Chapter 23

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.

No. of papers 1 9 32 39 43 22 6 2 Rated at 0 1 2 3 4 5 6 7 8 9

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.

THE HILLEGAS SCALE FOR ENGLISH COMPOSITION

====================================================== STEP No.|VALUE or SAMPLE|RANGE OF STEP |LENGTH OF STEP --------+---------------+--------------+-------------- 0. . . .| 0 | 0- .91[32] | .91 1. . . .| 1.83 | .92-2.21 | 1.30 2. . . .| 2.60 |2.22-3.14 | .93 3. . . .| 3.69 |3.15-4.21 | 1.07 4. . . .| 4.74 |4.22-5.29 | 1.08 5. . . .| 5.85 |5.30-6.30 | 1.00 6. . . .| 6.75 |6.30-7.23 | .93 7. . . .| 7.72 |7.24-8.05 | .81 8. . . .| 8.38 |8.05-8.87 | .82 9. . . .| 9.37 |8.88- | ======================================================

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]

====================================== INDIVIDUALS|No. OF |No. OF WORDS MEASURED |PROBLEMS|SPELLED CORRECTLY CORRECTLY | | -----------+--------+----------------- A | 1 | 2 B | 2 | 4 C | 3 | 6 D | 4 | 8 E | 5 | 10 F | 6 | 12 G | 7 | 14 ======================================

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:

(Sum x · y) r = ------------------------------ (sqrt(Sum x^2))(sqrt(Sum y^2))

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.

Sum = 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: =================================================================== |ARITH-| | | SPEL- | | | | |METIC | x | x^2 | LING | y | y^2 | x·y | --+------+---+------------+-------+---+-------------+-------------+ A | 1|-3 | 9| 2|-6 | 36| +18| B | 2|-2 | 4| 4|-4 | 16| +8| C | 3|-1 | 1| 6|-2 | 4| +2| D | 4| 0 | 0| 8| 0 | | | E | 5|+1 | 1| 10|+2 | 4| +2| F | 6|+2 | 4| 12|+4 | 16| +8| G | 7|+3 | 9| 14|+6 | 36| +18| | ___| | __| ___| | ___| __| | 7 |28| |Sum x^2 = 28| 7 |56| |Sum y^2 = 112|Sum x·y = +56| |Av. =4| | |Av. =8 | | | | =================================================================== Sum x · y +56 +56 r = ---------------------------- = --------------------- = ---- = +1 (sqrt(Sum x^2)(sqrt(Sum y^2) (sqrt(28))(sqrt(112)) 56

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:

=================================================================== |ARITH-| | | SPEL- | | | | |METIC | x | x^2 | LING | y | y^2 | x*y | --+------+---+------------+-------+---+-------------+-------------+ A | 1|-3 | 9| 14|+6 | 36| -18| B | 2|-2 | 4| 12|+4 | 16| -8| C | 3|-1 | 2| 10|+2 | 4| -2| D | 4| 0 | | 8| 0 | | | E | 5|+1 | 2| 6|-2 | 4| -2| F | 6|+2 | 4| 4|-4 | 16| -8| G | 7|+3 | 9| 2|-6 | 36| -18| | ___| | __| ___| | ___| __| | 7 |28| |Sum x^2 = 28| 7 |56| |Sum y^2 = 112|Sum x·y = -56| |Av. =4| | |Av. =8 | | | | ===================================================================

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.)

(Sum x·y) -56 -56 r = ------------------------------ = ------------------- = ---- = -1. (sqrt(Sum x^2))(sqrt(Sum y^2)) (sqrt(28)sqrt(112)) = 56

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:

================================================================= |ARITH- | | | | | | |METIC | x | x^2 |Spelling | y | y^2 | x.y --+-------+----+-----------+---------+----+-------------+-------- | | | | | | | - + A | 2 | -2 | 4 | 12 | +4 | 16 | -8 +6 B | 1 | -3 | 9 | 8 | 0 | | 0 +4 C | 4 | 0 | | 2 | -6 | 36 | 0 +4 D | 5 | +1 | 1 | 14 | +6 | 36 | -6 E | 3 | -1 | 1 | 4 | -4 | 16 | -14 +14 F | 7 | +3 | 9 | 6 | -2 | 4 | G | 6 | +2 | 4 | 10 | +2 | 4 | | ____| | | ___ | | | | |28 | |Sum x^2=28 | 7|56 | | Sum y^2=112 | x·y=0 | AV.=4 | | | AV.=8 | | | ===================================================================

(Sum x·y) 0 r = ---------------------------- = ------------------- = 0. (sqrt(Sum x^2)sqrt(Sum y^2)) (sqrt(28)sqrt(112))

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.

=============================================================== ARITHMETIC| x | x^2 |SPELLING| y | y^2 | ---+------+----+-----------+--------+----+------------+-------- A |1 | -3 |9 |4 | -4 | 16 |+ 12 B |2 | -2 |4 |2 | -6 | 36 |+ 12 C |3 | -1 |1 |8 | 0 | |+ 4 D |4 | 0 | |6 | -2 | 4 |+ 4 E |5 | +1 |1 |12 | +4 | 16 |+ 18 F |6 | +2 |4 |10 | +2 | 4 |Sx·y=50 G |7 | +3 |9 |14 | +6 | 36 | |Av. =4| |Sum x^2 =28|Av. = 8 | |Sum y^2= 112| ===============================================================

Sum x·y +50 r= -------------------------- = ---- = +.89. sqrt(Sum x^2)sqrt(Sum y^2) 56

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.

Quality 0 18 26 37 47 58 67 Frequency 2 68 73 3

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

No. problems 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Frequency 2 2 2 3 3 5 4 5 8 16 16 16 23 20 21 11 22 11 2

22 23 24 + 1

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?

|GEOMETRY |ALGEBRA |MARKS |MARKS 1. |80 |60 2. |68 |73 3. |65 |80 4. |96 |80 5. |59 |62 6. |75 |65 7. |90 |75 8. |86 |90 9. |52 |63 10. |70 |55 11. |63 |54 12. |85 |95 13. |93 |90 14. |87 |70 15. |82 |68 16. |79 |75 17. |78 |86 18. |79 |75 19. |82 |60 20. |70 |82 21. |52 |86 22. |94 |85 23. |72 |73 24. |53 |62 25. |94 |85

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.)

================================================== NO. SENTENCES | | COMPLETED | 10-YEAR-OLDS | 14-YEAR-OLDS --------------+--------------+-------------------- 24 |-- |-- 23 |-- |-- 22 |-- |-- 21 |1 |-- 20 |-- |-- 19 |-- |-- 18 |-- |-- 17 |-- |1 16 |3 |-- 15 |-- |2 14 |7 |4 13 |10 |3 12 |18 |7 11 |9 |10 10 |7 |9 9 |8 |10 8 |2 |10 7 |3 |10 6 |-- |2 5 |2 |3 4 |-- |2 3 |-- |-- 2 |-- |1 1 |-- |-- 0 |-- |-- ===========================================

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

============================== PUPIL|SPELLING|MULTIPLICATION -----+--------+--------------- A |9 |22 B |10 |16 C |2 |19 D |6 |14 E |13 |24 F |8 |22 G |10 |17 H |7 |20 I |3 |21 J |2 |21 K |14 |20 L |8 |18 M |7 |23 N |11 |25 O |8 |25 P |17 |24 Q |10 |21 R |4 |16 S |9 |15 T |6 |19 U |12 |22 V |14 |19 W |8 |17 X |3 |20 Y |11 |18 ==============================

* * * * *

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.