Class -. Book ,._r. / GopyiightN COPYRIGHT DEPOSIT. SPELLING ABILITY > ITS MEASUREMENT AND DISTRIBUTION B. R. BUCKINGHAM, Ph.D. TEACHERS COLLEGE, COLUMBIA UNIVERSITY CONTRIBUTIONS TO EDUCATION, No. 59 PUBLISHED BY ©farmers (Enilege, (Mumbta Unittr-ratfc NEW YORK CITY 1913 M Copyright, 1913, by B. R. Buckingham * a™ CONTENTS SEC. PAGE i Introduction. Purpose. Problem. Previous Investigation . . i 2 Limitations 5 3 The Original-List 6 4 The Selected List. 18- Word List 8 5 The Preferred Lists 11 6 Examination of the First Preferred List 16 7 Examination of the Second Preferred List 22 8 Conclusions Regarding the Preferred Lists 25 9 Ratings of Individual Pupils 27 10 Overlapping 31 1 1 Location of Grade Medians 34 12 Scaling the Words 40 13 The Use of the Scale. 51 14 The Zero-Point of Spelling Ability 55 15 Observations on the Distributions shown in Fig. 21 61 16 Supplementary Testing at Schools VI and VII 65 17 Arrangement of the Words of the Preferred List by Teach- ers' Judgments 69 18 Rice Sentence Test. Easy 50-Word Test 75 19 Derived Forms of Distribution 84 20 Conclusions no Appendix 113 111 INDEX OF TABLES NO. PAGE I Sample of word-ratings from Original (270) Word List. Schools I and II 8 II Sample of word-ratings (118 words) 30-86. Schools III, IV, and V 12 III Number correct, per cent correct, and rank of each word, First Preferred List 14 IV Number correct, per cent correct, and rank of each word, Second Preferred List 1 5 V Table to show method of deriving r-values by "foot- rule " method 1 8 VI r-values between grades of School I (3 methods) 19 VII Coefficients of Correlation, grade with grade, and each grade with all grades for each school (II, III, IV and V). First Preferred List 20 VIII Correlations of school with school and of each school with all schools for each grade. First Preferred List 21 IX Coefficients of Correlation, grade with grade and each grade with all grades for each school. Second Pre- ferred List 23 X Correlations of school with school and of each school with all schools for each grade. Second Preferred List 24 XI Distribution of individual ratings of pupils in Schools II, III, IV, and V 27 XII Distribution of individual ratings grouped to show modes 28 XIII Number and per cent of pupils in each grade whose ability equaled or exceeded that of the median pupil in every other grade 3 2 XIV Table of values of the Normal Probability Integral corresponding to values of P.E 35 XV The per cent of pupils in each grade whose ability equaled or exceeded that of the median pupil in every other grade with corresponding P.E. values.. 36 XVI Direct and derived values of median distances in terms of P.E 39 XVII Per cents and P.E. equivalents, Preferred List — all grades 45 XVIII Grade positions and average positions 48 XIX Words arranged in order of difficulty according to scale, their P.E. values and weights on a per cent basis S 1 XX A ten-point scale 5 2 XXI Distribution of individual ratings. Easy 50-Word Test 57 V VI Index of Tables NO. PAGE XXII Amount and per cent of overlapping with P.E. equiva- lents. Easy 50-Word Test 58 XXIII Values of median intervals and their derivation (2 a to 4th grade) 59 XXIV Median Intervals o-Sth grade 61 XXV Distribution of individual ratings, Schools VI and VII. Selected List 66 XXVI Comparison of results obtained in Schools VI and VII with those in Schools II, III, IV, and V 66 XXVII Number and per cent of pupils in each grade who equalled or exceeded the median of every other grade with P.E.'s. Schools VI and VII combined with II, III, IV, and V 68 XXVIII Median Distances derived from Table XXVII 69 XXIX Comparison of results by Record and by Teachers' Judgments. Preferred List 72 XXX Distribution of individual ratings. R. S. T 76 XXXI Per cent correct for each word in each grade with P.E. values. R. S. T 78 XXXII Per cent correct for each word in each grade with P.E. values. Easy 50-Word Test 80 XXXIII Percentages of Retention — Grades 3-8 87 XXXIV Plan of elimination and retention for each grade 89 XXXV Derivation of 6th-grade Modified Table of Frequency . . 92 XXXVI Modified Table of Frequency, 3d grade 93 XXXVII Modified Table of Frequency, 4th grade 94 XXXVIII Modified Table of Frequency, 5th grade 95 XXXIX Modified Table of Frequency, 6th grade 96 XL Modified Table of Frequency, 7th grade 97 XLI Modified Table of Frequency, 8th grade 98 XLII Number and per cent of pupils in each grade whose ability equalled or exceeded that of the median pupil in every other grade with the P.E. values corres- ponding to each per cent. Modified Distributions ... 10 1 XLIII Direct and derived values of Median Distances. Modi- fied Distributions 102 XLIV Comparison of Average Median Distances by Normal and Modified Distributions 103 XLV Per cent correct for each word of Preferred List with corresponding P.E. values by Normal Distribution and by Modified Distributions 104 XLVI Average position of each word by Normal Distribution and by Modified Distributions. Point of reference, 3d-grade median 108 XLVII P.E. values corresponding to given per cents of the Normal Surface of Frequency 116 INDEX OF FIGURES NO. PAGE i Distribution of Individual Ratings. Selected List, 3d Grade 29 2 Distribution of Individual Ratings. Selected List, 4th Grade 29 3 Distribution of Individual Ratings. Selected List, 5th Grade 29 4 Distribution of Individual Ratings. Selected List, 6th Grade 29 5 Distribution of Individual Ratings. Selected List, 7th Grade 29 6 Distribution of Individual Ratings. Selected List, 8th Grade 30 7 Distribution of Individual Ratings. Selected List, All Grades 30 8 Distribution of Individual Ratings. Rice Sentence Test, 6th Grade 33 9 Distribution of Individual Ratings. Rice Sentence Test, 7th Grade 33 10 Distribution of Individual Ratings. Rice Sentence Test, 8th Grade 33 n Showing the Overlapping of the 3d and 4th Grade Surfaces of Frequency 34 12 Normal Frequency Surface to Illustrate Word Placing 41 13 Showing the Placing of the first seven words of the Pre- ferred List, 3d Grade 43 14 3d Grade Scale. Preferred List 44 1 5 4th Grade Scale. Preferred List 44 16 5th Grade Scale. Preferred List 44 1 7 6th Grade Scale. Preferred List 44 18 7th Grade Scale. Preferred List 44 19 8th Grade Scale. Preferred List 44 20 General Scale, Preferred List 49 21 Series of Grade Distributions (normal) showing Median In- tervals and the Zero-point. Entire range of Spelling Ability in the Elementary School 62 22 Diagram showing difference in difficulty between words by Teachers' Judgments 73 23 Distribution of Individual Ratings. Rice Sentence Test, 4th Grade 77 24 Distribution of Individual Ratings. Rice Sentence Test, 5th Grade 77 2 5 Scales for the following grades and lists : 4th; Preferred, Rice Sentence, Easy 50-Word, 5th; Preferred, Rice Sentence, 6th; Preferred, Rice Sentence, > „ 7th; Preferred, Rice Sentence, j. j na 8th; Preferred, Rice Sentence J p. So vii viii Index of Figures NO. PAGE 26 2a Grade Scale, Easy 50-Word; 26 Grade Scale, Easy 50- ) Word Fac . 27 3d Grade Scale; Preferred and Easy 50 -Word I } n g 28 General Scale; (2a to 4th Grade combined) Easy 50-Word p. 82 29 General Scale; Preferred, Rice Sentence Test, Easy 50- Word J 30 Curves of Retention, 3d to 8th Grades 88 31-36 The amount and distribution of elimination and retention. Grades 3 to 8 90 37-42 Derived Forms of Distribution. Grades 3 to 8 99 43-47 Comparison of Grade scales according to (a) Normal Dis- tribution and (b) Modified Distributions 106-107 48 Comparison of General scales according to (a) Normal Dis- tribution and (b) Modified Distributions 109 SPELLING ABILITY— ITS MEASUREMENT AND DISTRIBUTION § i. Introduction The purpose of this dissertation is to derive a scale for the measurement of spelling ability and to show some of its uses and applications. Such a purpose relates itself closely to a general movement, which is now well under way, and which aims to place in our hands the means of stating with some- thing approaching the precision of objective measurement the amounts of each school ability possessed by an individual or a group. We received not long ago a scale for Handwriting (Thorndike, E. L., 1910) and still more recently a scale for English Composition (Hillegas, Milo B., 1912). The former consists in the use of selected specimens of handwriting each of which has been evaluated; the latter consists of a similar series of English compositions. It will be noticed that some of the conditions of objective measurement are met. We meas- ure given specimens of handwriting by comparing them with actual samples of handwriting of known value. We determine the quality of English composition by a like comparison with samples of actual English writing of known value. It seems clear, therefore, that if we are to measure ability in spelling at all it will be by reference to an evaluated standard or sample of spelling. If we can arrange a series of words on a linear projection in such a way that the words from the low end to the high end are placed at equal intervals determined by the difficulty of each word, and if we can determine a zero-point such that failure to spell the word fixed at that point under the required conditions indicates absence of spelling abil- ity, then we shall have constructed a scale by which we may measure the spelling ability of an individual, or by which we may through suitable tests determine the difficulty of any word in the language. Since the spelling of individuals may 1 2 Spelling Ability — Its Measurement and Distribution thus be rated, the spelling of classes, of schools, and of school systems may likewise be rated. It may be said that we have always rated pupils in spelling; and that schools and school systems have likewise been rated. Such is indeed the case. But there has always been a lack of precision in these ratings due to the inequality of the units em- ployed. Dr. Rice (Rice, '97), for example, in testing the pupils in 4th to 8th grades in twenty-one school systems used a list of words containing among others: disappoint, necessary, changeable, better, because, picture. The method of rating pupils was the usual one of deducting from 100 per cent the same per cent for each word. That is, all words were taken as equal measures of spelling ability. A moment's attention to the six words mentioned will lead us to suspect that this is not a true assumption; and an actual test of a group of 5th-year children with these words shows that our suspicion is correct. In such a test mistakes were made as follows : disappoint, 37 necessary, 42 changeable, 42 better, 3 because, 1 picture, o (Thorndike, '04, p. 8) To give these words equal weight in any test is to make inaccurate most of the conclusions based upon it. A pupil who spells all or nearly all of the list is a much better speller than the figures show; for he has probably spelled not only all the easy words but also many of the hard ones. On the other hand, a pupil who misses most of the words is a much poorer speller than his rating indicates because he has probably failed to spell all the hard words as well as most of the easy ones. Nor is this list of Dr. Rice's at all unusual. Cornman used the same list (Cornman, '02). Both used a composition test where pupils were rated according to the per cent of their correctly spelled words among the total number of words in a written exercise. Cornman also used a test in which school Introduction 3 children were required to write " as many words as they could " in 15 minutes. Of course in the composition test and in the 15-minute test no two children wrote the same words. More- over, the words written by each child must have varied widely in difficulty. The result for the 15-minute test, according to Cornman's best table, is as follows: School Year Median Percentage Average Variation 8th 97.9 .60 7th 96.2 .50 6th 95.2 .33 5tho 94.3 .36 5thb 94.3 .10 4tha 94.7 .66 4th& 93.7 .96 3da 93.5 .23 3db 93.0 1.43 One conclusion from this is that " pupils of the elementary school increase regularly from grade to grade in accuracy of spelling." This might almost be taken for granted. But in answer to the question, " How much does one grade surpass another?" the figures afford no information. Obviously from all we know of the elementary school, the difference between eighth-grade ability and low third-grade ability in spelling is far greater than the figures 97.9 and 93 indicate. Similarly the Composition Tests of Rice and Cornman are misleading when used to indicate spelling ability. According to the series of Composition Tests of the latter, 8th-year children on the average spelled 99.5 per cent of their words correctly, and children of the first half of the 3d year spelled 93.2 per cent correctly. The author draws conclusions from his figures as to the progress of each grade for the school year, as to the progress of the school and as to the effect of the suspension of instruction in spelling. Since in the series of eight tests the children wrote various kinds of lessons — Geography, History, Science, Language, Composition — each with its own peculiar words, and since each pupil used his own individual vocabulary, we cannot escape the conviction that while these figures may be suggestive of progress, or of the effect of change in method, or of grade differences, they are nothing more than suggestive. 4 Spelling Ability — Its Measurement and Distribution They leave unanswered the questions, — How much progress? How large an effect? How great a difference? As we grow more and more accustomed to quantitative thinking in our edu- cational work, we feel that these are precisely the questions that we ought in some way to be able to answer. These studies of spelling made by Cornman and Rice remain the most important statistical treatment of the subject. That they have not great value it would be presumptuous even to imply. Their results are in a general sense true. To a certain extent their lists, even though made up of words of various and undetermined difficulty, may be used, especially for com- parative purposes, as a total measuring device. They do, how- ever, undoubtedly suffer through lack of precision, while their statements of amounts of difference are in general misleading. The same thing may be said of later investigations. For example, Wallin's tables and his conclusions from them as to the transfer of spelling efficiency and its relation to age, grade, and sex are subject to the same limitations ( Wallin, 'n). Pear- son's " Experimental Studies in the Teaching of Spelling " (Pearson, '12), however, shows a recognition of the difficulty, although it offers no remedy. In his treatment of the relative values of the " together-method " and the " separate-method " of teaching homonyms this author says : " Owing to the in- equality of the units of measurements, it is impossible to deter- mine accurately from Table IV whether the together-method is superior to the separate-method. One cannot decide, for example, positively whether an improvement from 3.78 errors (median of a class) to 2.86 errors is greater or less than an improvement from 5.6 errors to 3.3 errors." If, however, the words used could have been evaluated through an independent test by reference to a scientifically constructed scale, the " in- equality of the units of measurement " would have disappeared. The further treatment of the foreshortening of the opportunity for improvement due to high initial performance is quite another matter. It will be clearly seen from the foregoing that in practically all work which has attempted to present the spelling situation statistically it is assumed as fundamental that one error equals Limitations 5 another and that to spell one word is the same as to spell another word. It will therefore be profitable to seek in this field as others have sought in other fields to devise an instrument which will more accurately measure that of which we are so often called upon to give a quantitative statement. § 2. Limitations The study here attempted is confined to the elementary school entirely. It covers the grades from the third to the eighth, both inclusive. The schools tested are all located in or near New York City. The cosmopolitan character of the population of the metropolitan area makes it extremely unlikely that results of a materially different character would have been obtained by testing schools in various sections of the country. It is believed that these schools are fairly typical within the limits of the area chosen. School I is a private school of high class whose pupils are mostly American born and from good homes. All the other schools are public schools. School II is in a German section of rather low class. School III is in a better neighborhood, foreigners predominating. School IV is in an Italian section. It has long had the benefit of high-class supervision and organization. School V is again predominantly American. It is located outside of the city system. School VI is in a good residential section of the city. School VII is a large school, most of whose pupils are of foreign parentage. Territorially, two schools are in Manhattan, one in the Bronx, one in Brooklyn, and two in Queens, while one is outside of the city entirely. In all 8,791 pupils were tested. It is thought that this is a sufficient number for practical purposes. In fact it was found that the returns from each additional school after the first three or four made almost no change in the results. It is probable that greatly increasing the number of pupils tested would have afforded little compensation for the additional labor. It has seemed wiser to limit the number to a moderate one and to spend considerable effort in making the statistical analysis as complete as possible. 6 Spelling Ability — Its Measurement and Distribution § 3. The Original List The preliminary testing was made with a list of 270 words. It will be called " The Original List." It was itself selected from a much larger list of graded words used by the author of this dissertation in his own school, the same having been secured by taking from five of the popular Spelling Books now in use a vocabulary of 5,000 words agreed upon by two or more of the books. The principles of selection for these 270 words were: (1) that all of them should be sufficiently common to be in the speaking vocabulary of third-grade children; and (2) that the spelling difficulty of many of them should be great enough to test the ability of eighth-grade children. As a matter of fact, the selection did not consist of 270 words at first. The list grew to that number only after the chosen words were put into sen- tences. The necessary helping words then swelled the total to the number given. The sentences were dictated during the fall term of 1910 to schools I and II. They were given to grades 3 to 8 in School II, and to grades 4 to 7 in School I. Their dictation consumed several periods for every class. The following are the sentences : There were forty birds on the bridge. Do not go until I come. On Wednesday an umbrella was found. Whose pencil is this? My uncle gave me a banana. The butcher gave the hungry dog a piece of meat. My answer is ninety. For a nickel I bought an orange, a peach, and a pear. A dollar is not too much money for so beautiful a picture. Learn to do right because it is right. The chicken ran across the road. The janitor sweeps every Tues- day afternoon. It is wrong to steal even a penny. It would be easy to watch for your cousin from the parlor window. It is the hour for recess. Smoke was coming out of their chimney. One summer evening my neighbor came into my kitchen. I did not know he was coming that night. To whom does this pair of scissors belong? I am almost sure they belong to the tailor. The doctor thought he ought to go at once. His bicycle was against the fence. But a carriage was stopping in front of his office. His friend was already beginning to speak to him. He said the soldier should have medicine this minute. Pshaw, there was neither a monkey nor an elephant at the circus. Get some The Original List 7 coffee, sugar, and soap at the grocery store. The soldier dropped his sword and pistol. Jack had a whistle and nineteen nails in his pocket. The pretty fairy had a sawry tongue. One (fay in February people saw a sleigh pass through the avenue. Shoes are made of leather and a /i^fe trow. A wee& from to-day there w7/ be a dance. Cut up a. tomato and an omow together. In my garden I j/m// mw? cabbage instead of &££&. The saucer was round like a circle. Make no noise; do not whisper or laugh. Nobody should be without a handkerchief. A straight line has length only. We shall believe the frw^. We have another piano at owr school. Is it frwe that there was grease on the towel? This animal has a /ar#e mouth. It is not o/tew co/d enough for the ocean to freeze. Guess what made me sneeze. Choose which one of the pigeons you like. Touch the button with your thumb. The American Indian had corw and tobacco. I have written the whole alphabet. I wear a number thirteen collar. If the mew quarrel, telephone me or .yewd a telegram. Our arithmetic lesson is in addition. We a/so subtract. A handful of corn was a// I had for supper. What is the £t£/e of the story f Did you /fc£ar the thunder last night? I am fyiwg up my shoe. A bcww of water sat on the fob/e. That sentence has twelve words in it. Those who dictated the sentences were directed to read them in whole or in part as many times as seemed necessary to secure their complete comprehension. Pupils were therefore not re- quired to retain in mind a long series of words. In rating the papers only the words printed in italics were considered. If a word occurred twice it was regarded only the first time it appeared. Omitted and illegible words were classed as wrong. All the papers here as well as elsewhere throughout this study were rated by the same person. They were rated from two points of view: (i) as to the number of times each word was correctly spelled, and (2) as to the per cent of the entire number of words each pupil spelled correctly. The former point of view is the only one to which attention is now directed. Table I is a sampling from the entire 270 words as given to schools I and II. At School I the grammar-school course is completed in seven years. It therefore has no 8th grade. As stated above, the test was not given to the 3d grade in this school. 8 Spelling Ability — Its Measurement and Distribution TABLE I Figures Indicate Per Cent Correct Table reads: "across" was spelled correctly in the 3d grade of School II by 17% of the pupils; in the 4th grade of School I by 60% of the pupils, and of School II by 40% of the pupils, etc. Grade . . . School . . across. . . . addition. . almost . . . alphabet . arithmetic bridge . . . button . . . choose . . . day guess .... handful . . pshaw . . . tomato.. . too whose. . . . 3d 4th II I II 17 60 40 2 38 26 16 62 41 25 13 1 27 89 53 29 59 42 14 50 35 6 25 10 97 100 98 6 29 17 36 47 33 1 4 6 34 83 49 10 3 17 49 15 5th I 76 60 73 63 100 87 70 37 96 67 46 29 67 17 40 II 58 28 65 12 72 52 49 31 100 30 19 6 43 4 29 6th 90 76 88 40 96 98 77 62 100 77 76 46 74 26 47 II 79 45 75 46 92 85 63 37 99 50 33 5 48 7 10 7th I 98 94 80 82 100 100 84 67 100 82 75 31 79 63 57 II 87 76 81 43 97 94 62 55 100 66 63 31 32 22 59 Sth II 93 83 87 68 98 97 83 65 100 85 57 18 38 27 66 § 4. The Selected List On the basis of the results for the Original List, a group of 100 words was chosen. It is here called the " Selected List/' In Table I are shown 15 words from the Original List. The word " across " is typical of the words taken for the Selected List. Since 17 per cent of the 3d-grade children spelled it cor- rectly, it was not so difficult in that grade as to offer no test of ability. It showed a steady increase throughout the following grades but did not reach so high a figure in the highest grades as to prevent its being a test of ability there. "Almost " and " button " were chosen for the same reason. "Addition " was not taken because it was too hard for 3d-graders. Only 2 per cent wrote it correctly. So small a number as two in a hundred might get it right by chance. Practically, therefore, the word is a zero word for the 3d grade; and such a word does not test ability. There may be — and in a given grade there certainly would be — wide differences in spelling ability, but such a word The Selected List 9 will not show them. "Alphabet " was rejected because though high in the 3d grade it was very low in the 4th, suggesting that in School II it was a word that the children had recently studied. "Arithmetic " was not taken because from the 6th grade on it offered practically no difficulty. As in the case of a word rated at zero or nearly zero, so in the case of a word rated at 100 or nearly 100, there is no test. Good spellers and poor spellers so far as the particular word is concerned behave exactly alike. " Bridge " was not taken for the same reason. " Choose " was too hard in the 3d grade. " Day " was too easy everywhere. In fact " day " is a type of word such that we may almost be warranted in saying that one who cannot spell it has no spelling ability. " Guess " was taken because although it is very seldom spelled correctly in the 3d grade, its form is so peculiar that the few who did write it correctly probably knew how to spell it, i.e., did not get it right by chance. " Handful " is a type of word taken because although it shows no regular increase from grade to grade it offers a real test for every grade. The later results in other schools, however, showed that its irregularity is not accidental in schools I and II but is a peculiarity of the word itself. " Pshaw " is a familiar word to the ear, but not to the eye. Very few get it right in any grade. It was rejected. " Tomato " is curious. On the whole neither school does any better with it in the highest than in the lowest grades. It was not taken. This word and the word " handful " strongly suggest the need of a greater number of pupils to test. The word " too " is a word which is misspelled with astonishing frequency. The difficulty is not so much one of spelling as of confusion with the other two words which have the same pronunciation. It was not used in the Selected List but was later included in a small supplementary list just to " try it out." " Whose " was taken although it shows a dip in the 5th and 6th grades. Pupils in these grades have learned the use of the apostrophe and their " little knowledge " proves a " dangerous thing " which the pupils of the earlier grades avoid. These words — each more or less typical in its way — show how from the Original List of 270 a better Selected List of 100 was chosen. Again the words were put into sentences, as follows : io Spelling Ability — Its Measurement and Distribution Whose answer is ninety ? If the janitor sweeps, he will raise a dust. You ought not to steal even a penny. Wait until the hour for recess to touch the button. Smoke was coming out of f/zew- chimney. Every afternoon the butcher gave the hungry dog a /uVce of meat. One evening a carriage was stopping in /ro«£ of my kitchen. I wear a number thirteen collar. Guess what made me sneeze. Send me a />air of leather shoes. I do not know, but I am almost sure they are mine. My wndc bought my cousin a pretty watch for /or/y dollars. The soldier dropped his sword. Jack had a whistle and a/so twelve nails. The ocean does not o/few freeze. You should speak to people zvhom you meet. It takes o«/y a minute to />a.w through the gate and across the road. Did you ever /zear a /airy laugh? The American Indian had a saucer without a cup. Neither a pear wor a peach was at the grocery store to-day. Cut up a whole onion with a handful of beans. My ^'owo lesson was easy. The animal ran wfo the road and straight against a tree. Give me another sen- tence which has the word " £i£/£ w in it. I believe true friends like to be together instead of apart. These sentences were dictated at schools III, IV, and V in the spring term of 191 1. They were later (fall of 1912) dictated at schools VI and VII. The following instructions were given to the examiners : Please read these instructions through before beginning to dic- tate the sentences. I. See that each sheet is headed with (a) the pupil's name, (b) the date, (c) the grade, (d) the name of the school. II. Give all the sentences during one session, i.e., either in the morning or the afternoon of the same day. III. In classes below the fifth year dictate in two periods, separated by at least half an hour, or by a recess period. IV. Each sentence may be dictated, either in whole or in part, as many times as may seem necessary to secure its complete understanding. This exercise is purely a test in spelling. It is not intended that pupils should be subjected to the added diffi- culty of an effort to recall the words dictated. V. Offer no explanation of words or sentences. If the mean- ing is not clear, repeat the sentence as a whole or in part. VI. Do not ask the children to underline words nor otherwise call their attention to the significant words of the sentences. The Preferred Lists n VII. After the children have written the sentences, read them again and allow pupils to insert words or make other corrections. VIII. Collect the papers. Subsequently at the same schools (III, IV, and V) was given a supplementary list of 18 words, again selected from the Original List (270 words). With the same directions to the examiners, these words were put into sentences as follows : Telephone me on Tuesday if the tobacco comes. The tailor sent a saucy telegram. Already the circus was beginning. Pigeons seem too beautiful to quarrel. I am trying to choose a towel. The chicken was fried in grease. Each of these 118 words was scored in each grade and for each school separately. Table II illustrates for a few of the words the manner in which this was done. The figures indicate per cent correct. 3a means third grade, first half ; 3Z? third grade, second half, etc. Ill, IV, and V refer to different schools. It will be seen at once that there is no steady progression for each word as we pass from the lowest to the highest grades. In fact for this and for other reasons it was found best to deal with grades by years rather than by half-years. It also seemed advisable to choose a few of these words and to make them the basis of study. § 5. The Preferred Lists From the data now in hand it was possible to select a few words which showed reasonably regular increase from grade to grade in the per cent of times they were spelled correctly. Two lists were made up, each containing twenty-five words. The first list is superior to the second in the testing power of the words in all grades and in the permanence of their relative diffi- culty throughout the grades. That is, to a somewhat greater extent than in the case of the second list, the words of the first list are found to be easy enough for low grades and hard enough for high grades. Also, a word occupying a certain serial posi- tion (say the 4th in point of difficulty for the third grade) tends more strongly in the first than in the second list to occupy the same position in all other grades. That both lists, however, , are reasonably satisfactory in these particulars will be abundantly shown. 12 Spelling Ability — Its Measurement and Distribution TABLE II Figures Indicate Per Cent Correct Grade 3a 36 4a 46 5a 56 6a 66 7a 76 8a 86 against III 20 5 8 12 10 11 22 37 33 56 15 8 7 10 3 60 19 77 3 4 16 39 28 40 61 38 61 25 11 2 6 45 4 10 16 45 45 54 26 10 37 17 24 44 38 45 63 68 62 65 54 28 13 74 9 17 16 31 15 34 43 29 15 54 36 14 72 58 38 75 71 60 46 19 14 31 50 32 52 36 19 58 27 85 60 70 30 53 32 28 78 72 55 73 78 60 60 59 36 24 63 61 45 49 21 64 54 18 62 61 52 44 22 55 83 82 68 70 81 84 65 49 52 61 22 50 40 51 52 46 40 26 60 92 54 35 50 76 91 97 84 88 85 80 70 61 64 66 57 82 33 73 66 14 10 36 91 78 74 57 52 50 95 93 74 82 93 79 86 60 61 49 67 64 73 60 45 60 11 41 73 95 63 73 62 77 98 97 68 93 97 79 71 87 62 83 81 50 78 83 66 24 33 18 93 94 81 93 78 71 97 96 90 97 98 88 80 82 71 64 74 89 83 72 79 14 33 26 97 100 88 73 83 84 100 100 86 97 98 100 97 83 79 96 84 80 80 78 81 65 53 51 95 IV 97 V 89 believe III 85 IV 78 V 72 cousin III 100 IV 100 V 91 know III 100 IV 100 V 96 ninety III 80 IV 86 V 74 pigeons III 86 IV 95 V 84 saucer III 75 IV 78 V 78 too III 3?, IV 45 V 39 The first list will be called the " First Preferred List." It contains the following words : i. even 10. forty- 18. saucer 2. lesson 11. pretty 19- stopping 3- ° nl y 12. wear 20. sword 4. smoke 13- button 21. freeze 5. front 14. minute 22. touch 6. sure 15- cousin 23- whistle 7. pear 16. nails 24. carriage 8. bought 17- janitor 25- nor 9. another The Preferred Lists 13 The second list, called the follows : Second Preferred List," is as 10. tailor 18. whole II. telegram 19. against 12. telephone 20. answer 13- tobacco 21. butcher 14. too 22. guess 15- towel 23- instead 16. Tuesday- 24. raise 17. tying 25. beautiful already beginning chicken choose circus 6. grease 7. pigeons 8. quarrel 9. saucy Table III gives for each word of the First Preferred List and for each grade the number of times the word was written, the number of times it was spelled correctly, and the per cent correct. Schools I, II, III, IV, and V are included. (Omitted words are considered as " written " and as wrong.) Table IV gives the same facts for the Second Preferred List. It will be seen from tables III and IV that for any given word the per cent correct in one grade is higher than it is in any lower grade. This is, of course, to be expected. But it is not sufficient. In order that this list should be of greatest value it should be so constituted that these increases in ' per- cents-correct ' so keep pace with the increase from grade to grade of general spelling ability that a word tends in all grades to maintain the same difficulty relative to all other words in the list to which it belongs. A word which is 20th in point of diffi- culty for the 3d grade ought to deviate as little as possible from the same rank in the other grades. The experience gained in making this investigation leads us to think that most words do not meet this condition even approximately. The span be- tween the 3d and the 8th grades is very wide. Accordingly a very large class of words is impossible for the earlier, yet easy for the later, grades. Still others are really difficult in the lower grades but of almost no difficulty in the upper grades. From our own Original List " coffee " and " people " are hard for 3d- and 4th-graders, but are almost always spelled correctly above the 6th grade. A third group of words breaks down in the middle. They appear to be easy in the lowest and highest 14 Spelling Ability — Its Measurement and Distribution 6? JJHBH >7 o o) <\ a u & ^.S 5? (5 ^.H 5s tJHO^h —< ^< 01 CM -I" t» o 01 O OXO l> O LO -^ O 01 C5 O rf co x 01 o o lo Ol i-H --H Ol iH NNM IN Ol (M rHOl CO- ICIO O N Ol 01 IMrtrt i-OlOli-i SO OS «5 LOt-H t^ o CO — ' m co — 1 oi ■* lo TCino 01 01 0) oi oi 01 01 OI O] Ol t^t^t^ f»i--i> CC (0 O 01O1OI (0(00 COtOO CCC CCCC CO CO CO CO Ol Ol CM Ol Ol Ol Ol O) Ol OS 01 Ol 01 01 OI OI oi^d LOO 00O5— 1 COCOCO hn eqiHH M-*t^ O^O rHTjlTH COOT* too 00303 00500 XCiOi 00O500 l> X X X^O OOO <-c*01 01 r^i> CO CO CO O f~ l> O0000 f^ O t--. O t^ O "* X >0 xcvoi o-< — 1 co co co xot> C: C 03 CO CO CO X X lO ■* X X X OO OlO'f LOO ICO MHiS rHOX PIhO Ol O •* CO t^ CT> ooro rami--. i>05X i-»ot> i-»i"»x 000 i-0 lo LO CMtJ<0 COrt o ■*-*co OSXX lo IQ uo Tfl^O oii-ico co^co COC0O5 cooco — xco CCCOX o — o oxen 000 oc 05 xoo aoxo O 1- 10 lO O LO LOO 10 o ill Unsa OJ4 Ml, o fe ° << CJ t. ^ rt h fl lO O lO o cooo 000 OOt>- COXO OOO OIOILO CO ^ Ol X Ol Tti CO XC5X OO' XXX XXO Ot>t> ot-o OOO C5 — < Ol C33 0105 OHN X' l^i-lO CO CO CO lOLOLO OOO OOO OOO NOOS CO CO CO OOO i-lOl 1-1 XOl Ol"~X f~ O ^< LO L.0 O LO C. >0 ■*> X CO ■* f — 1 X x o 01 cbco nosi* COCOOI OICOCO co co oi co OSt^LO XOM- t^ O 00 00 COCOCO COCOCO COCO'??? LO LO LO LO LO lO LO' L0 LO LO M"*0 Oi-iCO ■OOOf C35LOLO ■-1 1-1 Ol Ol 03 COOl-H CO-* LO LOOl^ Ol— (Ol O-hco 01 Ol 01 t^ LO O OO—l 05 1-1 O Ol O LO -1OI-H OICM NCOCJ LO 01 CO CO Ol 1-1 Ol Tf Ol 1-1 o COOO -H-^iOt^ — 10 X C1C1ISO —1 11 1-1 Ol O-fLO Ol Ol Ol H O H LO 10 ■* Ol 01 Ol CM Ol OI L0 O Ol Tf LO LO O LO T(< r Ol 01 Ol Ol Ol 01 01 01 01 01 ag>, o a o 5.as 2 3° £• o o 100 lO*OiO eq t^o-# IN rH-^CO 00IMO5 lOlOlO g-flis §p2 83 § %M£ So* S-p 1 6 Spelling Ability — Its Measurement and Distribution and hardest in the middle grades. " Whose " is a type of such a word. It presents no great difficulty until children learn the use of the apostrophe. Then they write " who's." Later they partly recover from this practice. There are quantities of words which show this dip in the middle grades. One homonym is often easy until the other has been consciously related to it. Analogies falsely assumed play a harmful role. The rapid en- riching of the vocabulary as new subjects and new phases of old subjects are taken up in the middle grades probably induces some confusion. To just what extent this is true we do not know but we are sure that it is true to a significant degree. Finally, we have the large class of words which — when a sufficient number of children are tested — do show an increase in correctness from grade to grade, but which do not advance in anything like a constant ratio to the advance from grade to grade in spelling ability. Even among the words chosen as most favorable this discrepancy may be seen. The word " sure " (Table III) is 47 per cent correct in the 3d grade. This gives it a rank of 6th among the 25 words of the First Preferred List for that grade. In the 4th grade it advances to 55 per cent correct, but this advance is not sufficient to maintain its position. It falls to a rank where it is tied with " whistle " for nth and 12th place — i.e., its rank is 11.5. In the 5th, 6th, 7th, and 8th grades its rank remains fairly constant with the 4th-grade rank. It is 10.5, 14.5, 10.5, and 12. If this sort of irregularity is found in a word chosen as among the most regular, it will easily be seen how much greater would be the irregularity of words which were rejected. § 6. Examination of the First Preferred List To further establish the value of the lists, we may investigate the behavior of the words as between grades in each school and in all schools combined. We shall begin with the First Preferred List. This list was chosen in the first instance from the returns of School I. It was taken to be the 25 words of closest correlation between the grades of that school. It was then referred to the other schools and the correlations were worked out for them. Examination of the First Preferred List 17 The method used was that suggested by Spearman in his article : " ' Foot-rule ' for Measuring Correlation " ( Spearman, '06). This method is explained and criticised by Brown in "The Essentials of Mental Measurement," pp. 71-76 (Brown, '11), and in a more elementary way by Whipple in his "Man- ual," pp. 34 and 35. (Whipple, '10.) The formula is R =1 1/6 (N 2 — 1) where %(g) denotes the sum of the "gains" in rank (sum of positive differences) of the second series on the first, and 1/6 (N 2 — 1) is the value of the sum of such gains which may be expected by chance. These i^-values were then expressed as r-values (Pearson Coefficients of Correlation) by means of a table of equivalents. This table (Whipple, '10, p. 36), has been 7T worked out from Spearman's conversion formula r = s'm (-R). 2 The method is illustrated for the 4th and 5th grades of School I in Table V. By the formula, R — 1 — 5(<7) R = .72. This is i/6(JV 2 — 1) equivalent to an r-value of .90. The correlation is therefore very satisfactory. Its Probable Error is .026, which is so small in relation to the obtained correlation, that the latter has a very high degree of reliability. The correlation of each grade with every other grade for School I was as follows: Correlation of 4th grade with 5th grade, 4th 4th 5th 5th 6th 6th 7th 6th 7th 7th Average. 90 88 88 92 93 88 90 (P. E. = .026) 18 Spelling Ability — Its Measurement and Distribution TABLE V School I. Coefficients of Correlation for 4th and 5th Grades Derived. Foot-rule Method even . . . lesson . . only. . . . smoke. . front . . . sure. . . . pear bought . another, forty . . . pretty. . wear . . . button . minute . cousin. . nails . . . janitor . saucer. . stopping sword . . freeze . . touch . . whistle . carriage . nor .... 4th Grade % Rank 90 80 77 77 70 70 69 67 65 63 63 56 50 49 42 42 40 40 30 25 21 21 21 17 13 1 2 3.5 3.5 5.5 5.5 7 8 9 10.5 10.5 12 13 14 15.5 15.5 17.5 17.5 19 20 22 22 22 24 25 5th Grade Rank 94 85 96 81 75 85 77 83 79 72 89 73 70 65 65 79 56 65 47 50 63 52 55 45 43 2 4.5 1 7 11 4.5 10 6 8.5 13 3 12 14 16 16 8.5 19 16 23 22 18 21 20 24 25 Gains, 5th on 4th 1 2.5 3.5 5.5 3 2^5 1.5 4' 2 29 = 2(0) It may be interesting before taking up the results in other schools to see to what extent the ' foot-rule ' method justifies 2 xy itself in this kind of work. The 'product-moment' (r = ) and the 'unlike signs' methods (r = cos ,™) were used as a check. Table VI shows the result. It is evident from Table VI that if the true numerical state- ment of the amount of correlation may be expected to be at or near the average of all three methods, the one which tends most nearly to approximate the true result is here the ' foot-rule ' Examination of the First Preferred List 19 TABLE VI /•-values Between Grades op School I as Found by Three Methods Pairs of Grades Foot-rule Product- moment Unlike signs Average 4th with 5th .90 .88 .88 .92 .93 .88 .92 .76 .73 .78 .79 .91 .99 .97 .81 .99 .88 .93 .94 4th " 6th .87 4th " 7th 5th " 6th 5th " 7th .81 .90 .87 6th " 7th .91 Averages .90 .82 .93 .88 method. It has therefore seemed justifiable in future computa- tions of this sort to save the threefold labor and to rely upon this method. It is to be expected that since the list of words was in the main selected from the results in School I, the correlation will prove to be higher in that school than in any other. Such is indeed the case as will be seen by an inspection of Table VII. It will be remembered that School I has no 8th grade and that the 3d grade in that school was not tested. We find that the correlations in School II are considerably lower than in schools III, IV, and V, even showing an apparent inverse relation in one instance. Yet the average of the coeffi- cients for School II is .42, the P.E. of which is less than .11. A correlation is entitled to scientific consideration if it is more than twice as large as its probable error. This one is nearly four times its probable error, and may therefore be regarded as satisfactory. Still more so are the relations between grades in the other schools ; while, for all schools combined, the coeffi- cients of grade-to-grade correlation range from .47 to ,93 with an average of .76, A.D=.i2. Since for these same values P.E. ranges from .10 to .02 with an average at .057, the reliability of these values is adequate. It appears therefore that this list of words possesses the advantage of maintaining practically the same order of difficulty throughout the grades from the 3d to the 8th. In any grade the hardest word, the easiest word, and the words which take rank between tend strongly to hold their positions in every 20 Spelling Ability — Its Measurement and Distribution other grade. Our list then is to a marked degree independent of fluctuations between grades. But this might be true and still leave something to be desired. Schools differ in many respects — in quality of teaching and supervision, in preferred methods, in word lists studied, in the character of the children as to economic and racial condition. The schools which we have under consideration differ widely in all those respects. These variations in local conditions may very likely produce considerable variation in the quality of the spelling output. TABLE VII Coefficients of Correlation. Grade with Grade, and Each Grade with All Grades for Each School. First Preferred List School 3d with 4th " " 5th " " 6th " " 7th " " 8th " " entire school 4th " 5th " " 6th " " 7th " " 8th " " entire school 5th " 6th " " 7th " " 8th " " entire school 6th " 7th " " 8th " " entire school 7th " 8th " " entire school 8th " entire school Average, grade with grade. Average, grade with school II III IV V .37 .78 .67 .69 .25 .55 .31 .71 — .07 .40 .45 .75 .40 .23 .34 .75 .01 .31 .47 .67 ■35 ■79 ■77 ■85 .81 .59 .61 .89 .54 .59 .62 .84 .37 .37 .60 .73 .20 .34 .62 .77 ■78 .84 .84 92 .72 .60 .90 .93 .60 .69 .88 .90 .52 .48 .83 .72 ■9i .84 .82 ■95 .56 .66 .94 .89 .62 .60 .85 .89 ■77 ■77 .84 ■93 .45 .76 .90 .80 ■75 .61 ■78 ■89 .60 ■57 .82 .81 .42 .53 .67 .80 .69 ■74 .81 .89 All schools .79 .71 .55 .47 .71 .82* .83 .69 .62 .72 .90* .90 .86 .93 .98* .90 .89 .88* .89 .86* .91* .76 * These r-values are for each grade with the grades of all schools Examination of the First Preferred List 21 TABLE VIII Correlations of School with School and of Each School with All Schools for Each Grade. First Preferred List School I is not included because of its different system of grading Grades. School II with III. II ' TV II ' V II ( All III ' ' TV III ' V III ' All IV ' ' V IV ' All V " All Average, school with school. Average, each school with all schools 3d grade .45 .32 .61 .62 .76 .54 .77 ■S3 ■83 .58 ■79 4th grade 5th grade 6th grade ■83 7th grade .82 8th grade .31 .56 .29 .61 .60 .50 .61 .64 ■83 .85 ■73 All grades .70 .80 .88 .gi* 88 81 87* 85 93* 93* 82 ■9 1 * These figures are for each school with all grades and schools combined, i.e., with all participants. A method in reading and word study which makes extensive use of the phonogram may possibly cause some words to become easy which are otherwise difficult. If the pupils in one school come from homes where English is not spoken, they may find difficult a set of words other than that which children of English- speaking parents find difficult. With the purpose of throwing some light on this point we shall consider what the correlation is between schools for each grade and for all grades with respect to the First Preferred List. Table VIII shows the correlation coefficients. The school-with- school average correlations range from .48 to .82 with a median at .69. The school-with-all-school averages range from .73 to .91 with a median at .83. A few of the coefficients throughout the table are low. There are, however, but six that are below .50. All but one of these are in the extreme grades (3d or 8th) and have to do with School II. The circumstances under which this school was examined account for this. The tests were given 22 Spelling Ability — Its Measurement and Distribution immediately after the long summer vacation and the test-material comprised the Original List (270 words). The other schools were tested considerably later in the school year and the pupils in those schools wrote the Selected List (100 words). Notwithstanding these few shortcomings the 70 coefficients of Table VIII form an impressive argument for the value of the First Preferred List. We may fairly contend that not only are the positions of the words of this list relatively stable as between grades (Table VII), but that this permanency holds as between schools. § 7. Examination of the Second Preferred List The second list of 25 words (see p. 13 or Appendix II) was examined in the same way that the first list was examined, i.e., with reference to correlations first between grades, and second between schools. At School I the correlations between grades were found to be as follows (Compare with similar tabulation for the First Pre- ferred List on page 17: 4th grade with 5th grade .87 4th " " 6th " .83 4 th " " 7th " .79 5th " " 6th « .95 5th " " 7th " .78 6th " " 7th " .83 Average 84 (P.E.=.o 4 ) For the other schools Table IX shows the correlations. It may be compared with Table VII (page 20). A comparison of Table IX with Table VII shows that although the word list to which Table IX refers was taken, on the basis of partial knowledge, to be somewhat inferior to the First Preferred List, these coefficients do not show it. The grade-with-grade averages (.66, .67, .60, .48, and .75) are higher for some schools than in Table VII and lower for others. Their central tendency is almost identical while the closeness of group- ing is greater for the second than for the first list. Of the Examination of the Second Preferred List 23 105 measures of correlation in Table IX only 13 are less than four times their probable error, and only 3 are less than twice their probable error. The grade-to-grade relationships for all schools (column 6) range from .40 to .95, average .75, A.D.== .14. This is satisfactory to a degree scarcely, if at all, less than is the showing for the first list. TABLE IX Coefficients of Correlation. Grade with Grade and Each Grade with All Grades for Each School. Second Preferred List School II III IV V All schools 3d with 4th ■ .60 .69 .61 .43 .51 ■73 .75 ..62 .55 .55 .52 .84 .55 .56 .55 .41 .35 •75 .38 .26 .11 .07 —.01 ■34 .74 " " 5th .73 " " 6th .55 " " 7th .40 " " Sth .41 .72* 4th " 5th .73 .74 .60 .57 .82 .82 .61 .62 .60 .88 .69 .44 .45 .51 ■78 .48 .40 .38 .38 .67 .90 " " 6th .80 " " 7th .69 " " 8th .62 ■93* 5th " 6th .75 .55 .62 .81 .76 .75 .73 .92 .74 .73 .72 .90 .84 .65 .83 .92 .90 " " 7th .83 " " 8th .80 " " entire school ■95* 6th " 7th .83 .82 ■93 .67 .71 .82 .79 .77 .84 .76 .84 .88 .94 " 8th .89 " " entire school .91* 7th " 8th .80 .88 .80 ■83 .74 .80 .86 ■78 .90 .87* 8th " entire school .89 .80 .81 .86 .80* Average, grade with grade .66 .67 .60 .48 .75 Average, grade with school . . . .84 ■87 .81 ■74 .86* * These r-values are for each grade with all grades of all schools. Table X (whose counterpart for the first list is Table VIII) reveals, as Table IX did not, the relative inferiority of the second list. There are 49 coefficients in Table X that are lower than the corresponding figures in Table VIII. Only 21 are 24 Spelling Ability — Its Measurement and Distribution higher. The school-with-school averages are lower in five instances and higher in but two. The order of difficulty of these words is therefore not so permanent as between schools. It is, however, sufficient abundantly to justify the list. There are but six of the 70 coefficients in the body of the table that are less than four times their probable error, and but three that are less than twice their probable error. In some respects, indeed, this list is superior to the first list. A comparison of TABLE X Correlations op School with School and of Each School with All Schools for Each Grade. Second Preferred List School I not included. Compare with Table VIII, p. 21 School II with III II " IV II " V II " All " III " IV " III " V " III " All IV " V " IV " All V " All Average, school with school Average, each school with all schools 3d grade 48 29 05 57 55 47 76 47 79 62 39 .69 4th grade 51 17 08 56 38 43 78 43 76 57 31 .68 5th grade 82 62 78 55 74 82 58 .80 6th grade 59 75 65 61 ■79 7th grade GO ■78 8th grade 62 .81 All .66 .43 .60 .59 .87 .65 ■77* .78* .63 .82* * These figures are for each school with all grades and schools combined, i.e., with all participants. Tables III and IV shows that the words of the second list are in general more difficult than those of the first. Doubtless the first list is a somewhat better test for lower grades, while the second is a better test for upper grades. This supposition is neatly supported by the figures in Table X. Sixteen of the 21 that are higher in this table than in Table VIII are in the three upper grades, while the two higher average correlations are in the 7th and 8th grades. If, therefore, some of the words in the first list are found to be too easy for the highest grades — as Conclusions Regarding the Preferred Lists 25 doubtless they may be — then the second list will supply a valuable supplement to the first. § 8. Conclusions Regarding the Preferred Lists Our lists therefore prove to be well selected. Success and failure in spelling them may be used with considerable confidence to measure the thing we call spelling ability. The establishment of this fact is of the utmost importance. In general when we are to measure mental traits or capacities the thing we directly measure is itself a physical phenomenon or fact. We measure fatigue by the number and height of lifts with the ergograph, or by the distance between points of the esthesiometer necessary to be recognized as ' two.' We measure attention and perception by counting dots or by cancellation; memory, by the number of digits reproduced; association by the number of words pro- nounced in a given time; and intelligence itself, by a series of tests which may be scored objectively. What we deal with directly is something, assumed to be functionally related to the trait in question, which can be measured in time or space or which can be counted. If this objective manifestation does not accurately register the subjective state to which it is supposed to correspond, it is impaired, to the extent of its inaccuracy, as an index to be directly measured. Now it is undoubtedly true that the misspellings of most words are unreliable as indicating lack of spelling ability in general ; and on the other hand it is probable that to spell them correctly often argues little more than that the subject can spell the particular words that he did spell. Most words are in some way special — and they are special (particularly for children) in ways that we do not realize. Very often they do not mean the same thing to one person that they do to another. They are frequently pronounced differently by different people. They suggest dis- similar imagery. They connote variously. They range from very easy to very hard ; and those that are easy for some people are hard for others. Moreover there are numerous ways of misspelling them, each indicating its own causal incoordination. An error may not be equal to an error even in misspelling the same word. 26 Spelling Ability — Its Measurement and Distribution It would be presumptuous to suppose that all these difficulties have been overcome in selecting our two preferred lists. Without doubt we have only roughly approximated the ideal conditions under which a physical fact may be the transcript of a mental trait. Probably nothing more satisfactory than an approxima- tion can be devised. But we have been at no small pains to secure a list of words which would be free from many of these variations, and we think we have done so. From an inspection of them we may be justified in believing that their pronunciation and meaning are fairly constant for everybody; and we may regard it as probable that their associative connections do not vary much for different people. From a statistical analysis of them we find that their behavior with elementary school children is constant to a marked degree, and in particular that it is relative- ly independent of maturity and of local conditions. Older children in higher grades spell them more frequently and in each grade more frequently than in the one before it. Children in schools under favorable circumstances do better with them than do children in less favorable situations. It is because they re- flect these conditions that they are valuable. By the use of them, conditions in a school, a class, or an individual may be revealed ; and conversely to a certain extent if the conditions are known (e.g., the grade) the results, by the use of them, are pre- dictable. These lists, then, tend strongly to remain intact under various conditions. As lists they appear to be reliable, and our numerical results give quantitative expression to this reliability. But as to the words themselves, we shall yet have much to say. There has been no attempt to secure lists composed of words of equal difficulty. The effort has rather been to choose words which differ widely in this respect. We shall now attempt to arrange these words on a scale which shall accurately represent their difficulty, assuming as true a certain supposition concerning the form of distribution of spelling ability within a school grade. The resulting scale will represent their difficulty approximately in so far as this supposition is approximately true. When this is done, their value for test purposes independently of the list which contains them will be established. Ratings of Individual Pupils 27 § 9. Ratings of Individual Pupils In addition to scoring words, the papers of individual pupils were rated. This was done in the usual manner, the ability to spell one word being scored as equal to the ability to spell any other word of the list. This procedure is subject to the criticism made in Section 1 above; but in the absence of any evaluation of the words, a system of weighting is not possible. The results will not be misused here. The test material consisted of the 100 words of the Selected List. The papers written at schools II, III, IV, and V were used. School I was not available because of its system of grad- ing. A few papers were incomplete in each of the schools ; these were rejected in this part of the work. In all, 2,487 papers were rated. Table XI shows for each grade the distribution of in- TABLE XI Distribution of Individual Ratings of Pupils in Schools II, III, IV and V Per- centage 3d Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade Correct No. % No. % No. % No. % No. % No. % 0- 5 9 2.0 1 .2 1 .2 6- 10 22 4.9 1 .2 2 .4 11- 15 30 6.7 10 2.1 1 .2 16- 20 38 8.5 12 2.6 2 .4 21- 25 44 9.9 13 2.8 6 1.2 2 .5 26- 30 47 10.6 23 4.9 12 2.3 31- 35 34 7.6 29 6.2 13 2.5 2 .5 2 .5 36- 40 38 8.5 27 5.8 11 2.1 41- 45 24 5.4 30 6.4 18 3.5 6 1.4 2 .5 46- 50 34 7.6 33 7.1 28 5.4 4 1.0 1 .3 51- 55 26 5.8 27 5.8 20 3.9 6 1.4 3 .8 56- 60 24 5.4 31 6.6 32 6.2 15 3.6 5 1.4 1 .4 61- 65 26 5.8 39 8.4 44 8.5 12 2.9 6 1.6 1 .4 66- 70 17 3.8 29 6.2 48 9.3 23 5.5 8 2.2 3 1.1 71- 75 13 2.9 45 9.6 49 9.5 30 7.2 18 4.9 8 2.9 76- 80 8 1.8 35 7.5 59 11.5 52 12.4 31 8.5 11 4.0 81- 85 4 .9 33 7.1 37 7.2 67 16.0 38 10.4 19 6.9 86- 90 4 .9 26 5.6 64 12.4 61 14.6 79 21.6 41 14.8 91- 95 3 .7 19 4.1 50 9.7 101 24.2 93 25.5 80 28.9 96-100 4 .9 18 3.5 37 8.9 79 21.6 113 40.8 Totals... 445 467 515 418 365 277 Medians. 35.8 60.70 73.10 84.90 90.50 94.68 A. D.... 18.0 20.9 10.4 10.0 7.9 5.8 28 Spelling Ability — Its Measurement and Distribution dividual ratings. It reads as follows: "In the 3d grade 9 pupils were rated between 0% and 5%, which was 2.0% of all the 3d grade pupils. In the 4th grade 1 pupil was rated between 0% and 5%, which was .2% of all the 4th grade pupils," etc. The striking characteristic of the distribution of these ratings is their extreme variability. Children of the 3d grade are repre- sented in every group but one, while children of the 4th and 5th grades are rated in every group. It appears that we may expect a few 6th- and 7th-grade children to spell not more than 20 or 30 of these hundred words, which is not quite as good as the typical ability of 3d-grade children for the same words. The 8th-grade pupils show the least variation. This is probably true of this grade in general. It is not, however, as marked as these figures indicate. The 100-word list as a whole, whatever may be true about some of the individual words, did not thorough- ly test this grade. A glance at Fig. 6 will show how sharply cut off at the high end is the curve of distribution. This is against all the facts which we know about eighth-graders in particular and human ability in general. A harder test would have shown a lower mode and a more gradual tapering off at the upper end of the curve. But even as this record stands we may look for a considerable number — between 7 and 8 per cent — of 8th-grade pupils to average no better than the typical performance of 5th- grade children. TABLE XII Distribution of Individual Ratings Grouped to Show Modes. Figs. 1-7 Percentage Correct 3d Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade 0- 10 6.91 \ 22.1 15. 2 J 20.5] 36.6 16. lj 13.01 24.2 11.2J 9.61 14.3 4.7J 1.81 .7) 2 ' 5 ■ 4 1 ... 4.7) 7.71 19.7 12. Oj 13.51 } 25.9 12.4J 14.61 \ 31.7 17. lj 12.71 17.7 5.0j 3 » 3.51 8.1 4.6J 8.91 \ 19.0 10. lj 17.81 38.8 21. OJ 19.61 32.8 13.2J • 5 ) ..0 • 5j 2.41 7.4 5.0J 8.41 28.0 19. 6 J 30.61 63.7 33. lj .5J .5 .81 \ 3.0 2.2J 3.81 f 17.2 13. 4j 32.01 79.1 47.1 11- 20 21- 30 31- 40 41- 50 ,\ < 51- 60 61- 70 1.51 71- 80 8.4 6.9J 81- 90 21.71 91.4 69. 7 j 91-100 Ratings of Individual Pupils 29 O 10 20 30 VO SO (A 10 Bo fo /oo hi id Zo 30 40 SO 10 70 80 90/00 10 Z0 30 40 -50 60 70 SO 90 W FIg.1 O 10 20 JO #0 SO 60 70 80 90 /oo Fi 3 A fi 3 .3. Frequency of each rating (of correctly) in Frequency of each rating (of correctly) in Frequency of each rating (of correctly) in Frequency of each rating (of correctly) in Fig. 5. Frequency of each rating (of correctly) in Fig. 1 Fig. 2 Fig. 3 Fig. 4 o io Zo jo 10 St 60 ro So 90 too Fig- 5. per cent of Selected List spelled grade 3. per cent of Selected List spelled grade 4. per cent of Selected List spelled grade 5. per cent of Selected List spelled grade 6. per cent of Selected List spelled grade 7. 30 Spelling Ability — Its Measurement and Distribution 70 2S- C5 CO 56 60 m 3S 30 2S 20 IS 10 5 Fig. 6 10 20 30 ft) 50 CO 70 80 10 loo O /0 2.0 3o 140 So £0 70 80 9o 100 Fig. 6. Frequency of each rating (of per cent of Selected List spelled correctly) in grade 8. Fig. 7. Frequency of each rating (of per cent of Selected List spelled correctly) in grades 3-8 combined. Overlapping 3 1 Another characteristic of the distributions shown in Table XI is the absence of clearly marked modes. Table XII is a group- ing of the per cent columns of Table XI into io's and 20's. From this grouping wide modes of marked character are shown. Figs. 1 to 7 show the same facts graphically. From the nature of these figures the test material appears to have been capable of revealing satisfactorily the spelling ability of grades 3, 4, and 5. Figs. 1 to 7 are the surfaces of frequency of spelling ability with the Selected List. In each of them the horizontal scale shows percentages correct; the vertical scale shows the per cent of children having each rating for percentage correct, by steps of 10. The number of children represented is 445 in grade 3, 467 in grade 4, 515 in grade 5, 418 in grade 6, 365 in grade 7, and 277 in grade 8. § 10. Overlapping It follows as a matter of course from the variability of these ratings that the overlapping of grade on grade is conspicuous. We have located the median abilities, of each grade, for the selected word list. They are: 3d grade, 35.8; 4th grade, 60.7; 5th grade, 73.1 ; 6th grade, 84.9; 7th grade, 90.5 ; 8th grade, 94.7 (See Table XI). Table XIII shows the number of pupils and the per cent of pupils in each grade whose ratings equalled or exceeded the medians of every other grade. The table reads as follows : In the 3d grade 76 pupils equalled or exceeded the median rating of the 4th grade which was 17.1% of all the 3d- grade pupils ; 27 equalled or exceeded the median rating of the 5th grade which was 6.1% of all the 3d-grade pupils. In the 4th grade 378 pupils equalled or exceeded the median rating of the 3d grade which was 80.9% of all the pupils of the 4th grade, etc. It will be noticed that there are two places where there is no overlapping. There are no 3d-grade children who equal the median rating of the 8th grade, although the 3 who exceed the 7th-grade median come very near it. Two of them are rated at 93 and one at 94, only 1.7 and .7 below the 8th-grade median. There is also no overlapping of the 8th grade on the 3d. All the pupils of the 8th grade exceed the median of the 3d grade. When, however, we say that at these points there is no overlapping, we do not mean that their surfaces of frequency do not enclose 32 Spelling Ability — Its Measurement and Distribution TABLE XIII Number and Per Cent of Pupils in Each Grade Whose Ability Equalled or Exceeded that of the Median Pupil in Every Other Grade 3d grade.. . N=445. . . . Med.=35.8 4th grade. . N=467. . . . Med.=60.7 5th grade. . N=515.... Med.=73.1 6th grade . . N=418. . . . Med.=84.9 7th grade. . N=365. . . . Med.=90.5 8th grade. . N=277. . . . Med.=94.7 3d Grade 4th Grade 5th Grade 6th Grade 7th Grade No. % 76 17.1 27 6.1 9 2.0 3 0.7 No. % 378 80.9 146 31.3 52 11.1 27 5.8 No. % 478 92.8 370 71.8 142 27.6 73 14.2 No. % 414 99.0 384 91.9 338 80.1 142 34.0 No. % 363 99.5 354 96.4 328 89.9 256 70.1 No. % 277 100 276 99.6 269 97.1 241 87.0 200 72.2 8th Grade 9 1.9 30 5. 57 13.6 99 27.1 common areas. If Fig. I is placed on Fig. 8 so that the zero points coincide, it is evident that there is considerable area com- mon to both. We mean that the upper part of the 3d-grade surface of frequency does not lap over the median point of the 8th-grade surface, and that the lower part of the 8th-grade sur- face does not reach down to the 3d-grade median. There are many 3d-grade children who do better than the poorest 8th- grade children. The fact is, then, that except as between the 3d and 8th grades, some pupils of each grade perform like typical children of every other grade. Since this is true, it serves to fix the location of the frequency curves and medians for each grade with reference to each other. For the purpose of doing so we shall for the pre- sent assume that the distribution of spelling ability in each grade is "normal," i.e., is correctly represented by the curve of error. There is some argument for this assumption. The fact that Overlapping 33 our surfaces of frequency (Figs. 1-6) do not, especially for upper grades, closely resemble the normal curve, only shows that the test material was not difficult enough to bring out a distribution in real accordance with spelling ability. The result of using a different list of words is shown for grades 6, 7, and 8 by Figs. 8, 9, and 10. The test material in this instance was Rice's "Sen- tence Test" : 396 children in the 6th grade, 367 in the 7th, and 244 in the 8th wrote this test in schools II, III, and VIII. The zs /$ /0 S O 10 20 to 40 SO 60 70 80 10 W Fiq.2 10 Z0 30 40 SO CO 70 80 90 /oo Fig.i 4to JS 30 25 20 IS- 10 s /o 20 30 *o so Co 70 &o 90 loo FIdO. Figs. 8, 9 and 10. Frequency of each rating (of per cent of Rice Sen- tence List spelled correctly) in grades 6, 7 and 8, respectively. iV=3g6 for grade 6; 367 for grade 7; and 244 for grade 8. The horizontal scale is for per cent spelled correctly; the vertical scale is for the percentage of children receiving each rating for percentage correct, by steps of 10. surface of frequency for the 6th grade is close to the "normal" surface. If that for the 7th and 8th grades is less so, it is still far more regular than the surfaces shown for these grades in Figs. 5 and 6 and might be made still more so by an appropriate selection of test material. There seems no good ground for as- 34 Spelling Ability — Its Measurement and Distribution suming that the distribution of spelling ability in any grade is not according to the normal curve or according to a curve which resembles it closely. § ii. Location of Grade Medians Upon the assumption, therefore, of a normal distribution we may proceed to locate the grade medians with reference to each other. In all cases we shall work with per cents instead of with numbers of pupils. This will reduce all surfaces of fre- quency to equal areas. We shall assume further that the real variability of any one of these grades in spelling ability is equal to the real variability of any other one of them. We have already seen (Table XIII) that 17.1% of the 3d- Fig. 11. Showing the overlapping of the 3d and 4th grade surfaces of frequency. grade pupils equal or exceed the median ability of 4th-grade children. Fig. 11 shows this fact by a diagram. The surface on the left (Axis OM) represents the 3d-grade distributions. M is its median point, MD (=MQ) is its probable error— i.e., figure NPQD is one-half its area, thus graphically representing one-half the cases in the 3d grade, which accordingly do not deviate from the median by an amount greater than MD. The surface on the right represents the 4th-grade distribution beyond whose median axis, O^M 1 , the 3d-grade surface extends to an amount represented by the shaded figure KCM 1 . This stands for the 3d-grade children who equal or exceed the 4th-grade median — i.e., it is 17.1% of the 3d-grade surface of frequency. Accordingly the area OKM 1 M represents 32.9% of the cases. The distance MM 1 may now be found in terms of P.E. It is the distance from the median point along the X-axis of the normal probability integral which includes 32.9% of the cases. Distances Location of Grade Medians 35 corresponding to different per cents of the total area of the curve have been worked out. Table XIV, which is taken from TABLE XIV Table of Values of the Normal Probability Integral Corre- sponding to Values of P.E. Total Area of the Surface of Frequency Taken as 10,000 X X X X No. A No. A No. A No. A P.E. Cases P.E. Cases P.E. Cases P.E. Cases 81 18 135 1.5 3441 80 3.00 4785 17 4.5 4988 .05 135 134 1.55 3521 76 3.05 4802 15 4.55 4989 .1 269 134 1.6 3597 74 3.1 4817 14 4.6 4990 .15 403 133 1.65 3671 71 3.15 4831 14 4.65 4991 .2 536 134 1.7 3742 69 3.2 4845 13 4.7 4992 .25 670 132 1.75 3811 65 3.25 4858 12 4.75 4993 .3 802 131 1.8 3876 63 3.3 4870 11 4.8 4994 .6 .35 933 130 1.85 3939 61 3.35 4881 10 4.85 4994.6 .6 .4 1063 130 1.9 4000 57 3.4 4891 9 4.9 4995.2 .5 .45 1193 128 1.95 4057 56 3.45 4900 9 4.95 4995.7 .5 .5 1321 126 2.0 4113 53 3.5 4909 8 5.0 4996.2 .4 .55 1447 124 2.05 4166 51 3.55 4917 7 5.05 4996.6 .5 .6 1571 124 2.1 4217 48 3.6 4924 7 5.1 4997.1 .3 .65 1695 121 2.15 4265 46 3.65 4931 6 5.15 4997.4 .3 .7 1816 119 2.2 4311 43 3.7 4937 6 5.2 4997.7 .3 .75 1935 118 2.25 4354 42 3.75 4943 5 5.25 4998.0 .2 .8 2053 115 2.3 4396 39 3.8 4948 5 5.3 4998 . 2 .2 .85 2168 113 2.35 4435 37 3.85 4953 4 5.35 4998.4 .2 .9 2281 111 2.4 4472 36 3.9 4957 4 5.4 4998.6 .2 .95 2392 108 2.45 4508 33 3.95 4961 4 5.45 4998.8 .2 1.0 2500 106 2.5 4541 32 4.0 4965 3 5.5 4999.0 1.05 2606 103 2.55 4573 29 4.05 4968 3 5.55 4999.1 1.1 2709 101 2.6 4602 29 4.1 4971 3 5.6 4999.2 1.15 2810 98 2.65 4631 26 4.15 4974 3 5.65 4999.3 1.2 2908 96 2.7 4657 25 4.2 4977 2 5.7 4999.4 1.25 3004 93 2.75 4682 23 4.25 4979 2 5.75 4999.5 .05 1.3 3097 91 2.8 4705 22 4.3 4981 2 5.8 4999.55 .05 1.35 3188 87 2.85 4727 21 4.35 4983 2 5.85 4999.6 .05 1.4 3275 85 2.9 4748 19 4.4 4985 2 5.9 4999.65 .05 1.45 3360 2.95 4767 4.45 4987 5.95 4999.7 36 Spelling Ability — Its Measurement and Distribution Thorndike ('13, p. 200), presents these distances in units of P.E. By reference to it we find that 32.9% corresponds to 1.4088 P.E. In a similar way, the 6.1% of 3d-grade children who equal or exceed the 5th-grade median (Table XIII) serve to locate that median at 2.2929 P.E. above the 3d-grade median. The 6th- grade median is 3.0441 P.E. and the 7th, 3.6429 P.E. above the 3d-grade median. The distance between the 3d- and 8th-grade medians cannot be directly calculated owing to the absence of sufficient overlapping. Wi AJ» *t» H,/^ I** B Suppose the line AB to represent the range of spelling ability in the elementary school. At a certain distance above A, the absolute zero-point, stands the 3d-grade median, M 3 . Above it and at distances to be determined are the medians of the 4th to the 8th grades, M i M 8 . For brevity we shall call the distance from the 3d- to the 4th-grade median M 3 _ 4 , etc. M 4 . 3 means the same distance as M 3 . 4 , but measured in the opposite or negative direction. TABLE XV The Per Cent op Pupils in Each Grade Whose Ability Equalled or Exceeded that of the Median Pupil in Every Other Grade; with the P.E. Values Correspond- ing to Each Per Cent 3d Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade 3d grade. . . . % P.E. 17.1 1.4088 6.1 2.2929 2.0 3.0441 0.7 3.6429 ? 4th grade. . . % P.E. 80.9 —1.2962 31.3 .7227 11.1 1.8111 5.8 2.3308 1.9 3.0767 5th grade . . % P.E. 92.8 —2.1663 71.8 —.8553 27.6 .8819 14.2 1 . 5888 5.8 2.3308 6th grade . . . % P.E. 99.0 —3.4500 91.9 —2.0735 80.1 —1.2532 34.0 .6117 13.6 1 . 6291 7th grade . . % P.E. 99.5 —3.8200 96.4 —2.6673 89.9 —1.8918 70.1 —.7818 27.1 .9041 8th grade . . % P.E. 100 ? 99.6 —3.9375 97.1 —2.8114 87.0 —1 . 6704 72.2 —.8730 Location of Grade Medians 37 Table XV gives all the distances between medians which our data permit us to calculate directly. The P.E. values, reading across the table, indicate that on the record of the pupils tested the 4th-grade median is found to be 1.4088 P.E. above the 3d- grade median, the 5th 2.2929 P.E. above it, the 6th 3.0441 P.E. above it, and the 7th 3.6429 P.E. above it; that the 3d-grade median is 1.2962 P.E. below the 4th-grade median, the 5th .7227 P.E. above it, etc. It will be seen that M 4 is given as 1.4088 above M z , while M s is given as only 1.2962 below 'M 4 . We shall have to adopt one or the other, or some value between them as the most probably correct distance, M z _±. Similarly for each of the other distances (except M 3 _ 8 ) we have two values, and these two values are in each case somewhat different one from the other. The following are the pairs of values which Table XV shows : ^3-4 1.4088 and 1.2962 ^3-5 2.2929 « 2.1663 M 3-6 3.0441 « 3.4500 M 3 _ 7 3.6429 a 3.8200 M 4 - 5 .7227 a .8553 ■^4-6 1.8111 u 2.0735 M 4 _ 7 2.3308 u 2.6673 ^4-8 3.0767 a 3.9375 ^5-6 .8819 u 1.2532 ^5-7 1.5888 a 1.8918 ^5-8 2.3308 a 2.8114 ^6-7 .6117 " .7818 ^6-8 1.6291 a 1.6704 M 7 « .9041 a .8730 The differences between these pairs of values is in most cases small. In all cases they afford data for the determination of the distances between medians which will be probably more accurate than either of them. We do not, however, need all these values. If we have five, namely, M 3 . 4 , M 4 _ 5 , M B ^, M 6 . 7 , and M 7 _ 8 , all the others may be obtained by adding these together. We shall therefore attempt to derive as accurately as possible these five values in terms of the unit, P.E. Each of them is represented directly by two quantities as shown above. But it is clear that if we use more of the data of Table XV we may obtain values whose 38 Spelling Ability — Its Measurement and Distribution accuracy will be much more satisfactory. We may, for instance, find for the distance between the 4th-grade median and the 5th-grade median (M 4 _ 5 ) a third value by subtracting from the distance between the 3d- and 5th-grade medians (M 3 _ 5 = 2.2929) the distance between the 3d- and 4th-grade medians (M 3 _ i = 1.4088). This gives .8841. Another value is the difference between the same two distances expressed negatively, i.e., accord- ing to our notation, between M 5 . 3 (2.1663) an d M 4 _ s (1.2962;, which is .8701. Again we may use the difference between M^ and M 5 _e, between M^ and M M) between ikf 4 _ 8 and M 5 _ 8 ; and for each of these differences between positive quantities we have a difference between corresponding negative quantities. This adds six more expressions, making ten altogether, for the same distance, M^ 5 . This is of course only a beginning of the great number of combinations which may be used to get expres- sions for the same distance. We think these few, however, since they use each of the 18 segments (nine counted both ways) which terminate at either M^ or M 5 will be sufficient to determine M 4 _ 5 with considerable accuracy. We doubt whether the remoter segments (e.g., M 6 _ 7 , M 6 _ 8 , and M 7 . s ) would, if used, increase the accuracy at all. Accordingly we have calculated 10 values for ikf 4 . 5 , M^, and M 6 _ 7 . Since we have no expression of direct relation between M 3 and M s , we have but 8 values for ikf 3 _ 4 and M 7 _ 8 . Table XVI gives all these values with the derivation of each. It also gives the averages, unweighted and weighted, of the values for each of the five median intervals. It was felt that to give each of these items the same weight was to fail to take account of their reliability. The direct values are, no doubt, most to be depended upon. Those computed by using a distance which passes over one median are less so. Those involving two or more of these " skips " are still less so and diminish in reliability as the number of " skips " increases. It will be found that in column 2 of Table XVI the first quantity .8841 is derived by using a value that involves one skip. M 3 _ 5 skips over M 4 , while M 3 _ 4 , which is taken from it, presents no skips. The second quantity, .7227, is direct, and there are no skips. .9292 involves one skip, .7420 three skips (M 4 _ 7 skips M 5 and M 6 , and M 5 . 7 skips M 6 ), etc. It will be found upon trial Location of Grade Medians 39 TABLE XVI Direct and Derived Values op Median Distances in Terms of P.E. ■Pay . 0.83t.... /-oSi .o.itr, C}/" . M* fl\* Ajr /fc »[s ^3-4 ^4-5 M 5 - b M 6 _ 7 M 7 _ 8 1 . 4088 (direct) 1 . 5704 (M 3 _-M 4 _ 5 ) 1.2330 (M 3 _ 6 — M 4 _ 6 ) 1.3121 (M 3 _— M 4 _ 7 ) ? (M 3 _ 8 — M 4 _ 8 ) 1.2962 (direct) 1.3110 1.3765 (M 6 _ 3 — Mg_ 4 ) 1 . 1527 (M 7 _~M 7 _ 4 ) ? (ikf 8 _ 3 — M 8 _ 4 ) .8841 (M 3 _ 5 -M 3 _ 4 ) .7227 (direct) :9292 (M 4 _ 6 -M 5 _ 6 ) .7420 (M 4 1 7 -M 5 _ 7 ) .7459 (M 4 _ 8 -M 5 _ 8 ) .8701 (M 5 _ 3 -M 4 _ 3 ) .8553 (direct) .8203 .7755 (M 7 _ 4 -M 7 ^) 1.1261 (M 8 _ 4 -M 8 _ 5 ) .7512 (M 3 _ 6 — M 3 _ 5 ) 1.0884 (M 4 _ 6 -M 4 _ 5 ) .8819 (direct) .9771 (M 5 _ 7 — M 6 _ 7 ) .7017 (M 5 _ 8 — M 6 _ 8 ) 1.2837 (M 6 _3— M 5 _ 3 ) 1.2182 (M 6 _ 4 -M 5 _ 4 ) 1.2532 (direct) 1.1100 (M 7 _ 5 — M 7 _ 6 ) 1 . 1410 (M 8 _ 5 — M 8 _ 6 ) .5988 (M 3 _ 7 -M 3 _ ) .5199 (M 4 _ 7 -M 4 _ 6 ) .7069 (M 5 _-M 5 _ 6 ) .6117 (direct) .7250 (M 6 _ 8 — M 7 _ 8 ) .3700 (M 7 _ 3 — M 6 _ 3 ) .5938 (M 7 _ 4 -M 6 _ 4 ) .6386 (M 7 _ 5 -M 6 _ 5 ) .7818 (direct) .7974 (M 8 _ 6 — M 8 _ 7 ) ? (M 3 _ 8 — M 3 _ 7 ) .7459 (M 4 _ 8 -M 4 _ 7 ) .7420 (M 5 _ 8 -M 5 _ 7 ) 1.0174 (M 6 _ 8 — M 6 _ 7 ) .9041 (direct) (M^-M 7 _ 3 ) 1.2702 (M 8 _ 4 -M 7 _ 4 ) .9196 (M 8 _ 5 -M 7 _g) .8886 (M 8 _ 6 — M 7 _ ) .8730 (direct) Average Weighted Average 1.3326 1.3505 .8471 .8363 1.0406 1.0505 .6344 .6608 .9201 .9101 that in all values there are either o, I, 3, or 5 skips. We have weighted them 6, 4, 3, and 2 respectively (ratio about 1.5). This is, of course, pure assumption, nor do we know of any convenient plan of weighting which would not be. All we can 40 Spelling Ability — Its Measurement and Distribution say is that to us the direct values seem to be quite one and one-half times as reliable as those involving a distance which passes over one median, that it seems reasonable the latter should be about as many times more reliable than those involving 3 skips, and that the derivation with 5 skips would be inferior in approximately the same ratio. Weighting therefore as above indicated, we obtain values for the median distances as given in the last line of Table XVI. These are the measures that will be used in this study ; but they differ so little from those obtained without weighting that the latter may serve almost as well. Concretely these results mean that if we represent the differ- ence between no spelling ability at all and the ability of typical 3d-grade children by x, the ability of typical 4th-grade children will be represented by #+1.351, of typical 5th-grade children by # + 1. 35 1 +.836 or x + 2.187, of typical 6th-grade children by x + 3.238, of typical 7th-grade children by x + 3.899, and of typical 8th-grade children by # + 4.809. (The determination of the value of x is not material in the present connection. We shall, however, have something to offer on this point in a later section.) The median distances suggest that so far as spelling is concerned the equal time intervals of one year between the grades do not at all correspond to the differences in ability. The difference between 3d-grade performance and 4th-grade performance is more than twice as great as the difference between 6th- and 7th-grade performance. Whether this is due to a more or less common failure in the 7th grade to give as much attention to spelling as in earlier grades or whether in general 6th and 7th grades are actually closer together than others, is a question which we cannot settle. That a lack of effort to instruct in spelling in the higher grades does not fully account for the differences is suggested by the fact that the 8th grade stands at a greater distance from the 7th than the 5th does from the 4th or the 7th from the 6th. § 12. Scaling the Words Assuming that the normal surface of frequency represents the distribution of spelling ability in each grade, we shall now seek to determine how difficult each one of the 50 words listed Scaling the Words 4i in Tables III and IV is for each grade. A word spelled by one hundred per cent of the pupils in, say, the 3d grade would have no difficulty for that grade. The ability of all pupils would be greater than the ability required to spell it, and the entire area of the frequency surface would lie above it — i.e., to the right of it. In Fig. 12, if OP represents the Probable Error, it would be located theoretically at an indefinite distance to the left of the point O, a distance, however, which we may for practical pur- poses call 5 or 6 times as great as OP — i.e., 5 P.E. or 6 P.E. A word spelled by 98 per cent of the pupils becomes more in- telligible. It would be located at a point K, a vertical at which (KL) would cut off 2 per cent of the area of the entire fre- quency surface. The point K will be found to be at a distance Fig. 12. Normal Surface of Frequency. of about 3 P.E. below the median O, i.e., at 3 P.E. A word spelled by nobody — i.e., a word rated at o — would be at, say, + 6 P.E., and a word spelled correctly by 50 per cent of the group would be located at the median O, that is, at a point above and below which are an equal number of cases. It will be interesting and will serve to show the misleading character of per cent ratings to observe what we mean by saying that one word is more difficult than another. Observe the two following groups of words taken from Tables III and IV for the 3d grade: (A) Per Cent Correct tailor 38 lesson 37 another 36 wear 35 (B) beautiful beginning telephone pigeons Per Cent Correct 42 Spelling Ability — Its Measurement and Distribution According to the ratings of these words the differences in point of difficulty between the words of group A are equal to the differences in group B, for the differences are all represented by i per cent. Habitually we are likely to think that this is true. But such a way of thinking quite neglects the form of distribu- tion of spelling ability. In fact it assumes that the frequency surface is a rectangle — i.e., that there are just as many very poor or very good spellers as there are spellers of medium ability. This we know is not true. The mediocre are always much more numerous than the dull or the gifted. A figure such as Fig. 12 takes account of this fact. Now the words in group A are much nearer the median (which would be a word 50% correct) than are those of group B. They are located on the base line at points such that between adjacent verticals drawn at these points one per cent of the area will lie. The words of group B, more remotely placed with reference to the median, are also so situated that between their adjacent verticals one per cent of the area will lie. But the points for group B stand at greater distances apart than do the points for group A because the verticals or ordinates are shorter for the remoter group. As a matter of fact, the differ- ence in difficulty between " beautiful " and " pigeons " is more than twice as much as the difference in difficulty between " tailor " and " wear," although each difference is represented by the same per cent. Bearing in mind the meaning of these per cent values we may readily place the 50 words of Tables III and IV along the x-axis or base line of a normal frequency surface. " Even," which is rated 59 per cent for the 3d grade, would be at a point below the median between whose ordinate and the median ordin- ate is 9% of the area of the surface. Calling the median zero and referring to Table XIV, we find that 9% of the cases (900 in 10,000) corresponds to a value of P.E. which lies between .3 and .35. By interpolation this value is found to be .338. There- fore the position of " even " is at — .338 P.E. This may be repre- sented on Fig. 13 by the point 1. " Lesson " (37% correct) will be at a point above zero between which and zero are 13% of the cases of a normal frequency surface. Table XIV locates this point at +.49 P.E. (Point 2, Fig. 13). "Only" (65% Scaling the Words 43 correct) is at — .572.P.E. (Point 3, Fig. 13) ; " smoke" (46%) at +.148 P.E. (point 4); "pear" (31%) at +.735 P.E. (point 5) ; " minute " (26%) at +.955 P.E. (point 6) ; " cousin " (19%) at + 1.300 P.E. (point 7), and so on. Words rated above 50% are located below the median ; those under 50% are above the median. Their distances from the median are negative in the first case and positive in the second. Assuming the same form of distribution for the 4th grade we find that "even" (79%) is located at — 1.20 P.E., "only" (75%) at precisely — 1.00 P.E., and " pear " (42%) at +.30 P.E. Similarly for each grade by using the per cents of Table III and IV and the P.E. equivalents of Table XIV we may " place " all ~^2$£. P£. 3 / V 2.ftfE.i +2J?E. Fig. 13. Showing the placing of the first 7 words of the Preferred List. 3d grade. the words. Table XVII gives the per cents and P.E. equivalents of the 50 words of the Preferred Lists which from now on will be treated as one list. Figs. 14, 15, 16, 17, 18 and 19 show how the words appear when arranged on a linear scale for each grade. For the meanings of the numbers, each of which refers to a word of the Preferred List, see Table XVII or Appendix II. Table XVII with its. corresponding figures (14 to 19) affords standards for grade performances. As will be observed, the P.E. values of all the words are calculated for each grade with refer- ence to the median of that grade, which is called zero. Their use may be illustrated with reference to the 4th grade. We may test a pupil of that grade by beginning with the easiest word and proceeding to the next hardest and the next and so on. The series would run : 1 even, 3 only, 2 lesson, or 5 front, 28 chicken, or 41 Tuesday, 4 smoke, 11 pretty, 8 bought. ... By the time 44 Spelling Ability — Its Measurement and Distribution i 4. P 4" [3 f 5 ~K£ «? *i,«oo-g Scaling the Words 45 TABLE XVII Per Cents Correct and P.E. Equivalents for Each Word of the Preferred List. Grades 3-8. See Figs. 14-19 No. of Word Words 3d Year 4th Year 5th Year 6th Year 7th Year 8th Year % P.E. % P.E. % 89 P.E. % 93 P.E. % P.E. % 97 P.E. 1 59 — .337 79 — 1 .196 — 1 .819 —2.188 93 —2.188 —2.789 2 37 4- .492 72 — .864 S3 — 1 .415 91 —1.988 94 —2 . 305 96 —2.597 3 65 — .571 75 — 1 .000 89 — 1 .819 95 —2.439 97 —2.789 99 —3.450 4 46 + .149 69 — .735 85 — 1 .537 94 —2.305 96 —2.597 99 —3.450 5 51 — .037 72 — .864 80 — 1 .248 90 —1.900 94 —2.305 97 —2.789 6 47 + .112 55 .187 69 . 735 78 —1.145 89 —1.819 94 —2.305 7 31 + .735 42 + .299 58 — .299 72 — .864 81 —1.302 94 —2.305 8 40 + .376 65 — .571 79 — 1 .196 91 —1 . 988 94 —2.305 97 —2.789 9 another .... 36 + .531 43 + .261 78 — 1 .145 86 —1.602 94 —2.305 96 —2.597 10 49 + .037 62 — .453 65 — .571 72 — .864 83 —1.415 87 —1.670 11 45 + .187 67 .652 76 — 1 .047 90 —1.900 90 —1.900 94 —2.305 12 35 4- .571 49 + .037 61 — .414 74 — .954 84 —1.475 93 —2.188 13 32 4- .693 52 — .074 61 — .414 73 — .909 74 — .954 87 —1.670 14 26 + .954 38 + .453 62 — .458 77 —1.096 86 — 1 . 602 92 —2.083 15 19 + 1.302 47 + .112 69 — .735 89 —1.819 89 —1.819 9b —2.439 16 43 + .261 58 .299 71 .820 87 —1.670 92 —2.083 96 —2.597 17 19 + 1.302 42 + .299 58 — .299 81 —1.302 81 —1.302 90 —1.900 18 11 + 1.819 29 + .820 42 + .299 58 — .299 79 —1.196 81 —1.302 19 stopping 27 + .909 39 + .414 55 — .187 71 — .820 76 —1.047 84 —1.475 20 13 + 1.670 46 + .149 57 — .261 78 —1.145 86 —1 . 602 93 —2.188 21 29 + .820 46 + .149 68 .693 83 —1.415 86 —1.602 94 —2.305 22 45 + .187 52 — .074 60 — .376 81 —1.302 84 —1.475 93 —2.188 23 22 + 1.145 55 — .187 56 — .224 64 — .531 75 —1.000 85 —1.537 24 carnage .... 13 + 1.670 40 + .376 50 .000 67 — .652 81 — 1 . 302 85 —1.537 25 63 — .492 61 — .414 65 — .571 68 — .693 77 —1.096 94 —2.305 26 already 16 + 1.475 42 + .299 43 + .261 62 — .453 44 + .224 77 —1.096 27 beginning. . . 9 + 1.988 25 + 1 .000 37 + .492 46 + .149 66 — .612 7 b —1.000 28 49 + .037 70 — .778 83 — 1 .415 90 —1 . 900 96 —2.597 99 —3 . 450 29 22 + 1.145 34 + .612 48 + .074 60 — .376 65 — .571 82 —1.357 30 circus 20 + 1.248 39 + .414 50 .000 72 — .864 76 —1.000 95 —2.439 31 11 + 1.819 18 + 1 .357 37 + .492 35 + .571 42 + .299 57 — .261 32 7 + 2.188 29 + .820 41 + .337 57 — .261 70 — .778 82 —1.357 33 15 + 1.537 39 + .414 53 — .112 75 —1.000 86 —1.602 94 —2.305 34 14 + 1.602 35 + .571 40 + .376 52 — .074 71 — .820 78 — 1 . 145 35 38 + .453 55 — .187 70 — .778 75 —1.000 81 —1.302 84 —1.475 36 telegram.. . . 15 + 1.537 31 + . 735 39 + .414 63 — .492 73 — .909 84 —1.475 37 telephone. . . 8 +2.083 35 + .571 48 + .074 67 — .652 83 —1.415 87 —1 . 670 38 tobacco .... 12 + 1.742 39 + .414 60 — .376 75 —1.000 88 — 1 . 742 96 —2.597 39 14 + 1.602 28 + .864 27 + .909 24 + 1.047 30 + .778 43 + .261 40 24 + 1.047 44 + .224 64 — .531 73 — .909 78 —1.145 94 —2.305 41 Tuesday 46 + .149 70 .778 67 .652 80 —1.248 87 —1.670 91 — 1 . 988 42 44 + .224 58 — .299 70 — .778 68 — .693 76 —1.047 87 — 1 . 670 43 whole 17 + 1.415 43 + .261 6J — .531 78 —1.145 84 —1.475 90 — 1 . 900 44 19 + 1.302 30 + .778 54 — .149 75 —1.000 84 —1.475 94 —2 . 305 45 27 + .909 47 + .112 67 — .652 86 — 1 . 602 90 — 1 . 900 97 —2.789 46 butcher .... 33 + .652 59 .337 69 .735 85 —1.537 90 — 1 . 900 97 —2.789 47 guess 20 + 1.248 32 + .693 49 + .037 67 — .652 77 —1.096 Xo — 1 . 537 48 21 + .693 + 1.196 48 54 + .074 .149 62 67 — .453 .652 86 84 — 1 . 602 —1 . 475 87 93 — 1 . 670 —2.188 91 94 — 1 . 988 49 —2 . 305 50 beautiful . . . 10 + 1.900 52 — .074 70 — .778 85 — 1 . 537 94 —2.305 96 —2.597 we have reached 13 button, 22 touch, 50 beautiful, 12 wear, and 48 instead, we are dealing with a group of words which 50 per cent of 4th-grade children spell correctly. The performance of 46 Spelling Ability — Its Measurement and Distribution a given 4th-grade pupil should approximate at least the standard set by these words. If we are asked, "What is 4th grade spelling ability ? " we may answer that it is the ability to spell these words that cluster about the median. Of course it is to be expected that any given pupil will miss some of the easier words and spell some of the harder words. We should test him by the whole series of 50 and his errors for words below the median may be balanced against correct spellings of words above the median at an approximately equal distance. He may miss 49 raise ( — .15 P.E.) but spell 20 sword or 21 freeze (+.15 P.E.). He may miss 16 nails, but spell 7 pear. In such cases he should be credited with having spelled the easier word. In a similar way, by using the words in the order in which they are placed for any other grade, we may determine whether a child is as good a speller as the median children of that grade. We do not need, however, to use the median as a standard unless we wish to. We may choose + 40 or + 60 and ascertain whether children are able to spell up to that point in the same manner as is indicated above for the zero-point. It is to be observed, however, that our series does not offer a very satisfactory test in the higher grades for such standards, because there are so few words that are placed as high or higher than +40 or + 60. The words, in short, are not difficult enough for this purpose. In a later section of the paper we shall introduce harder words into the series precisely with the object of affording a fuller test of ability for the higher grades. There remains, however, for the present one other use which may be made of our data. We may wish to disregard grades altogether and seek an answer to the question, In general, how hard are these words for children of the elementary school above the 2d year? or, with reference to a graphical representation, What is the average position of each word on a linear scale — that position from which the positions for each grade deviate by the smallest amounts? To answer such a question we shall have to use one point of reference for all grades instead of a different one for each grade. In the above treatment we have expressed each word- value as a deviation from the median of the particular grade we were considering. We shall now use this same data but Scaling the Words 47 transfer the point of reference to the third-grade median by- using the median intervals which were derived in Section 11. In Table XVI (page 39) we have given the results of our inquiry into the amounts of these intervals as follows : E. From M 3 to M i 1.351 p.: " M 3 « M 5 2.187 " " M z " M G 3.238 ( " M 3 " M 7 3.899 " " M, " M & 4.809 " Table XVIII gives, for each of the 50 words, its position for each grade when referred to the 3d-grade median as the zero- point or point of reference, together with the " average position " of each word. The method of securing these figures may be illustrated as follows: From Table XVII the P.E. values of the word " even " for each grade, referred to its own median, are shown to be 3d grade, — .337 4th " —1.196 5th " —1.819 6th " —2.188 7th " —2.188 8th " —2.789 The first of these values is of course already referred to the 3d-grade median. To refer the others to the same point we must increase each of them by the amount by which each grade median stands above the 3d-grade median, i.e., we must find the sum (algebraic) of — 1.196 and 1.351, of — 1.819 and 2.187, of — 2.188 and 3.238, of — 2.188 and 3.899, and of — 2.789 and 4.809. These sums give the figures of Table XVIII for the word " even." Their arithmetical mean is taken as the average position. Fig. 20 shows the averages of Table XVIII when reduced to a scale. The noticeable thing about these tabular and graphic representations is the fact that the words from easiest to hardest differ so little. The words " even " and " only " (No. 1 and No. 3), which are — .337 and — .571 respectively for the 3d grade, appear above the zero point at +.699 and +.569. Similarly the word "too" (No. 39) which for the 8th grade alone is + 5.07 becomes for all grades only + 3.491. It is a fact 48 Spelling Ability — Its Measurement and Distribution TABLE XVIII The Position of Each Word in Each Grade when Referred to the 3d- Grade Median as the Zero-point; and the Average Position of Each Word for All Grades, when so Referred. 1=P.E. Word 3d 4th 5th 6th 7th 8th Avern ge Num- ber Word Grade Grade Grade Grade Grade Grade Position 1 even — .337 .492 .155 .487 .368 .772 1.050 1.250 1.711 1.594 2.020 2.212 .699 2 1.135 3 —.571 .149 .351 .616 .368 .650 .797 .933 1.110 1.302 1.359 1.359 .569 4 .835 5 —.037 .112 .735 .376 .487 1.164 1.650 .780 .937 1.452 1.888 .991 1.338 2.093 2.374 1.250 1.594 2.080 2.597 1.594 2.020 2.504 2.504 2.020 1.057 6 1.568 7 pear 1.958 8 bought 1.169 9 .531 1.612 1.042 1.636 1.594 2.212 1.078 10 forty .037 .187 .898 .699 1.616 1.140 2.374 1.338 2.484 1.999 3.139 2.504 1.758 11 1.311 12 .571 .693 1.388 1.277 1.773 1.733 2.284 2.329 2.424 2.945 2.621 3.139 1.844 13 2.026 14 .954 1.804 1.734 2.142 2.297 2.726 1.943 15 cousin 1.302 1.463 1.452 1.419 2.080 2.370 1.681 16 nails .261 1.302 1.052 1.650 1.367 1.888 1.568 1.936 1.816 2.597 2.212 2.909 1.379 17 2.047 18 1.819 2.171 2.486 2.939 2.703 3.507 2.604 19 stopping .909 1.765 2.000 2.418 2.852 3.334 2.213 20 1.670 2.500 1.926 2.093 2.297 2.621 2.185 21 .820 1.500 1.494 1.823 2.297 2.504 1.740 22 touch .187 1.277 1.811 1.936 2.424 2.621 1.709 23 whistle 1.145 1.164 1.963 2.707 2.899 3.272 2.193 24 carriage 1.670 1.727 2.187 2.586 2.597 3.272 2.340 25 —.492 1.475 .937 1.650 1.616 2.448 2.545 2.785 2.803 4.123 2.504 3.713 1.652 26 already 2.699 27 1.988 2.351 2.679 3.387 3.287 3.809 2.917 28 .037 .573 .772 1.338 1.302 1.359 .897 29 1.145 1.963 2.261 2.862 3.328 3.452 2.502 30 1.248 1.765 2.187 2.374 2.899 2.370 2.141 31 1.819 2.708 2.679 3.809 4.198 4.548 3.294 32 2.188 2.171 2.524 2.977 3.121 3.452 2.739 33 quarrel 1.537 1.765 2.075 2.238 2.297 2.504 2.069 34 saucy 1.602 1.922 2.563 3.164 3.079 3.664 2.666 35 .453 1.164 1.409 2.238 2.597 3.334 1.866 36 telegram 1.537 2.086 2.601 2.746 2.990 3.334 2.549 . 37 telephone 2.083 1.922 2.261 2.586 2.485 3.139 2.413 38 1.742 1.765 1.811 2.238 2.157 2.212 1.988 39 1.602 1.047 2.215 1.575 3.096 1.656 4.285 2.329 4.677 2.754 5.070 2.504 3.491 40 1.978 41 .149 .573 1.535 1.990 2.229 2.821 1.550 42 tying .224 1.052 1.409 2.545 2.852 3.139 1.870 43 1.415 1.612 1.656 2.093 2.424 2.909 2.018 44 against 1.302 2.129 2.038 2.238 2.424 2.504 2.106 45 .909 1.463 1.535 1.636 1.999 2.020 1.594 46 .652 1.014 1.452 1.701 1.999 2.020 1.473 47 1.248 2.044 2.224 2.586 2.803 3.272 2.363 48 instead .693 1.425 1.734 1.636 2.229 2.821 1.756 49 1.196 1.900 1.202 1.277 1 . 535 1.409 1.763 1.701 1.711 1.594 2.504 2.212 1.652 50 1.682 Scaling the Words 49 to o > ft fcvS © "7! 4) K-H ^,d iS o 4) O O O o-.* """ to D r-KNCCtfiQcot^occiO r coeocooocococofoco-* "-,«; c« e— t 8 °w +* -■ top.. Ml © V. a ft „ ,2 M ,b'3 g © 8 s 3 fc ft © m.2 o g ft CS «8 ©,£) ©,8^.3 y Kf*** §^ --((McOTtHiOCOl^OOOiO (N >§aaaS©og§ * *T3 C— O anS o &0 c3«« t-HC<)C0-^tfitDt>00O© e 50 Spelling Ability — Its Measurement and Distribution then that for these words the influence of higher grades is to make easy words harder and of lower grades to make hard words easier. That is, grade considered, these words are harder for children of the upper grades than they are for those of the lower grades. There are at least two reasons for this condition. First, as a rule these particular words are taught in the lower grades. A popular speller, taken at random, presents 31 of the 50 words in the 3d year's work, 10 in the 4th, 2 in the 5th, and none in higher grades. There is nothing to lead one to suppose that this is peculiar. The words were among those chosen, it will be remembered, as at least in the speaking vocabulary of 3d-grade children. Most of them if taught at all will be taught in that grade. We may assume therefore that the 3d-grade record is somewhat affected by the recency with which these words have been presented. The succeeding grades will to some extent be discriminated against in the record. Second, the necessary basis of selection for these words from the larger lists would make it impossible for the words to take the same position on the scale for all grades. Consider a word which was spelled correctly by 50% of the 3d-grade children. Such a word would be at M z . In order to take the same position on the 8th-grade record it must be as far below M 8 as is the distance, already determined, between M 3 and M s or — 4.809 P.E. To do this it would have to be spelled correctly by 9994 pupils out of 10,000 (Table XIV), i.e., it would be 100% correct. But such a word would not have been selected, because it is not difficult enough in the 8th grade to be of any value as a test of ability. On the other hand, a word missed often enough in the 8th grade to be satisfactory as a test (say, 90% correct) would have to be less than 1 % correct on the 3d-grade record in order to take the same position on the scale. Such a word would have been of no use as testing 3d-grade ability and would have been rejected. The fact is that the span from 3d to 8th grade is — if our median distances be correct — too great for any list of words to be in all respects satisfactory. We need several lists each of which shall be given to three or four consecutive grades and overlapping on one another — e.g., one for 2d, 3d, and 4th grades, another The Use of the Scale Si for 3d, 4th, 5th and 6th grades, and another for 5th, 6th, 7th and 8th grades. An attempt will be made in a later section to do this and to show the results that may be expected. § 13. The Use of the Scale Meanwhile, however, we venture to think that the scale as ^hown in Fig. 20 is important and valid within its range. It may be used in several ways of which at least three are important. 1. It may be used just as it is without reference to the fact that the words are not separated from each other by equal intervals. We know the value or weight to assign to each word. We shall therefore not make the mistake of assuming that all the words are of the same value, as is the usual school practice. 2. Certain words of the series may be used which differ from each other by approximately equal steps. TABLE XIX Words Arranged in Order op Difficulty According to the Scale and Their P.E. Values No. on Scale 3 1 4 28 5 11 16 46 41 6 45 25 49 15 50 22 21 10 48 12 35 42 Word only even smoke. . . . chicken . . front another. . lesson. . . . bought. . . pretty nails butcher . . Tuesday. . sure answer. . . nor raise cousin. . . . beautiful . touch freeze. . . . forty instead.. . wear tailor tying P.E. x 100 57 70 84 90 106 108 114 117 131 138 147 155 157 159 165 165 168 168 171 174 176 176 184 187 187 No. on Scale 14 7 40 38 43 13 17 33 44 30 20 23 19 24 47 37 29 36 18 34 26 32 27 31 39 Word minute. . . pear towel tobacco . . whole .... button . . . janitor . . . quarrel. . . against. . . circus. . . . sword. . . . whistle. . . stopping . carriage . . guess telephone, choose . . . telegram . saucer saucy already. . . pigeons. .. beginning grease. . . . too P.E. x 100 194 196 198 199 202 203 205 207 211 214 219 219 221 234 236 241 250 255 260 267 270 274 292 329 349 52 Spelling Ability — Its Measurement and Distribution 3. Small groups of words may be so selected as to be equally difficult as groups ; or they may be so selected that their group-difficulties constitute an ascending series from easy to hard, differing by equal amounts. 1. By the first of these methods the entire series would be utilized or so much of it as in any given case would thoroughly test the subject. The order of the words of the series as given in Figure 20 is shown in Table XIX in the first column and in the second column the test values or weights of these words are given. 2. If it is desired to use a scale whose words differ in difficulty by equal steps, the arrangement as shown in Table XX will be found convenient. TABLE XX A Ten-point Scale No. of Word (Fig. 20) Word P.E. x 100 3 4 9 11 45 35 30 37 34 27 only smoke another. . pretty answer. . . tailor .... circus. . . . telephone, saucy .... beginning. 57 84 108 131 159 187 214 241 267 292 27 24 23 28 28 27 27 26 25 To this series may be added 39 " too " whose P.E. x 100 is 349 and which differs from " beginning " by 57 or approximately two steps. In the series of Table XX the average step is 26.2 with an A.D. of 1.3; or if the word " too" is included the average step is 26.6 with an A.D. of 1.4. This is quite accurate enough for any use to which the scale is likely to be put. If this conclusion The Use of the Scale 53 is accepted, these eleven words may be used to express our judgments of other words concretely and in terms that every- body can understand. We should not then have to resort to such terms as " hard," " easy," " rather difficult," " very hard," etc., but we may judge a word to be " as hard as ' another,' " " equal in difficulty to ' beginning,' " " as hard as ' answer ' but not as hard as ' tailor,' " etc. It is very desirable that other words should at some time be added to the scale at both ends. There are many words harder than " beginning " or " too " and there are others easier than " only," although the latter do not constitute much of a school problem. Neither set, however, could be used over a range as wide as 3d to 8th grades. 3 (a). It is often desirable to offer tests of equal difficulty, but of different words at various intervals of time to the same group or to the same individual. We may thus secure a progress record. In spelling, however, this has proved to be very difficult if not impossible. We can never be sure that the second or third test is equal in difficulty to the first test. In fact we may be pretty sure it is not. To give the same words over again is often valueless because of the added special familiarity with them. The following lists therefore are offered as lists of equal difficulty. The sum of the P.E. values in each is 976 or 977. In using them the words may be weighted as indicated, or may, with no great loss in precision, be each given a credit of 1. Number _ Number in Preferred in Preferred List Weight List Weight Group A Group B 41 Tuesday 16 45 answer 16 10 forty 18 48 instead 18 40 towel 20 43 whole 21 44 against 22 17 janitor 21 47 guess 24 24 carriage 24 Group C Group D 49 raise 17 21 freeze 18 22 touch 17 12 wear 19 42 tying 19 7 pear 20 14 minute 20 13 button 21 18 saucer 27 20 sword 22 Group E Group F 16 nails 14 8 bought 12 46 butcher 15 11 pretty 13 15 cousin 17 19 stopping 23 29 choose 26 37 telephone 25 32 pigeons 28 34 saucy 27 54 Spelling Ability — Its Measurement and Distribution (b) It may also be desirable to test, not with single words, which only in the long run may be expected to conform to the positions assigned to them, but with groups of words whose difficulties as groups differ by constant amounts. Such a series of groups arranged from easy to hard would themselves con- stitute a scale — a sort of Binet-Simon scale for measuring ability in spelling. On the analogy of the Binet-Simon scale we might easily fix upon a certain minimum performance for a group at which or better than which a subject might be allowed to have " cleared " that group and might pass on to the next. He might also be given additional credits for spelling words in groups above the highest one which he cleared. The groups are arranged in order of difficulty, Group I being the easiest and Group VII the hardest. Within each group the four words are also arranged in their order of difficulty begin- ning with the easiest. Since, however, within each group the words differ little in difficulty, they may be taken as having equal weights without material error. It is true that Group VII is not nearly as satisfactory in this respect as the others, differing between the first and fourth words by 1.08 P.E., whereas the first six groups have a range of but .225 on the average. Group I 3 only. . . . 1 even. . . . 4 smoke . . 28 chicken . Average. Group II 5 front 9 another 2 lesson 8 bought Average. Group III 16 nails 46 butcher 41 Tuesday 6 sure P.E. x 100 57 70 84 90 75 P.E. x 100 106 108 114 117 111 P.E. x 100 138 147 155 157 Group IV 10 forty 12 wear 42 tying 38 tobacco . . Average. Group V 33 quarrel 30 circus 24 carriage 47 guess 29 36 34 26 Average. Group VI choose telegram saucy already Average . 149 Group VII 37 telephone . 32 pigeons . . . 31 grease .... 39 too Average. P.E. x 100 241 274 329 349 P.E. x 100 176 184 187 199 186.5 P.E. x 100 207 214 234 236 223 P.E. x 100 250 255 267 270 260.5 Average . 298 The Zero Point of Spelling Ability 55 The average differences in difficulty between these groups in succession are 36, 38, 37.5, 36.5, 37.5 and 37.5. This is probably the most important use of the scale, for present school practice. If it is true that the general scale (Table XVIII and Fig. 20) may be used in these three ways — as a whole, by words selected to be at equal intervals, and by grouping words so that the groups are equal or differ by equal amounts — then it is also true that each of the grade scales (Figs. 14-19) may be used in like manner each for the grade to which it applies. It is probably true, moreover, that the grade scales will more closely fit real conditions in any given instance than will the scale for all grades. The labor of making selections and groupings of words for these scales is not great and may be made by any one on the analogy of the method used above. § 14. The Zero-Point of Spelling Ability As has been suggested in previous sections, we have only suc- ceeded in scaling by means of these 50 words a limited segment of the entire projection representing spelling ability. Our list is essentially an easy list, testing that ability only to a moderate degree. Words like " fatiguing," " guarantee," and " conscien- tious " (Rice Sentence Test) would stand much higher in the scale and require a considerable extension of it to the right; while such unfamiliar words as " eurycerous," " delitescence," and "gallinaceous" (Klein, '12, pp, 388, 389) would take still higher positions, passing quite beyond the range of the ability of elementary-school children. On the other hand, our scale is as certainly limited at the low end. There are many easier words than any we have used so far. Such words would reach far down on the scale towards the place where the absolute zero-point lies. But they would have been totally unfit for use in the higher grades. In fact, with the wide range of ability between 3d and 8th grades, it is surprising that we find any words at all which will afford a test at both extremes. Without seeking to determine the limit of the high end of the scale — perfect spelling ability — it is quite possible, and theoretically very desirable, to find the limit of the low end, i.e., 56 Spelling Ability — Its Measurement and Distribution to find the point where spelling ability just begins to be a positive quantity. How far, then, below the 3d-grade median, which has hitherto been our point of reference, is the absolute zero-point? In order to answer this question, a test was given to children of the 2d, 3d, and 4th grades. It consisted of 50 words in sentences. Nineteen of these had already been used in the Selected List (100 word list) ; and, of these, 6 had been chosen for the Preferred List. They had all been spelled, about 40 per cent or more correct, by the third-grade children. The remain- ing 32 words were thought to be among the easiest in the language: he, is, on, the, to, of, for, day, etc. They were put into sentences as follows and dictated at schools II and VIII: Easy 50-W0KD Test 1. You will hear him coming. 2. He is on the road and is almost sure to pass in front of me. 3. / send for him every day. 4. Go into the school. 5. But do not touch the table. 6. He also has only one pair of shoes. 7. They are not at all pretty. 8. No man ought to steal even a penny. It seems clear that a child who cannot spell any one of these words has substantially no spelling ability. Since our study is limited to written words we shall say, therefore, that for our purpose a child who does not, save by chance, write a single word of this list so that it can be recognized as correctly spelled has no ability. On account of the marked improvement in spelling of children in the latter half of the second school year over those in the first half of that year, we have treated the two half-years of the 2d grade separately, calling the lower half 2a and the upper lb. We shall proceed as follows. We shall find the distance between the 3d-grade median and the 2&-grade median and the distance between the latter and the 2a-grade median. Then if there are children of the 2a grade who utterly break down and The Zero Point of Spelling Ability 57 fail to write any word correctly we shall find their place in the 2a distribution. Table XXI shows the records of individual pupils according to their rating in the Easy 50-Word Test. Note the fact that no children of the 4th, 3d, or 2b grades wholly failed in this test. In 2a, however, 39 children were rated 10% or less, and of these there were 8 pupils who were actually marked zero. This is 4.6% of all the children of 2a. TABLE XXI Distribution of Individual Ratings. East 50-word Test Table reads: in 2a 39 children, or 22%, were rated between and 10%; 32 children, or 18%, were rated between 11% and 20%, etc. In 26 5 chil- dren, or 3%, were rated between 11% and 20%, etc. Per Cent Correct 2a Grade 26 Grade 3d Grade 4th Grade No. % No. % No. % No. % 0- 10 39 32 37 27 18 14 5 2 1 22 18 21 16 10 18 3 1 .6 5 9 29 26 47 31 14 7 1 3 5 17 15 28 18 8 4 .6 1 4 7 11 25 33 36 30 21 .6 2 4 7 15 20 21 18 13 4 13 29 50 86 134 11-20 21- 30 31- 40 41- 50 1 51- 60 4 61- 70 9 71-80 16 81- 90 27 91-100 42 Totals 175 169 168 316 Medians 26.50 56.17 72.50 88.12 The medians for the grades are as follows : for 2a, 26.50% ; for 2b, 56.17% ; for 3d grade, 72.50% ; and for 4th grade, 88.12%. The rapid rise of spelling ability from low second through the fourth grade is remarkable. It is much greater than the improvement during the next four years. Although the interval in time between 2a and 2b is but half a year, the medians suggest that the increase in ability between these grades is greater than it is between any consecutive yearly grades above the second. Further analysis will more precisely confirm this inference. Proceeding as in the case of grades 3 to 8, we show in Table XXII the amount and per cent of overlapping of each 58 Spelling Ability — Its Measurement and Distribution grade beyond the medians of the other grades, together with the corresponding linear segment in terms of the Probable Error as a unit. TABLE XXII Amount and Per Cent of Overlapping with P.E. Equivalents. Easy 50-word Test 2a Grade 2b Grade 3d Grade 4th Grade 2a grade. 26 grade. 3d grade. 4th grade No. % P.E. No. % P.E. No. % P.E. No. % P.E. 160 94.67 2.3932 165 98.21 3.1143 316 100 ? 17 9.71 1.9254 140 83.33 1.4341 308 97.47 2.8976 3 1.71 3.1429 22 13.02 1.6690 262 82.91 1 . 4094 ? 4 2.37 2.9395 30 17.86 1.3649 From these results, Table XXIII is computed. The object in this table is to show how, by using all the data of Table XXII, various values of the median intervals may be obtained whose averages will be the most probably correct values. The interval between the medians of 2a and 2b is written M2a-2b; that between the medians of 2b and the 3d grade is written M2&-3 ; etc. It may be remarked parenthetically that in the number 1.3771 of Table XXIII for the difference between the 3d- and 4th- grade medians, we have a striking confirmation of the substan- tial accuracy of our results as shown in Table XVI. The corresponding number is there given as 1.3505. That these should differ by so little when carried out upon different test material is exceedingly satisfactory. According to Table XXIII, the 2&-grade median is approxi- mately 1.35 P.E. below the 3d-grade median. The 2a-grade median is about 1.87 P.E. further below, or 3.22 P.E. below the 3d-grade median which we have thus far used as our origin or point of reference. But we have not yet reached the point of zero ability. Typical The Zero Point of Spelling Ability 59 TABLE XXIII Values of Median Intervals and Theie Derivation (2c-4th Grade) ■^20—26 -M2&-3 ^3-4 1.9254 (direct) 1.6690 (direct) 1.3649 (direct) 2.3932 (direct) 1.4341 (direct) 1.4094 (direct) 1.4739 (^2a- 3 — M 26- 3 ) 1 2175 (-^2a-3— M 2a-2b) 1.2705 (^2&-4— M 26- 3 ) 1.6802 (^3a-2— ^3-26) 1.5746 (^26-4-^3-4) .7211 (^3-2a— ^26-2a) 1.4882 (^4-26— M 4- 3 ) 1.4635 (^4-26— M 3-2b) Averages 1 . 8682 1.3518 1.3771 2a children have some ability, namely, according to our record, an ability to score 26.5% in the Easy 50- Word Test. The children of that grade who were unable to write any word correctly were 8 in number, representing 4.6 per cent. These 8 are included in the 39 rated between zero and 10% (Table XXI). Assuming that 2a children are grouped about their median according to the " normal " distribution, we find that in order to cut off 4.6% from the low end we must take a point 2.5 P.E. below the median, (See Table XIV). This brings the zero-point at 5.72 P.E. below the 3d-grade median (3.22 + 2.5). This figure, 5.72 P.E., can only be taken as approximately correct. It would be somewhat influenced by an increase of the number of children tested. There is, however, no reason to suppose that the children h schools II and VIII were unusual. The testing in grades 3 to 8 in all other schools shows that results in these two schools do not materially differ from the general results. In both central tendencies and variabilities they are a fair average. There seems, then, to be no good reason why we should not consider the ratings of children in these 60 Spelling Ability — Its Measurement and Distribution schools as typical. It must be borne in mind, however, that the classification of children into grades is a broad one. Just as we found it necessary to treat 2d-year children in half-yearly sections, so we should find that testing at the beginning even of a 20-week term would yield results, especially in the low grades, quite different from those obtained by testing toward the close of the term. Accordingly, the middle of the term is the best time at which to find typical conditions. Moreover, in order that the results may be comparable, the testing of all grades should be done at the same time. If 2a children were tested towards the end of their term in that grade, while 2& children were tested towards the beginning of theirs, the median interval would be unduly shortened. A considerable addition to the reliability of our results is found in the fact that all children were tested during the ioth week of a 20-week term. We may therefore conclude that the intervals between grades 2a, 2b, 3 and 4 are substantially as found in Table XXIII. But as to the distance of the zero-point below the 2a-grade median, we cannot be precise. Four and six-tenths per cent of the 2a children got no word right. As many as 22 per cent wrote less than 6 words correctly. Some of them probably spelled these few simple words correctly by mere chance. If this were true, they would have practically no spelling ability. The situation is more complicated than the above analysis indicates. If we were to assume that all the children who wrote 0-5 words correctly had practically no spelling ability (written), the zero- point would then be but 1.15 P.E. below the median instead of 2.5 P.E. If we were to assume that some of these children — say those who wrote no more than 3 words correctly — had zero ability, we should find that 29 of the 39 in Table XXI were included. Such an assumption would place our zero-point at 1.44 below the median. There are reasons for thinking that this is not far from the true position. The best judgment, there- fore, that we can make from our data and from our knowledge and experience of school conditions is, that the zero-point is about 1.5 P.E. below the 2a median, or about 4.72 P.E. below the 3d-grade median. We may summarize our results, then, in Table XXIV and Fig. 21, as follows: Observations on the Distributions Shown in Fig. 21 61 TABLE XXIV Median Intervals. Zero to 8th Grade Median Successive Distance Intervals Above 1.50 1.50 1.87 3.37 1.35 4.72 1.35 6.07 .84 6.91 1.05 7.96 .66 8.62 .91 9.53 2a grade 26 3d 4th " 5th " 6th " 7th " 8th " Fig. 21, page 62, shows these facts graphically. § 15. Observations on the Distributions Shown in Fig. 21 It is to be remembered that in Fig. 21 the eight surfaces of frequency constructed on each median vertical are theoretical and not according to the record. Moreover, they express the assumptions that for each grade the distribution of ability in spelling is strictly " normal " and that the real variability is alike in all grades. In a later section we shall take up the matter of applying to our results distributions which are not normal. Meanwhile, however, it will be interesting to observe how satis- factory a strictly normal form of distribution proves to be. To the extent that it expresses the same or nearly the same facts as the record (so far as it should, if valid, do so), it shows its value. 1. In Fig. 21 the 2a surface of frequency does not reach the 4th-grade median; but it only falls short a little. According to the record in the Easy 50-Word Test no 2a child did as well as the median 4th grade child. But the best 2a record was 82% which is only a little less than M 4 (88.12 by Table XXI). 2. In the graphic showing the 3d-grade distribution does not quite reach the 8th-grade median. Similarly the record shows that no 3d-grade child obtained a score equal to 94.68 which (Table XI, p. 27) is M 8 for the Selected 100- Word List; al- though 3 third-grade pupils had scores in the 91 to 95 group. (Table XI, column 2.) 3. By the figure we see that the low end of the 8th-grade distribution falls short of M z but not of M 4 . In the record the 62 Spelling Ability — Its Measurement and Distribution _2 * c 8 rt S3 c •rl w fe *H II *i •g u rt C *^ U^i >> o\ I' to C ... _ »^ G. ^ co ca o «° II »1 ^ O II ^ C8 C\ C^*0 rf >+- » IS 4- G^rCLcU 2^ IH J7 2» *7 36 <# J 2 31 21 il -to I I I l_l II II I 1 U_J LJil U l_l l_U 1 1 L_ I" U 19 3f 2 **» H V n tola n 28 !-V< oyaLxsV Mil 3i 3fl \5" CvoAe i.i i 1 i— j i < i i , I? so 2. 8 3t 28 3* 31 1 1 1 1 1 i "" ■ ■ i <^&mdU. 71 *W jp" i7,t 7t 6 ' ^ n & Si / 6*vc«.<:Le.. 8 Gcv SO i /00 . /*] ( /W '/to '/80 .200 Tt^wwaTL.Uv s it i // a ;o is TW Se>i*c~ t c Test * n, i ii *>Mlt M»/« .r; Ji t?j.« 7»a 2s Vis n 12^1, hi * 5 S taiM JO-Wor.CY.i'.V '/ fivtftil. ?r.V "V &«v« /fc/Hi2iw arw W3J1 J72onf+0) ■■ '<., ,.\. IS Ii 2v ;j t? II J>„ °yv "?T«W«A~L\.»V. "Yfc. S« to % „** 3« «&« (oil, I Jl* *A" i"" "'' 4* r.rn.t. 1W-*«aX'v»*: "^M.. $«.CVi.r.c.. '\t > <7 H[ Jz 1 ',,w"n ,7j , f 7r " ' "i" Completed Grade The ^ili grade si Scale. Grade lie should liavi Combining .54 of the the Rice Rice List Sentence List Table XXXI with the Preferred List. Table XVII. to which is also added for the 4 th grade, tbeEw at+Tss; the 6,hg?ade scale should have word 82 of the Rice List jo Word List Rii ■ lit ii Tabic '■:•■: \i l The c ( jt7 For thi ■ no) absolutely 11. 'I to ci Ippendia 1 1 Rice Sentence Test. Easy 50-Word Test 81 TABLE XXXII {Continued) Word 2a Grade 26 Grade 175 Pupils 169 Pupils % P.E. % P.E. 51.4 — .052 85.2 — 1 . 549 .6 +3.725 6.5 +2.245 1.1 +3.392 5.9 +2.321 48.0 + .074 78.7 — 1 . 181 1.1 +3.392 27.8 + .873 45.7 + .160 91.1 —1.997 12.0 + 1.742 27.8 + .873 25.7 + .968 64.5 — .551 54.3 — .160 84.6 —1.512 45.1 + .183 93.5 —2.245 22.3 + 1.130 25.4 + .982 34.3 + .600 78.7 —1.181 3.4 +2.706 39.6 + .391 36.6 + .508 89.3 —1.843 45.1 + .183 87.0 —1.670 34.3 + .600 71.0 — .820 16.0 + 1.475 60.4 — .371 13.1 +2.767 58.0 — .299 40.1 + .372 74.0 — .954 33.7 + .624 69.2 — .744 .6 +3.725 4.7 +2.483 1.7 +3.146 13.0 + 1.670 2.9 +2.811 15.4 + 1.512 36.0 + .531 65.1 — .575 5.7 +2.344 16.6 + 1.438 34.9 + .575 71.0 — .820 2.9 +2.811 11.2 + 1.803 1.7 +3.146 18.9 + 1.307 16.6 + 1.438 34.3 + .600 18.3 + 1.340 58.0 — .299 43.4 + .246 72.8 — .900 36.6 + .508 62.1 — .457 2.9 +2.811 20.1 + 1.243 23.4 + 1.076 71.0 — .820 50.9 — .033 74.0 — .954 1.1 + 3.392 12.4 + 1.713 1.7 + 3.146 12.4 + 1.713 5.1 +2.425 24.3 + 1.033 60.6 — .399 72.8 — .900 3.4 +2.706 13.6 + 1.629 3d Grade 168 Pupils % P.E. 4th Grade 316 Pupils % P.E. and . . , almost sure. . . to pa in front . . of me. . . . I send . . for. . . . every. , day . . . go ... . into. . . school . but . . . do ... . not . . . touch . table . . also . . . has . . . only. . . one . . . pair. . . shoes., they . . are. . . . at all ... . pretty, no ... . man. . . ought . steal . . even . . a penny . 97.0—2.789 42.9 + .265 48.8 + .044 93.5—2.245 51.2— .044 92.9 48.2 82.1 88.7 95.2 48.2 93.5 66.7 94.0 89.3 76.2 81.5 83.3 85.1 86.9 29.8 53.6 35.7 69.6 53.0 82.1 42.9 46.4 64.3 84.5 75.0 73.2 43.5 85.1 79.8 20.8 29.8 48.8 86.3 35.7 —2.177 + .067 —1.363 —1.795 —2.468 + .067 —2.245 — .640 —2.305 —1.843 —1.057 1.329 1.432 —1.543 —1.663 + .786 — .239 + .543 — .761 — .112 —1.363 + .265 + .134 — .543 —1.506 —1.000 — .918 + .243 —1.543 —1.238 + 1.206 + .786 + .044 —1 . 622 + .543 98.4 56.3 61.1 94.9 65.8 93.0 67.1 91.5 95.9 100.0 76.6 96.8 88.0 100.0 98.1 89.9 92.7 98.4 97.2 96.8 54.4 94.3 68.4 91.5 70.6 94.6 77.5 76.9 88.0 95.9 86.7 90.2 78.5 96.5 97.2 53.8 67.1 67.4 98.4 63.6 —3.182 — .235 — .418 —2.425 — .603 —2.188 .656 2.035 —2.579 ? -1.076 -2.746 -1 . 742 ? -3 .'077 -1.892 -2.155 -3.182 -2.155 -2.746 - .164 -2.344 - .710 -2.035 - .803 -2.384 -1 . 120 -1.091 -1 . 742 -2.579 -1.649 -1.918 -1 . 170 -2.686 -2.155 - .141 - .656 - .669 -3.182 - .516 82 Spelling Ability — Its Measurement and Distribution Figures 26, 27, and 28 give the scales for these words. In Figure 27 it is indicated below the line with the omission of the six words noted in the last paragraph. Above the line the words of the Preferred List are reproduced from Figure 14. Since the Easy 50- Word Test was also given to 4th-grade chil- dren it is likewise scaled for that grade omitting the same six words. (Fig. 25, 4th grade, lower line.) We have, therefore, scales for every grade from the first half of the second grade to and including the eighth. All of these scales above the 2d grade are much richer than were those given in Section 12. There are fewer gaps in them and their range is greater. They may be used to great advantage in testing the spelling ability of children in any grade of the elementary school in which children are supposed to have any such ability. If it is not convenient to use a whole scale, certain words differing in difficulty by approximately equal amounts may be selected. Groups of words may be made each of equal difficulty as a group, or each differing from the preceding group by a fixed amount. The position of each word shows the weight which ought to be assigned to it for test purposes. Each of these grade scales refers to the median of the grade as the zero-point. In Figure 29 is shown a scale for all grades referring, as in Figure 20, to the median of the 3d grade as the zero-point. Above the line is shown the Preferred List as in Fig 20. Below it are arranged the words of Rice's Test; and on a parallel scale the Easy 50- Word List. Caution, however, ought to be observed in accepting too literally the showing of the last two lists. Rice's Test was not given to the 3d grade, and the Easy 50-Word Test was given to the 2d grade and was not given above the 4th. They cannot, therefore, be closely compared with the Preferred List. The effect of high grades is to make the words harder, of low grades to make them easier. In the case of the Rice Test the words are probably a little — but only a little — too far to the right, — i.e., farther toward the high end than they would have been had they been used in the 3d grade — as Cornman used them. In the case of the Easy 50- Word Test the words would be a great deal too far to the left if set down as the record indicated. The six words common t-l&O ■ -200 +2.20 *2£0 7 r 35" h% »7 6,0 i 2. Utti 7 IZ 13 i 80 +/O0 S-/20 ft 23 Jo 2.1 WW j j0 ntim ■I] fO 670 690 710 80 +200 +220 + 240 /2*l /««tc«ca. Test. t 3 s. M 40-Wo»dl.\vl.T«We»'nil. XSSI »t-J ***» . lt,i-iJL ^ ^se^e-n^c-. it- &.<-«.de NUa'uvv (L.» t rt; r K) W ^overu ? f y, to Qb4oUTe 2.-.O. "?™V.vr«.a_~£.\.,v ■ 2& H ;« 4i J! 7» J3 j, « *7 l< J7 21 JI I' J* J! Xvce. ScoV.^,.,. '-V ca >c 31 KMJ7- 15 «« / fl » '4, H 2si a nn ?.r iss SIMSUU if ?V7/ 70 Figs. 26-29. For the various lists referred to, sec Appendix IT. Tlir scale for Fig. jo is not complete, word S2 of the Rice List being at +685 from the 3d-grade median, or +1135 from absolute zero. Rice Sentence Test. Easy 50-Word Test 83 to this list and to the Preferred List enable us to suggest a correction. Their P.E. values, when the averages of the grades writing them are taken, appear as follows : Easy 50- Preferred word List List Increase sure +.530 +1.57 1.04 front —.299 +1.06 1.36 touch +.652 +1.71 1.06 only —.089 —.57 .66 pretty —.024 +1.31 1.33 even —.097 + .70 .80 Average Increase 1 . 04 It appears, therefore, that in order to compare the words of the Easy 50-Word List with those of the Preferred List and to scale them together we ought to raise all the words of the former list about 1 P.E. In Figure 29, accordingly, all these words have been raised that amount. Fig. 29 shows our most complete scale. It has decided limita- tions, and it is impossible — in the case of the newly added words — to suppose that it is more than an approximation. A great deal more testing than we have been able to do will have to be done before these words and others with them can be precisely fixed beyond dispute. It is not claimed that the scale we give is final. We think, however, that, supposing the two fundamental assumptions upon which it is based to be valid, it may be used in its present form with substantially accurate results ; and we are confident that the general method by which it has been derived is the one by which a final scale may ulti- mately be secured. The top figures in Fig. 29 refer to the absolute zero-point, taken as 470 below the 3d-grade median. It enables us to state not only the difference in difficulty between words but their relationships. We may say, for instance, that school (No. 27 E. 50-W. L., scales at 428) is one-half as hard as grateful (No. 51 R. S. T., scales at 856). We may put certain facts in equation form as follows: in = y light = y pigeons = y fatiguing is = y 2 also = y occasion = % conscientious the = y chicken = y approval and = y penny = y peculiar 84 Spelling Ability — Its Measurement and Distribution Many more such statements may be made. It will, we think, surprise most people to learn that fatiguing is only four times as hard as he, or that to spell occasion shows but three times as much ability as to spell is. In fact it will, we think, be seriously questioned whether such words as at, of, on, do, etc., have difficulties anything like as great as is shown on our scale. It will be asked, What words can be easier than these? If a child cannot spell them does he not show zero ability ? The answer is that if one or more of these very easy words were isolated and pronounced to a group to be written, those who could not spell them would indeed show no spelling ability. But these words were not isolated, they were given in a context. It is one thing for children to write the word " at " when pronounced alone or in column dictation. It is quite another to write it in the sentence : " They are not at all pretty." Some will omit it, and this fault is not confined by any means to the lowest classes. Some will connect it with the word all, because they habitually do so in speaking. Some little children will quite break down on the whole sentence because they can't get over the word " they." In other sentences some will substitute a word (generally of similar meaning) for the one dictated. Each of these faults scores " wrong," and none of them would be made in column dictation. It is also true that children writing sentences more often write illegibly than they do when writing a few words in columns ; and this is particularly true with young children. It will therefore be clear that the decision as to how hard a word is, depends on how you use it in testing and when you call it " wrong." To verify the placings of the words given in this study one ought to use the same test material and the same method of scoring. In particular, column dictation will not do at all. § 19. Derived Forms of Distribution The foregoing treatment of the measurement of spelling ability has, as has been indicated frequently, proceeded upon the assumption that the distribution of ability is in all grades normal. Such an assumption has always been made in the investigation of school abilities by persons whose knowledge of Derived Forms of Distribution 85 the theory of statistics has enabled them to do so. In Section 10 I have said : " There seems no good ground for assuming that the distribution of spelling ability in any grade is not according to the normal curve or according to a curve which resembles it closely." By this alternative is suggested the possible applica- bility of certain curves not of normal form but resembling the normal form. Our problem will now be to derive and apply to some of our material such modifications of the type form of distribution as our present knowledge of grade conditions permits. In order that the frequency of measurements within a group may be distributed according to the Probability Integral it is necessary that the group be in no way selected on the basis of the characteristic that is measured. It must be a random sampling from a " total population." If the frequency distribu- tions of statures for adult males born in the City of New York may be expected to approximate the symmetrical type, the distributions of statures for adult males on the police force of New York City would not do so. Their curve will be of the "moderately asymmetrical type" being cut off at the low end because extremely short men are at a disadvantage in the group supposed to be measured. In other words, there is a selection on the basis of stature. The group " adult males on the police force of New York City " is not a random sampling of the total population " adult males of New York City." The question then is: To what extent does the membership of each grade of the elementary school fail of being a chance selection from a total population ? We may fairly assume, in the first place, that the pupils of the first and second grades are unselected. Practically all children attend school and none drop out in these grades. From the 3d grade on, however, each successive grade constitutes a group which is less and less a random sampling. Many influences are at work to eliminate a greater and greater number of individuals. Probably the most important of them is the inability of children to progress — i.e., lack of ability in the lines of work now required by the schools. The extent to which elimination takes place in the grades has been the subject of study by a number of investigators. The first of these was Thorndike ('07). He draws conclusions from 86 Spelling Ability — Its Measurement and Distribution conditions in 23 cities as they were about 1900. He estimates that out of 100 entering pupils, 97 remain till grade 3, 90 till grade 4, 81 till grade 5, 68 till grade 6, 54 till grade 7, and 40 till the last grammar grade (8th or 9th). Ayres ('09) sharply criticised these figures, stating that they were too small. He contended, particularly, that there was no dropping out before the 6th year — a conclusion which common observation and later investigation unite to disprove. Employment certificates are granted in great numbers to 5th-grade children. Mr. Ayres' figures for retention are as follows: Grades 1-5, 100 (i.e., no elimination) ; grade 6, 90; grade 7, 71 ; grade 8, 51. Thorndike, using later and better reports, subsequently derived figures a little higher than his former ones, but substantially in agreement with them (Thorndike, '10). They were no higher probably than 5 or 6 years of agitation would have led one to expect. Another important study of this question was made by Strayer ('11), the material being used from 318 cities. His conclusions tend to group with Thorndike's rather than with those of Ayres. Owing to the large number of cities whose returns were used, the uniform method of taking the census, and the recency of the conditions studied, this investigation is highly important. No single figures are given for retention in general, though they are easily found. Using the largest age group as the number of entering children, he gives the following as the median per cents in each grade. Cities of Over 25,000 Cities of Less than 25,000 Boys Girls Boys Girls 3d year 115 110 100 85 65 50 110 110 95 85 75 60 110 105 95 80 70 50 105 4th " 100 5th " 95 6th " . 85 7th " 70 8th " 60 Since in this study we group boys and girls together and consider general conditions, the average of these percentages will give figures for retention for each grade (subject to deduction for Derived Forms of Distribution 87 repeaters) as follows: 3d grade, no; 4th grade, 106; 5th grade, 96; 6th grade, 84; 7th grade, 70; 8th grade, 55. If, as Dr. Strayer says, a fair estimate of the number of repeaters in the 6th, 7th, and 8th grades would be 12%, 10%, and 8% of the pupils in each grade (p. 136), it is likely that the progression (8, 10, 12) may be carried back to the 5th, 4th and 3d grades without great violence to the facts. We estimate therefore that the number of repeaters in the 3d, 4th, and 5th grades is 18%, 16%, and 14% of the pupils in each grade. Making these de- ductions from the above percentages, we have for the retention : 3d grade, 92; 4th grade, 90; 5th grade, 82; 6th grade, 72; 7th grade, 60; 8th grade, 47. Weighing as best we can the results of these four studies, we have made the best estimate we can for the probable amount of retention at present in the grades. For reasons that will appear later we have expressed this estimate in numbers per 10,000 instead of per 100. Table XXXIII and Fig. 30 show the percentages we have adopted compared with those of Thorndike, Ayres, and Strayer (as derived). Fig. 30 gives only the earlier of Thorndike's percentages. TABLE XXXIII Percentages op Retention. Grades 3 to 8 8th Thorndike '07 Ayres '09 Thorndike '10 Strayer '11 (derived) Adopted 3d 4th 5th 6th 7th 97 90 81 68 54 100 100 100 90 71 91 81.5 70.9 56 92 90 82 72 60 97.25 95.46 88.40 70.87 57.44 40 51 41.2 47 48.21 Such an amount of retention for each grade having been adopted, the next question to consider is : What part of a normal distribution is thus eliminated? Obviously not all the poorest in ability drop out. Our results for spelling show that some very poor spellers are retained even in the highest grades. Yet the greatest elimination will no doubt be among those of lowest ability and will be progressively less among children of greater ability. How much this amounts to for successive incre- 88 Spelling Ability — Its Measurement and Distribution ments of ability we do not positively know. We are again forced to make as reasonable an estimate as we can, and this time without the help of any investigations. 100 ^0 70 £. 7p£ 12?L +3 P&. ■HfP-t. Figs. 31-36. The estimated amount and distribution of elimination and retention. See Table XXXIII. Derived Forms of Distribution 91 The next step was to apply the data of Table XXXIV to the normal distribution and to derive therefrom for each grade a modified distribution which should take account of the amount and range of elimination as estimated. In order that the validity of our method may be open to inspection, we shall illustrate for the 6th grade the manner in which these modified distributions were derived. We have adopted certain percentages of retention for desig- nated amounts of general ability (Table XXXIV, 6th grade), and these percentages must not only stand the test of reasonable- ness in themselves, but they must also when applied to a normal table of frequency (the sum of whose cases is, say, 1000), reduce the number of cases to an amount which represents a reasonable percentage of retention for the 6th grade (say, 70 or 71). That is, the derived table must show approximately 700 cases out of 1000, or 7000 out of 10,000. We shall see later to what extent this turns out to be true. Adapting the normal table of frequency (Table XIV, page 35) so as to include 1000 cases instead of 10,000 and taking intervals of 0.1 P.E., we have columns 1 and 2 of Table XXXV. In column 3 we increase the percentages of retention from o at — 2.7 P.E. to 40 at — 1.7 P.E. by increments of 4 for each of the ten steps; then by increments of 2.5 until 65 is reached at — 0.7 P.E. ; then by increments of 2 to 85 at + 0.3 P.E. ; and so on as required by Table XXXIV, col. 4. Taking these percentages of the frequencies in column 2 gives the derived frequencies of column 4. The sum of the entries in this column being 708.7, the plan gives an amount of elimination which is reasonable for the 6th grade. (See Table XXXIII.) The amount and distribution of elimination and retention are shown by diagram for each grade in Figs. 31 to 36. Fig. 34 in particular shows these facts for the 6th grade, and is the graphic representation of the series of frequencies in column 4 of Table XXXV. Fig. 31 shows the same facts for the 3d grade, Fig. 32 for the 4th grade, etc. The progressive increase in elimination and the extension of it to higher and higher parts of the normal curve are the facts to be noticed. But we have not in column 4 of Table XXXV, a frequency table for the 6th grade in the most useful form. The area of its 92 Spelling Ability — Its Measurement and Distribution TABLE XXXV Sixth Grade. Derivation of Modified Table of Frequency Below Normal Median Above Normal Median Per- Per- Normal cent- Derived Same on Normal cent Derived Same on X Fre- ages Fre- basis of X Fre- ages Fre- basis of quen- of quen- 10,000 • quen- of quen- 10,000 P.E. cies Reten- tion cies cases P.E. cies Reten- tion cies cases 0—1 27 79 21.3 301 0— .1 27 81 21.9 309 .2 27 77 20.8 293 .2 27 83 22.4 312 .3 26 75 19.5 275 .3 26 85 22.1 316 .4 26 73 19.0 268 .4 26 85.5 22.2 316 .5 26 71 18.5 261 .5 26 86 22.4 313 .6 25 69 17.3 244 .6 25 86.5 21.6 308 .7 25 67 16.8 237 .7 25 87 21.8 305 .8 23 65 15.0 212 .8 23 87.5 20.1 284 .9 23 62.5 14.4 203 .9 23 88 20.2 285 1.0 22 60 13.2 186 1.0 22 88.5 19.5 275 1.1 21 57.5 12.1 171 1.1 21 89 18.7 264 1.2 20 55 11.0 155 1.2 20 89.5 17.9 252 1.3 19 52.5 10.0 141 1.3 19 go 17.1 241 1.4 18 50 9.0 127 1.4 18 90.5 16.3 230 1.5 16 47.5 7.6 107 1.5 16 91 14.6 206 1.6 16 45 7.2 102 1.6 16 91.5 14.6 206 1.7 14 42.5 6.0 85 1.7 14 92 12.9 182 1.8 14 40 5.6 79 1.8 14 92.5 13.0 183 1.9 12 36 4.3 61 1.9 12 93 11.2 158 2.0 11 32 3.5 50 2.0 11 93.5 10.3 145 2.1 11 28 3.1 44 2.1 11 94 10.3 145 2.2 9 24 2.2 31 2.2 9 94.5 8.5 121 2.3 9 20 1.8 26 2.3 9 95 8.6 120 2.4 7 16 1.1 16 2.4 7 95.5 6.7 95 2.5 7 12 .8 11 2.5 7 96 6.7 95 2.6 6 8 .5 7 2.6 6 96.5 5.8 82 2.7 6 4 .2 3 2.7 6 97 5.8 82 2.8 5 2.8 5 97.5 4.9 69 2.9 4 98 3.9 55 3.0 4 98.5 3.9 55 3.1 3 99 3.0 42 3.2 3 99.5 3.0 42 3.3 2 100 2.0 28 !• 3.4 2 100 2.0 28 3.5 2 100 2.0 28 3.6 2 100 2.0 28 3.7 1 100 1.0 14 ' 3.8 1 100 1.0 14 3.9 1 100 1.0 14 4.0 1 100 1.0 14 etc. to etc. to etc. etc. to etc. to tal No. 6.0 .02 .02 .28 Tc 1000 708.7 9999.72 Derived Forms of Distribution 93 curve is no longer iooo, but only 708.7. In order to express the several frequencies in the form of per cents, we shall have to divide each of them (column 4) by their total (708.7). Express- ing these quotients on the basis of 10,000 instead of 1000, we have column 5. These are the numbers in the columns 3 and 5 of Table XXXIX (p. 96) ; and when their sums are taken begin- ning at o they constitute the Modified Table of Frequency for the 6th grade (Table XXXIX). TABLE XXXVI Modified Table of Frequency, 3d Grade. Median =+0.051 P.E. Plan of elimination: —4 P.E., 100%; —3 P.E., 40%; —2 P.E., 0% Total area of the surface of frequency taken as 10,000. See Fig. 37. X Low High X P.E. Low High X Low High P.E. % A % A % A % A P.E. % A % A 278 278 109 113 5.1 .1 278 278 278 278 2.1 4334 85 4338 93 4.1 5116.1 5.1 .2 556 267 556 267 2.2 4419 81 4431 93 4.2 5121.2 5.1 .3 823 267 823 267 2.3 4500 60 4524 72 4.3 5126.3 4.1 .4 1090 267 1090 267 2.4 4560 58 4596 72 4.4 5130.4 2.1 .5 1357 257 1357 257 2.5 4618 47 4668 62 4.5 5132.5 2.1 .6 1614 257 1614 257 2.6 4665 44 4730 62 4.6 5134.6 1.0 .7 1871 236 1871 236 2.7 4709 35 4792 51 4.7 5135.6 1.0 .8 2107 236 2107 236 2.8 4744 26 4843 41 4.8 5136.6 1.0 .9 2343 226 2343 226 2.9 4770 25 4884 41 4.9 5137.6 1.0 1.0 2569 216 2569 216 3.0 4795 17 4925 31 5.0 5138.6 .51 1.1 2785 206 2785 206 3.1 4812 15 4956 31 5.1 5139.11 .51 1.2 2991 195 2991 195 3.2 4827 9 4987 21 5.2 5139.62 .31 1.3 3186 185 3186 185 3.3 4836 7.4 5008 21 5.3 5139.93 .31 1.4 3371 165 3371 165 3.4 4843.4 6.2 5029 21 5.4 5140.24 .31 1.5 3536 165 3536 165 3.5 4849.6 4.9 5050 21 5.5 5140.55 .31 1.6 3701 144 3701 144 3.6 4854.5 1.9 5071 10 5.6 5140.86 .21 1.7 3845 144 3845 144 3.7 4856.4 1.2 5081 10 5.7 5141.07 .21 1.8 3989 123 3989 123 3.8 4857.6 0.6 5091 10 5.8 5141.28 .21 1.9 4112 113 4112 113 3.9 4858.2 5101 10 5.9 5141.49 .21 2.0 4225 4225 4.0 5111 6.0 5141.70 94 Spelling Ability — Its Measurement and Distribution TABLE XXXVII Modified Table of Frequency, 4th Grade. Median— +0.087 P.E. Plan of elimination: —3.7 P.E., 100%; —2.7 P.E., 40%; —1.7 P.E., 10%; — 1.2 P.E., 0%. Total area of the surface of frequency taken as 10,000. See Fig. 38. X Low High X Low High X Low High P.E. % A % A P.E. % A % A P.E. % A % A 283 283 93 115 5.2 .1 283 283 283 283 2.1 4316 74 4421 94 4.1 5210.2 5.2 .2 566 272 566 272 2.2 4390 71 4515 94 4.2 5215.4 5.2 .3 838 272 838 272 2.3 4461 53 4609 73 4.3 5220.6 4.2 .4 1110 271 1110 271 2.4 4514 51 4682 73 4.4 5224.8 2.1 .5 1381 262 1381 262 2.5 4565 41 4755 63 4.5 5226.9 2.1 .6 1643 262 1643 262 2.6 4606 40 4818 63 4.6 5229 1.05 .7 1905 241 1905 241 2.7 4646 31 4881 52 4.7 5230.05 1.05 .8 2146 241 2146 241 2.8 4677 23 4933 42 4.8 5231 . 10 1.05 .9 2387 230 2387 230 2.9 4700 20 4975 42 4.9 5232.15 1.05 1.0 2617 220 2617 220 3.0 4720 13 5017 31 5.0 5233.20 .52 1.1 2837 210 2837 210 3.1 4733 11 5048 31 5.1 5233 . 72 .52 1.2 3047 199 3047 199 3.2 4744 6 5079 21 5.2 5234.24 .31 1.3 3246 185 3246 189 3.3 4750 5 5100 21 5.3 5234.55 .31 1.4 3431 161 3435 168 3.4 4755 4 5121 21 5.4 5234.86 .31 1.5 3592 158 3603 168 3.5 4759 3 5142 21 5.5 5235.17 .31 1.6 3750 135 3771 147 3.6 4762 1 5163 10.5 5.6 5235 . 48 .21 1.7 3885 132 3918 147 3.7 4763 5173.5 10.5 5.7 5235.69 .21 1.8 4017 109 4065 126 3.8 5184 10.5 5.8 5235.90 .21 1.9 4126 97 4191 115 3.9 5194.5 10.5 5.9 5236.11 .21 2.0 4223 4306 4.0 5205 6.0 5236.32 Derived Forms of Distribution 95 TABLE XXXVIII Modified Table of Frequency, 5th Grade. Median= +0.215 P.E. Plan of elimination: —3.2 P.E., 100%; —2.2 P.E., 50%; —1.2 P. E., 20%; — 0.2 P.E., 0%. Total area of the surface of frequency taken as 10,000. See Fig. 39. X Low High X P.E. Low High X Low High P.E. % A % A % A % A P.E. % A % A 305 305 70 124 6 .1 305 305 305 305 2.1 4093 54 4772 102 4.1 5627 6 .2 610 294 610 294 2.2 4147 51 4874 102 4.2 5633 6 .3 904 28S 904 294 2.3 4198 36 4976 79 4.3 5639 5 .4 1192 282 1198 294 2.4 4234 32 5055 79 4.4 5644 2 .5 1474 266 1492 283 2.5 4266 24 5134 68 4.5 5646 2 .6 1740 260 1775 283 2.6 4290 20 5202 68 4.6 5648 1 .7 2000 234 2058 260 2.7 4310 14 5270 57 4.7 5649 1 .8 2234 229 2318 260 2.8 4324 9 5327 45 4.8 5650 1 .9 2463 214 2578 249 2.9 4333 7 5372 45 4.9 5651 1 1.0 2677 200 2827 238 3.0 4340 3 5417 34 5.0 5652 .6 1.1 2877 186 3065 226 3.1 4343 2 5451 34 5.1 5652.6 .5 1.2 3063 172 3291 215 3.2 4345 5485 23 5.2 5653.1 .3 1.3 3235 157 3506 204 3.3 5508 23 5.3 5653.4 .3 1.4 3392 134 3710 181 3.4 5531 23 5.4 5653.7 .3 1.5 3526 129 3891 181 3.5 5534 23 5.5 5654.0 .3 1.6 3655 108 4072 158 3.6 5577 11 5.6 5654.3 .2 1.7 3763 103 4230 158 3.7 5588 11 5.7 5654.5 .2 1.8 3866 84 4388 136 3.8 5599 11 5.8 5654.7 .2 1.9 3950 73 4524 124 3.9 5610 11 5.9 5654.9 .2 2.0 4023 4648 4.0 5621 6.0 5655.1 96 Spelling Ability — Its Measurement and Distribution TABLE XXXIX Modified Table of Frequency, 6th Grade. Median =+0.418P.E. Plan of elimination: —2.7 P.E., 100%; —1.7 P.E., 60%; —0.7 P.E., 35%; +0.3 P.E., 15%; +1.3 P.E., 10%; +2.3 P.E., 5%; +3.3 P.E., 0%. Total area of surface of frequency taken as 10,000. See Fig. 40. X Low High X P.E. Low High X P.E. Low High P.E. % A % A % A % A % A % A 301 309 44 145 7 .1 301 293 309 312 2.1 3602 31 5235 121 4.1 6268 7 .2 594 275 621 316 2.2 3633 26 5356 120 4.2 6275 7 .3 869 268 937 316 2.3 3659 16 5476 95 4.3 6282 6 .4 1137 261 1253 313 2.4 3675 11 5571 95 4.4 6288 3 .5 1398 244 1566 308 2.5 3686 7 5666 82 4.5 6291 3 .6 1642 237 1874 305 2.6 3693 3 5748 82 4.6 6294 1.4 .7 1879 212 2179 284 2.7 3696 5830 69 4.7 6295.4 1.4 .8 2091 203 2463 285 2.8 5899 55 4.8 62968 1.4 .9 2294 186 2748 275 2.9 5954 55 4.9 6298 . 2 1.4 1.0 2480 171 3023 264 3.0 6009 42 5.0 6299.6 .7 1.1 2651 155 3287 252 3.1 6051 42 5.1 6300.3 .7 1.2 2806 141 3539 241 3.2 6093 28 5.2 6301 .4 1.3 2947 127 3780 230 3.3 6121 28 5.3 6301.4 .4 1.4 3074 107 4010 206 3.4 6149 28 5.4 6301.8 .4 1.5 3181 10? 4216 206 3.5 6177 28 5.5 6302.2 .4 1 fi 3283 4422 3.6 6205 5.6 I 6302.6 85 182 14 .28 1.7 3368 79 4604 183 3.7 6219 14 5.7 6302.88 .28 1.8 3447 61 4787 158 3.8 6233 14 5.8 6303.16 .28 1.9 3508 50 4945 145 3.9 6247 14 5.9 6303 . 44 .28 2.0 3558 5090 4.0 6261 6.0 6303.72 Derived Forms of Distribution 97 TABLE XL Modified Table of Fkequency, 7th Grade. Median— + 0.669 P.E. Plan of elimination: —2.3 P.E., 100%; —1.3 P.E., 70%; —0.3 P.E., 50%; +0.7 P.E., 20%; +1.7 P.E., 10%; +2.7 P.E., 5%; +3.7 P.E., 0%. Total area of frequency surface taken as 10,000. See Fig. 41. X Low High X P.E. Low High X P.E. Low High P.E. % A % A % A % A % A % A 277 290 17 176 9 .1 277 263 290 306 2.1 2844 9 5846 145 4.1 7096 9 .2 540 240 596 308 2.2 2853 fl 5991 145 4.2 7105 9 .3 780 226 904 322 2.3 2858 6136 114 4.3 7114 7 .4 1006 217 1226 334 2.4 6250 114 4.4 7121 3 .5 1223 200 1560 336 2.5 6364 99 4.5 7124 3 .6 1423 191 1896 348 2.6 6463 99 4.6 7127 ?, .7 1614 169 2244 326 2.7 6562 83 4.7 7129 ?, .8 1783 160 2570 326 2.8 6645 66 4.8 7131 ?, .9 1943 146 2896 318 2.9 6711 68 4.9 7133 2 1.0 2089 132 3214 306 3.0 6779 50 5.0 7135 .9 1.1 2221 118 3520 296 3.1 6829 50 5.1 7135.9 ,9 1.2 2339 106 3S16 283 3.2 6879 35 5.2 7136.8 .5 1.3 2445 94 4099 273 3.3 6914 35 5.3 7137.3 .5 1.4 2539 75 4372 246 3.4 6949 35 5.4 7137.8 .5 1.5 2614 66 4618 246 3.5 6984 35 5.5 7138.3 .5 1.6 2680 50 4864 220 3.6 7019 17 5.6 7138.8 .35 1.7 2730 43 5084 220 3.7 7036 17 5.7 7139.15 .35 1.8 2773 31 5304 190 3.8 7053 17 5.8 7139.50 .35 1.9 2804 23 5494 176 3.9 7070 17 5.9 7139.85 .3 2.0 2827 5670 4.0 7087 6.0 7140.15 98 Spelling Ability — Its Measurement and Distribution TABLE XLI Modified Table of Frequency, 8th Grade. Median= +0.746 P.E. Plan of elimination: — 2 P.E., 100%; — 1 P.E., 70%; P.E., 50%; +1 P.E., 30%; +2P.E.,20%; +3 P.E., 10%; + 4 P.E., 6%; +5 P.E., 2%; + 6 P.E., 0%. Total area of frequency surface taken as 10,000. See Fig. 42. X Low High X P.E. Low High X P.E. Low High P.E. % A % A % A % A % A % A 280 290 184 10 .1 280 270 290 303 2.1 5834 165 4.1 7195 10 .2 550 249 593 303 2.2 5999 145 4.2 7205 10 .3 799 237 896 313 2.3 6144 130 4.3 7215 8 .4 1036 226 120t 323 2.4 6274 116 4.4 7223 4 .5 1262 207 1532 323 2.5 6390 108 4.5 7227 4 .6 1469 197 1855 332 2.6 6498 108 4.6 7231 2 .7 1666 172 2187 320 2.7 6606 91 4.7 7233 2 .8 1838 162 2507 319 2.8 6697 75 4.8 7235 2 .9 2000 145 2826 319 2.9 6772 75 4.9 7237 2 1.0 2145 131 3145 309 3.0 6847 56 5.0 7239 1 1.1 2276 112 3454 299 3.1 6903 56 5.1 7240 1 1.2 2388 95 3753 288 3.2 6959 38 5.2 7241 .6 1.3 2483 79 4041 276 3.3 6997 38 5.3 7241.6 .6 1.4 2562 60 4317 260 3.4 7035 37 5.4 7242.2 .6 1.5 2622 50 4577 242 3.5 7072 37 5.5 7242.8 .6 1.6 2672 35 4819 235 3.6 7109 19 5.6 7243.4 .4 1.7 2707 27 5054 215 3.7 7128 19 5.7 7243.8 .4 1.8 2734 14 5269 197 3.8 7147 19 5.8 7244.2 .4 1.9 274S 6 5466 184 3.9 7166 19 5.9 7244.6 .4 2.0 2754 5650 4.0 7185 6.0 7245 In this manner each of the Modified Tables of Frequency was made up. They are given in tables XXXVI to XLI. They are intended to take the place, each for the grade to which it applies, of the Table of Frequency for the normal distribution. Since they are asymmetrical, the lower and upper parts have to be given separately. For the same reason there is no P.E., the use of the Probable Error as a unit of amount being properly confined to normal curves only (Yule, '11, p. 147). The quartile deviation (Q3 — Qi) might be used instead, but its Derived Forms of Distribution 99 Figs. 37-42. Derived Forms of Distribution. Grades 3 to 8. ioo Spelling Ability — Its Measurement and Distribution value differs for each of the six tables. In order therefore to employ a unit which should be the same for all, including the normal distribution, we have retained the P.E. of the Proba- bility Integral. It is now no longer a function of the modified dis- tributions, but a mere unit of length. Likewise in order to have a common point of reference the median of the normal distribution has been retained, the terms " low " and " high " in the tables referring to parts below or above that point. The real median of each modified distribution, however, is given, being expressed as a deviation from the old median. Figs. 37 to 42 are to be considered in connection with tables XXXVI to XLI, of which they are the graphic expressions (the curves being " smoothed," to represent an indefinite number of cases) . They are also to be considered in connection with Figs. 31 to 36, from the " retention " parts of which they are derived by making the areas 10,000. In Figs. 37 to 42, the curve extend- ing farther to the left is in each case the normal curve and OM is its median vertical. 1 M 1 is the median vertical of the modified surface and OO 1 is the distance between medians. The values of these are as follows: 3d grade, 0.051 P.E. ; 4th grade, 0.087 P.E. ; 5th grade, 0.215 P.E. ; 6th grade, 0.418 P.E. ; 7th grade, 0.669 P - E - '> 8tri g ra de, 0.746 P.E. Is it worth while to use these tables instead of the normal one? Will the same material when analyzed by the skew and normal distributions yield differences that are important? With the purpose of throwing some light on this question we have used the modified tables to interpret the results of testing with our Preferred List, and the rest of the present section will be devoted to this matter. The differences will not In many cases be found to be large. This is, of course, particularly true when the early grades are concerned, the curves being for those grades almost normal. It may be remarked, however, that the applica- bility of these tables does not rest upon the results here shown. It is general ability rather than spelling ability that tends strongly to keep children in school. Spelling ability does not correlate as highly with general ability as do the abilities in most other school subjects. It is quite likely, therefore, that Derived Forms of Distribution 101 the use of these tables for the statistical treatment of other subjects may be more satisfactory than it is for spelling. They are given here primarily to illustrate the method. TABLE XLII Number and Per Cent of Pupils in Each Grade Whose Abilitt Equalled or Exceeded that of the Median Pupil in Every Other Grade with the P.E. Values Corresponding to Each Per Cent. Selected List. Modified Distributions. Compare with Table XV (p. 36) 3d Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade 3d grade. . . iV=445 No. % P.E. 76 17.1 1.3858 27 6.1 2.2736 9 2.0 3.0029 3 0.7 3.607 ? 4th grade. . iV=467 No. % P.E. 378 80.9 1.1949 146 31.3 .6965 52 11.1 1.7616 27 5.8 2.2778 9 1.9 3.0076 5th grade. . iV=515 No. % P.E. 478 92.8 1.7916 370 71.8 .7341 142 27.6 .8136 73 14.2 1.5172 30 5.8 2.2104 6th grade . . iV=418 No. % P.E. 414 99.0 2.5043 384 91.9 1.6747 338 80.1 1.0451 142 34.0 .5386 57 13.6 1.4812 7th grade. . iV=365 No. % P.E. 363 99.5 2.6038 354 96.4 2.0360 328 89.9 1.5096 256 70.1 .6586 99 27.1 .7564 8th grade . . #=277 No. % P.E. 227 100 ? 276 99.6 2.4719 269 97.1 2.0197 241 87.0 1.4424 200 72.2 .635 In Section n we located the grade medians assuming normal distribution. In Tables XLII and XLIII the same data have been subjected to analysis using the modified distributions. These tables are to be compared with Tables XV (page 36) and XVI (page 39). The median intervals are considerably less than they were found to be by using the normal distribution. Note the comparisons in Table XLIV (page 103). On the average, the intervals by the present method are less than the same intervals found by using the normal distribution by 0.1247 P.E., or about half a step in the 10-point scale (Table XX, page 52). Since this occurs five times the entire range io2 Spelling Ability — Its Measurement and Distribution TABLE XLIII Direct and Derived Values of Median Distances. Modified Distribu- tions. Selected List ^3-4 M 4 . 5 ^5-8 M^ M 7-S 1.3858 (direct) .8878 (M 3 _ 5 -M 3 _ t ) .7293 (M 3 _ 6 — ilf 3 _ 5 ) .6041 (M 3 _— M 3 _J ? (M 3 _ 9 —M 3 _ 7 ) 1 . 5771 (M 3 _ 5 -M 4 _ 5 ) .6965 (direct) 1.0651 (M 4 _ 6 -M 4 _ 5 ) .5162 (M 4 _ 7 -M 4 _ 6 ) .7298 (M 4 _ 8 -M 4 _ 7 ) 1.2413 (M 3 _ 6 — M A _ 6 ) .9478 (M 4 _ 6 -M 5 _ 6 ) .8136 (direct) .7036 (M 5 _ 7 -M 6 _ 8 ) .6932 (M 5 _ s -M 5 _ 7 ) 1.3292 (M 3 _ 7 — ilf 4 _ 7 ) .7606 (M 4 _ 7 -M^ 7 ) .9786 (M 5 _—M^_ 7 ) .5386 (direct) .9426 ? (M 3 _ 8 —M 4 _ 8 ) .7972 (M 4 _ 8 -M 5 _ 8 ) .7292 (M 5 _ 8 — M„_ 8 ) .7248 (M 6 _ 8 — M 1-9> .7564 (direct) 1.1949 (direct) .5967 (M 5 _ 3 -M 4 _ 3 ) .7127 (M 6 _ 3 — M 5 _ 3 ) .0995 (M 7 _ 3 -M_ 3 ) ? (M 8 _ 3 — M 7 _ 3 ) 1.0575 .7341 (direct) .9406 (M^-JI^) .3613 (ilf 7 _ 4 -M^ 4 ) .4359 (M 8 _ 4 -M 7 _ 4 ) .8296 (Jlf^-M^) .9406 (ilf 6 _ 4 -M 6 _ 5 ) 1.0451 (direct) .4645 .5101 (M 8 _ 5 -itf 7 _ 5 ) .5678 (M 7 _ 3 -M 7 _ t ) .5264 (M 7 _ 4 -M 7 _ 5 ) .8510 .5386 (direct.) .7838 (M 8 _ 6 — M 7 _ 6 ) (M 8 _ 3 -M s _ 4 ) .4522 (M s _ 4 -M s _ 5 ) .5773 (M 8 _ 5 — M 8 _ 6 ) .8074 (M 8 _ a — M 8 _ 7 ) .6350 (direct) Average.. 1.1479 .7340 .8443 .5359 .6858 Weighted Average 1.2008 .7483 .8685 .5606 .7065 from M 3 to M 8 is contracted by 0.7235 P.E., an amount which is more than some grade intervals (M 6 _ 7 by normal dis., M" 6 _ 7 and M 7 _ 8 by modified dis.). This is an important difference. In the matter of scaling the words, there is, as might be supposed, very little difference for the 3d grade — so little as to be quite negligible. For the 4th grade there is some difference, and for each successive higher grade the difference between Derived Forms of Distribution 103 the placings of the same word by the two methods becomes greater and greater as the asymmetry of the modified curves becomes more and more pronounced. TABLE XLIV Comparison of Averages op Median Distances by Normal Distribu- tion and by Modified Distributions Normal Distribution Modified Distributions Unweighted Averages Weighted Averages Unweighted Averages Weighted Averages M, 1.3326 0.8471 1.0406 0.6344 0.9201 1.3505 0.8363 1.0505 0.6608 0.9101 1 . 1479 0.7340 0.8443 0.5359 0.6858 1 . 2008 M A = 0.7483 Vr 4— 5 M c ~ 0.8685 Ma n 0.5606 M , ' 0.7065 Table XLV compares the deviations from grade medians of the words of the Preferred List by Normal Distribution and by Modified Distributions. Figs. 43 to 47 give the same facts in graphic form. Words spelled by 50 per cent of pupils are of course always at o. Words spelled by more than 50 per cent of pupils do not deviate from the median as much by modified as by normal distribution. The same is true of those spelled by less than 50 per cent of pupils. The easier a word is and the harder a word is, the greater, accordingly, is the difference in placing. The effect therefore of the modified distributions is to shorten the range of the grade scales. In using the scales, especially for pupils of the higher grades, all differences in ability between individuals or groups would tend to be decreased. It seems likely that these differences are in reality more nearly what the modified distributions show them to be. The wide range of the normal curve especially when its spread is assumed to be the same for all grades would seem to extend too far, particularly towards the low end. 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CO OJ -^J P 15 +j S e O fc rt en G rt 1) ^ a! On H CO h .g 00 fr io8 Spelling Ability — Its Measurement and Distribution Table XLVI and Fig. 48 show a comparison for all grades combined. The same shortening of the range is evident but, whereas the contraction in the grade scales was more pro- nounced at the low ends, it is now in the general scale more TABLE XLVI The Average Position of Each Word According to Normal Distribu- tion and According to Modified Distribution. Point of Reference is 3d Grade Median. See Fig. 48 Word Average Position Word No. Word Average Position Word No. Nor- mal Distri- bution Modi- fied Distri- bution Nor- mal Distri- bution Modi- fied Distri- bution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 even lesson only smoke front sure pear another .... forty pretty wear button minute cousin nails janitor saucer stopping . . . sword freeze touch whistle carriage. . . . nor .699 1.135 .569 .835 1.057 1.568 1.958 1.169 1.078 1.758 1.311 1.844 2.026 1.943 1.681 1.379 2.047 2.604 2.213 2.185 1.740 1.709 2.193 2.340 1.652 .753 1.018 .604 .831 .949 1.349 1.697 1.057 1.287 1.477 1.137 1.587 1.724 1.687 1.491 1.226 1.773 2.256 1.894 1.766 1.517 1 . 465 1.870 2.022 1.397 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 beginning. . . chicken .... choose circus grease pigeons quarrel saucy tailor telegram .... telephone. .. tobacco .... too towel Tuesday. . . . lying whole against answer butcher. . . . guess instead raise beautiful . . . 2.699 2.917 .897 2.502 2.141 3.294 2.739 2.069 2.666 1.866 2.549 2.413 1.988 3.491 1.978 1.550 1.870 2.018 2.106 1.594 1.473 2.363 1.756 1.652 1.682 2.305 2.525 .872 2.143 1.872 2.838 2.378 1.816 2.294 1.579 2.204 2.101 1.767 2.998 1.718 1.313 1.578 1.751 1.847 1.425 1.308 2.038 1.517 1.456 1.519 evident at the high end. There are also differences in arrange- ment, as there could not be in the grade scales. If two words which take the same position on the normal scale by ratings markedly different in upper and lower grades, but balancing each other in the aggregate, these words would not take the same position on the modified scale. The one which had Derived Forms of Distribution 109 <* ■ s 5$- 7r* rt bo re 1— < O f\ ft o h no Spelling Ability — Its Measurement and Distribution relatively low ratings in the lower grades would take a position above the other whose ratings were relatively high in the lower grades. This is because the modified distributions in the upper grades are such that counting in from the high end more rapidly approaches the median than does counting in the same per cent of the area of the normal curve. Take for example the words 25 nor and 45 answer. Compare the per cent ratings in Table XLV. Nor is easy in low grades and hard in upper grades, relative to answer. With the same normal distribution for all grades nor is a little harder than answer. By the modified dis- tributions it is easier. Other words may easily be selected in Fig. 48 which show differences in arrangement. It is therefore true that the use of modified forms of distribution makes a difference which is worth noting in the scaling of words. A question to be decided on the evidence of more complete testing and a wide use of these forms of distribution is whether the differences here shown to exist impair the usefulness of the scales we have previously derived. Our judgment at present is that they do not. § 20. Conclusions We have now certain data in hand and we may make a few general statements from them. We have selected from a school list of about 5000 words a list for test purposes in grades 3-8 which, when put in sentences, yielded a list of 270 words. As a result of testing in two schools a selected list of 100 words was chosen and to it were added, at a later time, 18 more. These were dictated at three schools and the 100 words alone subsequently at two more schools. From the 118 were chosen two lists of 25 each. The three successive selections were made with the purpose of securing words which were easy enough in the 3d grade and hard enough in the 8th grade to afford a test in those and therefore in intei- mediate grades, and which showed regular increases in per cent correct from grade to grade. The two 25-word lists were then subjected to analysis and found to have high correlations between grades and between schools. Using the entire test material and the ratings of individual pupils and assuming normal distribution and equal variability, Conclusion in the differences between typical grade abilities were found and expressed as median intervals. The 50 words which had been derived by a threefold selective process and subjected to close inspection for permanency as between grades and schools were scaled for each grade and for all grades combined. By using an " Easy 50- Word List " an expression was derived for the zero-point; and, by further test- ing under rigidly controlled conditions, previous grade-intervals were verified. To fill in and extend the scale, the Rice Sentence Test was dictated and the word-scores for the Easy 50-Word List were used. It is to be understood, however, that neither of these lists was subjected to the scrutiny that was made of the Preferred List. Accordingly we cannot regard the placing of these words as very reliable. Finally we have derived and applied tables of frequency more or less asymmetrical in character according to the amount of retention for each grade and its estimated distribution. By using them, results have been obtained which in some instances differ considerably from those obtained on the basis of a normal distribution. Such differences as appear are, we are convinced, differences in the direction of a truer representation of the facts. On the whole, however, the differences are not sufficient to impair our previous results for any practical use which is likely to be made of them. It has become evident to us that there is a lack of knowledge of the spelling problem not only among teachers but also among those who direct their work. This is unfortunate, considering the relative definiteness of the subject and the comparative ease with which results in it may be scored. Nor is there any special consciousness of the need of more insight in this matter. Almost, if not quite, all the studies that have hitherto been made have dealt with individual performances. The behavior of words has received no attention. It is our belief, however, that a powerful improvement in the teaching of spelling may be derived from a more critical knowl- edge and more accurate judgment on the part of teachers and supervisors of the material of the subject — i.e., of the words of the language. If in a list of 50 words the one word that is incontestably hardest is by more than one-fourth of a representa- ii2 Spelling Ability — Its Measurement and Distribution tive group of teachers judged to be the easiest, or the easiest but one, that fact in itself is a very good reason why the word is so hard. Pupils misspell it because their teachers do not realize the need of teaching it. If text-book makers disagree so widely as to put the same words in grades that are three, four, and even five years apart, it is proof of the confusion that exists as to how hard words are, and when they should be taught. There are various types of words, and each type requires different treatment. There is the type that does not need to be taught at all. There is the type which appears easy in the lower classes and (grade considered) hard in the upper classes. Such may have been prematurely taught in the lower classes. There is the type that appears to possess special difficulty for the middle grades. This is due to a constant cause — e.g., in the case of whose, to the learning of the use of the apostrophe in possessives. There are types of errors; there is the problem of substitution, of illegibility, and of omission. To obtain any accurate notion of " word behavior " we must rate for words as distinct from individuals. Moreover we must give our per cent ratings thus obtained an interpretation for difficulty which takes account of the distribution of spelling ability. When we do so we shall find how unreliable percentages are as indicating differences in difficulty. We shall find, for instance, that a difference of 10 per cent between two words rated 89 and 99 means more than four times as great a difference in difficulty as is that between two words rated at 45 and 55, although the percentage difference is in both cases the same. Table XLVII (See appendix) is a ready reckoner for the con- version of percentages into units that take account of the form of distribution, assuming it to be ' Normal.' If this study does no more than show the need of word criticism and indicate a method, it may be worth while. Every school affords a place and every day a time at which something may be done to help throw light on the nature of the material we deal with in spelling. All such work should be collected and made generally available. If teachers, principals, or superin- tendents who have made or who hereafter make a study of the difficulty of words, will submit them to the author of this study, the data will be gratefully received and utilized to disseminate a larger and more accurate knowledge. APPENDIX I. List of Authors and Titles Specifically Referred to in the Text Thorndike, E. L. ('io). Handwriting. Teachers College Record, Vol. XI, No. 2. Hillegas, Milo B. ('12). A Scale for the Measurement of Quality in English Composition by Young People. Teachers College Record, Vol. XIII, No. 4. Rice, J. M. ('97). The Futility of the Spelling Grind. Forum, Vol. XXIII, pp. 163-172; 409-419. Thorndike, E. L. ('13). An Introduction to the Theory of Mental and Social Measurements. Second Edition. Teachers College, New York. Cornman, O. P. ('02). Spelling in the Elementary School. Ginn and Co., New York. Wallin, J. E. Wallace, ('ii). Spelling Efficiency in Relation to Age, Grade, and Sex, and the Question of Transfer. Warwick and York, Baltimore. Pearson, Henry C. ('12). Experimental Studies in the Teaching of Spelling. Teachers College Record, Vol. XIII, No. 1. Spearman, C. ('06). ' Foot-rule ' for Measuring Correlation. Brit. Joum. of Psych., Vol. II, Pt. I, July, 1906. Brown, William, ('ii). The Essentials of Mental Measurement. Put- nam, New York. Whipple, Guy Montrose, ('10). Manual of Mental and Physical Tests. Warwick and York, Baltimore. Klein, Linus W. ('12). A Study in the Psychology of Spelling. Joum. of Ed. Psych., Vol. Ill, No ; 7. Thorndike, E. L. ('07). The Elimination of Pupils from School. Bureau of Education, Bulletin No. 4, 1907. Ayres, Leonard P. ('09). Laggards in our Schools. Russell Sage Foundation, New York. Thorndike, E. L. ('10). Promotion, Retardation, and Elimination. Psych. Clinic, Vol. Ill, No. 8 and 9. Strayer, George Drayton ('ii). Age and Grade Census of Schools and Colleges. Bureau of Education, Bulletin No. 5, 191 1. Yule, G. Udney ('ii). An Introduction to the Theory of Statistics. Lippincott, Philadelphia. 113 H4 Spelling Ability — Its Measurement and Distribution II. Lists Referred to in the Text and Used in the Scales Preferred List Easy 50-Word First List Rice Sentence List I even 1. you 1. running 44- deceive 2 lesson 2. will 2. slipped 45- driving 3 only 3- hear 3- listened 46. surface 4 smoke 4- him 4. queer 47- rough 5 . front 5- coming 5- speech 48. smooth sure 6. he 6. believe 49- hopping 7 pear 7- is 7- weather 50. certainly 8 bought S. on 8. changeable 51. grateful 9 another 9- the 9- whistling 52. elegant 10 forty 10. road 10. frightened 53- present 11 pretty 11. and 11. always 54- patience 12 wear 12. almost 12. changing 55- succeed 13 button 13. sure 13- chain 56. severe 14 minute 14- to 14. loose KV. accident i5 cousin 15- pass IS- baking 58. sometimes 10 nails it. in 16. piece 59- sensible 17 janitor ]/• front 17- receive 60. business 18 saucer 18. of 18. laughter 6l. answer 19 stopping 19. me 19. distance 62. sweeping 20 sword 20. I 20. choose 1 63. properly 21 freeze 21. send 21. strange 64. improvement 22 touch 22. for 22. picture 65. fatiguing 23 whistle 23- every 23. because 66. anxious 24 carriage 24. day 24. thought 67. appreciate 25 nor 25- go 25- purpose 68. assure 26. into 26. learn 69. imagine Second 27. school 27. lose 70. peculiar 26 already 28. but 28. almanac 7i. character 27 beginning 29. do 29. neighbor 72. guarantee 28 chicken 30. not 30. writing 73- approval 29 choose 31. touch 3i. language 74- intelligent 30 circus 32. table 32. careful '73- experience 31 grease 33- also 33- enough 76. delicious 32 pigeons 34- has 34- necessary 77- realize 33 quarrel 35- only 35- waiting 78. importance 34 saucy 36. one 36. disappoint 79- occasion 35 tailor 37- pair 37- often 80. exceptions 36 telegram 38. shoes 38. covered 81. thoroughly ?,7 telephone 39- they 39- mixture 82. conscientious 38 tobacco 40. are 40. getting 83. therefore 39 too 41. at 41. better 84. ascending 40 towel 42. all 42. feather 85- praise 4i Tuesday 43- pretty 43- light 86. wholesome 42 tying 44. no 43 whole 45- man 44 against 46. ought 45 answer 47- steal 46 butcher 48. even 47 guess 49. a 48 instead 50. penny 40 raise 50 beautiful Appendix 115 III. Memorandum on the Method of Computing with Modified Frequency Tables. (Tables XXXVI-XLI.) 1. Derivation of Median Intervals. Table XLII, lines 4 and 5, gives for the 4th grade the number and per cent of pupils who equal or exceed the median pupil of each of the other grades. In line 6 the correspond- ing P.E. values are shown. These are obtained by using Table XXXVII as follows: (a) Since 80.9% of 4th-grade pupils surpass the median 3d-grade pupil, deduct 8090 cases from the high end of the 4th-grade distribution. Since there are 5236.32 above M 4 (nor. dis.), 2853.68 more must be taken, extending to a point which is 1.1079 P.E. below M 4 (nor. dis.). But M 4 (nor. dis.) is itself .087 P.E. below M 4 (mod. dis.). Correcting for this, we have 1.1949 P.E. below M 4 (mod. dis.). ( — 1.1079 — .087 = — 1.1949.) This is the first entry in line 6 of Table XLII. (b) Deduct 3130 from 5236.32, leaving 2106.32. By interpellation this corresponds to +.7835 P.E. Subtracting .087 P.E. as before, we have +-6965, the second entry in line 6 of Table XLII. (c) 5236.32 less 11 10 gives 4126.32, corresponding to -f- 1.8486 P.E. Again subtracting .087 P.E., we have -f- 1.7616 P.E., which is the third entry in line 6, Table XLII. 2. Scaling the Words. For " even," Table XLV, columns headed " Modified Distributions," the figures are derived as follows, using for each grade its proper frequency table : Third Grade. 59% correct. Count out the 5900 highest cases. There are 5141.7 above M3(nor. dis.). We must, therefore, take 758.3 cases below that point. This brings us to — .276 P.E. Subtracting (algebraically) .051 P.E., in order to refer this to ilfs(mod. dis.). we have — .327 P.E., as in Table XLV. Fourth Grade. 79% correct. Counting out 7900 cases from the high end, we take all the "highs" and 2663.32 of the "lows," reaching as far as — 1.0212 P.E. But M 4 (mod. dis.) is .087 P.E. above M 4 (nor. dis.). Subtracting this amount, we have — 1.108 P.E. as in Table XLV. Fifth Grade. When percentages are high, it is generally easier to count out their complements from the low end. " Even " is in this grade 89% correct. We may there- fore count 8900 cases from the high end or 1100 from the low end. In either case we reach the 3245th case of the " lows," which corresponds to — 1,306 P.E. Correcting for the deflection of the median from its "normal" position (.215 P.E.), we have — 1,521 P.E. as given. Sixth Grade. 3696 — 700 = 2996. The 2996th case corresponds to — 1-339 P-E. Median displacement = .418 P.E. Subtracting from — 1-339 P.E., we have — 1-757 P-E., as given. The 7th and 8th grade positions are derived in the same way, care being taken to use the proper grade table of fre- quency in each case. Table XLVI. The average position for each word as given in the column headed " Modified Distributions " was computed as follows : Add to the P.E. value of "even" for each grade (Table XLV) the distance which the grade median is above the 3rd-grade median. From Table XLII these distances are shown to be: M s -4, 1.148 P. E. ; M 3 - 5 , 1.S82 P.E.; Ms-e, 2.726 P.E. ; M 3 -7, 3.262 P.E. ; M 3 - s . 3.948 P.E. Adding these values to those of Table XLV, beginning with the 4th grade and writing the 3rd grade as given, we have the following P.E. values : — .327, +.040, +.361, +.969, + i-54i> and -f- I -93 2 - The average of these is +-753 P-E., as given for the word " even " in Table XLVI. The average positions of the remaining words were computed in the same way. n6 Appendix IV. TABLE XLVII — P.E. Values Corresponding to Given Per Cents of the Normal Surface of Frequency, Per Cents Being Taken from the Median .1 .2 .3 .4 .5 .6 .7 .8 .9 .000 .004 .007 .011 .015 .019 .022 .026 .030 .033 1 .037 .041 .044 .048 .052 .056 .059 .063 .067 .071 2 .074 .078 .082 .085 .089 .093 .097 .100 .104 .108 3 .112 .115 .119 .123 .127 .130 .134 .138 .141 .145 4 .149 .153 .156 .160 .164 .168 .172 .175 .179 .183 5 .187 .190 .194 .198 .201 .205 .209 .213 .216 .220 6 .224 .228 .231 .235 .239 .243 .246 .250 .254 .258 7 .261 .265 .269 .273 .277 .2S0 .284 .288 .292 .296 8 .299 .303 .307 .311 .315 .318 .322 .326 .330 .334 9 .337 .341 .345 .349 .353 .357 .360 .364 .368 .372 10 .376 .380 .383 .387 .391 .395 .399 .403 .407 .410 11 .414 .418 .422 .426 .430 .434 .437 .441 .445 .449 12 .453 .457 .461 .464 .468 .472 .476 .480 .484 .489 13 .492 .496 .500 .504 .508 .512 .516 .519 .523 .527 14 .531 .535 .539 .543 .547. 551 .555 .559 .563 .567 15 .571 .575 .579 .583 .588 .592 .596 .600 .603 .608 16 .612 .616 .620 .624 .628 .632 .636 .640 .644 .648 17 .652 .656 .660 .665 .669 .673 .677 .681 .685 .689 18 .693 .698 .702 .706 .710 .714 .719 .723 .727 .731 19 .735 .740 .744 .748 .752 .756 .761 .765 .769 .773 20 .778 .782 .786 .790 .795 .799 .803 .807 .812 .816 21 .820 .825 .829 .834 .838 .842 .847 .851 .855 .860 22 .864 .869 .873 .878 .882 .886 .891 .895 .900 .904 23 .909 .913 .918 .922 .927 .931 .936 .940 .945 .949 24 .954 .958 .963 .968 .972 .977 .982 .986 .991 .996 25 1.000 1.005 1.009 1.014 1.019 1.024 1.028 1.033 1 .038 1 .042 26 1.047 1.052 1.057 1.062 1.067 1.071 1.076 1.081 1 .086 1 .091 27 1.096 1.101 1.105 1.110 1.115 1.120 1.125 1.130 1 .135 1 .140 28 1.145 1.150 1.155 1.160 1.165 1.170 1.176 1.181 1 .186 1 .191 29 1.196 1.201 1.206 1.211 1.217 1.222 1.227 1.232 1 .238 1 .243 30 1.248 1.253 1.259 1.264 1.269 1.275 1.279 1.286 1 .291 1 .296 31 1.302 1.307 1.313 1.318 1.324 1.329 1.335 1.340 1 .346 1 .351 32 1.357 1.363 1.368 1.374 1.380 1.386 1.391 1.397 1 .403 1 .409 33 1.415 1.421 1.427 1.432 1.438 1.444 1.450 1.456 1 .462 1 .469 34 1.475 1.481 1.487 1.493 1.499 1.506 1.512 1.518 1 .524 1 .531 35 1.537 1.543 1.549 1.556 1.563 1.569 1.576 1.582 1 .589 1 .595 36 1.602 1.609 1.616 1.622 1.629 1.636 1.643 1.649 1 .656 1 .663 37 1.670 1.677 1.685 1.692 1.699 1.706 1.713 1.720 1 .728 1 .735 38 1.742 1.749 1.757 1.765 1.772 1.780 1.788 1.795 1 .803 1 .811 39 1.819 1.827 1.835 1.843 1.851 1.859 1.867 1.875 1 .884 1 .892 40 1.900 1.909 1.918 1.926 1.935 1.944 1.953 1.962 1 .971 1 .979 41 1.988 1.997 2.007 2.016 2.026 2.035 2.044 2.054 2.064 2.074 42 2.083 2.093 2.103 2.114 2.124 2.134 2.145 2.155 2 .166 2 .177 43 2.188 2.199 2.211 2.222 2.234 2.245 2.257 2.269 2 .281 2 .293 44 2.305 2.318 2.331 2.344 2.357 2.370 2.384 2.397 2 .411 2 .425 45 2.439 2.453 2.468 2.483 2.498 2.514 2.530 2.546 2 .562 2 .579 46 2.597 2.614 2.631 2.648 2.667 2.686 2.706 2.726 2 .746 2 .767 47 2.789 2.811 2.834 2.857 2.881 2.905 2.932 2.958 2 .986 3 .015 48 3.044 3.077 3.111 3.146 3.182 3.219 3.258 3.300 3 .346 3 .395 49 50 3.450 3.506 3.571 3.643 3.725 3.820 3.938 4.083 4 .275 4 .600 OCT \3A3