In the discussion and tables which follow:

Q stands for Quotient, which will mean a Subject Age divided by a Chronological Age. R stands for Ratio, which will mean a Subject Age divided by a Mental Age.

AQ means Woody-McCall Arithmetic Age divided by Chronological Age, and AR means this AA divided by Mental Age.

VQ means Thorndike Vocabulary Age divided by Chronological Age, and VR means this VA divided by Mental Age.

RQ means Alpha 2 Reading Age divided by Chronological Age, and RR means this RA divided by Mental Age.

CQ means Kelley-Trabue Completion Age divided by Chronological Age, and CR means this CA divided by Mental Age.

SQ means any Subject Quotient, that is, any Subject Age divided by Chronological Age, and SR means any Subject Ratio, that is, any SA divided by Mental Age.

EQ means the average of all Subject Quotients and AccR, the Accomplishment Ratio, means the average of all Subject Ratios.

All r’s are product-moment correlation coefficients, uncorrected. As the reliabilities (Table 4) are almost what the other coefficients are in June, 1920 (Table 5), it is apparent that the corrected coefficients, when Grade III is excluded, would all be very near unity at that time.

THE QUOTIENTS

In Table 1 are presented all the quotients for all periods of testing, grouped by children. The table, a sample of which is included here,⁠[9] shows clearly how all SQ’s approach IQ as special treatment continues. The grades indicated in this grouping are as of June, 1920. Inasmuch as many double and triple promotions were made in an effort to get maximum product for intelligence invested, no conclusion can here be formed of the grade to which these children belonged at any time except June, 1920. The correspondence between IQ and the SQ’s in June, 1920 is further shown in Table 2. In this table the 48 children who took all tests at all periods are ranked from high to low IQ and their SQ’s are listed opposite. The high correspondence is readily apparent.

TABLE 1⁠[10]
Intelligence Quotients for All Periods Grouped by Children

The children are arranged by grade as they were in June, 1920, and alphabetically within the grade. The periods of testing are lettered in their chronological sequence; a is November, 1918, b is June, 1919, c is November, 1919 and d is June, 1920. * = Zero Score

Grade 3

TABLE 2⁠[11]
Group Taking All Tests at All Periods Arranged in Order of Magnitude of Intelligence Quotients

The intercorrelations of the quotients of these 48 cases for all periods may be seen in Table 3 (page 21). The correlations with IQ and the intercorrelations of the SQ’s have increased toward positive unity or rather toward the limits of a correlation with tools of measurement such as we have used. This limit is a function of the reliability of the tests employed. It is customary to use a formula to correct for attenuation in order to find the percentage which the correlation is of the geometric mean of the two reliability coefficients. This is tantamount to saying that any correlation can go no higher than the geometric mean of the reliability coefficients of the tests used. It is better to assume that an r can go as high as the ∜(r₁₁⋅r₂₂) since an r can go as high as the square root of its reliability coefficient. Dr. Truman L. Kelley has shown that the correlation of a test with an infinite number of forms of the same test would be as the square root of its correlation with any one other form.

The reliabilities and limits defining a limit as the fourth root of the multiplied reliability coefficients are in Table 4.

Correction for attenuation is often ridiculously high because the reliability coefficient of one of the measures used is so low. If an element is included in the two tests which are correlated, but not in the other forms of each test used to get reliability, the “corrected coefficient” is corrected for an element which is not chance. Whenever the geometric mean of the reliabilities is less than the obtained r, the corrected r is over 1.00 and hence absurd.⁠[12]

Therefore we use here instead, a comparison to the maximum possibility in a true sense. Since a test correlates with the “true ability” √(r₁₁), ∜(r₁₁⋅r₂₂) is the limit of an r, its optimum with those tools. Although these limits apply, strictly speaking, only to the total correlations, since the reliability correlations are with all the data; we may assume that the same facts hold with regard to the correlations of each of the grades, that is, the reliability is a function of the test not of the data selected.

TABLE 3
Intercorrelation of All Quotients for All Periods of the 48 Children Who Took All Tests

TABLE 4
Reliability Coefficients

Limits of the r’s = ∜(r₁₁ × r₂₂)

The limits of the June, 1920 r’s are naturally somewhat larger than the others since two forms of tests (except the Binet) were used; the unreliability of the quantitative indices is therefore lower and hence the correlation with IQ may be larger.

The correlations in 1920 of another group—the whole school except Grade III—are reproduced in Table 5. Grade III was excluded since here there had as yet been little chance to push the r’s. Partials were obtained with these data (Table 6). Little faith may be placed in the relative sizes of these partials, much because the r_VQ.RQ is here only .73 and, in the data presented in Table 3, it is .87. This is due to the fact that the data in Table 3 cover all periods (2 years) while those in Table 5 cover only one. This difference has comparatively slight influence on our general conclusions; but it makes a huge difference in the correlation of RQ and VQ when IQ is rendered constant, whether the one or the other set of data is used. Moreover, the whole logic of arguing for general factors by reduction of partial correlations from the original r has been called gravely into question in Godfrey H. Thomson’s recent work on this subject: “The Proof or Disproof of the Existence of General Ability.” Thomson shows that partial correlation gives one possible interpretation of the facts, but not an inevitable one. Thus we cannot say that because RQ and IQ and RQ and AQ are highly correlated, correlation of IQ and AQ is dependent upon RQ. We can say, however, that it is likely to be. IQ and AQ may be correlated by reason of inclusion of some element not included at all in RQ. The higher the correlations which we deal with the less we need worry about this, and of course correlations of unity exclude any such consideration.

TABLE 5
Intercorrelation of All Quotients in June, 1920. All Children Exclusive of Grade 3 are Here Represented

The P.E.’s are all less than .05

N = 81

I therefore draw no conclusions from the comparative size of these partials, nor do I get partials with any of the other data, and rest the case mainly on the high r’s between IQ and SQ’s in 1920; increase in correspondence of the central tendencies and range of the SQ’s by grade with the central tendency and range of the IQ’s of the same data; small intercorrelation of SR’s and negative correlation of AccR with IQ.

The general lowness of the partials (Table 6) does, however, indicate the great causative relation between IQ and disparity of product. The elements still in here are common elements in the tests and the mistreatment of intelligence.

TABLE 6
Partial Correlations of Quotients Irrespective of Intelligence Quotients

N = 81

What happened by grade in 1918-1919 is summarized in Table 7. What happened by grade in 1919-1920 is summarized in Table 8. Since there were many changes in personnel from 1918-1919 to 1919-1920, we need expect no continuity from Table 7 to Table 8. For the continuous influence of the two years, see Table 3, which includes 48 children taking all tests at all periods.

TABLE 7
All Correlations, Means, and Standard Deviations by Grade, Showing Progress from November, 1918 to June, 1919

• I stands for Intelligence Quotient
• V stands for Vocabulary Quotient
• R stands for Reading Quotient
• C stands for Completion Quotient

TABLE 8
All Correlations, Means, and Standard Deviations of Quotients by Grade, Showing Progress from November, 1919 to June, 1920

• I stands for Intelligence Quotient
• V stands for Vocabulary Quotient
• R stands for Reading Quotient
• C stands for Completion Quotient
• A stands for Arithmetic Quotient

Note—Totals without Grade III are much higher than these (Table 5). Grade III has many children in it who have not been long enough in an academic situation to allow their SQ’s to go as high as they may.

It is proper to note here that not much can be expected from Grades III and VIII and from totals including Grade III, since children in Grade III have not been there long enough to be pushed, and children in Grade VIII have been pushed beyond the limits which the tests used will register. Our logic is one of pushed correlations. If the association of IQ and the SQ’s is what we are attempting to establish, it is necessary to show:

1. That the r comes near unity;

2. That the central tendencies come near coincidence;

3. That the S.D.’s come near coincidence.

The value of the r is obvious; the value of coincidence of means becomes clearer if we think of Σ(IQ-EQ)/n, the average difference of potential rate of progress and actual rate of progress. This average of differences is the same as the difference of the averages, which is more readily calculated. Obviously, if we wish to use an AccR, it is necessary to show more than correspondence when differences in average and spread are equated as they are by the correlation coefficient. Besides, coincidence of M’s, correspondence of S.D.’s is also necessary since a correlation might be positive unity, the M’s might be equal, and still the spread of one measure might be more than the spread of the other. If the spreads are the same and the M’s are the same, and the correlation is positive unity, each x must equal its corresponding y. Then b₁₂ = b₂₁ = 1.00; and the M’s being equal, the deviations are from the same point. Therefore, we will attempt to measure similarity of M’s and S.D.’s as well as r.

It will be observed that both Tables 7 and 8 give evidence of each of these tendencies in all grades. In Table 8 marked progress in arithmetic is apparent. This is due to re-classification in terms of the Woody-McCall test, which was not done in 1918-1919. In 1918-1919 no arithmetic test was given and all re-classification was in terms of reading, being done on the basis of both reading tests. Spelling re-classification was done each year, but the data were not treated in this manner. It can be said that wherever re-classification in terms of intelligence and pedagogical need was undertaken the desired result of pushing the SQ’s up to IQ was hastened. Of all the remedial procedure, such as changing teachers and time allotment and books and method, all of which were employed to some extent, it is my opinion that the re-classification was more important than everything else combined.

It is noticeable that when r’s approach the limit which the unreliability of the test allows them, they drop down again. This is probably due to continued increase of SQ’s over IQ. Of course, for some SQ’s to be greater than IQ out of proportion to the general amount lowers the correlation as much as for some to lag behind. When the SQ’s of the children of lower intelligence reach their IQ they continue above. This, of course, is due to errors in establishment of the age norms. The norms are not limits of pushing, though an attempt was made by correction for truncation to get them as nearly so as possible. It is to be noted, however, that these norms are up the growth curve, that is, reading age of 10 means a score such that the average age of those getting it is 10, not the average score of children whose mental age is 10. The average reading achievement of children all ten years old chronologically is higher than that of a group all mentally ten, since many of the mentally advanced have not been pushed in product. The group used here to establish norms gives more nearly pushed norms than the others would.

The tendency of the low IQ’s to go over unity in their SR’s is apparent in Table 1 and in Table 12 and also in the negative correlation between AccR and IQ.

In both years some second grade children were advanced to Grade III during the year. This accounts for the low r’s in June, 1919, but in 1919-1920 the Grade III correlations are raised and the means raised toward the M_IQ, even though some second grade children were put in this group during the year.

TABLE 9
Summary of Progress in Arithmetic by Increase in r, Decrease in M_IQ-M_AQ and Decrease in Difference of Standard Deviations Irrespective of Direction