STATISTICAL TREATMENT OF THE EXPERIMENT
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
=============+======+==========+==========+========+==========
Intelligence| Test |Arithmetic|Vocabulary|Reading |Completion
Quotient |Period| Quotient | Quotient |Quotient| Quotient
-------------+------+----------+----------+--------+----------
| _a_ | | | |
101 | _b_ | | | |
| _c_ | 64 | 58 | | 43
| _d_ | 106 | 88 | | 93
| | | | |
| _a_ | | | |
128 | _b_ | | | |
| _c_ | 80 | 102 | | 81
| _d_ | | 152 | 124 | 153
| | | | |
| _a_ | | | |
116 | _b_ | | | |
| _c_ | 56 | 90 | * | 49
| _d_ | 94 | 95 | 77 | 89
| | | | |
| _a_ | | | |
87 | _b_ | | | |
| _c_ | 90 | 40 | 35 | 54
| _d_ | 72 | 74 | 61 | 52
| | | | |
| _a_ | | | |
112 | _b_ | | | |
| _c_ | 90 | 137 | 133 | 112
| _d_ | 112 | 113 | 121 | 131
-------------+------+----------+----------+--------+----------
TABLE 2[11]
GROUP TAKING ALL TESTS AT ALL PERIODS ARRANGED IN ORDER OF MAGNITUDE OF
INTELLIGENCE QUOTIENTS
=============+============+==========+==========+===========
Intelligence | Arithmetic |Vocabulary| Reading |Completion
Quotients | Quotients |Quotients |Quotients |Quotients
-------------+------------+----------+----------+-----------
146 | 111 | 154 | 164 | 150
142 | 129 | 135 | 137 | 136
141 | 109 | 118 | 107 | 121
139 | 124 | 141 | 124 | 134
138 | 101 | 112 | 105 | 106
| | | |
138 | 121 | 130 | 110 | 109
130 | 107 | 139 | 135 | 136
122 | 127 | 130 | 124 | 121
122 | 113 | 121 | 117 | 124
122 | 112 | 102 | 114 | 129
| | | |
121 | 128 | 125 | 128 | 128
120 | 100 | 116 | 102 | 119
118 | 117 | 123 | 114 | 125
117 | 131 | 111 | 118 | 124
117 | 106 | 122 | 112 | 111
| | | |
114 | 105 | 126 | 110 | 114
109 | 83 | 113 | 117 | 103
107 | 103 | 112 | 95 | 103
107 | 94 | 126 | 94 | 123
104 | 99 | 117 | 96 | 104
| | | |
104 | 103 | 110 | 94 | 116
103 | 108 | 113 | 112 | 106
101 | 100 | 114 | 109 | 106
100 | 90 | 103 | 92 | 92
100 | 109 | 118 | 108 | 113
| | | |
99 | 114 | 104 | 106 | 110
99 | 114 | 119 | 117 | 115
98 | 102 | 101 | 108 | 104
98 | 99 | 106 | 107 | 106
97 | 95 | 109 | 107 | 105
| | | |
97 | 108 | 101 | 102 | 105
97 | 95 | 104 | 89 | 110
96 | 90 | 104 | 91 | 91
95 | 84 | 99 | 93 | 100
95 | 90 | 107 | 99 | 105
| | | |
95 | 85 | 117 | 114 | 103
94 | 106 | 57 | 89 | 108
94 | 103 | 103 | 106 | 104
92 | 96 | 86 | 94 | 85
87 | 83 | 88 | 92 | 87
| | | |
87 | 95 | 96 | 94 | 102
84 | 85 | 87 | 93 | 87
83 | 106 | 91 | 87 | 104
80 | 77 | 91 | 80 | 84
80 | 84 | 75 | 79 | 84
| | | |
80 | 89 | 107 | 88 | 86
78 | 87 | 90 | 93 | 85
60 | 69 | 56 | 71 | 77
-------------+------------+----------+----------+-----------
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 ∜(_r_₁₁⋅_r_₂₂) 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” √(_r_₁₁),
∜(_r_₁₁⋅_r_₂₂) 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
NOVEMBER, 1918
IQ VQ RQ S.D. M
IQ 19.12 105.15
±1.32 ±1.86
VQ .72 20.54 102.52
±.05 ±1.41 ±2.00
RQ .64 .64 19.09 95.90
±.06 ±.06 ±1.31 ±1.86
CQ .63 .71 .77 19.34 99.44
±.06 ±.05 ±.04 ±1.33 ±1.88
JUNE, 1919
IQ VQ RQ S.D. M
IQ 19.12 105.15
±1.32 ±1.86
VQ .73 20.80 113.54
±.05 ±1.43 ±2.02
RQ .65 .58 14.73 101.31
±.06 ±.06 ±1.01 ±1.43
CQ .62 .68 .77 19.76 101.04
±.06 ±.05 +.04 ±1.36 ±1.92
NOVEMBER, 1919
IQ AQ VQ RQ S.D. M
IQ 19.12 105.15
±1.32 ±1.86
AQ .46 14.08 102.90
±.08 ±0.97 ±1.37
VQ .86 .23 17.07 109.17
±.03 ±.09 ±1.18 ±1.66
RQ .65 .56 .71 13.91 101.42
±.06 ±.07 ±.05 ±0.96 ±1.35
CQ .79 .47 .83 .82 17.53 105.21
±.04 ±.08 ±.03 ±.03 ±1.21 ±1.71
JUNE, 1920
IQ AQ VQ RQ S.D. M
IQ 19.12 105.15
±1.32 ±1.86
AQ .73 14.10 101.79
±.05 ±0.97 ±1.37
VQ .81 .60 18.89 108.94
±.03 ±.06 ±1.30 ±1.84
RQ .79 .68 .87 16.43 104.94
±.04 ±.05 ±.02 ±1.13 ±1.60
CQ .84 .77 .78 .84 15.87 108.08
±.03 ±.04 ±.04 ±.03 ±1.09 ±1.54
TABLE 4
RELIABILITY COEFFICIENTS
One Form Two Forms One Form Two Forms
of Each of Each with an with an
Test Test (by Infinite Infinite
Brown’s Number Number
Formula) of Forms of Forms
_r_₁₁ _r_₁₁ √_r_₁₁ √_r_₁₁
Intelligence Quotient .888 .942
(by Brown’s Formula)[13]
Arithmetic Quotient .824 .904 .908 .951
Vocabulary Quotient .820 .901 .906 .949
Reading Quotient .866 .928 .931 .963
Completion Quotient .883 .938 .940 .968
Limits of the _r_’s = ∜(_r_₁₁ × _r_₂₂)
Nov. 1918,
June and Nov. 1919 June 1920
IQ and AQ .925 .946
IQ and VQ .924 .946
IQ and RQ .936 .953
IQ and CQ .941 .955
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
Arithmetic Vocabulary Reading
IQ Quotient Quotient Quotient
Arithmetic Quotient .733
Vocabulary Quotient .837 .628
Reading Quotient .758 .694 .734
Completion Quotient .821 .770 .825 .801
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
Arithmetic Vocabulary Reading
Quotient Quotient Quotient
Vocabulary Quotient .04
±.07
Reading Quotient .31 .28
±.07 ±.07
Completion Quotient .43 .44 .47
±.08 ±.06 ±.06
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
GRADE _r_ M S.D.
Nov. June Nov. June Nov. June
I V .467 .633 I 109.89 113.20 I 12.83 15.49
±.12 ±.07 ±1.98 ±1.91 ±1.40 ±1.35
III I R .541 .492 V 96.11 109.90 V 21.21 18.69
±.11 ±.09 ±3.28 ±2.30 ±2.32 ±1.63
I C .641 .386 R 82.26 101.40 R 22.58 15.85
±.09 ±.11 ±3.49 ±1.95 ±2.47 ±1.38
C 86.89 108.40 C 22.76 15.79
±3.52 ±1.94 ±2.49 ±1.37
_N_ = 19 30
-----------------------------------------------------------------
I V .724 .819 I 105.90 104.82 I 18.08 18.21
±.07 ±.05 ±2.73 ±2.98 ±1.93 ±2.11
IV I R .665 .845 V 97.20 108.53 V 17.26 24.92
±.08 ±.05 ±2.60 ±4.08 ±1.84 ±2.88
I C .596 .717 R 91.06 107.82 R 27.85 10.35
±.10 ±.08 ±4.20 ±1.69 ±2.97 ±1.20
C 101.45 108.12 C 21.53 17.75
±3.25 ±2.90 ±2.30 ±2.05
_N_ = 20 17
-----------------------------------------------------------------
I V .887 .822 I 101.64 99.42 I 24.76 17.63
±.04 ±.05 ±3.56 ±2.73 ±2.52 ±1.93
V I R .799 .832 V 100.59 111.58 V 26.71 19.78
±.05 ±.05 ±3.84 ±3.06 ±2.72 ±2.16
I C .818 .890 R 94.59 101.42 R 22.10 12.56
±.05 ±.03 ±3.18 ±1.94 ±2.25 ±1.37
C 97.00 102.68 C 22.52 17.71
±3.24 ±2.74 ±2.29 ±1.94
_N_ = 22 19
-----------------------------------------------------------------
I V .793 .772 I 109.90 115.90 I 23.45 24.38
±.08 ±.09 ±5.00 ±5.20 ±3.54 ±3.68
VI I R .497 .726 V 108.00 126.80 V 30.20 25.25
±.16 ±.10 ±6.44 ±5.39 ±4.55 ±3.81
I C .798 .891 R 103.10 107.20 R 13.77 20.62
±.08 ±.04 ±2.94 ±4.40 ±2.08 ±3.11
C 108.90 117.10 C 15.23 18.81
±3.25 ±4.01 ±2.30 ±2.84
_N_ = 10 10
-----------------------------------------------------------------
I V .625 .504 I 99.29 98.92 I 11.11 11.45
±.11 ±.14 ±2.00 ±2.14 ±1.42 ±1.51
VII I R .622 .709 V 109.43 115.23 V 14.07 17.43
and ±.11 ±.09 ±2.54 ±2.95 ±1.79 ±2.31
VIII
I C .782 .730 R 97.00 98.85 R 12.59 15.77
±.07 ±.09 ±2.27 ±3.26 ±1.61 ±2.09
C 102.43 95.85 C 13.49 17.72
±2.43 ±3.31 ±1.72 ±2.34
_N_ = 14 13
-----------------------------------------------------------------
I V .685 .680 I 105.07 106.88 I 19.34 18.45
±.04 ±.04 ±1.41 ±1.32 ±1.00 ±0.93
I R .568 .626 V 101.12 112.67 V 22.83 21.58
TOTAL ±.05 ±.04 ±1.67 ±1.54 ±1.18 ±1.09
I C .639 .702 R 92.40 102.91 R 22.65 15.27
±.04 ±.04 ±1.66 ±1.09 ±1.17 ±0.77
C 98.08 106.27 C 21.48 18.19
±1.57 ±1.30 ±1.11 ±0.92
_N_ = 85 89
-----------------------------------------------------------------
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
_r_ M S.D.
Nov. June Nov. June Nov. June
I A .413 .709 I 102.00 105.53 I 9.60 10.89
±.16 ±.08 ±1.87 ±1.68 ±1.32 ±1.19
III I V .649 .667 A 82.75 97.84 A 15.88 18.62
±.11 ±.09 ±3.09 ±2.88 ±2.19 ±2.04
I R .651 .609 V 94.00 103.47 V 33.44 27.66
±.11 ±.10 ±6.51 ±4.28 ±4.60 ±3.03
I C .612 .719 R 87.59 93.88 R 32.06 19.02
±.12 ±.07 ±6.24 ±3.21 ±4.41 ±2.27
C 90.17 96.84 C 28.82 25.59
±5.58 ±3.96 ±3.95 ±2.80
_N_ = 12 19
-----------------------------------------------------------------
I A .426 .725 I 111.48 113.00 I 14.73 15.04
±.10 ±.06 ±1.85 ±1.93 ±1.30 ±1.36
IV I V .635 .772 A 94.07 111.08 A 12.34 15.02
±.075 ±.05 ±1.55 ±1.99 ±1.09 ±1.40
I R .316 .569 V 109.79 115.61 V 16.97 18.39
±.11 ±.09 ±2.13 ±2.34 ±1.50 ±1.66
I C .594 .837 R 99.31 110.11 R 17.89 14.67
±.08 ±.04 ±3.24 ±1.67 ±1.58 ±1.32
C 108.14 118.14 C 15.51 12.70
±1.94 ±1.62 ±1.37 ±1.15
_N_ = 29 28
-----------------------------------------------------------------
I A .698 .713 I 103.72 98.83 I 19.57 18.84
±.07 ±.07 ±2.69 ±2.65 ±1.91 ±1.87
V I V .881 .908 A 87.58 99.71 A 12.43 16.47
±.03 ±.02 ±1.71 ±2.27 ±1.21 ±1.60
I R .773 .891 V 109.00 105.17 V 15.58 19.97
±.06 ±.03 ±2.14 ±2.81 ±1.52 ±1.99
I C .786 .923 R 104.46 103.00 R 16.99 17.07
±.05 ±.02 ±2.34 ±2.40 ±1.65 ±1.70
C 107.00 103.48 C 16.12 14.51
±2.22 ±2.04 ±1.57 ±1.44
_N_ = 24 23
-----------------------------------------------------------------
I A .533 .805 I 102.43 105.39 I 11.61 13.56
±.13 ±.06 ±2.09 ±2.16 ±1.48 ±1.52
VI I V .774 .858 A 91.43 104.53 A 11.43 11.31
±.07 ±.04 ±2.06 ±1.75 ±1.46 ±1.24
I R .420 .661 V 106.07 112.94 V 11.93 10.94
±.15 ±.09 ±2.15 ±1.74 ±1.52 ±1.23
I C .739 .620 R 96.64 106.20 R 12.38 11.88
±.08 ±.10 ±2.23 ±1.79 ±1.58 ±1.27
C 100.36 107.61 C 13.95 10.55
±2.51 ±1.68 ±1.78 ±1.19
_N_ = 14 18
-----------------------------------------------------------------
I A .740 .795 I 107.27 100.58 I 23.29 19.78
±.09 ±.07 ±4.74 ±2.85 ±3.35 ±2.72
VII I V .867 .718 A 100.00 99.31 A 9.26 11.00
±.05 ±.09 ±1.86 ±2.06 ±1.33 ±1.45
I R .862 .799 V 114.36 108.75 V 19.15 14.42
±.05 ±.07 ±3.89 ±2.81 ±2.75 ±1.98
I C .833 .677 R 101.73 98.58 R 12.28 11.56
±.06 ±.11 ±2.50 ±2.25 ±1.77 ±1.59
C 105.82 101.42 C 17.41 16.02
±3.54 ±3.12 ±2.50 ±2.21
_N_ = 11 12
-----------------------------------------------------------------
I A .663 .796 I 104.83 108.79 I 15.46 18.25
±.11 ±.07 ±3.01 ±3.29 ±2.13 ±2.33
VIII I V .828 .750 A 92.92 93.86 A 10.20 9.74
±.06 ±.08 ±1.99 ±1.76 ±1.40 ±1.24
I R .775 .722 V 111.67 117.21 V 16.44 14.02
±.08 ±.08 ±3.20 ±2.53 ±2.26 ±1.79
I C .838 .868 R 100.83 104.38 R 11.52 20.62
±.06 ±.04 ±2.24 ±3.72 ±1.59 ±2.63
C 104.92 109.64 C 18.11 17.41
±3.53 ±3.14 ±2.49 ±2.22
_N_ = 12 14
-----------------------------------------------------------------
I A .576 .686 I 106.02 105.87 I 16.73 16.87
±.05 ±.03 ±1.12 ±1.07 ±0.79 ±0.75
TOTAL I V .679 .727 A 91.35 102.01 A 13.22 15.61
±.04 ±.03 ±0.88 ±0.98 ±0.62 ±0.69
I R .529 .609 V 107.95 110.54 V 19.76 19.57
±.05 ±.04 ±1.32 ±1.24 ±0.93 ±0.87
I C .678 .731 R 99.22 103.65 R 18.85 17.12
±.04 ±.03 ±1.26 ±1.08 ±0.89 ±0.76
C 104.06 108.00 C 18.87 18.11
±1.26 ±1.14 ±0.89 ±0.81
_N_ = 102 114
-----------------------------------------------------------------
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 _b_₁₂ =
_b_₂₁ = 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
Average Intelligence Difference of
GRADE _r_ Quotient Minus Standard Deviations
Average Arithmetic Irrespective of
Quotient Sign (of IQ and
Arith. Q)
Nov. June Nov. June Nov. June
III .413 .709 19.25 8.16 6.27 6.63
±.16 ±.08 ±2.87 ±2.05 ±2.04 ±1.45
IV .426 .725 7.41 0.46 2.39 0.47
±.10 ±.06 ±1.84 ±1.50 ±1.29 ±1.02
V .698 .713 16.14 0.54 7.14 2.06
±.07 ±.07 ±1.93 ±1.84 ±1.37 ±1.30
VI 5.33 .805 11.00 3.00 0.19 1.63
±.13 ±.06 ±2.01 ±1.19 ±1.42 ±0.85
VII .740 .795 7.27 0.62 14.03 8.15
±.09 ±.07 ±3.58 ±2.33 ±2.53 ±1.63
VIII .663 .796 11.92 [14]14.93 5.26 [14]8.53
±.11 ±.07 ±2.25 ±2.69 ±1.59 ±1.54
Total .576 .686 14.67 3.72 3.51 1.16
±.05 ±.03 ±0.94 ±0.81 ±0.67 ±0.57
TABLE 10
SUMMARY OF PROGRESS IN READING, NOVEMBER, 1918 TO JUNE, 1919, BY INCREASE
IN _r_, DECREASE IN M_{IQ}-M_{RQ}, AND DECREASE IN DIFFERENCE OF STANDARD
DEVIATIONS IRRESPECTIVE OF SIGN
Average Intelligence Difference of
GRADE _r_ Quotient Minus Standard Deviations
Average Reading Irrespective of
Quotient Sign (of IQ and RQ)
Nov. June Nov. June Nov. June
III .541 .492 27.63 11.80 9.75 0.36
±.11 ±.09
IV .665 .845 14.84 -3.00 9.77 7.86
±.08 ±.05
V .799 .832 7.05 -2.00 2.66 5.07
±.05 ±.05
VI .497 .726 6.80 8.70 9.68 3.76
±.16 ±.10
VII .622 .709 2.28 0.07 1.48 5.98
3 of VIII ±.11 ±.09
Total .568 .626 12.67 3.97 3.31 3.18
±.05 ±.04
TABLE 11
SUMMARY OF PROGRESS IN READING, NOVEMBER, 1919 TO JUNE, 1920, BY INCREASE
IN _r_, DECREASE IN M_{IQ}-M_{RQ}, AND DECREASE IN DIFFERENCE OF STANDARD
DEVIATIONS IRRESPECTIVE OF SIGN
Average Intelligence Difference of
GRADE _r_ Quotient Minus Standard Deviations
Average Reading Irrespective of
Quotient Sign (of IQ and RQ)
Nov. June Nov. June Nov. June
III .651 .609 14.41 11.57 22.46 8.62
±.11 ±.10 ±5.22 ±2.55 ±3.69 ±1.81
IV .316 .569 12.17 2.43 3.16 0.76
±.11 ±.09 ±2.41 ±1.78 ±1.70 ±1.26
V .773 .891 -0.74 -4.17 2.58 1.77
±.06 ±.03 ±1.72 ±1.20 ±1.22 ±0.85
VI .420 .661 5.79 0.90 0.77 0.87
±.15 ±.09 ±2.33 ±1.53 ±1.65 ±1.09
VII .862 .799 5.54 0.92 11.00 8.31
±.05 ±.07 ±2.88 ±2.54 ±2.03 ±1.80
VIII .775 .722 4.00 4.43 3.94 2.41
±.08 ±.09 ±1.90 ±2.64 ±1.92 ±1.87
Total .529 .609 6.80 2.86 2.12 0.06
±.05 ±.04 ±1.16 ±0.30 ±0.82 ±0.67
The changes in rates of progress are expressed in summaries by subject
matter in Tables 9, 10, and 11. Approach of Arithmetic Quotient to
Intelligence Quotient is measured in Table 9 by:
1. Comparison of _r_ in June with _r_ in November.
2. Comparison of M_{IQ}-M_{AQ} in June and M_{IQ}-M_{AQ} in November.
3. Comparison of S.D.’s of Arithmetic and Intelligence Quotients in June
and November.
The P.E.’s of each of these differences were obtained by
P.E._{diff}² = P.E.₁² + P.E.₂² - 2 _r_₁₂ P.E.₁ P.E.₂
The only M_{IQ}-M_{SQ} in Table 9 which does not show a decrease at
least two times as large as the P.E. of either of the elements involved,
is the 8th grade; and this is due to the limits of the test used. As
mentioned before, the 8th grade did not register its true abilities in
June since a perfect, or nearly perfect, score in the test was too easy
to obtain. The small arithmetic S.D.’s in Grade 8 and consequent great
S.D._{IQ}-S.D._{SQ} is due to the same cause.
Tables 10 and 11 present the summary of facts with regard to Thorndike
Reading Quotients, the first and second years respectively.
THE RATIOS
The discussion which follows concerns _Ratios_, not _Quotients_.
TABLE 12
INTELLIGENCE QUOTIENTS AND SUBJECT RATIOS FOR ALL PERIODS GROUPED BY
CHILD. THE ORDER OF ENTRIES IS JUST AS IN TABLE 1
GRADE III
Intelligence Arithmetic Vocabulary Reading Completion
Quotient Ratio Ratio Ratio Ratio
_a_
101 _b_
_c_ 63 57 43
_d_ 105 87 92
_a_
128 _b_
_c_ 62 80 63
_d_ 119 97 120
_a_
116 _b_
_c_ 48 78 * 42
_d_ 81 82 66 77
_a_
87 _b_
_c_ 103 46 40 62
_d_ 83 85 70 60
_a_
112 _b_
_c_ 80 122 119 100
_d_ 100 101 108 117
_a_
101 _b_
_c_ 84 93 37 55
_d_ 90 110 98 92
_a_
90 _b_
_c_ 76 58 72 89
_d_ 68 121 77 102
_a_
105 _b_
_c_ 60 43 * 57
_d_ 104 95 83 66
The remainder of this table is filed in Teachers College Library,
Columbia University.
TABLE 13
Nov., 1918 June, 1919 Nov., 1919 June, 1920
MEANS
Arithmetic Ratio 89.02 97.16
±1.05 ±1.07
Vocabulary Ratio 98.96 111.44 106.20 107.61
±1.48 ±1.61 ±0.90 ±0.93
Reading Ratio 96.47 101.96 98.98 100.60
±1.19 ±1.18 ±1.03 ±0.97
Completion Ratio 99.76 101.83 101.67 103.10
±1.11 ±1.23 ±0.93 ±0.85
STANDARD DEVIATIONS
Arithmetic Ratio 12.03 12.53
±0.74 ±0.76
Vocabulary Ratio 15.71 16.58 10.34 10.84
±1.05 ±1.14 ±0.64 ±0.66
Reading Ratio 12.63 12.14 11.82 11.36
±0.84 ±0.84 ±0.73 ±0.69
Completion Ratio 12.34 12.63 10.85 9.90
±0.82 ±0.87 ±0.67 ±0.60
CORRELATIONS OF RATIOS
Arithmetic and Vocabulary .60 .30
±.06 ±.08
Arithmetic and Reading .70 .64
±.04 ±.05
Arithmetic and Completion .48 .61
±.07 ±.05
Vocabulary and Reading .34 .32 .57 .47
±.08 ±.09 ±.06 ±.07
Vocabulary and Completion .45 .36 .53 .54
±.07 ±.08 ±.06 ±.06
Reading and Completion .61 .65 .67 .67
±.06 ±.06 ±.05 ±.05
In Table 12 are presented the Subject Ratios in the same order as the
Quotients appear in Table 1.[15] There plainly is a rapid rise of SQ⁄IQ
from period to period, excluding all pupils who did not take all tests
and excluding Grade III; which includes all children taking all tests who
were in school in June, 1920, and were Grade IV and above in November,
1918. The average AccR is 98.24 in November, 1918, and 102.78 in June,
1920. The average IQ for these children is 105.22. The S.D_{AccR₁₉₁₈} is
11.17; the S.D._{AccR₁₉₂₀} is 9.09; the S.D._{IQ} is 19.24. It is obvious
that the average amount of product per intelligence has increased, that
the range of AccR’s has decreased (which means that factors causing
disparities, other than intelligence, have been removed), and that the
S.D. of the AccR’s is about one half the S.D. of the IQ’s. M’s are about
equal so it is not necessary to use coefficients of variability. The
variability of children, intelligence aside, is only one half what the
variability is otherwise. The correlations when IQ = _X_, AccR₁₉₁₈ = _Y_
and AccR₁₉₂₀ = _S_ and when AccR = average of Vocabulary, Reading and
Completion Ratios, are:[16]
_r__{X.Y.} = -.602
_r__{X.S.} = -.493
_r__{Y.S.} = +.549
The remaining disparity is then due to something which is in negative
correlation with intelligence.
The number of cases here is only 48.
The P.E.’s are then as follows:
P.E._{M} P.E._{S.D.}
_X_ 1.91 1.35
_Y_ 1.11 0.79
_S_ 0.90 0.64
P.E._r__{X.Y.} = .06
P.E._r__{X.S.} = .08
P.E._r__{Y.S.} = .07
The differences between the M’s and between the S.D.’s of our 1918 and
our 1920 AccQ’s; namely, 102.78 - 98.24 = 4.54 and 11.17 - 9.09 = 2.08,
have formed a step in the argument. We must have the P.E.’s of these
amounts in order to establish the reliability of the quantitative indices
we employ:
P.E._{diff} = √P.E._{X}² + P.E._{Y}² - 2 _r__{XY} P.E._{X} P.E._{Y}
P.E._{M₂₀-M₁₈} = 0.94
P.E._{S.D.₁₈-S.D.₂₀} = 0.47
These differences are then reliable. If the same data were accumulated
again in the same way with only 48 cases, the chances are even that the
4.54 would be between 3.50 and 5.48 and the 2.08 between 1.61 and 2.55.
That there would be positive differences is practically certain, since
the difference between the means is over four times as large as its P.E.,
and the difference between the S.D.’s over four times as large as its P.E.
To make still more certain this observation of positive amount in M of
second testing minus M of first testing and in S.D. of first testing
minus S.D. of second testing (AccR), which means an increase in central
tendency of AccR’s and a decrease in spread of AccR’s under special
treatment, we have listed in Table 13 the means and standard deviations
of Subject Ratios of each test for each period and the intercorrelations
of these Subject Ratios. These do not include exactly the same children
in each period but are inclusive of all grades for all periods. They
are a measurement of increased efficiency of the school as a whole,
rather than of any one group of children; though, of course, the bulk
of the children have representation in each of these indices. Too much
continuity is not to be expected from June, 1919, to November, 1919, as
the children are different. Comparison should always be from November to
June.
These tables bear out the fact presented by AccR. It is clear that
there is a marked development in the S.R.’s, both by increase of M.
and decrease of S.D. The decrease of correlation between S.R.’s is not
so marked, but neither is the negative correlation between AccR and
IQ much less in June, 1920, than in November, 1918. The association
of achievements in terms of intelligence is very probably due to
mistreatment, since it is in negative correlation with IQ, as a general
inherited ethical factor could not be.
We will note that the Arithmetic Ratios are in as high positive
association with the Reading Ratios as the Vocabulary Ratios are with the
Reading Ratios. This makes it highly improbable that the intercorrelation
of these remnants is due, to any large extent, to common elements in
the test or to specific abilities. The common interassociation of all
Ratios seems to point to the operation of some common factor other than
intelligence as a determinant of disparity in school progress. It would
be easy to identify this as the part of Burt’s “General Educational
Factor” which is not intelligence—that is, industry, general perseverance
and initiative—were it not for the fact that this same influence _stands
in negative association to intelligence_. It is our belief that it is the
influence of a maladjusted system of curricula and methods which accounts
for these rather high interassociations of achievements, irrespective of
intelligence.
SUMMARY
The association of abilities in arithmetic, reading, and completion with
intelligence is markedly raised by special treatment. Disparities of
educational product are therefore to a great extent due to intelligence.
(Tables 2, 3, 5, 7, 8, 9, 10 and 11.)
The remnants (intelligence being rendered constant by division of each
SQ by IQ) intercorrelate about .5. If there were specialized inherited
abilities, these intercorrelations would not all be positive nor would
they be as uniform. (Tables 6 and 13.)
The averages of these remnants, for reading, vocabulary, and completion,
correlate -.61 in 1918 and -.49 in 1920 with IQ. These remnants are in
negative association to intelligence. If the intercorrelations of these
remnants were due to a “General Factor,” this correlation would not be
negative.
Therefore intelligence is far and away the most important determinant of
individual differences in product.
As part of the relation between tests, irrespective of intelligence, is
due to common elements in the tests, this reasoning becomes still more
probable.
General factor in education, as distinct from intelligence, has not
been separated here from inherited bases of ambition, concentration,
and industry. It seems out of our province to conjure up some inherited
complex of abilities other than intelligence, specialized inherited
abilities, or proclivities and interests tending to thorough prosecution
of school work. I have therefore meant this last by the general factor.
McCall has correlations varying continually in size from -.63 to +.98
between various measurements of a group of 6B children.[17] The abilities
involved were not pushed as are those considered here. Some of the low
correlations are no doubt indications of low association because of the
way children _are_, not the way they _might be_ by heritage; still
others, such as handwriting and cancellation (unless bright children
do badly in cancellation tests because they are _more bored_ than the
others), are correlated low or negatively with intelligence when the
correlation is at its maximum. Such results as those of McCall serve as a
guide not to argue about other tests by analogy. It is necessary to find
which traits and abilities can be pushed to unity in their relation to
intelligence and which, like handwriting, are practically unrelated to
general mental power.
It is well to know about music tests and such tests as Stenquist’s
mechanical ability test _when the correlation with intelligence is
pushed_, before we decide whether the quality measured is a manifestation
of specific talent or general intelligence.
Cyril Burt obtained data much like that presented here except that
instead of getting rid of the influence of intelligence and finding
determinants for the remnants of disparity, he built up a hierarchy of
coefficients as they would be if they were due entirely to a common
factor and compared these with his obtained _r_’s. I will present his
conclusions with regard to a general factor which are in substantial
though not complete agreement with those advanced here.
“Evidence of a Single Common Factor.
“The correlations thus established between the several school
subjects may legitimately be attributed to the presence of
common factors. Thus, the fact that the test of Arithmetic
(Problems) correlates highly with the test of Arithmetic
(Rules) is most naturally explained by assuming that the same
ability is common to both subjects; similarly, the correlation
of Composition with Arithmetic (Problems) may be regarded as
evidence of a common factor underlying this second pair; and
so with each of the seventy-eight pairs. But is the common
factor one and the same in each case? Or have we to recognise a
multiplicity of common factors, each limited to small groups of
school subjects?
“To answer this question a simple criterion may be devised.
It is a matter of simple arithmetic to reconstruct a table
of seventy-eight coefficients so calculated that all the
correlations are due to one factor and one only, common to
all subjects, but shared by each in different degrees. Such
a theoretical construction is given in Table XIX. In this
table theoretical values have been calculated so as to give
the best possible fit to the values actually obtained in the
investigation, and printed in Table XVIII. It will be seen that
the theoretical coefficients exhibit a very characteristic
arrangement. The values diminish progressively from above
downwards and from right to left. Such an arrangement is termed
a ‘hierarchy.’ Its presence forms a rough and useful criterion
of the presence of a single general factor.
“On turning to the values originally obtained (Table XVIII.)
it will be seen that they do, to some extent, conform to this
criterion. In certain cases, however, the correlations are far
too high—for instance, those between Arithmetic (Rules) and
Arithmetic (Problems), and again Drawing and both Handwork and
Writing (Quality). Now these instances are precisely those
where we might anticipate special factors—general arithmetical
ability, general manual dexterity—operating over and above
the universal factor common to all subjects. These apparent
exceptions, therefore, are not inconsistent with the general
rule. Since, then, the chief deviations from the hierarchical
arrangement occur precisely where, on other grounds, we
should expect them to occur, we may accordingly conclude that
performances in all the subjects tested appear to be determined
in varying degrees by a single common factor.
“Nature of the Common Factor.
“What, then, is this common factor? The most obvious
suggestions are that it is either (1) General Educational
Ability or (2) General Intelligence. For both these qualities,
marks have been allotted by teachers, quite independently of
the results of the tests. The correlations of these marks with
performances in the tests are given in the last two lines of
Table XVIII.
“Upon certain assumptions, the correlation of each test with
the Hypothetical Common Factor can readily be deduced from the
coefficients originally observed. These estimates are given in
the last line but two of the table. They agree more closely
with the observed correlations for General Educational Ability,
especially if the latter are first corrected for unreliability.
(Correlations: Hypothetical General Factor coefficients and
General Educational Ability coefficients .86; after correction
.84. Hypothetical General Factor coefficients and General
Intelligence coefficients .84; after correction .77.) We may,
therefore, identify this hypothetical general factor with
General Educational Ability, and conclude provisionally that
this capacity more or less determines prowess in all school
subjects.
“The high agreement of the estimated coefficients with the
intelligence correlations suggest that General Intelligence is
an important, though not the only factor in General Educational
Ability. Other important factors are probably long-distance
memory, interest and industry. It is doubtless not a pure
intellectual capacity; and, though single, is not simple, but
complex.”[18]