TABLES. I. Variation in units of spelling ability. II. Variation in opinions concerning units of arithmetical ability. III. Variation in opinions concerning units of ability in controlled association of ideas. IV. Variation in opinions concerning units of ability in controlled association of ideas. V. Variation in opinions concerning units, of ability in controlled association of ideas. VII. Accuracy of discrimination of length of E. H. VIII. IX. The condition of labor in the case of the Amalgamated Society of Engineers. Reaction-time of H. X. Quickness of movement of T. XI. Ability in addition of J. S. XII. XIII. XIV. XV. XVI. XVII. Efficiency of perception of S. The condition of labor in the case of the Friendly Society of Attendance of a school. Sub-treasury daily receipts from banks. Pulse of B. Discrimination of length in 11 individuals. XVIII. Two abilities differing only in variability. XIX. XX. Illustration of the trustworthiness of an average of a group, calculated from a few records for each member. Illustration of the trustworthiness of an average of a group, calculated from a few records for each member. XXI. Illustration of the trustworthiness of a distribution, calculated from only a few records for each member. XXII. Table of frequencies of the normal probability surface in terms of A. D. XXIII. XXIV. Table of frequencies of the normal probability surface in terms of σ. Combinations of 4 causes. XXV. Combinations of 5 causes. XXVII. Combinations of 10, 15 and 20 causes. XXVIII. Cost of education per pupil for full year's attendance in XXIX. American cities. 100 boys ranked by their serial order with respect to a trait. XXX. Relative frequencies of equidistant abilities of the surface of frequency of Fig. 69. XXXI. Average distance from the average of each serial per cent. of the cases in any normal distribution. XXXII. XXXIII. Average distance from the average of any continuous group in a normal distribution. Abilities in the A test of boys and girls. XXXIV. Growth of 25 boys from the 13th through the 17th year. XXXVIII. XXXIX. The relation between two perceptive abilities, each being expressed in terms of deviations from the central tendency as a zero point. The relationship of Tables XXXVI. and XXXVII. expressed in a series of measures each with the average of its related array. The relationship of Tables XXXVI. and XXXVII. expressed as a series of individual ratios. XL. Table XXXIX., with allowance made for the variability of each trait. XLI. Calculation of r, using averages of arrays. Calculation of r, using individual records. XLIII. Table of values of the normal probability integral, correspond ing to values of x/o. XLIV. Table of values of the normal probability integral, corresponding to values of x/A. D. XLV. Table of values of the normal probability integral, corresponding to values of x/P. E. XLVI. XLVII. = 24, and Table of frequencies of normal surface with Av. CHAPTER I. INTRODUCTION. Mathematics and Measurements. THE power to follow abstract mathematical arguments is rare and its development in the course of school education is rarer still. For example, few of us are able to understand the symbols or processes used in the quotation on the following page. Yet it is a rather easy sample of the discussions from which the student is expected to gain insight into the theory of measurement appropriate to the variable phenomena with which the mental sciences have to deal. It would be unfortunate if the ability to understand and use the newer methods of measurement were dependent upon the mathematical capacity and training which were required to derive and formulate them. The great majority of thinkers would then be deprived of the most efficient weapon in investigations of mental and social facts, and adequate statistical studies could be made only by the few students of psychology, sociology, economics and education who happened to be also proficient mathematicians. There is, happily, nothing in the general principles of modern statistical theory but refined common sense, and little in the technique resulting from them that general intelligence can not readily master. A new method devised by a mathematician is likely to be expressed by him in terms intelligible only to those with mathematical training, and to be explained by him through an abstract derivation which only those with mathematical training and capacity can understand. It may, nevertheless, be possible to explain its meaning and use in common language to a common-sense thinker. With time what were the mysteries of the specialist become the property of all. To aid this process in the case of certain recent contributions to statistical theory is one of the leading aims of this book. Knowledge will be presupposed of only the elements of arithmetic and algebra. Artificial symbols will be used only when they are really convenient. Concrete illustrations will always accompany and often replace abstract laws. Deduction of Equation of Curve of Error, from A. L. Bowley's 'Elements of Statistics,' p. 275 f. We can now proceed to the determination of the equation of the curve of error. The chance of r successes is greatest when r is the greatest integer in pn; this is found by the ordinary method of determining the maximum term in a binomial expansion. Let P be this maximum value = "Cp" 9", taking the supposition for brevity that pn is integral, which will not affect the proof. = /n /pn/qnppqq", for pn + qn = n. Let P be chance of pn + x white balls. Then . gn (gn 1)... (gn x + 1) (pn + 1)(pn + 2) · · · ( pn + x) (1) (1) (1) 1.1 =Px gn 2 Taking logarithms of both sides = log P, - log P+ log (1-1)+ log (1-2)+.. gn + log (121) - log (1+1) - log (1+ 2) -( 2 2 1+2+ ...+(x-1) 12 + 22 + ··· + (x − 1)2 1+22++ x2 2q2n2 2p2n2 + (x − 1) · x · (2x — 1) 12q2n2 Let no one suppose that the foregoing statements imply that mathematical gifts and training are useless possessions for a student of quantitative mental science. On the contrary, the assumption of their absence in the reader' will necessitate long descriptions, roundabout arguments and awkward formula. If this book were written by a mathematician for the mathematically minded it would not need to be one fifth as long. If it is read by such a one, it may well seem intolerably clumsy and inelegant. General Information about Measurements. There are, in addition to the recent studies of the general theory of mental measurements, a number of matters concerning the quantitative treatment of human nature which sufficient experience teaches thoughtful workers everywhere, but which have not been stated simply and conveniently in available form for study and reference. At present one must learn these gradually and with difficulty by himself or acquire them from the oral traditions of the laboratory or class-room. They are, for the most part, extremely simple. But that one sees them at the first glance when they are presented does not imply that he would not in nine cases out of ten fail to discover them if they were not presented. To put these at the service of all ▾ who need to know about them is the second aim of this book. The Technique of Measurements. Although the formulæ used in expressing and comparing mental measurements are in most cases straightforward and simple, they are often so foreign to the habits acquired in connection with the arithmetic and algebra of one's school days that ready and sure use of them can be acquired only by practice. Convenient and accurate manipulation of figures is one of the many things which one learns to do by doing. A mere statement of a rule leaves one uncertain. Only after applying it a number of times does he really possess it. For example, I doubt if any one of my readers is sure that from a mere reading he understands the following, which is an accepted short method of determining the average of a number of measures: "Arrange the numbers in the order of their amount; choose any number likely to be nearest the average; add together, regarding signs, the deviations from it of all the numbers; divide this result |