. In the present case where the groups are equal in size the group averages might be obtained and then averaged with the same result ; but where the groups are of unequal size this will not do.\nThe usual way of computing the average when the figures are large is to assume first an approximate figure. Cursory inspection shows that the average is about 245. By mental calculation we find that the individual measurements differ from this by -f 6, +12, etc., and we add or subtract directly as we proceed. The eye does not need to notice the first figure 2 as itis constant. Running over the figures 51, 57, etc., we note + 6, -f 18, + 38, + 33, + 7, \u2014 7, \u2014 12, \u2014 15, \u2014 23, \u2014 23, \u2014 27,\n-\t37, \u2014 39, \u2014 47, \u2014 S3, -55, \u201451, \u201457, \u201451, \u2014 61, \u2014 52,\n\u2014\t60, \u2014 42, \u2014 24, \u2014 32. That is, the sum of all the measurements exceeds 25 times 245 by \u2014 32. Dividing the excess by the number of measurements we find that 245 must be corrected by \u2014 32-^25= \u2014 1.3 to give the average. The average is thus 243.7. Computation with long columns of figures can thus be carried on with ease and accuracy. The addition may advantageously be performed by an adding machine or an oriental abacus. This latter is a simple device used by the Chinese and Japanese. After some practice with an abacus or an oriental counting frame interminable additions can be carried on with great accuracy and with very little mental labor.\nComputation of the variations from the average.\nThe variations of the individual results from the average can generally be obtained mentally. The eye does not need to notice the first figure 2. The results are placed in the second column. It is frequently convenient to first fill in the signs in this column ; thus all measurements larger than 243.7 will yield all smaller ones will yield \u2014. Likewise the decimals may be filled in : .3 where there is a + and . 7 where there is a \u25a0\u2014\u2022. When this has been done the computation may readily be carried out by subtracting 44 from the figure with a -f and by subtracting from 43 for that with a \u2014.\nWe next find the sum of the positive errors and the sum of the negative errors ; they should be equal in size. The difference is found to be + 0.5 which is 25 times the amount by which 243.7 is smaller than the average; thus the average would be 243.72 if the second place were to be regarded.","page":116},{"file":"p0117.txt","language":"en","ocr_en":"Computation of a set of simple direct measurements.\t117\nComputation of the probable error.\nThe errors are now to be squared with the aid of a table of squares such as is to be found in every book of tables. The results are placed in the third column. The addition of this column may be aided by the use of an adding machine or an abacus. The sum of the squares is 2802.05. Dividing by 24 we get 116.75, of which the square root is 10.8r. The probable error is fi of this, or 7.21. Thus the average time is stated to be 243.717 \u00b1 7.21\u00ae'.\nThis means (aj) that upon making other measurements under exactly the same circumstances we can expect the result of any one measurement to fall within the region bounded by 243. 7\u00ae + 7.21\u00ae and 243.7\u00b0' \u2014 7.210' just as often as it will fall outside of it ; (b) that, since all instrumental and external sources of variation were eliminated, the value 7.21'\u2019\u2019 represents the internal uncertainty of Subject A\u2019s mental process in the given case.\nThe great value of the characteristic errors as measures of mental activity seems to have been generally overlooked by psychologists. It has been in constant use in the Yale laboratory for a number of years and has furnished very important data, for example in the researches of Gilbert on school children1 and of Nadler on alcoholism and other nervous troubles.2\nTesting the average.\nIt is desirable to get some idea of how well the average represents the set of measurements. The choice of the average as a representative value rests upon the assumptions : (x) that in an infinite series of measurements there would be as many values larger than the average, as there were smaller; (2) that extreme values are less frequent than those grouped around the average. In the present set there are 9 values larger and 16 smaller than the average ; if in further sets of measurements the case is found to be similar, the conclusion would be reached that even in an infinite number of measurements the disproportion would probably still prevail and that the average is not a very good representative value. In such a case it would be well to recompute the results, using the median as the representative value. The median is the middle value in the series according to size ; in a set of 2 5 measurements it is the 13th from either end when the values are arranged in order of size. In the set we are computing the\n1\tGilbert, Researches on the mental and physical development of school-children, Stud. Yale Psych. Lab., 1894 II 43.\n2\tNadler, Reaction-time in abnormal conditions of the nenjous system, Stud. Yale Psych. Lab., 1896 IV I.","page":117},{"file":"p0118.txt","language":"en","ocr_en":"ii8\nE. W. Scripture,\n13th figure, or the middle value, is 240. But the 12th value is also 240 ; we therefore change 240 by a small amount. This amount is calculated as follows: There are 11 measurements smaller than 240 and 12 larger. We take the excess of the smaller over the larger, or \u2014 1, and divide it by twice the number of values identical with the middle value ; this gives\n---\u2014 = \u2014 Adding the result to the middle value, we have 240\n2X2\n\u2014\t= 239^ or, since only the first decimal is considered, 239.8 as\nthe median.\nThe fact that the median is smaller than the average indicates a grouping of the measurements around small values with a few extremely large ones. This is a very frequent case in psychological work ; if it were not for the almost universal use of the average in the other sciences, it might be desirable to use the median as the regular representative value for much psychological work. In any case it is very desirable to use the median as well as the average whenever hypothesis (1) is not fulfilled. When it is fulfilled, the median and the average are the same.1 2\nThis lack of symmetry in the distribution of the results often indicates some one factor or some group of factors entering overpoweringly into the phenomenon measured. Thus, if the results from Subject A have been steadily symmetrical for several occasions and then show asymmetry, we would strongly suspect that\u2014instrumental and bodily conditions remaining the same\u2014some new mental factor had entered into his reaction ; for example, his method of listening for the signal may have\u2014 perhaps unknown to himself\u2014become quite different. The second hypothesis is not well supported in the above example. There are 8 or 9 values out of 25 that can be considered as very large or very small. This indicates that the phenomenon we are measuring is lacking in unity. By unity we mean that the phenomenon is composed of parts of constant characters in constant relations. Thus simple reaction may be usually assumed to consist mentally of perception and volition f there are, however, undoubtedly very many other elements involved which act differently and irregularly in some cases. The prevalence of extreme values tells against the reliability of the average or any other representative value, and indicates the need of more careful study of the conditions of the experiment. Our example is quite typical of the usual results ; it is undoubtedly true that there are unrecognized mental elements in simple reaction-time which await the discoverer.\n1\tLiterature and discussion on these points may be found in Scripture, Mean values for direct measurements, Stud. Yale Psych. Lab., 1894 II I.\n2\tScripture, New Psychology, 152, London 1897.","page":118},{"file":"p0119.txt","language":"en","ocr_en":"Computation of a set of simple direct measurements.\n119\nTesting the law of distribution.\nIt is next desirable to have an indication of the law of distribution for the results. The assumption of the average as the most probable value in addition to the hypotheses ( 1 ) and ( 2 ) above leads to the well-known law of probability\n__ h _nixt\nAccording to this law the following relations would hold good for an infinite number of measurements :\n(a) Average = median ;\n(l3) Mean error = 1 % times average error ;\n(7) Probable error = median error.\n(<5) Number of successions of the same sign (+ + or---------) before\nthe errors in the order of occurrence = number of changes ( -)------or\n.\n(a) In the 25 measurements of our example we have already found that a cannot be considered to hold good.\n(/?) The average error can be readily calculated by adding the sums of the + and \u2014 errors already found, disregarding the sign. We get 208.9, and, dividing by ^25 x 24 = 24.5 we have an average error of 8.5. The mean error 10.8 is 1.27 times the average error. This is very close to the relation 1.25 required by the ideal law of probability.\n(y ) If we consider all the errors regardless of sign we find that there are 11 larger and 14 smaller than the probable error 7.2.\nThis would not be a bad indication if it were not for the fact that two of the 11 are 7.3, thus differing very little from 7.2 ; the relation is more nearly 9 larger to 14 smaller. The curve of probability is, therefore, apparently fairly\u2014but only fairly\u2014smooth and regular if we treat all the errors as positive. This gives us some confidence in its steadiness of the phenomenon under measurement.\n(<5) There are 13 cases of succession (+ + or-----) and 11 of change\n(H----or-----(-), which would be a favorable sign if the successions and\nchanges were more evenly distributed.\nThe conclusion which we must draw from these tests is that the phenomenon we are measuring is fairly under control but probably contains more than one factor or close group of factors. This indicates the possibility and desirability of further investigation aiming at the analysis of \u201csimple\u201d reaction-time into factors more nearly elementary.","page":119},{"file":"p0120.txt","language":"en","ocr_en":"120\nE. W. Scripture,\nSearch for systematic errors.\nIt is frequently possible to determine directly from the results some of the factors undergoing change. Since the measurements were made in somewhat independent groups, the average and the probable error for each group may be considered.\nThe averages are 246, 239, 239, 243, 251 (p. 116). Owing to the small number of measurements for each average no reliance can be placed on the tenths arising by division and these averages are therefore not carried beyond the unit\u2019s place. The deviations from the general average are + 2, \u2014 5, \u2014 5, \u2014 i, +7. This would indicate a systematic influence tending at first to shorten the reaction-time and afterwards to lengthen it as the experiments are made in succession. The early shortening is usually termed the \u201cinfluence of practice,\u201d and the later lengthening the \u201ceffect of fatique.\u201d\nThe probable errors are 12, 3, 2, 5, 8 (p. 116) or, as percentages of the respective averages, 5%, 1 \u00b0J0, 1 %, 2 %, 3%. These indicate that the subject was in the first group quite irregular, that in the second he became much more regular and remained so in the third, whereas he lost some regularity in the fourth and still more in the fifth. The early increase in regularity is also termed the \u201cinfluence of practice\u201d and the later loss in regularity the \u201ceffect of fatigue.\u201d\nThere are thus two sources of systematic error known as practice and fatigue. To investigate these there should have been no intervals of rest between groups of five and the whole set might have been longer. If the object of the present set, however, was to measure the reaction-time in a constant condition of practice and with no effect of fatigue, the grouping was not fully effective. To improve it the first group might be omitted if it appeared as a regular thing in all cases that the first group differed from the others. The last group might perhaps also be omitted. The intervals between the groups might possibly have been longer with good effect.\nA search for systematic errors within a single group might be attempted by averaging all the first values, then all the second values, etc., as indicated in the computation on p. 122. The averages are 244, 244, 250, 246, 235. The probable errors are \u00b16.1, \u00b1 6.1, \u00b1 8.8, \u00b1 7.5, \u00b16.5. There is first a lengthening and then a shortening of the average time, and first a decrease in regularity and then an increase. These are directly opposed to the course of the systematic errors mentioned above as \u201c influence of practice\u201d and \u201ceffect of fatigue.\u201d If this same peculiarity should appear in most of the other sets of measurements, a new source of systematic error would have to be looked for.","page":120},{"file":"p0121.txt","language":"en","ocr_en":"Computation of a set of simple direct measurements.\nI 2 I\nV\tV*\n251\t+ 7\t49\n231\t\u2014 13\t169\n241\t\u2014 3\t9\n243\t\u2014 i\ti\n254.\t+ 10\t100\n244\t\t328\n257\t+ 13\t169\n240\t\u2014 4\t16\n235\t\u2014 9\t81\n249\t+ 5\t25\n237\t\u2014 7\t49\n244\t\t340\n265\t+ iS\t225\n242\t\u2014 8\t64\n243\t\u2014 7\t49\n239\t\u2014 IX\t121\n263\t+ 13\t169\n250\t\t628\n240\t\u2014 6\t36\n237\t\u2014 9\t81\n237\t\u2014 9\t81\n251\t+ 5\t25\n263\t+ 17\t289\n246\t\t512\n219\t\u201416\t256\n245\t+ 10\t100\n239\t+ 4\t16\n235\t0\t0\n237\t+ 2\t4\n235\t\t376\nM\n9.1\n9-2\n\u201d\u20225\nXI-3\n97\nP\n6.1\n6. i\n8.8\n7-5\n6-5\nReliability of the average.\nAssuming that the measurements followed the ideal law of probability we have the average as the most probable value. The assumption excludes asymmetry and systematic errors.\nWe have already said that, on making another single measurement under the same conditions, its value ought to be just as likely to fall within the limits 243.7 \u00b17.21 as to fall outside of them.\nBut suppose we make another series of measurements under the same circumstances ; their average ought to be more likely to fall within the limits than outside them. It is found that this likelihood increases as","page":121},{"file":"p0122.txt","language":"en","ocr_en":"122\nE. W. Scripture,\nthe square root of the number of measurements used for the average. Thus it is as likely as not that one single measurement will fall within the limits 243.7 \u00b17.21 but it is just as likely as not that a similar set of 23\n7 2 1\nmeasurements will fall within the limits 243.7 \u00b1\t__= 243.7 \u00b1 1.44.\n^25\nAn average derived from 25 results should have a probable error one-fifth as large as one of its single measurements. We can therefore say of the average 243.7 that, if the probable error for a single measurement be taken as unity, its probable error should be one fifth of that amount. The probable error for a single measurement in this example is 7.21 and that for the average will be 1.44.\nThe average and its probable error are therefore stated to be 243.7 \u00b1 1.44.\nThis probable error has quite a different meaning from that discussed on page 118. The other one served as an indication of the unity of the process involved ; this one shows how much value is to be attached to the average as a numerical figure. If the measurements had been stopped at the fifteenth, the average and the original probable error might possibly have been found to be exactly the same, but the probable error for the average must be larger because the average was derived from a smaller number of measurements and the original probable error would be divided by ^15 instead of v/2 5>\nFor psychological work it is nearly always necessary to give the probable error for a single measurement and the probable error for the series. The use of the latter may readily be illustrated by requiring a result pre-\nI.44\ncise to a certain percentage. The average 243.7 is precise to ---or\n0.7 %. If the required percentage had been 2 %, an unnecessary number of experiments was taken; if it had been 1%, there were no! enough.\nTo illustrate let us consider the following problems : Subject A is tested for the results of a change in one of the factors of his mental condition. In one set of experiments a all the conditions are exactly as in the example just discussed ; in another set b he receives the instructions : \u201c Make an intense effort to act quickly.\u201d The sets are repeated a number of times in the order abbabaab, etc. Let us suppose that the results when averaged show for a 45 measurements with an average of 243.6 and a probable error of 8.1 and for b 57 measurements with an average of 212.2 and a probable error of 11.2. These results would show that the extra effort of will shortened the reaction-time and made it less regular. The two results do not, however, possess the same reliability.","page":122},{"file":"p0123.txt","language":"en","ocr_en":"Computation of a set of simple direct measurements.\n123\nThe probable error for the first average is -7= = 1.2 or 0.49% and that\n^45\n11.2\nfor the second \u2014\u2014 = 1.5 or 0.77%. The average for set b is, there-</58\nfore, less reliable than that from a and the number of experiments should have been still further increased.\nThe probable error of the average may conveniently be indicated by p, to distinguish it from the immediate probable error P (p. 115).","page":123}],"identifier":"lit28758","issued":"1900","language":"en","pages":"109-123","startpages":"109","title":"Computation of a set of simple direct measurements","type":"Journal Article","volume":"8"},"revision":0,"updated":"2022-01-31T15:26:17.164086+00:00"}