THE UNIVERSITY OF TEXAS BULLETIN
No. 3606: February 8, 1936
1lc i
THE TEXAS MATHEMATICS TEACHERS'
BULLETIN
Volume XX
thi"' r 1t, .t 1. "'l'.'.~·
Fulf' f." i.C J
PUBLISHED BY
THE UNIVERSITY OF TEXAS
AUSTIN
Publications of The University of Texas
Publications Committees :
GENERAL:
J. T. PATTERSON R. H. GRIFFITH LOUISE BAREKMAN A. SCHAFFER FREDERIC DUNCALF G. W. STUMBERG FREDERICK EBY A. P. WINSTON
ADMINISTRATIVE:
E. J. MATHEWS L. L. CLICK
C. F. ARROWOOD C. D. SIMMONS
E. C. H. BANTEL B. SMITH
The University publishes bulletins four times a month, so numbered that the first two digits of the number show the year of issue and the last two the position in the yearly series. (For example, No. 3601 is the first bulletin of the year 1936.) These bulletins comprise the official publications of the University, publications on humanistic and scientific subjects, and bulletins issued from time to time by various divisions of the University. The following bureaus and divisions distribute bulletins issued by them; communications concerning bulletins in these fields should be addressed to The University of Texas, Austin, Texas, care of the bureau or division issuing the bulletin: Bureau of Business Research, Bureau of Economic Geology, Bureau of Engineering Research, Bureau of Industrial Chemistry, Bureau of Public School Interests, and Division of Extension. Communications concerning all other publications of the University should be addressed to University Publications, The University of Texas, Austin.
Additional copies of this publication may be procured from the
University Publications, The University of Texas,
Austin, Texas
THE UNIVERSITY OF TEXA5 PRl!SI
~
THE UNIVERSITY OF TEXAS BULLETIN
No. 3606: February 8, 1936
THE TEXAS MATHEMATICS TEACHERS'
BULLETIN
Volume XX
PU•Ll•HllD •Y THIE UNIVIEUITY POUR TIMU A MONTH AND IENTl:RKD A8
•IECOND-CLA99 MATTIER AT THIE POSTOP'P'ICIE AT AU•TIN, TDAS,
UNDIER THIE ACT OP' AUGU•T &4, 191&
The benefita of education and of
uaeful knowledce, senerally dilluaecl
tbrouch a community, are eaaential
to the preaerTation of a free covern•
meat.
Sa.m Houaton
Cultivated mind ia the cuardian ceniua of Democracy, and w hi I e guided and controlled by virtue, the nobleat attribute of man. It ia the only dictator that freemen acknowledge, and the only aecurity which freemen desire.
Mirabeau B. Lamar
THE UNIVERSITY OF TEXAS BULLETIN
No. 3606: Februar:v 8, 1936
THE TEXAS MATHEMATICS TEACHERS'
BULLETIN
Volume XX
Edited by
P. M.BATCHELDER
Auociate Profeuor of Pure Mathematica
and
MARY E. DECHERD Assistant Professor of Pure Mathematic&
MATHEMATICS STAFF OF THE UNIVERSITY OF TEXAS
R. E. Basye R. N. Haskell
P. M. Batchelder F. B. Jones
H. Y. Benedict E. G. Keller
J. W. Calhoun L. J. B. La Coste
C. M. Cleveland R. G. Lubben
A. E. Cooper R. L. Moore
H. V. Craig Mrs. G. H. Porter Mary E. Decherd M. B. Porter
E. L. Dodd W. P. Udinski
H. J. Ettlinger H. S. Vandiver
0. H. Hamilton C. W. Vickery
This bulletin is open to the teachers of mathematics in Texas for the expression of their views. The editors assume no responsibility for statements of facts or opinions in articles not written by them.
TABLE OF CONTENTS Numbers or Mathematics____________________________________ H, J. Ettlinger_________ 5 An Insight into the Mathematical Situation in Texas________________________________
__________
_______________Bee Grissom._____________ 10 Means or Averages and their Uses_____ __
____________E, L. Dodd________________ 19 Why do Students Fail?-------------------------------------Mary E. Decherd_____ 35 The Brown University Prize Examination_________________________________ _________ 39
NUMBERS OR MATHEMATICS*
BY H. J. ETTLINGER
The University of Texas
As the University representative in mathematics of the State Curriculum Revision Committee, it is my purpose to survey in brief the mathematics part of the work of this Committee. On this program there will be several reports with respect to particular grades and subdivisions such as the junior and senior high school. The teachers themselves are sometimes expected to "look up" toward the work which is to follow and which may possibly culminate in work at the University. May I say at the outset that I agree most heartily with those who maintain that possibility of the student's future attendance at a university or college should not dominate, nay, even affect in any way the content of the course in hand in the lower schools. I would be willing to defend this statement not alone with respect to the internal construction of the mathematics curriculum throughout the grades but also with reference to the definite place that mathematics should have in the curriculum itself. I accept whole-heartedly the platform laid down some years ago by the National Committee on Mathematics Requirements that each grade or each year of mathematics should be based entirely on what the student is prepared to receive at that stage, without reference to what he will do later. In a program which is planned to develop to the fullest the social adaptability of each child at every stage, number work must have a prominent place. Number situations are so intertwined with everyday life and experience, with necessities and comforts, with intelligent voting, with personal requirements involving the elementary physical situations as well as the problem of living and working together happily, that there can be
*Report addressed to the Mathematics Section of the Texas State Teachers Association, San Antonio, Texas, Nov. 29, 1935.
only one answer to the question "Should number work or mathematics have a place in the curriculum?"
The objections to the study of mathematics, which have now become jokes almost of the horse-chestnut variety, are aimed at weaknesses which are present partly in content and partly in the teaching of the subject. I shall content myself with pointing out some simple examples. The old courses of formal manipulation and "dry as dust" mechanical technique belong to the middle ages of education. The learning of proofs by memory and the acquirement of any other kind of information, such as formulas, by rote as an end in itself, must go the route of the extinct dodo bird into oblivion.
Without any further preliminaries, we will state then that it is the purpose of mathematics to train students to think quantitatively and spatially. These are the lowest terms to which one can reduce the objectives of mathematics. There is no clash between the above statement and the assertion that studies should have social ends or should be concerned with real life situations, or that education should be functional, or that courses of study should be interesting. It is a trite saying to maintain that every person at every stage of existence is confronted by number situations. You may be a poet by temperament, you may be an artist by inclination, you may be a literary genius by heredity, but you cannot escape the everyday number situations which confront you as a human being, as a member of a community, as an individual with problems involving an understanding of numbers. The history of the individual as well as the history of the race could be told in numbers not necessarily limited to the seven ages delineated by Shakespeare. One can easily parallel the entire history of the development of the number system with the human history either of the whole race or of a single person. Similarly, but in a no less manifest manner, the sense of appreciation of form or spatial relationships is very important in the everyday life of the individual. These two notions of number relationships and spatial relationships
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must have their threads or strands completely shot through our courses of study. We must eliminate memory work, we must eliminate tedious manipulation, we must translate everything into such terminology as can be understood by the student at each particular stage, but the teacher who would successfully impart mathematical knowledge must at all times be conscious of the fact that mathematics is numbers and form.
The structure of our courses, as well as their content, should reveal the nature of the study of numbers itself. We are forced to expand our remarks particularly concerning the drill side of number work. The vast mass of criticism levelled against mathematics is based on the drill or manipulative side of mathematics. Particularly at the high school stage, algebra and geometry draw the fire of those who themselves are completely untutored in the fundamental concepts of number work. They claim that there are no relationships between real life situations and factoring in algebra and little or none in the demonstrations usually given in plane geometry. These skillful teachers will, however, definitely show the very close relationship between the number situations or spatial relationships involved and those of everyday life. For after all, in order to know mathematics, one must know how to do it. The only way to learn to do is by doing it. Consequently, drill work must be given in order that skill may be developed. One may give the following illustrations from the very beginning in the lowest grades. It is of significance that in the beginning we count objects such as one apple, two apples, three apples, etc. ; or one sheep, two sheep, three sheep, etc. ; or one window in a courthouse, two windows in a courthouse, three windows in a courthouse, etc. It is easy to see how tedious this sort of thing may become. In order to avoid losing the interest of the pupil we are forced to simplify a drill of the above kind to the familiar succession of one, two, three, etc., and thereby we gain in pupil interest. This, perhaps the simplest mathematical situation, is typical of a great many others. The acquirement of skill is impossible without drill. Drill should be simplified so as not to become too involved or tedious, but even here a great deal depends upon the atmosphere that has been created as to whether even the simplest kind of drill remains a mumbo-jumbo performance or becomes meaningful in terms of common experience. The kind of problems which may be used for drill can easily be made to include both the abstract kind and the concrete kind. A proper mixture will help to conserve the number background behind all abstract situations.
A word of caution is necessary from the point of view of simplification, applications, and concreteness. Any one of these may be overdone to the consequent detriment of any given group of students. The attempt to over-simplify sometimes results in involving the situation, because of length of explanation. At times the applications brought in may become mere repetition and, therefore, very boresome. Concreteness may wander so far away from the student's experience as to introduce new difficulties much weightier than those involved in the numbers themselves, for, after all, a number situation or a space relationship is very, very simple compared with the complications involved in the most elementary so-called life situation. Most life situations involve the conduct of human beings. Who is there who will say that the problem of two and two is not infinitely simpler than the problem of predicting human behavior, even though all the motives and influences be given?
Considerable attention should be given to ways and means for articulating the complete set of courses mapped out from the first to the eleventh or twelfth grade. This problem is especially important at the junior or senior high school stage where the old line courses in algebra and geometry mark off these two subjects as separate and distinct. Complete articulation may be obtained by the use of the so-called spiral method which consists of a return to the same topic at several stages but at a higher level
The Texas Mathematics Teachers' Bulletin
each time. This conception is completely in line with the dynamic or developmental point of view with regard to numbers.
Finally it should be pointed out that a serious problem confronting the teachers in this day of many classes and large classes is to appeal to all the members of the class and not merely to the average member. The talented pupils in the class have every right to have their capabilities stimulated, and the work should be so designed as to afford the widest opportunities for developing the best students in the class. I close with sounding a warning that there is grave danger in building the course entirely around unit projects, for that may make it merely vocational with no depth at all and only a fair amount of surface.
AN INSIGHT INTO THE MATHEMATICAL
SITUATION IN TEXAS
BY BEE GRISSOM
Austin High School
I know of no better way in which to present an insight into the mathematical situation in Texas to the teachers of mathematics and to other persons interested in mathematics than by offering a resume of the program presented at the meeting of the Mathematics Section of the Texas State Teachers Association on November 29, 1935. The program presented at that time was rather unusual in that it not only consisted of reports from the State Production Committee in the Field of Mathematics, but it also included the experiences, opinions, and ideas from the faculties of some 200 schools in Texas.
In November, 1934, the mathematics section realized its responsibility with regard to curriculum revision, its obligation to the teachers of mathematics to lend assistance and encouragement in the Texas Curriculum Revision Movement, and caused a committee to be appointed to study the mathematical situation in the public schools in Texas. The committee had for its purposes the securing of some idea as to the existing conditions in the mathematics curriculum, what is beiing done in the way of curriculum revision in mathematics in our high schools, and the securing of some expression from the teachers of mathematics as to what changes, if any, should be made. A questionnaire was prepared and mailed to approximately 200 high schools of all sizes and located in all sections of the state.
The questions were answered and returned by 188 schools. The returned questionnaires were first grouped according to the number of students graduating each year-that is, according to the enrollment. There were 106 schools graduating 49 students or fewer, 42 schools graduating from 50 to 99 students, 22 schools graduating
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from 100 to 199 students, and 18 schools graduating over 200 students each year. However, after the tabulation of answers was made by groups, the similarity between the various groups was such that it did not warrant their separation in the final report.
In a survey of this type one of the first questions usually considered is the credits required for graduation from high school. Although more than half of the schools require either 2 units of algebra and 1 unit of geometry (73 schools to be exact), or 1 unit of algebra and 1 unit of geometry ( 41 schools), there is considerable variation in the graduation requirements of the remainder of the schools. Several schools offer different types of diplomas, namely, a diploma issued to students who have prepared themselves to enter a college or university, another diploma for students who have prepared for a trade, and another for vocational or commercial work. It is interesting to note that this cross-section of our schools reveals 5 schools which do not require any mathematics at all for graduation and 9 schools which require only 1 unit for graduation.
"Are you contemplating a change in the graduation requirements in mathematics?" This question yielded interesting results. Only 27 per cent of the schools canvassed by the questionnaire planned to continue the requirement of 3 units in mathematics for graduation. The remaining schools varied considerably in their requirements for graduation, with 29 per cent requiring 2 units and 11 per cent requiring 21;2 units. It is interesting to note that 7 schools plan to raise requirements (one school requires as many as 4 units) while 29 per cent of the schools plan to lower requirements, with the remaining schools having a variation from no units required to 31;2 units required. The very large number of schools requiring less than 3 units is due to the fact that many of these schools lowered graduation requirements simultaneously with the lowering of entrance requirements by The University of Texas and other institutions of higher learning.
The most popular new courses are general mathematics and advanced or commercial arithmetic. Most of the general mathematics courses are to be added in the eighth grade, while there is considerable variation in the grade placement of the arithmetic. Each of these new courses in the majority of cases is to be considered as an optional course by the students.
In answer to the question as to how much of the text used in first-year algebra is actually taught, we find a surprising amount of variation. Half of the schools teach through linear systems while only 11 per cent of the schools teach through ratio and proportion. Attention is called to the fact that students taking only one year of algebra in these schools which teach only through linear systems do not have an opportunity to gain any facility in the use of exponents, radicals, or in the solution of quadratic equations-some of the most important concepts in algebra.
Whether plane geometry should be offered to students who have had only one year of algebra is a prevalent question in some sections of our state. Eighty-four per cent of the schools in Texas, according to this questionnaire, which have tried the plan of presenting plane geometry immediately following a one-year course in algebra recommend it and continue to use it. In practically all of these schools the algebra course is offered in the eighth grade and the plane geometry course in the ninth grade. Hence, in view of our present trend toward lowering graduation requirements in mathematics, this is a very interesting and important bit of information. We believe the plan should be very practical for use in the schools of Texas provided the one-year course in algebra is made sufficiently inclusive.
Eighty-seven per cent of the schools answering the ques
tionnaire favored making our high school mathematics
more practical. The most frequent suggestions as to how
this might be accomplished, which I merely submit to you
without comment, were:
The Texas Mathematics Teachers' Bulletin
1.
To use problems met by the child in actual life situations.
2.
To give more practical problems.
3.
To offer more courses in arithmetic and general mathematics.
4.
To use the laboratory method and field trips.
5.
To obtain better textbooks.
6.
To correlate mathematics with other subjects in the high school.
The majority of the "permanent drops" from school occur in the eighth and ninth grades ; this, we believe, is a very strong argument for placing the more "practical courses" such as general mathematics and commercial arithmetic in the early high school grades.
The per cent of high school graduates who attend institutions of higher learning, of course, varies largely with the requirements of the school and its proximity to such an institution; also, large numbers of schools report that the per cent of graduates attending colleges and universities was considerably higher in more prosperous times. However, 55 per cent of the schools which answered this questionnaire now send less than 40 per cent of their students to institutions of higher learning. This, it seems, is another argument for making our mathematics courses fit the needs of those students who will never attend a college or university. Nevertheless, we must at the same time make our mathematics courses fit the needs of those students who will attend a college or university.
Attention is called to the fact that only 11.7 per cent of the schools have a detailed course of study and only 15 per cent are now writing or plan to write detailed courses of study. A large majority of the schools, then, teach by outline only and seemingly are content to continue this practice. We believe that the Texas Curriculum Revision Movement will tend to help the teachers of mathematics to write useful, detailed courses of study.
Finally, we asked for any suggestions or criticisms which would tend to improve the mathematical situation in Texas.
The University of Texas Bulletin
The responses were many and varied and space will not permit our presenting them all to you. However, they were mostly centered about the following themes:
1.
More uniformity in the state course of study with agreement among the teachers as to what should be taught and how it should be taught.
2.
Continue the teaching of arithmetic along with other courses in mathematics in the high school.
3.
Make the course more practical, that is, more suited to the needs of the student.
4.
Require teachers of mathematics to be majors in that field.
5.
Make more of the course elective, especially plane geometry.
6.
Revise the entire curriculum of mathematics from elementary grades through the high school.
7.
Differentiate between the students preparing for college and those preparing for trades or vocations.
8.
Work for the complete cooperation of the teachers of mathematics over the entire state.
It seems then, that the teachers of mathematics are anxiously awaiting a curriculum revision program. That program is here in the form of the Texas Curriculum Revision Movement which has been functioning for more than a year and a half. The influence of this movement upon our present program will be in direct proportion to the teacher participation in it. Many teachers are already at work in this cooperative movement and many others are being enrolled in this voluntary endeavor each day. It is to be hoped that a large majority of the teachers will contribute not only a generous portion of their time and energy but many accounts of successful teaching units gleaned from their vast store of experience. It is only through teacher participation in such a cooperative plan ag outlined and promulgated by the State Department of Education and the Texas State Teachers Association
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through the Executive Committee that a real Texas program for Texas schools can be successfully realized.
The teachers of Texas are urged to keep before them the point of view upon which this dynamic program was conceived and is being developed. It is well for each teacher to review frequently the general basis upon which this program is being built and which is so capably expressed by Dr. Fred C. Ayer in his article on the "Major Purposes and Guiding Principles in the Curriculum Revision Movement" and which is so clearly adapted to the various fields of education by the State Production Committees.
The program of the Mathematics Section was of particular importance, and its value was greatly enhanced by the willingness of the speakers to make last minute adjustments and additions to their papers in order to bring to the teachers present a full report of the work of the State Production Committee, even to the very last meeting of that committee which was held in San Antonio just before the convening of the Association.
The viewpoint of the State Committee was carefully reviewed. Teachers of mathematics were especially urged to develop as many units of teaching as possible during the remainder of the present school year and to submit as many contributions toward the development of this revision movement as possible. Realizing that certain portions of the mathematics field could not be satisfactorily adapted to the "Account of Unit of Teaching" form as suggested by the State Department of Education, the fol
lowing steps were suggested as an alternative outline for reporting a unit of teaching:
I. General Information.
1.
School.
2.
County.
3.
Post Office address.
4.
Grade.
5.
Teacher.
6.
Subject.
7.
Title of unit.
8.
Date begun.
9.
Date completed.
10.
Total number of days covered by unit.
11.
What college training in mathematics has the teacher had?
12.
How much teaching experience in mathematics has the teacher had?
II. The setting (type of school situation in which unit was developed, type of students, previous work, etc.).
III. Initial planning or bases for unit.
IV.
How unit was started and methods of securing general group interest.
V.
Description of development of unit (with reference to subject matter and procedure in considerable detail, narrative or outline).
VI. Statement of expected outcomes of unit.
VII. Evaluation of unit in terms of attainment and outcomes.
VIII. Articulation with other subjects:
1.
Mathematics.
2.
Language Arts.
3.
Science.
4.
Physical and Health.
5.
Fine Arts.
6.
Industrial education and homemaking.
7.
Social studies.
8.
Other subjects.
9.
Vocabulary.
IX.
Leads to other units.
X.
Bibliographies and materials (pupils and teacher, separate) .
XI. Recommendations for:
1.
Other units that may have been used.
2.
Other approaches to unit.
3.
Changes for the future.
As a suggested aid to the articulation of mathematics with mathematics, and of mathematics with the other "strands" involved in our curriculum, a tentative summary of the strands in mathematics was offered to the teachers for their consideration. The following suggested strands in mathematics were presented and will appear in the
The Texas Mathematics Teachers' Bulletin .
"Curriculum News Bulletin" in the form of a chart for all grades with the hope that the teachers of mathematics as well as the curriculum makers will carefully consider at all times both the "vertical and horizontal" educational growth of the child :
I. Numeration.
II. Notation.
III. Fundamental Operations (addition, subtraction, multiplication, and division).
1.
Integers.
2.
Fractions.
IV.
Numeration.
V.
Graphs.
VI. Use of Tables.
VII. Functional Concepts.
VIII. Language Arts.
IX.
Home and Vocational Arts.
X.
Social Relations (including the civic).
XI. Nature and Science.
XII. Creative and Recreative Arts (individual development).
"We should never think of mathematics as a mastery of facts and skills, isolated and independent of life situations, but rather as a meaningful growth in abstract modes of thought which can be used for more thorough understanding and appreciation of life itself at the present time as well as in later experiences. We possess this growth only when we are able to grasp and understand mathematical situations and to cope with them with confidence, sureness, and quantitative intelligence. . . . Every educational field involves mathematical ideas, notions, or concepts. May the present trend in education not overlook the value of both the quantitative and -social aspect of the subject of mathematics. Neither aspect will follow the other, but both must be taught separately and together. That is, a conscientious effort must be made on the part of all the teachers to present mathematics as a method of quantitative thinking, mathematics as mathematics, and as a means of social participation with distinct correlation at every step of the educational progress."
This article was made possible by the very able assistance and cooperative spirit of K. L. Carter, Austin High School, Austin, Texas;
J. W. O'Banion, Assistant State Supervisor and Director of Supervision; and the following members of the State Production Committee: H. J. Ettlinger, The University of Texas, Austin, Texas; Miss Minnie Behrens, Sam Houston State Teachers College, Huntsville, Texas; W. Wingo Hamilton, Greenville, Texas; Dr. Lorena Stretch, Baylor University, Waco, Texas; Mrs. Margaret G. Savage, Beaumont, Texas; Miss Edna McCormick, Southwest Texas State Teachers College, San Marcos, Texas; Mrs. W. W. Hair, Belton, Texas.
MEANS OR AVERAGES AND THEIR USES*
BYE. L. DODD
The University of Texas
§ 1. Introduction. Common average, median, mode
What I have to say in the earlier part of this talk is fairly well known, I am sure, to many members of the Science Club. But a mathematician should be careful to keep his airplane in the lower altitudes, at least for a while. Otherwise, he may soon be lost in the clouds. So I shall try to begin at the beginning.
Most of you received early this year a 60-page booklet from the Office of the President, entitled: "The University of Texas." Herein are presented statistics selected as suitable for consideration by the Legislature. Pages 28 and 29 are devoted to salaries; and the three forms of means or averages most frequently employed by statisticians to describe what is called central tendency are here used; viz., the common average-.technically known as the arithmetic me.an-the median, and the mode. On page 29, the average full-time salary for professors at The University of Texas in 1934-35 is set down at $3550. On page 28 appear statistics taken from Pamphlet No. 58, United States Office of Education, 1934. Here the "most common salaries" are given for "deans and professors at twenty-eight state universities, 1934-35." Among these, The University of Texas ranked sixteenth-a little below the middle-with $3,325 as the "most common" salary for professors. This is what statisticians would call the modal salary. Here the mode, $3,325, is less than the average or arithmetic mean, $3,550. The last figure given on page 28 is $3,434, given as the median salary of professors in 71 universities and colleges. By the median salary we understand the middle salary
*Presented t.o the Science Club of The University of Texas, November 4, 1935.
when all salaries are arranged in the order of their magnitude. Thus this median salary $3,434 as obtained from the records of 71 colleges and universities happens to be just about midway between the mode and the arithmetic mean of salaries for professors at The University of Texas in 1934--35.
Everyone would understand what an average salary isthe total budget available for salaries payable to a given group, divided by the number of the group. Perhaps not everyone is so familiar with the median or the mode. But statisticians use these forms of averages extensivelyespecially in unsymmetrical distributions, and distributions with considerable irregularity in the small sizes or the large sizes.
§ 2. Measures of dispersion or scattering
A good deal of information about a set of numbers or variates or measurements is given by specifying some central or average value, such as median, mode, or common average. But another question which naturally arises is this: Just how do the various values of the set cluster about this central value? Are they all close to the central value, or is there great scattering or dispersion? To illustrate the treatment of this problem, let us consider a simple diagram that is commonly used in connection with artillery fire. Suppose a hundred shots fired at a target large enough so that all the hundred shots are recorded as bullet holes. Draw a vertical line so that 50 shots lie to the left, and 50 shots to the right of this vertical line. This line is then the vertical center of fire. It may not pass through the bull's-eye aimed at, as there may be a constant error or deflection caused perhaps by the wind; but this line is what is called the center of fire. It turns out that the distribution of shots is usually so symmetrical about this line that this line represents at the same time average, median, and mode-although primarily it is a median. Now place to the right of this center of fire, AA', a parallel line BB' so that 25 shots or 25 per cent of the total number of shots fall between AA' and
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BB'. And put a similar line CC' to the left of AA'. The distance from AA' to BB' will in practice be essentially that from AA' to CC'. This distance r is the so-called probable error. The name probable error is not very well chosen, but it has probably come to stay.
The idea is this: If the 100 bullet holes are numbered 1, 2, ... 100, and if from 100 tickets numbered 1, 2, ... 100, one is drawn at random, there is an equal chance that the corresponding hole will lie inside the band from CC'
to BB' or outside. Or, if just one man fired the 100 shots, and he were to fire again, it would be fair to assume that he had equal chances of hitting inside or outside the band from CC' to BB'. If we call the horizontal distance from a bullet hole to the line AA', taken positively, its "error," then the so-called probable error is the median error. For half of the errors are less than the probable error and half are greater. It is interesting to note how shots outside the band from CC' to BB' are distributed. Let a system of parallel lines be drawn at intervals of r. Just outside a 25 per cent band there is a 16 per cent band; then a 7 per cent band; and finally a 2 per cent band. This is in accord with the so-called "normal" distribution-which artillery fire is
found to follow closely. These numbers can be easily remembered; for 25 = 5 X 5; 16 = 4 X 4; and 9 = 3 X 3, with 7 +2 = 9.
The diagram indicates that practically all shots fall within a range extending 4r on each side of the center of fire. More accurately, assuming the normal law of distribution, there should be about 7 shots per thousand falling outside this range-roughly 113 of 1 per cent to the left of the left vertical line, and 113 of 1 per cent to the right of the right vertical line. Some statisticians use 4 times the probable error to characterize an "unusual" event. To them an unusual event is one that does not occur more frequently than 7 times in a thousand trials. The error of a shot will exceed 3 times the probable error about 43 times in 1000 trials. This may be regarded as fairly unusualsufficiently so to form the basis of a "significance test."
Thus far, we have been considering the median error or deviation taken positively, called the probable error. The average deviation with deviations taken positively is called the mean deviation. This is generally somewhat larger than the probable error. Indeed, in the case of a normal distribution, as illustrated by artillery fire, it is fairly easy to see why the average deviation should exceed the probable error. If all the 25 per cent of shots to the right of BB' were to be consolidated uniformly in the band from BB' to DD', then as BB' would lie in the middle of the band extending from AA' to DD', we might well expect that the distance of BB' from AA' would be just about the average distance of all shots to the right of AA'. But with shots actually spreading out to the right of DD' we would expect an average deviation greater than r.
The probable error and the mean or average deviationwith deviations considered as positive-are both measures of dispersion or scattering. But there is another measure of dispersion, more important than either of these two, viz. : the standard deviation. To illustrate this, consider the following very simple numerical case.
The Texas Mathematics Teachers' Bulletin
Deviations Absolute
Measurements x from Aver. x Deviation !xi Sqµaresx2
21 -2 2 4
22 -1 1 1
26 3 3 9
- -
3)69 3)0 3)6 3)14
Aver.= 23 0 2 S2 = 4.6667
s = 2.16
Here the standard deviation is 2.16.
1-21+1-11+131
The mean deviation is 1 = 2. 3
The probable error also is 2, since in the sequence of values of the absolute deviations, viz., 1, 2, 3, arranged in numerical order, the number 2 is in the middle.
Now the assumed measurements 21, 22, 26, above, do not form a "normal distribution." But if, in an analogous manner, we make calculations from a fairly large number of measurements of a physical quantity, we would find that the probable error is about % the standard deviation ; and the mean deviation is about 4fr> the standard deviation. Likewise, these relations would be found valid for shots in artillery fire, and for many biological measurements-such
as measurements of stature for a group of individuals of like race and sex. As a matter of fact, it is customary to obtain first the standard deviation, and then find the probable error from:
Probable Error = .67 45 X Standard Deviation. This assumes normality in the distribution. It may not give a good approximation when the distribution is known to be not normal.
The question often arises: Why not compute the average or mean deviation, and then take:
Probable Error = .84533 X Mean Deviation.
It was pointed out by Helmert* that if a distribution is normal, there is a greater probability that a computed standard deviation will be close to the theoretic standard deviation that it represents than that the computed mean deviation should be close to its corresponding theoretic mean deviation.
Statisticians do rather tenaciously cling to the standard deviation in preference to the mean deviation. It may have been proved somewhere that the standard deviation is preferable even in cases where the distribution is obviously and grossly abnormal, but I have not seen a reference to such proof.
§ 3. Power means
If we let x be the arithmetic mean or common average of X., X 2, • ,Xm the standard deviation s is given by s = [ (X1 -x)2 + (X2 -x)2 + ... + (Xn -x) 2]1/2.n-112. It is the square-root of the average squared deviation from the arithmetic mean-briefly, it is a root-mean-square. Likewise, statisticians use the cube-root of the mean cube to measure the skewness or lack of symmetry of a distribution, and the fourth-root of the mean fourth power to measure the peakedness or bluntness of the distribution. These are all special cases of the power mean. Given a set of numbers x1, x 2, • • • , Xn, and a number p not zero; then, if p is a whole number, the power mean is
[ (x/ +Xl + ... +XnP) )1/P,
If p is not a whole number, the absolute or positive value of each x, is taken before raising it to the pth power.
§ 4. Indirect averages. Geometrical interpretation
A great many means, including these power means, are what may be called indirect averages. An averaging
*"Ober die Wahrscheinlichkeit der Potenzsummen usw., Zeitschrift fur Mathematik und Physik 21 (1876)-as cited in CzuberWahrscheinlichkeitsrechnung I (1914) p. 312.
The Texas Mathematics Teachers' Bulletin
process is performed, not on the data or given numbers, but on some specified function of the data. Let us consider the root-mean-square from the geometric standpoint. Construct the curve y =X2 Here the ordinate is the square
•
of the abscissa. Let a number of points be laid off on the X-axis at distances of Xu x2, • • • , ri,., to the right of the origin 0. Erect ordinates to the curve y = x2• From the points found thus on the curve extend horizonal lines to the Y-axis, marking the points thus found Yu Y2• ..• , Yn· Find the actual average ii of these y points:
ii= (Y1 +Y2 + · ·· +Yn) / n. From ii carry a line horizontally to the curve y = x2, and then vertically down to the X-axis. The point m thus obtained is the root-mean-square of the data, x1 , x2, • • • , Xn· The curves to find the root-mean-cube and root-meanf ourth-power look much like that for root-mean-square, but rise more rapidly to the right of x = 1.
§ 5. Averages in insurance and fiMnce
In the mathematics of finance and insurance there appear means of a different character, exponential means, involving an exponential function, instead of a power function.
The subject of life insurance, annuities, and pensions is not so very difficult as long as we are concerned with but a single life. But when we consider joint life insurance, payable upon the first death in a group-e.g., partners in a firm-or a life annuity payable as long as any one of a certain family survives, some rather complicated formulas appear. The symbol ax denotes the cash value to a man, now of age x, of $1 per year to be paid to him as long as he lives. To a man of age x, then, $1000ax is the cash value of $1000 payable at the end of each year that he survives. Likewise, axy is the cash value of $1 per year payable as long as two men, of ages x and y, both survive; a