A STUDY OF SUCCESS IN AN INTRODUCTORY EXAMINATION IN HIGH SCHOOL ALGEBRA AS RELATED TO SUCCESS IN COLLEGE ALGEBRA By SARAH HELEN TAYLOR A. B. Illinois College, 1920 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS IN MATHEMATICS IN THE GRADUATE SCHOOL OF THE UNIVERSITY OF ILLINOIS, 1922 URBANA, ILLINOIS J UNIVERSITY OF ILLINOIS THE GRADUATE SCHOOL . — v .' 0 1 92_£- I HEREBY RECOMMEND THAI' THE THESIS PREPARED UNDER MY SUPER VI SION B Y lara U Eel ?. i Caylor , . ENTITLED A_ Study o f Su cc ess in an Introductory u>y;y y. __ i n Hi ;i School Al-oErr m Related to Success in dol Ip-p Algebra BE ACCEPTED AS FULFILLING THIS PART OF THE REQUIREMENTS FOR THE DEGREE OF "r 3 c r o f / : i y Recommendation concurred in* Committee on Final Examination* •Required for doctor's degree but not for master’s Digitized by the Internet Archive in 2016 https://archive.org/details/studyofsuccessinOOtayl Contents (a) Chapter I. The Preliminar y T est. Page Introduction 1 The Questions 2 Method of Grading 5 Distribution of Grades 6 Average or Mean 7 Median 8 Quart ile Deviation 8 Influence of the Class Interval on the Mean 8 Stanuard Deviation 11 Histogram and Probability Curve 13 Errors 16 Conclusions 18 Chapter II. The Final Exa min ation. The Questions 19 Distribution of Grades £0 Mean, Median and Stanuard Deviation 22 Probability Curve 24 Discussion of Semester Grades 26 Influence of the Examination on the Semester Grace.... 28 Chapter III Co) Correlation of Grades of the Two Examinations. Page Correlation Defined.. 30 Correlation Table 30 Coefficient of Correlation 31 Probable Error 32 Regression Lines 32 Correlation Ratios 33 blakeman Test 34 Scatter Diagram 35 Conclusions and Suggestions 41 Chapter IV. C o rrelation with Respect to the Tim e Element . Time Element 46 Correlation toy Years 47 Discussion of and Values 52 Conclusions 53 Summary 54 bibliography 56 1 Chapter I. The Preliminary Examination. Introduction . The examination which furnished the data for this investiga- tion was given in a two and one half hour period, early in October, 1921, to those students taking the three hour course in college algebra at the University of Illinois. The prerequisite for this course is one and one half years of high school algebra and one year of plane geometry. Preceding this examination, the classes had eight review lessons over high school algebra, each instructor following the outlined review suggested by the depart- ment which is here included. The text used was Bietz and Cra- thorne . Lesson I. Introductory lecture. Explain review and test. Give an analysis of high school algebra with emphasis placed on tractions . Lesson II. Theory of exponent s , sections 6- 12. Lesson III. . Parenthe sis, complex fractions , sections 13-15. Lesson IV. Pact or ing , radicals, sections 16 -18. Lesson V. Radicals , sections 19- 20 . Lesson VI. Equat ions ana iaentit ies. Solut ions of s imp 1 e . ions , £ sect i Lons 30-35. Lesson VII. Elimination by addition, subtraction, and substitution, (without determinants). . ■ . ’ 2 The _2ue §,t i ons . The examination was made up through the cooperation of the whole staff. In the spring of 1921, a letter was sent to each of the professors and instructors who had classes in algebra asking for a list of fifteen problems suitable for an examination over high school work. With each problem the instructor was asked to state the principle on which the problem examined so that each problem would involve a single item , --definition , process, princi- ple or ability, — rather than a complex item. Two members of the department who are especially interested in the teaching of Freshman Mathematics, listed twenty-five principles which, in their judgment, the examination should cover and then chose from the lists submitted by the instructors the problem which best tested each principle. Questions involving single principles are most suitable for test use in that a student can classify and work such problems in the short time allowed for the test much more readily than he could think through and solve problems involving complex principles. A student may be able to handle all the algebraic processes singly but not be able to connect them by solving a problem which involves several of them and so he will completely fail on the problem just as does the student who can use them neither singly or together. The wide difference in student ability to use algebraic processes shows up much more clearly when questions involving single rather than complex principles are used. The resulting list of twenty-five questions is given below. The examination included no problem in quadratics simply because the review period was too short to include this subject, and it . . . 3 was left for a subsequent review. The examination was the result of a thoughtful and cooperative effort of the teaching staff. On careful inspection, one nay question the length of the examination, the difficulty or simplicity of certain questions or the correct- ness of emphasis on certain types of problem. However, a criticism of the questions can better be considered at the end of the chapter when the purpose and possible uses of such an examination are dis- cussed. MATHEMATICS 3. October, 1921. Answer all questions. 1. Write the following English statements in algebraic language: (a) The perimeter of a square equals four times a side. (b) The area of a square is equal to the square of' a side. 3 2 2 3 2 o ' ; |7 2. If p=x + 3x y + 3xy t y and Q = 2x + 5 x^y - 4xy^ - y find 2P - q. 3. What should be considered as negative if the following are considered as positive: (a) west longitude. (b> dollars gain. (c) miles northeast. (d> cubic inches of expansion. (e) excess of water pumped from a well over that flowing into it in the same time. i 2 4. s - ■£ gt t at , in which g =. 16. 8 , t=lO, a =65. 3. Find s . 5. Obtain an expression equal to (x(l-x)~ - x (x-1) (x+lV) x 3 2 2 6. Factor x -2xy+ y w - z . 7. Factor ax + ay -t-bx +by. 2a 8. Show that 2b(b-a) = - — — — — Give a reason for each step. - b (a-b ^ . . . . ■ . . . 9. Simplify (h + o-a > ("b - a ^° D ) ( ) ° + a 10. Seduce to a simple fraction: a- x _ a+x a+x a-x a-x a+x a+x a-x 11. Solve and check 3(2x-l) - 4 (6x-5) = H* to X 1 5) - 22. 12. Define and illustrate the solution of an equation. 13. Solve for x: 2x+3 2x-3 48 2x-3 — 2x-+3 4x 2 -9 ~ u Check the result. 14. Solve (x-+5y) = (3x-lly) = 95 15. State three laws of exponents and prove one for positive integers . 16. Find the value of 12^ - 4^ + 3 ^+ 0^ - 8 \ 17. Simplify and write with positive exponents. 18. Simplify 3VT47 - 3 vT 3 IS. Add: VT + Y27 + 3VT8 - Y?6 + W. 30. Simplify 7 +■ /2 7 _ yfT vT“ v—n#" 21. A man of 4o has a son lo years old; in how many years will the father he three times as old as the son? 22. A square grass plot would contain 73 sq. ft. more if each side were one foot longer. Find the length of one side of the plot . 23. Which of the following equalities are true for all values of x and which are true for only one value: . . . ■ ' . ~ VCM) *' fat I . . : . . . : 5 3x^1 = x 4-2 x t-1, x + 4x = 5 X - X 2x -7 - 6 (x+2) (x- 2) ss x 2 - 4 24. (a+b) 2 - 2 2 a +■ 2ab + b . State this law in your own words. 3 4 25. Which is the largest? 'f*. 1~7. Why? Method of Grading . The examination was graded in committee; twenty-five members of the staff working at the same time so that each instructor corrected only one and the same question, on each paper. No grade was placed on the question; instead the instructor put a long dash below the problem he graded in order that the next person might rind the place more readily, and then turned the book over and placed the grade for that problem on the back cover. The grades on the back were written in a column in the order in which the questions occurred. The method of committee grading resulted in greater accuracy and less variation, it is thought. ±$y such a method each one has to keep in mind only one problem and the evaluation or dirrerent types or mistakes in that problem. The ract that no grades were placed on the questions themselves tenuea to do away with any suggestion or ability or lack of ability which might influence the instructor in his grading ir he saw numerous fours or zeros on the preceding problems. In grading the papers, each problem was considered to be of . . 6 the same value as any other. The problems were inarmed on a per- centage basis, each one correctly solved counting four. Ir the algebraic process involved was correctly stated and used but a mistake was made in some arithmetic process, three-fourths credit was allowed and a grade of three given on the problem. Since some problems had two and some four parts, grades of zero, one, two, three and four were possible on practically every problem. The distribution ox grades for the nine hundred and seventy- seven students taking this examination is shown in the following frequency table: Table showing number or students receiving each grade . Grade Frequency Gr ade Frequency Grade Frequency 100 1 84 28 68 17 99 0 83 17 67 22 98 4 82 15 66 26 97 3 81 22 65 11 96 2 80 26 64 21 95 6 79 20 63 28 94 8 78 21 62 23 93 13 77 22 61 17 92 6 76 16 60 14 91 13 7 5 25 59 18 90 14 74 26 58 14 89 11 73 16 57 18 88 16 72 20 56 16 87 14 71 16 55 8 86 18 70 17 54 8 85 14 69 27 53 9 . 7 Grade Frequency Grade Fre quency 52 16 31 3 51 16 30 5 50 13 29 2 49 14 28 6 48 13 27 2 47 10 26 2 46 9 25 0 45 9 24 3 44 14 23 1 43 12 22 1 42 8 21 0 41 2 20 3 40 3 19 1 39 9 18 1 38 5 17 1 37 11 16 1 36 6 15 0 35 11 14 2 34 7 13 1 33 3 12 1 32 2 11 1 Total977 1 The Mean. The average or arithmetic mean is obtained by finding the sum of the numerical values of the measures multiplied by the 1. Rugg, Statistical Methods, p. 117 8 frequency of their occurence, and dividing this sum by the total number of measures. The formula is: Z f .m N where m equals the grade or score, f the frequency of that grade and N the total number of measures, which is 977 in this case. The average or mean for this table, obtained in this way, is So. 96. 1 The Median The median or that point on the scale of the frequency dis- tribution on each side of which one-half the measures falls, is here the grade of the person whose order of ran*c is four hundred eighty-nine. This score falls between 67 and 68 and by interpola- tion is found to be 67.35. It is to be noted that the median is above the mean which indicates that the frequency is greater in the lower part of the range than it is in the upper part. 2 Q. uartile Deviatio n The quartile deviation, defined as one-half the distance between the first and third quarter points in the distribution, equals 18.39. Influence of Class Interval on the Me an . Since the range of grades is from 11 to 1JQ, or a range of 89 units, the data must be grouped into some sets of intervals 1. Hugg , Statistical Methods, p. 109. 2. Rugg, Statistical Methods, p. 155. . . . . . 9 greater than one for convenience in statistical work. After using various intervals to group the data for a histogram, the interval of five units was chosen as the best since it seemed, to fit this particular data "better than intervals of four, 3ix, eight or ten. The intervals of both four and eight units give very irregular histograms. The interval of ten units actually gives the most regular histogram but that interval divides a range of 89 units into only nine classes and it is usually considered best to work 1 with at least twelve to fifteen intervals. Since an interval of five units groups the grades in nineteen classes and since the histogram for the unit five is fairly regular, it was chosen as an apparently good interval. A later study of the mean and the standard deviation for each of these several intervals shows the interval of five to be the best. Taking five as the class interval, the grao.es arrange them- selves in the frequency distribution as shown below. The mean is 2 found by a short method in which an assumed mean is found by inspection and a correction made by adding to it the sum of the positive and negative deviations around the assumed mean divided by the sum of the f r equenc ie s . This may be stated in formula: M = + D - D 1 F + m Where M is the true mean, m the assumed mean, D the deviations from the assumed mean, ana F the frequency of any measure. In the table below the mean is seen by inspection to lie in the interval between 62.5 and 67.5 and is assumed to be 65, which is the mid- point of the interval. 1 . Ib id . p . 86 . 2. Ibid. p. 121. . . ■ I 10 Distribution Table . Class Interval Fre quency Deviation (d) from Assumed mean f x d 7.5 - 12.5 2 -11 - 22 12.5 17.5 5 -10 -50 17.5 22.5 6 - 9 -54 22.5 27.5 8 -8 -64 27.5 32. 5 18 -7 -126 32.5 37.5 38 -6 -228 37.5 42.5 27 -5 -135 42.5 47.5 54 -4 -216 47.5 52.5 72 -3 -216 52. 5 57.5 59 -2 -118 57.5 62.5 86 -1 -86 62.5 67.5 108 0 -1315 67.5 72.5 97 1 97 72.5 77.5 105 2 210 77.5 82.5 104 3 312 82.5 87.5 91 4 364 87.5 92.5 60 5 300 92. 5 97.5 32 6 192 97.5 102. 5 5 7 35 977 1510 - 1315 195 Assumed mean a 65. Correction 195 = 97? = .995 65 t .995 = 66 - 995 s true mean. 11 . When the grades were grouped by intervals or six, eight, ana ten, the mean was found in each case and the results are here set down in tabular form. Interval Mean Variation from true mean 1 65 . 960 .000 5 65.995 .035 6 66.195 .235 8 66.617 .657 10 66 . 390 .430 The mean obtaineu when the class interval is five units varies less from the true mean than uo the other three. 1 Standard Deviation In verifying the choice of the class interval the standard deviation was next considered. The standard deviation cT', is a unit measure or variability. If on a probability curve, a distance equal to the standard deviation be laid off on the base line on each side of the mean, and if ordinates be erected from these points on the base line and extended to cut the curve, then Between the base line, the ordinates and the curve, there will be included in this area 68.26 percent oi the measures represented by the total area. The standard deviation is given by the f ormula : when f is the frequency of occurence of any class interval, d is the deviation of the interval from the mean, and w is the total number of measures. 1. Ibid. pp. 167-173 . • ■ < 12 The work is much shortened oy assuming a mean at the midpoint of some clas3 interval instead oi' finding the actual deviations about the true mean which is a number involving three decimal places. Then if the deviations are laid off in units oi class intervals instead of one, the arithmetic work is reduced to a minimum. The table showing the standard deviation for the interval five is shown at this point but only the results are stated for the intervals one, six, eight, and ten. Computation of Standard Deviation. Interval Frequency (f) Deviation JAl _ * f3T fd 7.5-12. 5 2 -li 121 242 -22 12.5-17.5 5 -10 loo 500 -50 17.5-22.5 6 -9 81 486 -54 22. 5-27.5 8 -8 64 512 -64 27.5-32.5 18 -7 49 882 -126 32.5-37.5 38 -6 36 1368 -228 37.5-42.5 27 -5 25 675 -135 42.5-47 . 5 54 -4 16 864 -216 ^ • O • <2 • D 72 -3 9 648 -216 52.5-57.5 59 — C, 4 236 -118 57.6-62.5 86 -1 1 86 -86 62. 5-67.5 108 0 0 0 -1315 67.5-72.5 97 1 1 97 97 72.5-77.5 105 2 4 420 210 77.5-82.5 104 3 9 936 312 82.5-87.5 91 4 16 1456 364 87.5-92.5 60 5 25 1500 300 92.5-97.5 32 6 36 1152 192 97.5-102.5 5 7 49 245 35 (T - C o-l/vjl nx u / 5 ~ /0 -/ / > ^ cr ■ • 4 1 . 242 13.4610 „ s~^ n -x H . .1505 7 . 2430 H .087 4.8285 2 .054 2.997 z\ .032 1.776 si .018 .999 2f .0095 .527 3 .004 . 222 si .002 .111 The values corresponding to^rwere found in Shephard's Table. The column oi' y values was obtained by multiplying the values by 55.511. Plotting these values the probability curve results. The accompanying graph shows the histogram, with the curve obtained by smoothing it, and also the true probability curve. The curve which is founa by smoothing the histogram is shewed slightly. The three curves shown are for the interval of five. 1 . Ib id . p . 60 . 16 It has teen found that many traits for a sufficiently large unselected group arrange themselves in a curve which we call the normal curve. Grades made in the upper classes in high schools and in college classes usually do not distribute themselves in this fashion, for such groups are selected, groups and the graaes are more frequent along the upper half of the range. It is un- usual to find the graaes in a college class distributing them- selves so nearly along the normal curve as they do here. Ty pes of Problems Misse d . The types of problems on which the most students failed can be easily read off from the next table. For each problem the number of papers having zero, one, two, three or four is stated. Problem. .Frequency. .Frequency. .Frequency. .Frequency. .Frequency of zeros of ones of twos of threes of fours I 41 3 II 229 23 III 9 4 IV 32 55 V 285 7 VI 347 1 VII 56 82 VIII 119 91 IX 397 77 X 294 82 XI 52 4 XII 294 107 XIII 132 188 35 38 890 8 8 709 12 80 872 105 62 7 23 46 111 528 7 2 6 20 3 8 828 280 198 289 67 66 370 67 67 467 180 82 659 162 152 262 36 68 553 ... Table cont 1 d . 17 Problem. . Frequency . of zeros .Frequency. .Frequency, of ones of twos .Frequency, of threes . Frequency of fours XIV 36 13 85 49 794 XV 181 278 240 202 76 XVI 362 158 95 80 282 XVII 315 65 170 139 288 XVIII 376 147 76 79 299 XIX 194 27 85 229 442 XX 452 97 87 44 297 XXI 226 12 15 6 718 XXII 343 12 29 20 573 XXIII 156 25 41 97 658 XXIV 137 30 131 347 332 XXV 193 21 515 70 178 Totals o258 1609 2577 2274 12707 In noting the totality of zeros, fours ana the grades in o e twe en , it is seen that 21.52 percent of all the problems were graded zero, 6.58 percent graded one , lo . 55 percent graded two , 9.31 percent graded three and 52.0 2 percent graded four or absolutely correct. The questions which were ausolutely missed by over three hundred people were those numbered VI, IX , XVI, XVII, XVIII, XX and XXII. These are with the exception of VI and XxII problems involving fractions or radicals. By comparing the failure s as listed in the preceding table with the questions, one sees that the other questions on fractions and radicals all had a large number of failures. i . . ■ . . . . . 18 One sees by reference to the questions that a large number of the questions, ten in fact, involve fractions, radicals, or fractional exponents. The questions no doubt put more emphasis on that type of question than a committee of high school teachers would. Many high school teachers would consider this list of questions too difficult as a test of a student’s knowledge of high school work. But as a test of ability or a test for measure- ment purposes it is good because the results do snow accurately the wide range of ability amoung the students and allow of a possible reorganization of the classes which a shorter, easier test would not do. The test was probably not too long because the majority of students worked on all the questions, and there was no large decrease in the number solving correctly the last part of the list. C one lus ions . The results of the first test show a great range in student ability in Freshman Algebra. The fact that one-thiru of the students scored less than 60 in such an examination following a review, shows a need for some reorganization. Whether this should be by means of a redistribution of the students in classes, such as zero sections, star sections, and average sections, or by a method of entire elimination from the course is left for discussion in the chapter on correlation. , . IS Chapter II. Th e__Fi nal E xa mi nat ion . The semester examination was conducted in essentially the same manner as the preliminary examination. The questions, fewer in number but more complex in character, are given below. Mathematics 2. Answer any ten questions. Time, three hours. 1. Simplify 1 - x^ x 1 + - 1 +■ x 2 x i-x~ Solve f or y the fol low ing sys tern by determinant s ( x - 7 - z -6 i 2x +■ y + z — 0 ( 3x - 5y-*-8 z = 13 Find t he solut ions of 2 X - bx + 2 x^ - 5x 1 10 — -2 Date rm ine K so that the e quat ion x J +- 4x - 2k = 0, sat i s f i 8 s the gi van condi t i on in each c ase : : (a) one root is i; w the tw o r 00 t s are e qual ; (c> the product of the r oot s is 3 . 5. Solve for x and y x - 2 xy+ b = 0 (x -y)~ - 4 - 0 6 . Obtain the 1st, 2 nd, 3rd and ?th terms in the expansion of ( ^r- y) 8 . 7. The second term of a geometric progression is 4 and the fifth term is -32. Find the first term and the sum of the first seven terms. 8 . Find all the roots of x 3 .2 - x -3x -f- 4 x - 4 — 0 . . . • . 20 9. Given log 2 = 0.3010, log 3 = 0.4771. Find 10. Answer one or the following: (a) From 11 men how many committees of 4 men can be selected, when one man is always included on the committee? (b) If 2 balls are drawn from a bag containing 3 black ana 5 white balls, what is the chance that they will both be black? 11. Answer one of tfce following: (a) Derive the formula for the roots of ax 6 f bx t c = 0. (b) Derive the formula for the sum of n terms in arithmetic progression. 12. A rectangular lawn is 9o feet long and 60 feet wide. How wide a strip must be cut around it when mowing the grass to have cut half of it? D ist ri bution of Grade s . The number of students who took the semester examination was nine hundred eight, or sixty nine less than took the October examination. The grades for the nine hundred eight stuaent3 are arranged in the following frequency table. Grade Frequency Grade Fre quency Grade Frequency 100 11 90 20 80 21 99 5 89 16 79 10 98 15 88 17 78 14 97 14 87 17 77 12 96 15 86 19 76 9 95 11 85 18 75 15 94 2 84 15 74 19 93 17 83 15 73 14 92 10 82 13 72 17 91 10 81 7 71 12 . . * . . . 21 Frequency taole cont'd. Grade Frecjuency Grade Frequency Grade Frequency ?0 16 46 9 22 0 69 14 45 11 21 4 68 12 44 6 20 3 67 21 43 12 19 0 66 17 42 11 18 3 65 21 41 9 17 2 64 19 40 7 16 3 63 12 39 5 15 1 62 11 38 5 14 1 61 14 37 2 13 3 60 20 36 6 12 2 59 10 35 6 11 2 58 14 34 5 10 4 57 16 33 3 9 4 56 12 32 7 8 2 55 11 31 1 7 0 54 12 30 2 6 3 53 8 29 3 5 1 52 13 28 5 4 1 51 10 27 3 3 3 50 7 26 1 2 2 49 9 25 3 1 2 43 6 24 2 0 13 47 8 23 2 Total 908 ' 22 It i s to be noted that there is an increase at the extremities of the range in this table as compared with the other examination. In the first examination the lowest graae was 11, and only nine graa.es fell below 20. Likewise the graaes over 90 were few, seventy in fact. hut in the preceding table, show- ing the distribution of grades for the second examination, there are thirty-nine grades below 12 and fifty-two below 20, but on the other hand there is an increase near the upper limit, for one hundred thirty students have grades of SO or above. It is interesting to discover just how the change in distrioution affects the mean, median and standard deviation. Mean , Median an d Standard Deviation . Since the total frequency is nine hundred eight, the median lies between the four hundred fifty fourth and four hundred fifty fifth score when they are arranged in order of ranx. by interpolat ion we find: Median «■ 67.881 The calculation used for computing the mean and the standard deviation follows. The results obtained are: Mean ** 64.879 Standard Deviation — 23.45 . 23 2 Ulase Interval . . .Frequency. .Deviation. . .fd. . . .fd 100 - 9?. 5 31 7 217 1519 97 . 5--92 . 5 59 6 354 2124 92.5- 87.5 73 5 365 1825 87.5- 82.5 84 4 336 1344 82.5- 77.5 65 3 195 585 77.5- 72.5 69 2 138 276 72.5- 67.5 71 1 71 71 67.5- 62.5 90 0 1676 62.5- 57.5 69 -1 -69 69 57.5- 52.5 59 -2 -118 236 52.5- 47.5 45 -3 -135 405 47.5- 42.5 46 -4 -184 736 42.5- 37.5 37 -5 -185 925 37.5- 32.5 22 -6 -132 79 2 32.5- 27.5 18 -7 -126 882 27.5- 22.5 11 -8 -88 704 22.5- 17.5 10 -9 -90 810 17.5- 12.5 10 -10 -100 1000 12.5- 7.5 14 -11 -154 1694 7.5- 2.5 8 -12 -96 1152 2.5- 0 17 -13 -221 2873 908 -1698 20022 1676 -22 : 1AAJ? Ccx V '■ & V V u. — (o ^ (T* ^ - ©a; I* r <2 00^2. t t. x'y' is obtained by multiplying the frequency of each class by its deviation from the horizontal axis and then by its deviation from the vertical axis, and finding the sum of these products. C and C v are the corrections to be made about the assumed means in the x-array and the y-array, respectively. The horizontal axis is drawn through the assumed mean of the y-array and the vertical axis through the assumed mean of the x-array. C Tjt and <7y are the standard deviations of the two arrays. These terms C , C , have been used in the calculation of x y the mean and of the standard deviation. P. E. is the probable error, to be defined in the next topic. The ar ithrae t ical work involved in the calculation of r is given in detail with the correlation table and the resulting value i3: r = .673 p.E. 1. Rugg , Statistical Methods Applied to Education, p. 269 . . . * . 32 . Probable Error The probable error in any result may be defined as that deviation from the determined value such that it is an even wager that the true value lies within this amount of the determined value 2 The formula for the probable error in the case of the correlation coefficient is: P. E . of r = .6744 9 (l-r *S i~ir where r = coefficient of correlation, w = frequency of distribution. Eor this data p £. m .£ 7Y?7 ( / - ' L - .0 13 Ou+4 T -.i. 73 31.0 13 3 The Regression Lines. The degree of correlation in the data is indicated by the variation or the means from array to array. The variation in the means of the arrays is shown graphically by the curve of the means which is called a regression curve. There are two regression curves, one corresponding to each set of arrays. A straight line fitted to the means of the arrays is called a line of re- gression. If the regression curves approximate straight lines, the regression is said to be linear. The slope of the regression lines depends on the standard deviations of the two arrays. The equations of the lines of regression are: T fy 1. Rietz and Shade. U. of J. 2. Rugg , Statistical Methods 3. West, Introduction to Mathematical Statistics, p. ^4. Studies applied Vol. Ill p. 17. to Education, p. 272. ' . - • - . 33 In this correlation and by substituting become the values or r, C^-and 1 have been determined these values in the auove equations, they - JC Jr which are the equations of the two lines of relationship. 1 Corr e lation Ra t ios . If the means of the correlation table do not accord fairly well with a straight line, the coerficient r and the regression equations will not describe the relationship of the two traits unaer consideration, for the coefficient depends upon the fact that the means of the arrays lie not far from the line of regression. If the means of the table do not fall approximately on a straight line, the relationship is expressed by the correla- tion ratio >[, which is the ratio of the standard deviation of the arithmetic means of each of the columns (or rows) of the table to the standard deviation of the whole table itself. There are two values of one for each array. expresses the dependence of y upon x, or in this table the dependence of the preliminary examination upon the final; ^expresses the dependence of x upon y, or of the final examination upon the preliminary. The latter ratio is more significant in this study because if the second examination depends very strongly on the first, the preliminary examination will have a prognostic value. 1. Rugg, Statistical Methods, p. 276-282. 1. West, Mathematical Statistics, p. 76. ■V . . . . . 34 To calculate the values, aaa two rows and two columns to the correlation taole. One the ( JE? X 0 column is merely the square of the XXao lumn already in the table. The second is obtained by dividing this £ -X^column by the frequencies or the y arrays. The formulae for the correlation ratios are: y Vr (7 ■3T" y The calculation is included in the correlation table. The values obtained are: = .6766 yl = .6372 K Blakeman Test . In order to determine whether or not the correlation table exhibits linear regression, that is, whether we can use the pro- duct moment formula, we use the Blakeman criterion for linearity. 2 It is that if J\T - r V <21 then the distribution shows linear regression. The test in this correlation shows that the regression is non linear for the value, or the dependence of the first examination on the second. M -.r*) - Y, Jf - r’) - /<*■ >U 1. Bugg , Statistical Methods, p. 283. 2. Professor A. h . Grathorne suggested this simplified form : . ■ 35 Since the regression is nofi linear the regression lines cLo not i'it the curve through the means or the arrays. This is shown by plotting the means of the arrays ana noting how far they fall from the regression lines. The values to be plotted are shown in the last column ana last row or the table. S c atter .Diagram . The correlation table is given in much detail. The worf in rinaing r, and , follows rather closely the methods used by Hugg ana by West except for some s impl ir ica r ion which was suggested oy Professor Urathorne. The correlation ratios can be obtained directly from the table, oy the addition or columns 7 and 8, and the corresponding rows. The results of the work have been stated in the preceeaing topics hut they can be more readily understood when the correlation table is put in the form of a scatter diagram. Such a diagram is shown in the next table. The numbers representing the frequency in each interval are replaced by the correct number of dots. The diagram contains eight hundred sixty seven dots each one located correctly with reference to its position in the two examinations. The axes are two lines drawn tnrough the means of the x ana y arrays,. Taking the intersection of these two lines as the center or origin, the regression lines are plotted from the equations previously formulated. The means of the arrays through which the regression curves should pass are plotted from the values in column 9 and row 9, respectively. In the diagram the positive direction is to the right and upwards and the negative to the left and downwards. ' . . 36 N £ 5 * v \*\j n 2 C: ^ XA ^ nr n yj^ nr n ^ 22 nr *i n A ^ X X X •Y^ "~ nr A *3 £. -3 r« o- NS IN. N to '^3 ? „ Ni Q ^ N ^ ^ £7 N N c^ is. o n x to ns -o 7 q cn |C 'N <; n nS N Y N x !_ n®i L. D v& n cn. n*. >. >S ^ ^ £> ° ^ N v n 1 i n/ no o* Oq oq X I i l O - K^ & fc £ ni ;n C^ T sS ^5 ^ ^ ^ N»^fe TYi 1 >777?' 1 o? ns CIN O o \ nS q, y 2 7 7 0t> N C* N <0 X X nr nr 'J'n X nr n cy nr nr l nr oM q or c i I On on (N n in- ^ ^US ^ b' 77 | ON i^o n; ^ v> \J| bo • I I II II ? q' *) t>e ^ ^ ni o co cV rt» r~^. cn Cr-» Qi I ^ oo CN s n> A n q> Q> 'n / 'O k) NS N IN nr n) A n, n(- h> n^> O'- n- \ ^T be n3 O- N n( ci v^N '—' Z> '-^. 0^ nS bo n> bei -», *s V\ IN N) ^ n- ^ ^N. - nr n< on on £-- On N- be> O- />. ■nS ^0 X \ CN bj) X N) bo (>, X -S> 7 /" X ^ qr nr s \ Vi O Vj r Tv- vXi Q) Xje • ^ V) *1 ^ N> W c^ '-- q X -3 ^ '- *2> IN Co JO c^ x c^c) w ro; -- nr t£- ^ On oo n nr — ^ H ^ ^ n. ^ ^b 7S -N,qq : N* °i N3 X Z°)b- jftQt- S3 1 1 IT f {, U~ 6S 8 sTT ; J7 7 AI 4' AST : 7 1 7 2r 172! 7 °) / 77 \ OA (7 $Aq if- j: S3 / - :d - : }T7 f - it/- nr L? &~ ty/- 4/> i,- HI- S C — i ?£&- {,& 7- •'O °Y cY nr n^ ^ nf N > V> o '. x nr n o n q ' q n n 7 i Yjii fO csz* TV Mi TZ j/S’ £Z LOSS/ &V‘U t Lbl IM 7 JjzeAof fc‘ 1 0 14 a WEC L?S 1 ? 47' _cs 4 oc~ 00 b °)QL'Lj Lv' 4m Ll QIC ll 4 w? xx S£/ scat? oos IS ;'V* Loa fcV LMf hoH s - jjj H te- 74 'S 1/Lff hCQ 7 nr/- 4 r?‘ ti - y.ih MM ML Li v?/- Mo/ /X l$£ lol? Mb' Lf £i‘C- Mo/ Ll- oss WqM 2/‘i~ a/2 M- ML oo Mb IT- 2~if- oof 00°)/ LUW // T- 0l? / is - /L£ hb sn'% (7/T- 0/o A 00 - OCL 00 0£ nnsc ■$■■% - />/<}/ oy ~ 0/£. 20 L ■nr/s/ s~ X ■X 4 NS X ns r VJ \ X va X ll ACL£ } /<*/- Lib! lOiTo/ U]2‘7fS /P'S*- -■> Jh sU S / -z\ ' ’ > *t II rJ U V3 is ' , . . ... ' . . . 40 If each student made exactly the same grade on each examina- tion, the two show an absolute dependence, one on the other and a perfect correlation is said to exist. In such a case all the frequencies or the dots in the scatter diagram would fall on the main diagonal, a line making 45 degree angles with the horizontal and vertical axes. The higher the correlation the more nearly do the dots arrange themselves on the main diagonal. In a perfect correlation, the value of r or >| is one; if no correlation exists the value of r or i| is zero. Between these two lie all degrees of correlation. There are many opinions as to what constitutes a high correlation as is evident from a study of educational reports 1 involving correlation problems. Rietz found that the coefficient of correlation for stature of father and son is .596 and for stature of brother and brother is .591. We think of stature of father and son as closely related variates so Rietz concludes that anything greater than .4 is a high correlation. A reference to a series of correlations made by Professor Crathorne for students in Mathematics courses in the University gives further standards for an interpretation of the magnitude of correlation which is to be considered significant. A correlation between grades made in Freshman Algebra and Trigonometry taken under the same instructor and taken in the same semester for two thousand cases shows a value of J\J = .?3?±.Q07 1. Rietz and Shade. U. of I. Studies. Vol.III. p. 19 * 41 A correlation between Freshman Algebra taken in the same semester under different instructors for four hundred students gives a value of r = .620 J-.Q21 A correlation for Mathematics 7 and 9, two successive courses in Calculus, under the same instructor and with the same text, for two hundred and fifty students gives a value r = . 330 ± .013 These three cases are ones in which the relationship is known to be very great so that the resulting correlations are considered high. Students taking Calculus for two semesters unaer the same instructor would in most cases receive identically the same grade and a correlation ootained in such a case is to be thought of as unusually high. Likewise a correlation obtained for parallel courses in Algebra and Trigonometry is thought of as high. Since the correlation between the preliminary and final examinations in this present discussion is .673 ana lies between the correlation of .620 for Algebra and Trigonometry under different instructors, and that of .737 for Algebra and Trigonometry under the same instructors, it must be a high correlation. C onclu s ions and Suggestions . A correlation of .673 such as we have in this table is very high when compared with the correlations cited aoove. In the scatter diagram the slope of the regression lines shows the broad general tendency of the traits. The rate at which one trait increases with the other depends upon the slope of its regression line, so the relationship can be read from these lines. ’ . . ' . . 42 When the true means are used instead of the regression means, in this diagram, the regression curves would pass through the dots indicated by crosses for the one case and by circles in the second case, which do not coincide with our regression lines, as one sees at once from the diagram. In the consideration of the prognostic value of the preliminary examination it is essential to consider all the data as set forth in the scatter diagram, rather than to draw sweeping conclusions from the high correlation coefficient as is so frequently done in educational studies. The correlation coefficient or ratios express a relationship for the group as a whole but they say nothing certain regarding the individuals. There may be a number of cases which show absolutely no relationship in the two traits, out on inspection of our correlation ratios will fail to indicate this. Inspection of the diagram shows that of the seventy-seven students scoring less than 42.6 on the first examination only three scored more than 67.6 on the other examination. One hundred seventeen scored less than 47.6 on the first examination and of these only thirteen scored above 67.6 on the second. From the one hundred seventy nine who were graded below 52.5 on the first examination, only twenty-four were above 67.5 in the final. The indications then are that if the students who made the very low grades, say below 50, on the first examination, were eliminated from the course at that time, or transferred to some course adapted to their needs, the number of failures in the Freshman Algebra classes would be reduced by half. . . . . . . 43 Another consideration here adds its weight to the argument for the prognostic value of this preliminary test. That is the consideration of the record of those students who withdrew before the final examination. hy reference to the table of semester grades in Chapter II, it is found that one hundred sixty-eight students withdrew before the final examination. Of these ninty-nine with- drew even before the preliminary test, and so do not enter into this study in any way. Of the sixty nine who withdrew in the period between the two examinations the tabulation of results from the first test shows that in a majority of cases these were weam students who eliminated themselves. So it appears that the stuaents who did very poor work in the October examination, did almost as poor work in the final, or in many cases eliminated themselves before the final. The table shows the record of such students. First Examination Record of Stuaents Withdrawing before final examination. Grade Frequency Grade Frequency 90-100 4 40-50 15 80-90 1 30-40 10 70-80 6 20-30 4 60-70 10 10-20 4 50-60 15 0-10 0 Total . . . The method to be adopted in a reorganization of the Freshman Algebra course following an examination over high school algebra, . 44 is to a large extent an administrative problem. There might be merely a shirting or students within the classes, so that the very poor ones form "zero” sections, and the very superior students "star” sections. The Benefits of such a division are debatable, but such an arrangement is not without successful precedent. Those who do not need the algebra as a prerequisite for other courses and can secure their group requirements in another way might well be eliminated entirely if their first examination grades are unusually low. If there was no way to enter them in some other course at that time, such students could well oe allowed to devote their time to the remaining thirteen or fourteen hours which they are attempting to carry. Another problem is that of taxing care of those students who are weak in algebra and yet must carry the course because it is a prerequisite for their later courses. If a student from the engineering school fails in algebra, he is in most cases so delayed, that his college course is prolonged one year. Almost the only suggestion here is that these students should be given five hours of algebra a week with three hours credit. This method was formerly used and the chief objection to it was that the weaker students were required to do more work than the others, for the preliminary examination subsequent to registration meant an additional two hours work on an already normal schedule. But since such a high correlation is seen to exist between a preliminary and a final examination, there is nothing to be said for allowing the poor students to go on through the semester to almost certain failure, if it can be prevented by some other method. In order that elimination or reorganization work an injustice to the fewest . •! . 45 number, the standard for elimination should he low. For instance, we have shown that an elimination of all students scoring less 1 than 42.5 woulu at most he unfair to three. A number of studies show that the correlation between entrance examinations and college work is much lower than a correlation between high school work and college work and conclude that entrance examinations are to he seriously opposed. Professor Lincoln of Harvard states that the correlation between entrance examinations and college worm is .*7 and that between high school work and college work it is .69. The preliminary examination of this discussion was not an entrance examination for it was given after eight review lessons under college instructors, ana the coefficient of correla- tion was .673, much higher than in the case of entrance examina- tions as cited by Lincoln. This difference largely removes the objectionable features of an entrance examination , but to be absolutely fair in the matter, the elimination standard should be made lower than the usual passing mark. A careful considera- tion of the scatter diagram should be the basis of a decision in this matter and in this case the conclusion is that a score of 50 would be approximately the correct standard for elimination. 1. Thorndike, Educational Review. Vol. XXXI. 1 . Lincoln, E. A., Relative Standing in High School. Early College and Entrance Examinations. School and Society, Vol.V. . ■ . . ' . * I 46 Chapter IV. Correlation with Respect to the Time Element. The previous chapter on correlation does not consider the element of time which is also a variable for this data. Many of the students had taken high school algebra several years ago, while others had taken such courses very recently. It would seem that the length of time intervening between the last high school course and this examination over high school algeora would be a factor of decided influence. To accurately determine this factor the students cards were divided into seven groups, according to the length of time which had elapsed since their high school work in algebra. The stuaents were asked to give this data on their final examination books and only six hundred eleven did so. How- ever, this is a large numoer and is representative in its distribution over the seven periods as is shown in this table. Date of last course Time since .frequency 1921 1 yr . ago 74 1920 2 yrs . ago 172 1919 3 yrs . ago 177 1918 4 yrs . ago 83 1917 5 yrs . ago 46 1916 6 yrs . ago 25 1903-1915 7- 14 yrs, ago 34 Total 611 . . • 4 ? If the time element is non-essential approximately the same correlation would be reached in each of the seven groups, when the results of the two examinations are correlated separately for each group. If the time element is noticeable, a gradual increase or decrease in the correlation coefficient and ratios will indicate it. Correlation tables are given in each case, but the values of >| , and ¥ together with the probable error ana the Blakeman test are shown in the one table. Correlation by years. Since last course r P.E. > blakeraan Test (yj y ) Blakeman Test ()j x ) 1 year .7015 .040 .771 7 . 57 < 1 1 .765 6.89 <11 2 " . 703 . 026 .745 10.45 < 11 .718 3.66 <11 2 « .679 .027 . 692 3.01 <11 .728 12.04 > 11 4 " . 674 .040 . 804 16.01 >11 . 733 6.89 <11 5 " .608 .062 .903 19.6 > 11 .741 8.25 < 11 6 " .533 .096 .866 11.60 > 11 .746 6.81 < 11 7-14 " . 749 .051 .771 6.46 < 11 .765 4.31
  • 5fc> / r / &T / / / / / / £ £ / $o / -? / / / 7^ / / cj / IS 2 1 70 / •» / <2./ N = 74- / / / / / •3 60 / / / / / / / SS / / / st / / / jy / *#- jo tS /O c S SC /cr-7o oPyS ^^(5 SS~ 9c 9ir fC yTj" Oc &? ?o P'S gc ?S 9& 9s ' 49 /M> ^0 $S 7S 7 0 L,S jS so Y c 3S l &> / JS 2®\ !$ /O :js 30 JS 20 IS 10 / / / / / 3^3// / <2 r-r % / ! / A / / <2 / -5 3 / / 3 / / / / / / FA / / / / / / / / / / 4 / / / / // // / / / / / / / / / / n i* N--%3 0 s / c 'Sr j-o a-s •JO 3> yc >“^T > 4T2>~ ^(9 &C> 7C xr - FO -?o zs /tn> / ?S / 4 / JD / / 3 / / & <=R / J °2 Z c 3 */ / <5" 33 / *? nn 7 3 / 3 / 3 3 * 3 / Y NrH7 yc <2 / / ^ f / «R / 3 L*> / / / £ / ¥ / / IpO / / / -2 / y a / ss / 1 a £ / sc / / / / ¥■ / / / / ?s / 12 J <5 A 2 / / 70 ■ / / / ^ 45 " /£? /«r Jo ^r36 3tf-*o XI- 5® ^5" OF?* ?,T £0 ^ *? Y-S' '0V . ' /S 52 D iscuss i on of r and ')] va lue s . In reviewing a large numoer of eaucational stuuies which involve statistical methods I have notea that the conclusions have been based solely on the coefficient oi correlation. If only the values of T were used from the preceeding table, we could reach conclusions entirely contrary to those depending on a considerat ion of the y values. It is they! values which are significant for the hlakeman test shows that in most of the cases the regression is non-linear, and that the correlation coefficient is not an accurate measure of relationship. From the T 1 value s , which show a gradual aecrease with the increase in time, one would conclude that those students who took their high school work some years ago did much poorer work on the first examination than those wno had taken it only recently, and hence were open to a greater degree or improvement during the semester, so that a lower correlation existea between the two examinations. how- ever, a consideration of the correlation ratios, which are the more accurate measure of relationship here, shows no such gradual decrease with the years. Since the degree of dependence of the second examination upon the first is the important relationship, the values of are to be studied. There seems to oe no special significance in the size of these ratios, although with the exception of 1921, the >/ x -values increase slightly with the time, instead of decreasing. The high V( value for the stuuents in the 1908-15 group is at first surprising but if can he easily explain- ed in view of the fact that students who return to school work - ■ . ■ 53 after a long period of time are no doubt exceptional students , or at least more mature and more apt to maintain the same stanuara of work at all times, leading to a high correlation between the examination graces. Conclusi o ns . Prom the correlation as set forth in this chapter, one can draw little positive conclusion as to the effect of the time variable in the investigation. The wor.£ in this chapter is of considerable value, however, in that it furnishes a striking example of the fallacy of establishing the correlation coefficient and drawing sweeping conclusions therefrom, unless one has first studied the type of regression and found that this coenicient is a true expression of relationship for the problem at hanu. One who is acquainted with the matnemat ical questions involved in the interpretation of the coefficient of correlation is continually appalled by the manner in which writers of little mathematical training, who have obtaineu correlation results oy biinuly following a given formula, araw sweeping conclusions from such results. With the present trend of modern educational metnods toward the use of tests, measurements and statistical representa- tion, there is much room in this field for investigators with a thorough icno'.vleuge of statistics who will sanely interpret the results of their studies. ■ » 54 Summa ry. Certain definite conclusions can be made from this study. In the first place, the results of the preliminary test show that the students taking Freshman Algebra are essentially a non-selected group; that about one-third of them fall below 60 in such a test and that there is need, from the very first of tne semester of some sort of readjustment. Next, we found that in the final examination more than one-third fall uelow 60; that the average and median grades approximate those of the preliminary test, so that there seems to be no general improvement in the group. In the third place, a correlation of the two sets of grades gives a coefficient equal to .673 which shows that the second examination is highly dependent on the preliminary test. A study of the scatter diagram shows that of one hundred seventeen students scoring less than 47.5 on the first examination only thirteen score above 67.5 on the final, and that of one hundred seventy-nine scoring less than 52.5 on the first examination only twenty-four were above 67.5 on the final. So if 50 were taken as the elimina- tion standard, the failures in Freshman Algebra could be reduced by one-half. Lastly, a consideration of the time elapsing since the student's last course in high school algebra, indicates that this is not a decided factor in the investigation ana that with a review period previous to the first test, this time element is of little significance. The question of a reorganization method . > H . . . - 55 is yet to be decided. The poorer students who do not need mathe- matics as a prerequisite to other courses, might well be eliminated entirely. Those who must have it to enable them to take other courses, should be put in a special course designed to meet their needs. This study clearly indicates that a very high prognostic value may be put on such a preliminary test and that it should be given with such a purpose and the results used for a complete reorganization of the Freshman work. ' . 56 it .bib 1 iog raphy . Beferences to Books . Rugg , Harold 0. --Statistical Methods Applied to Education. West, Carl J . Intr oauct ion to Mathematical Statistics. Henderson, J. L. --Admission to College by Certificate. Whipple, G. M. --Manual of Mental and Pxiysical Tests. References to J o urnal s . Burris, W. P . --Correlat ion of Abilities Involved in Secondary School Work. Columbia University Contributions to Education Vol. Al. February 1903. Broome, E. C.--A Historical and Critical Discussion of College Admission Requirements. Columbia University Contributions to Education. Vol. XI April , 1903. Dodson, E. C. --Study of Student Achievement in Mathematics. School Science and Mathematics. Vol. XIV. May 1914. Frailey, L. E. , and Crain, C. M. — Correlation of Excellence in Different School Subjects. Hancock, H.--Crit icism on Mathematics Teaching. School and Society, Vol VI, September 1917. Lincoln, E. A. --Relative Standing in High School, Early College and Entrance Examinations. School ana Society. Vol.V April 1917. Moritz, R. E. --Mathemat ics as a Test of Mental Efficiency. School ana Society Vol. VII, January 1918. * . . . . I _ M ■ ; • «r* ; ' “ . "y i * i 57 Rietz ana Shade. --A Study of University of Illinois Grades. University of Illinois Studies, Vol. III. Thorndike, E. L.--The Future of the College Entrance Examina tion Board. Educational Review, Vol XXXI, May 1906. Werrmeyer, It. W. --Rel iao il it y of Test Grades in Mathematics. School Science and Mathematics, Vol. XIV. May 1914.