How to Teach

Many tests have been devised which have been thought to have more general application than those which have been mentioned above for the particular subjects. One of the most valuable of these tests, called technically a completion test, is that derived by Dr. M.R. Trabue.^[29] In these tests the pupil is asked to supply words which are omitted from the printed sentences. It is really a test of his ability to complete the thought when only part of it is given. Dr. Trabue calls his scales language scales. It has been found, however, that ability of this sort is closely related to many of the traits which we consider desirable in school children. It would therefore be valuable, provided always that children have some ability in reading, to test them on the language scale as one of the means of differentiating among those who have more or less ability. The scores which may be expected from different grades appear in Dr. Trabue's monograph. Three separate scales follow.

Other standard tests and scales of measurement have been derived and are being developed. The examples given above will, however, suffice to make clear the distinction between the ordinary type of examination and the more careful study of the achievements of children which may be accomplished by using these measuring sticks. It is important for any one who would attempt to apply these tests to know something of the technique of recording results.

In the first place, the measurement of a group is not expressed satisfactorily by giving the average score or rate of achievement of the class. It is true that this is one measure, but it is not one which tells enough, and it is not the one which is most significant for the teacher. It is important whenever we measure children to get as clear a view as we can of the whole situation. For this purpose we want not primarily to know what the average performance is, but, rather, how many children there are at each level of achievement. In arithmetic, for example, we want to know how many there are who can do none of the Courtis problems in addition, or how many there are who can do the first six on the Woody test, how many can do seven, eight, and so on. In penmanship we want to know how many children there are who write quality eight, or nine, or ten, or sixteen, or seventeen, as the case may be. The work of the teacher can never be accomplished economically except as he gives more attention to those who are less proficient, and provides more and harder work for those who are capable, or else relieves the able members of the class from further work in the field. It will be well, therefore, to prepare, for the sake of comparing grades within the same school or school system, or for the sake of preparing the work of a class at two different times during the year, a table which shows just how many children there are in the group who have reached each level of achievement. Such tables for work in composition for a class at two different times, six months apart, appear as follows:

A study of such a distribution would show not only that the average performance of the class has been raised, but also that those in the lower levels have, in considerable measure, been brought up; that is, that the teacher has been working with those who showed less ability, and not simply pushing ahead a few who had more than ordinary capacity. It would be possible to increase the average performance by working wholly with the upper half of the class while neglecting those who showed less ability. From a complete distribution, as has been given above, it has become evident that this has not been the method of the teacher. He has sought apparently to do everything that he could to improve the quality of work upon the part of all of the children in the class.

It is very interesting to note, when such complete distributions are given, how the achievement of children in various classes overlaps. For example, the distribution of the number of examples on the Courtis tests, correctly finished in a given time by pupils in the seventh grades, makes it clear that there are children in the fifth grade who do better than many in the eighth.

If the tests had been given in the fourth or the third grade, it would have been found that there were children, even as low as the third grade, who could do as well or better than some of the children in the eighth grade. Such comparisons of achievements among children in various subjects ought to lead at times to reorganizations of classes, to the grouping of children for special instruction, and to the rapid promotion of the more capable pupils.

In many of these measurements it will be found helpful to describe the group by naming the point above and below which half of the cases fall. This is called the median. Because of the very common use of this measure in the current literature of education, it may be worth while to discuss carefully the method of its derivation.^[30]

^[31]The median point of any distribution of measures is that point on the scale which divides the distribution into two exactly equal parts, one half of the measures being greater than this point on the scale, and the other half being smaller. When the scales are very crude, or when small numbers of measurements are being considered, it is not worth while to locate this median point any more accurately than by indicating on what step of the scale it falls. If the measuring instrument has been carefully derived and accurately scaled, however, it is often desirable, especially where the group being considered is reasonably large, to locate the exact point within the step on which the median falls. If the unit of the scale is some measure of the variability of a defined group, as it is in the majority of our present educational scales, this median point may well be calculated to the nearest tenth of a unit, or, if there are two hundred or more individual measurements in the distribution, it may be found interesting to calculate the median point to the nearest hundredth of a scale unit. Very seldom will anything be gained by carrying the calculation beyond the second decimal place.

The best rule for locating the median point of a distribution is to take as the median that point on the scale which is reached by counting out one half of the measures, the measures being taken in the order of their magnitude. If we let n stand for the number of measures in the distribution, we may express the rule as follows: Count into the distribution, from either end of the scale, a distance covered by *n/2 measures. For example, if the distribution contains 20 measures, the median is that point on the scale which marks the end of the 10th and the beginning of the 11th measure. If there are 39 measures in the distribution, the median point is reached by counting out 19-1/2 of the measures; in other words, the median of such a distribution is at the mid-point of that fraction of the scale assigned to the 20th measure.

The median step of a distribution is the step which contains within it the median point. Similarly, the median measure in any distribution is the measure which contains the median point. In a distribution containing 25 measures, the 13th measure is the median measure, because 12 measures are greater and 12 are less than the 13th, while the 13th measure is itself divided into halves by the median point. Where a distribution contains an even number of measures, there is in reality no median measure but only a median point between the two halves of the distribution. Where a distribution contains an uneven number of measures, the median measure is the (n+1)/2 measurement, at the mid-point of which measure is the median point of the distribution.

Much inaccurate calculation has resulted from misguided attempts to secure a median point with the formula just given, which is applicable only to the location of the median measure. It will be found much more advantageous in dealing with educational statistics to consider only the median point, and to use only the n/2 formula given in a previous paragraph, for practically all educational scales are or may be thought of as continuous scales rather than scales composed of discrete steps.

The greatest danger to be guarded against in considering all scales as continuous rather than discrete, is that careless thinkers may refine their calculations far beyond the accuracy which their original measurements would warrant. One should be very careful not to make such unjustifiable refinements in his statement of results as are often made by young pupils when they multiply the diameter of a circle, which has been measured only to the nearest inch, by 3.1416 in order to find the circumference. Even in the ordinary calculation of the average point of a series of measures of length, the amateur is sometimes tempted, when the number of measures in the series is not contained an even number of times in the sum of their values, to carry the quotient out to a larger number of decimal places than the original measures would justify. Final results should usually not be refined far beyond the accuracy of the original measures.

It is of utmost importance in calculating medians and other measures of a distribution to keep constantly in mind the significance of each step on the scale. If the scale consists of tasks to be done or problems to be solved, then "doing 1 task correctly" means, when considered as part of a continuous scale, anywhere from doing 1.0 up to doing 2.0 tasks. A child receives credit for "2 problems correct" whether he has just barely solved 2.0 problems or has just barely fallen short of solving 3.0 problems. If, however, the scale consists of a series of productions graduated in quality from very poor to very good, with which series other productions of the same sort are to be compared, then each sample on the scale stands at the middle of its "step" rather than at the beginning.

The second kind of scale described in the foregoing paragraph may be designated as "scales for the quality of products," while the other variety may be called "scales for magnitude of achievement." In the one case, the child makes the best production he can and measures its quality by comparing it with similar products of known quality on the scale. Composition, handwriting, and drawing scales are good examples of scales for quality of products. In the other case, the scales are placed in the hands of the child at the very beginning, and the magnitude of his achievement is measured by the difficulty or number of tasks accomplished successfully in a given time. Spelling, arithmetic, reading, language, geography, and history tests are examples of scales for quantity of achievement.

Scores tend to be more accurate on the scales for magnitude of achievement, because the judgment of the examiner is likely to be more accurate in deciding whether a response is correct or incorrect than it is in deciding how much quality a given product contains. This does not furnish an excuse for failing to employ the quality-of-products scales, however, for the qualities they measure are not measurable in terms of the magnitude of tasks performed. The fact appears, however, that the method of employing the quality-of-products scales is "by comparison" (of child's production with samples reproduced on the scale), while the method of employing the magnitude-of-achievement scales is "by performance" (of child on tasks of known difficulty).

In this connection it may be well to take one of the scales for quality of products and outline the steps to be followed in assigning scores, making tabulations, and finding the medians of distributions of scores.

When the Hillegas scale is employed in measuring the quality of English composition, it will be advisable to assign to each composition the score of that sample on the scale to which it is nearest in merit or quality. While some individuals may feel able to assign values intermediate to those appearing on the Hillegas scale, the majority of those persons who use this scale will not thereby obtain a more accurate result, and the assignment of such intermediate values will make it extremely difficult for any other person to make accurate use of the results. To be exactly comparable, values should be assigned in exactly the same manner.

The best result will probably be obtained by having each composition rated several times, and if possible, by a number of different judges, the paper being given each time that value on the Hillegas scale to which it seems nearest in quality. The final mark for the paper should be the median score or step (not the median point or the average point) of all the scores assigned. For example, if a paper is rated five times, once as in step number five (5.85), twice as in step number six (6.75), and twice as in step number seven (7.72), it should be given a final mark indicating that it is a number six (6.75) paper.

After each composition has been assigned a final mark indicating to what sample on the Hillegas scale it is most nearly equal in quality, proceed as follows:

Make a distribution of the final marks given to the individual papers, showing how many papers were assigned to the zero step on the scale, how many to step number one, how many to step number two, and so on for each step of the scale. We may take as an example the distribution of scores made by the pupils of the eighth grade at Butte, Montana, in May, 1914.

All together there were 154 papers from the eighth grade, so that if they were arranged in order according to their merit we might begin at the poorest and count through 77 of them (n/2 = 154/2 = 77) to find the median point, which would lie between the 77th and the 78th in quality. If we begin with the 1 composition rated at 0 and count up through the 9 rated at 1 and the 32 rated at 2 in the above distribution, we shall have counted 42. In order to count out 77 cases, then, it will be necessary to count out 35 of the 39 cases rated at 3.

Now we know (if the instructions given above have been followed) that the compositions rated at 3 were so rated by virtue of the fact that the judges considered them nearer in quality to the sample valued at 3.69 than to any other sample on the scale. We should expect, then, to find that some of those rated at 3 were only slightly nearer to the sample valued at 3.69 than they were to the sample valued at 2.60, while others were only slightly nearer to 3.69 than they were to 4.74. Just how the 39 compositions rated on 3 were distributed between these two extremes we do not know, but the best single assumption to make is that they are distributed at equal intervals on step 3. Assuming, then, that the papers rated at 3 are distributed evenly over that step, we shall have covered .90 (35/39 = .897 = .90) of the entire step 3 by the time we have counted out 35 of the 39 papers falling on this step.

It now becomes necessary to examine more closely just what are the limits of step 3. It is evident from what has been said above that 3.69 is the middle step 3 and that step 3 extends downward from 3.69 halfway to 2.60, and upward from 3.69 halfway to 4.74. The table given below shows the range and the length of each step in the Hillegas Scale for English Composition.

From the above table we find that step 3 has a length of 1.07 units. If we count out 35 of the 39 papers, or, in other words, if we pass upward into the step .90 of the total distance (1.07 units), we shall arrive at a point .96 units (.90 × 1.07 = .96) above the lower limit of step 3, which we find from the table is 3.15. Adding .96 to 3.15 gives 4.11 as the median point of this eighth grade distribution.

The median and the percentiles of any distribution of scores on the Hillegas scale may be determined in a manner similar to that illustrated above, if the scores are assigned to the individual papers according to the directions outlined above.

A similar method of calculation is employed in discovering the limits within which the middle fifty per cent of the cases fall. It often seems fairer to ask, after the upper twenty-five per cent of the children who would probably do successful work even without very adequate teaching have been eliminated, and the lower twenty-five per cent who are possibly so lacking in capacity that teaching may not be thought to affect them very largely have been left out of consideration, what is the achievement of the middle fifty per cent. To measure this achievement it is necessary to have the whole distribution and to count off twenty-five per cent, counting in from the upper end, and then twenty-five per cent, counting in from the lower end of the distribution. The points found can then be used in a statement in which the limits within which the middle fifty per cent of the cases fall. Using the same figures that are given above for scores in English composition, the lower limit is 2.64 and the limit which marks the point above which the upper twenty-five per cent of the cases are to be found is 5.08. The limits, therefore, within which the middle fifty per cent of the cases fall are from 2.64 to 5.08.

It is desirable to measure the relationship existing between the achievements (or other traits) of groups. In order to express such relationship in a single figure the coefficient or correlation is used. This measure appears frequently in the literature of education and will be briefly explained. The formula for finding the coefficient of correlation can be understood from examples of its application.

Let us suppose a group of seven individuals whose scores in terms of problems solved correctly and of words spelled correctly are as follows:^[33]

From such distributions it would appear that as individuals increase in achievement in one field they increase correspondingly in the other. If one is below or above the average in achievement in one field, he is below or above and in the same degree in the other field. This sort of positive relationship (going together) is expressed by a coefficient of +1. The formula is expressed as follows:

Here r = coefficient of correlation.

x = deviations from average score in arithmetic (or difference between score made and average score).

y = deviations from average score in spelling.

Σ = is the sign commonly used to indicate the algebraic sum (i.e. the difference between the sum of the minus quantities and the plus quantities).

x · y = products of deviation in one trait multiplied by deviation in the other trait with appropriate sign.

Applying the formula we find:

If instead of achievement in one field being positively related (going together) in the highest possible degree, these individuals show the opposite type of relationship, i.e., the maximum negative relationship (this might be expressed as opposition--a place above the average in one achievement going with a correspondingly great deviation below the average in the other achievement), then our coefficient becomes -1. Applying the formula:

It will be observed that in this case each plus deviation in one achievement is accompanied by a minus deviation for the other trait; hence, all of the products of x and y are minus quantities. (A plus quantity multiplied by a plus quantity or a minus quantity multiplied by a minus quantity gives us a plus quantity as the product, while a plus quantity multiplied by a minus quantity gives us a minus quantity as the product.)

If there is no relationship indicated by the measures of achievements which we have found, then the coefficient of correlation becomes 0. A distribution of scores which suggests no relationship is as follows:

In a similar manner, when the relationship is largely positive as would be indicated by a displacement of each score in the series by one step from the arrangement which gives a +1 coefficient, the coefficient will approach unity in value.

Other illustrations might be given to show how the coefficient varies from + 1, the measure of the highest positive relationship (going together) through 0 to -1, the measure of the largest negative relationship (opposition). A relationship between traits which we measure as high as +.50 is to be thought of as quite significant. It is seldom that we get a positive relationship as large as +.50 when we correlate the achievements of children in school work. A relationship measured by a coefficient of ±.15 may not be considered to indicate any considerable positive or negative relationship. The fact that relationships among the achievements of children in school subjects vary from +.20 to +.60 is a clear indication of the fact that abilities of children are variable, or, in other words, achievement in one subject does not carry with it an exactly corresponding great or little achievement in another subject. That there is some positive relationship, i.e., that able pupils tend on the whole to show all-round ability and the less able or weak in one subject tend to show similar lack of strength in other subjects, is also indicated by these positive coefficients.

QUESTIONS

1. Calculate the median point in the following distribution of eighth-grade composition scores on the Hillegas scale.

2. Calculate the median point in the following distribution of third-grade scores on the Woody subtraction scale.

3. Compare statistically the achievements of the children in two eighth-grade classes whose scores on the Courtis addition tests were as follows:

Class A--6, 5, 8, 9, 7, 10, 13, 4, 8, 7, 8, 7, 6, 8, 15, 6, 7, 0, 6, 9, 5, 8, 7, 10, 8, 4, 7, 8, 6, 9, 5, 7, 2, 6, 8, 5, 7, 8, 7, 8, 5, 8, 10, 6, 3, 6, 8, 17, 5, 7.

Class B--10, 4, 8, 13, 11, 9, 8, 10, 7, 9, 11, 10, 18, 7, 12, 9, 10, 8, 11, 10, 12,
9, 2, 11, 8, 10, 9, 14, 11, 7, 10, 12, 10, 6, 11, 8, 10, 9, 10, 17, 8, 11,
9, 7, 9, 11, 8, 12, 9, 13.

4. If the marks received in algebra and in geometry by a group of high school pupils were as given below, what relationship is indicated by the coefficient of correlation?

5. Compare the abilities of the 10-year-old pupils in the sixth grade with the abilities of the 14-year-old pupils in the same grade, in so far as these abilities are measured by the completion of incomplete sentences.

(Note: 5 = 5.0-5.999.)

6. From the scores given here, calculate the relationship between ability to spell and ability to multiply. Use the average as the central tendency.

INDEX

Achievements of children, measuring the,
    and examinations,
    in English composition,
    in arithmetic,
    arithmetic scale,
    reasoning problems in arithmetic,
    distribution of hand-writing scores,
    handwriting scale,
    spelling scale,
    scale for English composition.
Æsthetic emotions,
    appreciation and skill,
    appreciation, intellectual factors in.
Aim of education, I
Analysis and abstraction, III.
Angell, J.R.
Appreciation,
    types of,
    passive attitude in,
    development in,
    value of,
    lesson.
Associations, organization of,
    number of.
Attention,
    situations arousing response of,
    and inhibition,
    breadth of,
    to more than one thing,
    concentration of,
    span of,
    free,
    forced,
    immediate free,
    immediate and derived,
    derived,
    forced,
    and habit formation,
    focalization of,
    divided.
Ayres, L.P.

Ballou, F.W.
Bread-and-butter aim.

Classroom exercises, types of.
Coefficient of correlation,
    calculation of,
    values of.
Comparison and abstraction, step of.
Concentration, of attention.
    habits of.
Conduct, moral social.
Consciousness, fringe of.
Correlation, coefficient of.
Courtis, S.A.
Culture as aim of education.
Curriculum, omissions from.

Deduction lesson, the,
    steps in.
Deduction, process of.
Dewey, John.
Differences, individual,
    sex.
Disuse, method of.
Drill,
    lesson, the,
    work, deficiency in.

Education, before school age.
Effect, law of.
Emotions, aesthetic.
Environment and individual differences.
Examinations,
    limitations of.
Exceptions, danger of.

Fatigue and habits.
Formal discipline.

Gray, W.S.

Habit formation,
    and attention,
    laws of,
    and instinct,
    complexity of,
    and interest,
    and mistakes.
Habits, of concentration,
    modification of the nervous system involved,
    and fatigue,
    and will power,
    and original work.
Harmonious development of aim.
Heck, W.H.
Henderson, E.N.
Heredity and individual differences.
Hillegas, M.B.

Illustrations, use of.
Imagery, type of,
    and learning,
    productive, types of.
Images,
    classified,
    object and concrete.
Imagination.
Individual differences,
    causes of,
    and race inheritance,
    and maturity,
    and heredity,
    and environment,
    and organization of
    public education
    in composition
    in arithmetic
    in penmanship
Induction and deduction
    differences in
    relationship of
Induction, process of
Inductive lesson, the
Inquiry in school work
Instinctive tendencies
    modifiability of
    inhibition of
Instincts
    transitoriness of
    delayedness of
    of physical activity
    to enjoy mental activity
    of manipulation
    of collecting
    of rivalry
    of fighting
    of imitation
    of gregariousness
    of motherliness
Interest
    an end

Judd, C.H.
Junior high school, the

Kelly, F.J.
Knowledge aim

Learning
    incidental
    and imagery
    curves
Lecturing
    and appreciation
Lesson
    the inductive

McMurry, F.M.
Maturity and individual differences
Measurement of group
    comparison of seventh-grade scores in composition
    comparison of scores in arithmetic
Measuring results in education
Median
    calculation of
    point
    step
    measure
Memorization
    verbatim
    whole-part method illustrated
Memory
    factors in
    and native retentiveness
    and recall
    part and whole methods
    practice periods
    immediate
    desultory
    rote
    logical
    and forgetting
    permanence of
Miller, I.E.
Moral conduct
    development of
Morality
    defined
    and conduct
    and habit
    and choice
    and individual opinion
    social nature of
    and training for citizenship
    and original nature
    and environment
    stages of development in
    and habit formation
    transition period in
    direct teaching of
    and classroom work
    and service by pupils
    and social responsibility
    and school rules
Morgan, C.L.

Openmindedness
Original nature
    of children
    and racial inheritance
    and aim of education
    utilization of
    and morality
Original work and habits

Payne, Joseph
Physical welfare of children
Play
    theories of
    types of
    complexity of
    characteristics of
    and drudgery
    and work
    and ease of accomplishment
    and social demands
    supervision of
Preparation
    steps of
Presentation
    steps of
Problems as stimulus to thinking
Punishment

Questioning
Questions
    types of
    responses to
    number of
    appeal of

Reasoning and thinking
    technique of
Recapitulation theory
Recitation
    social purpose of
Recitation lesson, the
Repetition
Retention
    power of
Review
Review lesson, the
Roark, R.N.

Satisfaction
    result of
Scales of measurement
School government
    participation in
Sex differences
    education
Social aim of education
    and curriculum
    and special types of schools
Stone, C.W.
Study
    how to
    types of
    and habit formation
    and memorization
    and interest
    necessity for aim in
    and concentrated attention
    involves critical attitude
    general factors in
    for appreciation
    involving thinking
    use of books in
    supervised
Substitution
    method of

Thinking defined
Thinking
    stimulation of
    and problematic situations
    by little children
    and habit formation
    essentials in process of
    for its own sake
    and critical attitude
    laws governing
    and association
    failure in
    and classroom exercises
Thorndike, E.L.
Thought
    imageless
Trabue, M.R.
Training
    transfer of
    identity of response
    probability of
    amount of
Transfer of training

Will power and habits
Woody, Clifford
Work, independent
Work and play

1. The nervous system is composed of units of structure called neurones or nerve cells. "If we could see exactly the structure of the brain itself, we should find it to consist of millions of similar neurones each resembling a bit of string frayed out at both ends and here and there along its course. So also the nerves going out to the muscles are simply bundles of such neurones, each of which by itself is a thread-like connection between the cells of the spinal cord or brain and some muscle. The nervous system is simply the sum total of all these neurones, which form an almost infinitely complex system of connections between the sense organs and the muscles."

The word synapses, meaning clasping together, is used as a descriptive term for the connections that exist between neurone and neurone.

2. This is synonymous with James's Involuntary Attention, Angell's Non-Voluntary Attention, and Titchener's Secondary-Passive Attention.

3. Educational Psychology, Briefer Course, pp. 194-5.

4. Thorndike, Psychology of Learning, p. 194.

5. How We Think, p. 6.

6. The Psychology of Thinking, p. 98.

7. How We Think, p. 66.

8. How We Think, pp. 69-70.

9. Psychology of Thinking, p. 291.

10. How We Think, p. 79.

11. Thorndike, Educational Psychology, Briefer Course, p. 172.

12. Introduction to Psychology, p. 284.

13. Thorndike, Origin of Man, p. 146.

14. Racial Differences in Mental Traits, pp. 177 and 181.

15. Thorndike, Educational Psychology, Briefer Course, p. 374.

16. Thorndike, Educational Psychology, Vol. III, p. 304.

17. Moral Principles in Education, p. 17.

18. For a fuller discussion of this topic see next chapter.

19. For a discussion of these scales see Chapter XV.

20. The Courtis Tests, Series B, for Measuring the Achievements of Children in the Fundamentals of Arithmetic, can be secured from Mr. S.A. Curtis, 82 Eliot Street, Detroit, Mich.

21. Measurements of Some Achievements in Arithmetic, by Clifford Woody, published by the Teachers College Bureau of Publications, Columbia University, 1916.

22. Reasoning Test in Arithmetic, by C.W. Stone, published by the Bureau of Publications, Teachers College, Columbia University, 1916.

23. A Scale for Handwriting of Children, by E.L. Thorndike, published by the Bureau of Publications, Teachers College, Columbia University.

24. A scale derived by Dr. Leonard P. Ayres of the Russell Sage Foundation is also valuable for measuring penmanship, and can be purchased from the Russell Sage Foundation.

25. Copies of the Spelling Scale can be secured from the Russell Sage Foundation, New York, for five cents a copy.

26. A Scale for the Measurement of Quality in English Composition, by Milo B. Hillegas, published by the Bureau of Publications, Teachers College, Columbia University.

27. The Harvard-Newton Scale for the Measurement of English Composition, published by the Harvard University Press, Cambridge, Mass.

28. Scale Alpha. For Measuring the Understanding of Sentences, by E.L. Thorndike, published by the Bureau of Publications, Teachers College, Columbia University.

Scales for measuring the rate of silent reading and oral reading have been derived by Dr. W.S. Gray, of the University of Chicago, and by Dr. F.J. Kelly, of the University of Kansas. Reference to the use of Dr. Gray's scale will be found in Judd's Measuring Work of the Schools, one of the volumes of the Cleveland survey, published by the Russell Sage Foundation. Dr. Kelly's test, called The Kansas Silent Reading Test, can be had from the Emporia, Kansas, State Normal School.

29. Completion Test Language Scales, by M.R. Trabue, published by the Bureau of Publications, Teachers College, Columbia University.

30. The student who is not interested in the statistical methods involved in measuring with precision the achievements of pupils may omit the remainder of this chapter.

31. This explanation of the method of finding the median was prepared for one of the classes in Teachers College by Dr. M.R. Trabue.

32. The third decimal place is omitted in this table.

33. In order to discover the relationship which exists between two traits which we have measured we would use many more than seven cases. The illustrations given are made short in order to make it easy to follow through the application of the formula.