In defining the borderline of feeble-mindedness it will be found that
certain assumptions are usually tacitly made as to the form of the
curves of normal and retarded development. These assumptions which are
often based on vague conceptions of mental measurements should be
brought clearly to mind if we are to compare the relative merits of
different scales of mental tests or different ways of stating the
borderlines of deficiency. With this in view it is proposed to take up
in this second part of the monograph a brief technical discussion of the
units of mental measurement, the equivalent individual differences at
different ages, and the curves of mental development. The bearing of
these conceptions on the various quantitative definitions of tested
deficiency, including the percentage definition, will then be discussed
in the following chapter. Practical advice as to individual diagnosis or
group comparisons has been confined to Part One, so that those who are
not concerned with the theoretical assumptions on which the conception
of mental development and the interpretations of tested deficiency are
based should omit Part Two.
[Illustration: FIG. 3. _Hypothetical Development Curves (Normal
Distribution)_]
When we try to picture to ourselves the significance of individual
differences and mental development we are at once forced to think in
terms of graphs showing the distribution of abilities at particular
periods of life and the changes from one life-age to another. To
simplify the discussion I have presented in Fig. 3 the graphic picture
of the conditions on the simplest hypothesis, namely, that mental
capacity at each age is distributed in the form of the normal
probability curve extending to zero ability and that individuals retain
their same relative capacity on the scale of objective units.
A. COMPARISON OF UNITS AND SCALES FOR MEASURING INDIVIDUAL DIFFERENCES.
(a) EQUIVALENT UNITS OF ABILITY WHEN THE DISTRIBUTIONS ARE NORMAL.
In considering the curves of development it is desirable first to notice
the differences between measurement in equal physical units and
measurement in equivalent units of ability or of development. The
difference in the point of view of the two forms of measurement is so
pronounced that I can hardly hope to make myself clear to those who are
not somewhat familiar with such terms as “distribution curves,”
“frequency surfaces,” “standard deviation,” and other phrases connected
with the theory of probability, which are treated at length in such
books as Thorndike's “Mental and Social Measurements” and Yule's
“Introduction to the Theory of Statistics.” We often, by mistake, regard
the growth of an inch in height, for example, as always representing an
equivalent unit of growth. This will lead us into rather serious
misconceptions unless we are careful, for it is perfectly evident that
the growth of an inch in height has a very different significance for
the three-year-old boy than for the eight-year-old. Half of the
three-year-old boys grow about 3 inches during a year while at eight
years of age not more than about one in seven grow that much. Moreover
it is not always satisfactory to regard the same _relative_ increase in
physical size as an equivalent unit of development. To say that a boy 20
inches tall who grows 1-10 in height shows an increase in development
equivalent to a boy of 50 inches who grows one-tenth, may be quite
misleading. Nearly every 20-inch child grows one-tenth in height in a
year while not one in fourteen of the boys who are 50 inches in height
may grow at that physical rate. In considering human traits, and
especially developmental traits, it would seem to conduce to more
significant thought if we gave up at times our habit of thinking in
terms of equal or relative physical units and thought instead in terms
of more equivalent biological units.
In the measurement of mental ability, moreover, it is exceedingly
difficult to utilize equal physical units. Most of the objective units
which are commonly called alike are clearly not equal even in the
physical sense. “Spelling one word,” for example, is not equal to
spelling another “one word;” but only equal to spelling the same word.
Out of such units of amount accomplished, it is, of course, not possible
to build a satisfactory scale without referring to some other concepts
of measurement. Some tests, however, are scored in equal units. When the
measurements for example, are in the units of time it takes to perform
the same task under the same outward conditions we have the possibility
of a scale of equal objective units. Such a scale is approached by the
results with the form board test which give the number of seconds it
takes children to place blocks of different shapes in their proper
openings.
Even the unit of time may be deceptive in name, as it is with the Binet
scale. A year of time is, of course, the same physical unit and the task
proposed with the Binet scale is always the same, but the other
essential with this scale, the children of each age who pass the tests
at each age norm, varies decidedly. “Test-age five,” for example, means
44% of the children pass and “test-age eleven” means 88% pass, even with
approximately random samples of children of these life-ages. This
question of the equality of the Binet age units will have to be
considered further, therefore, in connection with the other concept of
equivalence used in psychology.
In order to determine equivalent units of activity we find that a number
of different concepts have been utilized. With some of the scales for
measuring educational products, such as Thorndike's Scale for
Handwriting, equal units of merit in handwriting mean differences judged
equal by relatively the same proportion of competent judges. This form
of unit has not been used, however, in any scale of mental development
thus far proposed.
In the measurement of mental ability the most commonly accepted idea of
equivalent units is that they are provided by the units of standard
deviation for a series of measurements which distribute in the normal
form. The meaning of these units may be understood by referring to Fig.
3 which shows Gaussian or normal distributions of abilities of
individuals at various periods of life in curves A, B, C, D and E. The
straight lines of the measurement scales form the bases of these
distribution curves. These graphs represent the normal form of
distribution usually expected when any fundamental ability is measured
in a random group. If the number of cases at each unit of measurement
are plotted by a point placed relatively as far above the scale, used as
a base line, as the number of cases found at that unit of the scale, it
will be discovered that these points arrange themselves in the form of a
symmetrical curve high at the middle and flaring out along the base-line
scale. This bell-shaped curve, known as a normal probability curve,
shows that the largest number of cases occurs at the middle or average
measurement. From this middle point on the scale the number of cases
falls off gradually and symmetrically in both directions. Distances
along the base line of this distribution surface may then be measured in
terms of the standard deviation regarded as unity. This S. D. is the
best measure of the scatter of the deviations. It is the square root of
the average of the squares of the deviations of the separate
measurements from the average of all the measurements. There are
approximately four units of the standard deviation between the average
and either extreme when the distribution is normal, as in Fig. 3. Only
six cases in one hundred thousand fall outside these limits.
The studies of biological traits suggest that a unit of the standard
deviation is the most important measure we have for equivalent degrees
of any trait which distributes normally. It measures the same portion of
the total distance from the lowest to the highest ability on any
objective scale so long as the distribution of measurements is in the
_normal_ form. It thus affords the best interchangeable unit from
measurements at one life-age to those at another, provided that the
distributions keep close to the form of the normal probability curve.
This is the assumption on which practically all the developmental scales
have been based. The difference in ability between an individual at the
average and at -1 S. D. (standard deviation) below the average is
equivalent to that between the last individual and one at -2 S. D. The
same distances along the base line of different distribution surfaces
measured in terms of their respective deviations set off equivalent
portions at each age so long as the distributions are normal. For
example individuals measuring between -2 and -3 S. D. in any
distribution in Fig. 3 are equivalent in ability to those lying between
-2 and -3 S. D. in any other of these normal distribution surfaces.
Later we shall consider equivalent units when the form of the
distribution of ability is not normal or is unknown.
We may now compare the relations of the units in the physical scale,
shown at the left of the figure, to units of the scales for adults or
for the immature of any age, expressed in units of the standard
deviation from the averages of these groups. Relative ability measured
on the physical scale or any one of the distribution scales in Fig. 3
will be found identical since they all start from the same zero point
and the distributions are all normal. But the ability of an individual
in one distribution can hardly be compared with that of an individual in
another distribution in a biologically significant way by their actual
positions on the physical scale. A physical unit, does not measure the
same sort of fact of development in a scale for the immature that it
measures in the scale for adults or that it measures in another dynamic
scale for the immature. This can be seen when a physical unit is
compared with the amount of standard deviation which it measures in the
different scales. Moreover, the correspondence of relative distances on
the physical scale and any one of these other scales will not hold the
moment the distributions do not start from the same point or are
unsymmetrical.
It does not seem seriously wrong to suppose that there are some
individuals at any age who have no more mental ability than the baby of
the poorest mental ability at birth. At any rate our intelligence scales
are hardly fine enough to measure the difference in intellectual
capacity between the dullest adult idiots and the dullest idiot babies.
We shall, therefore, here assume that mental capacity extends to zero at
each age. The importance of this will be evident when we consider the
question whether the distributions of ability are symmetrical around the
average point at each age. Postponing for the present the discussion of
unsymmetrical or skewed distributions, we may consider the several
meanings of stages of development.
In applying the concept of the probability curve we should distinguish
between individuals who have attained their mature mental capacity and
those who are still maturing. The former would be represented by a
random group of adults (Distribution E, Fig. 3) the latter by a group of
nine-year-olds (Distribution C). If we say, for example, that a child
has reached a certain stage of development we might have in mind the
final distribution of mature capacity or the distribution of capacity
among those of his particular age or of all ages. When we compare stages
of development we must, therefore, be careful to indicate the
distribution surface to which we are referring.
An increase in development may refer to at least five different things
depending upon the scale of measurement to which reference is made.
Besides an increase measured by the physical scale, the scales for
adults, for the immature or for all ages, to which we have already
referred, it may mean an increase judged by the distribution of
increases which individuals of the same life-age and capacity make in
the same period of time. This last meaning may be the most significant,
although it has never been used. It has reference to a distribution
surface of _increases_ such as is represented in Distribution F, Fig. 3.
This is intended to show the increases in one year of all two-year-old
children who had average ability at 2 years, on the assumption that at 3
years these children would on the average equal the average of all
three-year-olds. It is clear that when these increases are measured in
objective units the latter have a still different significance from that
assigned to them in connection with other scales. An increase of one
objective unit here might represent twice the standard deviation, while
it only represents 0.2 of the standard deviation in another
distribution.
(b) THE YEAR UNIT OF THE BINET SCALE.
A sharp disagreement of opinion as to whether the Binet year units can
be regarded equivalent has arisen between Karl Pearson, Director of the
Galton Laboratory in London, and certain psychologists who have used the
Binet scale. Cyril Burt, for example, says, as quoted by Pearson:
“Except for rough and popular purposes, any measurement of mental
capacity in terms of age is unsatisfactory.... The unit fluctuates in
its significance all along the scale. When the child is just beginning
to walk and talk, when he is 7 or 8, when he is 10 to 11, when he is on
the verge of puberty—at these different periods a retardation of a
single year means very different things” (_164_, p. 36).
A number of good psychologists including Yerkes, Terman, and Kuhlmann,
agree with Burt in maintaining that a year of retardation at different
ages has very different significance.
With this statement of Burt, Pearson takes issue, saying:
“Can the psychologist to the London County Council ever have seen the
growth curves of children, or would he write thus?... There is no valid
reason to suppose that a year's growth in mental power may not be taken
for all practical purposes to mean the same unit for ages of 6 to 15,
the period for which Binet and Jaederholm have used the tests” (_164_,
p. 44).
Like many other apparently opposite statements both contain truth. The
conflict arises apparently, first from a disagreement between the data
obtained with the Jaederholm form of the scale, on which Pearson bases
his statement, and data obtained with other forms of the scale; second,
from a discrepancy in the points of view. Pearson stresses the fact that
the mental year-marks equal average growth increment with the Jaederholm
scale (_167_). He shows that the regression of years of mental excess
(or deficiency) on increase of life-age is a straight line, just as he
found it with physical measurements. Moreover, the standard deviation of
the mental measurements for the entire group of normal school children,
6-14 years of age, was found to be about one year of mental age (.96
year for the corrected data) (_167_). To which Pearson's opponents might
reply, these facts are of comparatively little significance unless the
_deviations for the separate ages_ are alike in terms of these year
units on the scale. Neither linear regression nor the balancing of years
of excess by years of deficiency at each age indicates that the
deviations of the separate ages are alike in terms of the year units.
The new Stanford scale, for example, shows both of these conditions and
yet the range of months of life-ages which sets off the middle 50% of
the children of the different tested ages increased decidedly from 6 to
14 years of age. The middle half of the tested ages, for example, at age
VI on the scale include a randomly selected group of six-year-old
children whose range of life-age is ten months, at age VIII on the scale
this range is 13.4 months, at X it is 16 months, at XII, 20 months, and
at XIV, 26 months. “The number of 6-year-old children testing 'at age'
is approximately twice as great as the number of 12-year-olds testing at
age, and 50% greater than in the case of the 9-year-olds” (_196_, p.
557).
To this argument Pearson might reply that he had not overlooked the
question of variation in the deviations from one age to the next for he
has a footnote in which he states regarding the Jaederholm data: “There
are, however, relatively little differences in these mental age
standard-deviations of the normal children beyond what we may attribute
to the effect of random sampling” (_164_, p. 46). In this respect, then,
the Jaederholm data differ notably from Terman's data obtained with
random groups with the Stanford scale and, as I shall show, from data
obtained by Goddard with the 1908 Binet scale, the two largest groups of
Binet test data which have been collected. Even with the Jaederholm data
on efficient school children, although the largest difference between
the standard deviations of different age groups is only about twice its
probable error, it is notable that 24 of his 39 7-year-olds are included
within an interval of the middle year of tested age, while only 9 of his
35 11-year-olds are included within the same middle year interval.
Taking Goddard's data for the 1908 scale for the separate ages from 5-11
at which probably the factor of selection for his groups may be
neglected, I have calculated the standard deviations from his Table I
and find them as follows:
────────────────────────────────────┬──────────────────────────────────
│ Life-Ages
────────────────────────────────────┼────┬────┬────┬────┬────┬────┬────
│ 5 │ 6 │ 7 │ 8 │ 9 │ 10 │ 11
────────────────────────────────────┼────┼────┼────┼────┼────┼────┼────
Standard deviations in Mental Excess│1.10│.98 │.93 │.99 │1.04│1.23│1.19
or Deficiency │ │ │ │ │ │ │
────────────────────────────────────┴────┴────┴────┴────┴────┴────┴────
The differences between the deviations for ages 7 and 11 or between ages
8 and 10, are more than three times their standard errors, so that we
would not be justified in assuming that the standard deviations of the
separate ages measured in terms of years of excess are equivalent. There
seems to be a tendency for the deviations to increase, at least from age
7 to 10 and 11.
The comparison of the year units on the Binet scale with the diagrams in
Fig. 3 shows that if the scale at each life-age shut out the same lowest
proportion, say half, of the children of that age, then the year units
might be regarded as equal in the sense of equal average growth
increments, as Pearson suggests. A child 7 years of age testing VII
would be at least one annual average-growth unit higher in mental
development than one of 6 years testing VI, and so with each age until
the limit of development had been reached. This is the condition
approximated closely for children by the new Stanford scale and the
corrected Jaederholm data. Since there is little prospect, however, even
with a scale perfected so far as its age norms are concerned, that the
total distributions for each of the different years would be the same
multiple of the year-units, the main significance of the age units is in
permitting the statement that a child had reached the tested development
normal for the children of a certain age.
It is also legitimate to use years of retardation as a short way of
expressing rough borderlines when they happen thus to afford an easy
method of empirically describing equivalent borderlines for a particular
scale. This is what I have done for convenience in Part One of this
book. I certainly do not mean to contend that four-years retardation has
theoretically the same significance at different ages, in terms of the
deviation of the separate ages. To me the Binet years are no more than
names for certain positions on the scale.
To most psychologists who have been dealing with the measurement of
mental development, I believe that the most significant concept of
equivalent units would be in terms of the deviations for each age
provided that the form of the distributions remained normal. But the
deviations vary so much in the terms of the year units that it is not
likely that they will be willing to accept a _year of excess or
deficiency_ as an equivalent unit for different ages with the common
forms of the scale in use in English-speaking countries. Moreover, below
the age of 6 and above 15, the limits which Pearson discusses, there is
good reason to expect the year unit to vary still further. This Pearson
recognizes for the complete developmental curve. It is only at the
intermediate years, in which the average increases are most constant in
relation to the deviations of the separate ages, that the year unit may
be at all serviceable in measuring the deviation of a child from the
norm of his age.
With the scales in use in this country the Binet year units are not
equivalent in the sense in which they are usually spoken of as
equivalent. We should recognize this and emphasize it. Even if the norms
at each age marked off the same proportion of the individuals, as shown
in A and B of Fig. 4, unless we knew that the forms of distribution were
always alike, we should not know that the distance between successive
age norms was the same on any sort of objective scale other than average
age increments. Moreover, we would not have an objective scale of equal
units applicable to measuring the deviation of children of any one age.
The average annual increments would not necessarily represent the same
proportion of the total distance from the lowest to the highest ability
at different ages even if the distributions were all normal. With normal
distributions it would also be necessary to demonstrate empirically that
the annual average growth increment between successive ages always bore
a constant relation to the deviations at these adjacent ages as shown in
B of Fig. 4 where the increment is equal to 1 S. D. at each age. This
could not possibly hold when the increment lessened near maturity.
[Illustration: FIG. 4. _The Question of Equivalence of Year Units._]
If the distributions of ability were variously skewed, the year units of
excess or deficiency would not be shown to be equivalent at the
different ages even if the proportion of individuals one year
accelerated was equal to the number one year retarded, two years
accelerated equal to those two years retarded, etc., at each age and the
norm at each age shut out the same proportions of the age group. This is
shown in C of Fig. 4 in which the year units are clearly not equal steps
from lowest to highest ability even for the same age and yet the usual
criteria which have been suggested for discovering the equivalence of
the units are fulfilled. Whether the actual distribution of ability is
skewed or normal cannot be determined by the Binet scale, of course, on
account of the uncertain and probably varying size of its year units in
measuring deviations at any age.
With the empirical evidence against the equivalence of the year units
and the impossibility of determining their equivalence unless we first
know that ability is distributed normally at each age, it is certainly
hazardous to assume that individual deviations measured in terms of year
units are equivalent at different ages.
It may be noted that it is quite as hazardous to suppose that the units
of the Point scale are equivalent in any theoretical or practical sense.
This question will be discussed later in Chap. XIII, B, (b).
(c) IS TESTED CAPACITY DISTRIBUTED NORMALLY?
Before leaving the question of the significance of units on a scale
described in terms of the standard deviation we should ask whether
tested mental abilities have been found to distribute normally, _i. e._,
in the form of the symmetrical Gaussian curve with each extreme the same
distance from the middle measurement. Contrary to the usual supposition
in this matter, it seems as if the evidence was somewhat against this
assumption, although neither position can be asserted at all
dogmatically on the basis of our present data. A résumé of this evidence
which I have given below makes it appear that the assumption of a normal
distribution will not conflict with a practical use of normal
probability tables for medium degrees of ability, but may quite
seriously interfere with such use for the borderline of deficiency.
There is little doubt, as Pearson believes, that the bulk of the
children now in special classes for the retarded in the public schools
would fall within the lower range of a normal distribution fitted to the
general population. On the other hand, there is likely to be a
respectable minority of the deficients which will be beyond such a
normal curve. These facts are sufficiently evident, I believe, to make
it impossible to base quantitative descriptions of borderline of
deficiency on a hypothesis of normal distribution.
The best evidence on this point is probably the data of Norsworthy with
eleven tests on groups of 100 to 150 feeble-minded children in
institutions and special classes and 250 to 900 normal children. She
expressed the position of each child in terms of the deviation of the
group of normal children of his age for each test. Pearson has presented
her data graphically on the assumption that her defective group
represented 0.3% of a general population of 50,000 children, and then
fitted a normal distribution curve to her data with her normal group.
The result makes it evident, especially for the intelligence tests, that
the defective group would better be described as part of a skewed
distribution. To less extent this is also true for the maturity and
memory tests (_15_, p. 30). Norsworthy's own table of data show that 43
of the 74 feeble-minded taking the intelligence tests were over -5 times
the probable error of their ages below the averages of the normal
children, a criterion which she proposes as indicating ability outside
of that included in the normal species. Moreover, 9 children score
between -22 P. E. and -32 P. E. which is far beyond any conceivable
extension of the normal curve. Her figure for the composite results of
all her mental tests is also manifestly skewed toward deficiency
although she hesitates to adopt this conclusion, and was content with
showing that they grade off into the distribution of normal children.
The other data, which I have found, that indicate that tested ability,
when measured in equal physical units for the same task, is skewed
toward deficiency, have to do with tests that are pre-eminently for
psychomotor activities rather than intellectual. They consist of
Sylvester's and Young's results with the form board test on Philadelphia
school children, Stenquist's results with his construction test, and
Smedley's results with the ergograph test on Chicago school children.
Here we may apply the better criterion of the distance of the quartiles
above and below the median of the group. These positions would be less
likely, through extreme records, to be affected by chance conditions
during the testing.
It is to be remembered that if the records of school pupils appear to be
normally distributed this would not settle our problem, since it is
apparent that idiots and many imbeciles are not sent to the public
schools at all. The lowest children at any age would not be represented
in the regular school groups. On the other hand, the brightest children
are not generally drawn away from the public schools at least before 14
years of age in this country. We shall confine ourselves, therefore, to
school-children 6-13 years of age. If we find that they show ability
skewed toward deficiency the results will underestimate rather than
over-estimate the skewness.
Sylvester (_191_) tested with the form board a group of 1537 children in
the Philadelphia public schools, from 80 to 221 at each age from 5 to 14
inclusive. “Except that no especially backward or peculiar children were
included there was no selection.” This study gives, with the complete
distribution tables, the number of seconds required for the same task by
the children at each age. If we find that the limit of the lower 25
percentile was farther from the median than the limit of the upper 25
percentile we can be reasonably sure that the difference would be still
greater if the excluded deficient and backward children were also
included. By calculating the quartiles and their differences from the
medians at each age, I find that for only two of the eight ages is the
upper quartile farther from the median than the lower quartile. The
average excess of the distances of the lower quartile is .64 of a
second. At only age 7 is the difference three times its probable error,
2.1 seconds, P. E. .67. The form board distributions thus tend to be
slightly skewed toward deficiency. The errors of the quartiles were
found by the method given in Yule's _Introduction to the Theory of
Statistics_, Chap. XVII, which assumes normal distribution, so that they
are too small. The skewness is more manifest when the extreme
measurements are compared with medians at each age. It is not possible,
unfortunately, to compare his group of normal children with those in the
special classes since he did not use the same method of giving the test.
Since it was not important to compare the amounts of skewness in
different data, I have not attempted the more elaborate calculations of
coefficients of skewness. These would give the results a more elegant
statistical expression. The simpler method I have here used affords more
convincing evidence of asymmetry for the non-mathematical reader.
Young has published the results with Witmer's form board test on
approximately two hundred Philadelphia children for each age, giving the
results for the sexes separately for each half year of life-age (_227_).
This affords 36 different groups in which he gives the median and upper
and lower quintiles for the shortest time records. The lowest quintile
is farther from the median in 25 cases, equal in 6 and less than the
upper quintile in only 6 of the 36 comparisons. This skewness would have
been even greater if children of the special classes had not been
excluded from his groups.
Stenquist's results (_54_) with his construction test are scored in
arbitrary units in which allowance is made for the quality of the score,
but we should expect no constant effect on the form of the distribution
from the character of these units of measurement. At ages 6 to 13 he
tested from 27 to 74 pupils randomly selected from the public schools, a
total of over 400. For six of these eight ages the lower quartile is
farther from the median than the upper quartile, when calculated from
his distribution table. The number of cases at each age, however, is so
small that the largest difference, 15 units, is not three times its
probable error, 6.
Smedley gave his ergograph test to about 700 school children of each of
the ages we are considering. Since he tested so many more subjects than
any other investigator this should provide the most valuable data on the
question of distribution with a test recorded in the same physical units
for the same task. Unfortunately, his results for two succeeding years
are so directly contradictory to each other that they seem to have no
significance for our problem. The simplest explanation of this
contradiction is that the groups tested may have been selected on a
different basis each year.
A casual observation of his standard percentile curves for the
ergograph test at the different ages gives the impression that the
distributions are decidedly skewed toward deficiency, but this
impression is not justified by a careful analysis of his results
(_51_). In the table which accompanies his standard percentile
curves, giving his total results for the two years, we find that
there is a sharp disagreement between the distributions of the
boys and the girls. The distributions for the boys at each age
between 6 and 13 years show a greater distance, measured in
kilogram-centimeters, from the median to the 80-percentile than
from the median to the 20-percentile, in 5 ages out of 8. The
total difference is also slightly greater between the median and
the upper 80-percentile. On the other hand, the table for the
girls at these ages shows the 20-percentile farther from the
median in 5 out of 8 ages, with a total difference considerably
greater than that shown for the boys. Usually the differences were
small compared with their errors. With the boys only at age 13 was
the difference in favor of the 80-percentile three times its
probable error, while with the girls the four oldest ages show the
distance of the 20-percentile greater by three times its probable
error.
A comparison with the reports of Smedley on this test for the
previous year (Report No. 2), leaves his results still more
uncertain. While he does not give the medians at each age, we may
make less satisfactory comparisons between the distance of the
10-percentile from the 25-percentile and the distance of the
90-percentile from the 75-percentile. If we do this, we find the
distance is uniformly greater at the upper end of the distributions
for each age both for the boys and girls. The Smedley results are,
therefore, decidedly contradictory. The first year shows
distributions skewed toward excellence and total results for two
years show distributions skewed mainly toward deficiency.
Broadly considered, the Binet records with school children point to a
skewed distribution toward deficiency when large allowance is made for
the difference in value of the year units. It is extremely rare to find
a child testing 4 years in advance of his life-age, while 15-year-old
idiots are presumed to test 12-year-units or more under a mature
standard.
Pearson believes that “the Gaussian curve will be found to describe
effectively the distribution of mental excess and defect” for
intermediate ages as measured by Jaederholm's form of the Binet scale.
The data on which Pearson places reliance are Jaederholm's results in
testing 261 normal children 6-14 years of age in the Stockholm schools
and 301 backward children in the special help classes of the same city.
The best fit of a normal curve to the data was obtained with a group of
100 8-year-old children, in which case the chances were even that
samples from a normal distribution would fit. With his larger normal and
backward groups combined in proper proportions in one population the
chances were 20 to 1 that such a distribution as was actually found
would not fit into the Gaussian distribution. He admits that “this is
not a very good result,” although it is better than when the Gaussian
curve is fitted to either the normal or the backward group alone. In a
subsequent paper he gives each child a score relative to the standard
deviation of the normal child _of his own age_, a method comparable to
his treatment of Norsworthy's data. He then finds that “10% to 20% or
those from 4 to 4.5 years and beyond of mental defect could not be
matched at all from 27,000 children” (_164_, p. 46). In each case the
distributions actually found were skewed somewhat toward deficiency.
Furthermore, when he suggests that -4 S. D. may be used as a borderline
for tested deficiency, he recognized that the mental ability of children
is skewed so far as the empirical data are concerned. With a normal
distribution there would not be two children in 100,000 who would fall
below this borderline. Nevertheless, the normal curve serves for most
practical purposes to describe the middle ranges of ability.
Pearson thinks that the skewed distributions of his data may possibly be
explained by the drawing off of older children of better ability to the
“Vorgymnasium,” or to the higher-grade schools, by the incompleteness of
the higher age testing, or by the “possibility of the existence of a
really anomalous group of mental defectives, who, while continuously
graded _inter se_, and continuously graded with the normal population as
far as intelligence tests indicate, are really heterogeneous in origin,
and differentiated from the remainder of the mentally defective
population” (_164_, p. 34). The last hypothesis, of course, supposes
that mental ability is skewed and suggests the cause. He supplements
this explanation by stating that the heterogeneous cause of the “social
inefficiency” of the deficients may not be connected directly with the
intellect but affect rather the conative side of the mind. A skewed
distribution under biological principles of interpretation supposes a
single cause or group of causes especially affecting a portion of the
population.
It is also to be noted that the apparent form of distribution may be the
result of the nature of the test and the units in which it is scored.
Some tests might not discriminate equally well a difference in ability
at the lower and at the upper ranges of ability. If the test were too
easy the group might bunch at the upper portion of the scale and the
distribution appear to be skewed toward the lower extreme where there
were only a few cases. If too difficult a test were used the form of
distribution might shift in the opposite direction, most of the group
ranking low. It is extremely difficult to formulate mental tests so that
they will equally well measure differences at each degree of ability.
This objection should not hold, however, if the scoring were in units of
time for the same task, as with the form board test. The essential
characteristics of a test in order that it may indicate the form of a
distribution is that the units of scoring shall be objectively equal
under some reasonable interpretation and that they shall be fine enough
to discriminate ability at each position on the scale. Under such
conditions the variations in the difficulty of tests should not obscure
the form of the distribution of the ability tested.
Turning to the analogy of measurements of physical growth, a strong
argument may be made for the hypothesis of shifting forms of
distribution. As Boas points out regarding measurements of the body at
adolescence, owing to the rapid increase of the rate of growth the
distribution of the amounts of growth is asymmetrical, “the asymmetry of
annual growth makes also all series of measurements of statures,
weights, etc., asymmetrical.” Moreover, “acceleration and retardation of
growth affects all the parts of the body at the same time, although not
all to the same extent.... Rapid physical and rapid mental growth go
hand in hand” (_80_). There is no reason to suppose that the brain is
free from this phenomenon of asymmetrical distribution of annual
increments of growth among children of the same age when the rate of
growth is changing as at adolescence. It is therefore to be expected
that the separate age distributions would be skewed at early ages and at
adolescence even if the distribution should be normal with a static
population. The presumption from physical measurements is that the form
of distribution shifts with age.
Again we may note that if some of the idiots reach an arrest of
development before any of the normal individuals, as several
investigators contend, this would imply that the distributions must be
skewed unless there is a curious corresponding acceleration of growth on
the part of geniuses to balance this lagging by idiots.
In spite of these arguments and the evidence of asymmetry of
measurements at least at some periods of life it is to be noted that
current opinion is probably contrary to this hypothesis, although, as I
believe, because it has been concerned mainly with those who are not of
extreme ability. For all large medium ranges of ability slight skewness
might well be negligible. It is interesting to note that Galton says
that “eminently gifted men are raised as much above mediocrity as idiots
are depressed below it” (_159_, p. 19). Measured by intelligence
quotients with the Stanford scale, Terman finds among school children
that deviations below normal are not more common than those above
(_197_, p. 555). Burt, following a suggestion of Cattell as to college
men, however, seems to incline to the opinion that the general
distribution of ability, like wages, is skewed toward the upper end. He
adds, “In crude language, dullards outnumber geniuses, just as paupers
outnumber millionaires” (_85_).
(d) EQUIVALENT UNITS OF DEVELOPMENT WHEN THE FORM OF DISTRIBUTION IS
UNCERTAIN.
For our problem of units and scales of measurement, an asymmetrical
distribution sets a very difficult problem. It may be that this very
difficulty has been one of the main reasons for slowness in recognizing
the drift of the evidence. In order to set forth the difference in the
conception of measurement when distributions become asymmetrical I have
presented this hypothesis in connection with the curves of development
in Fig. 5. It will be noted that if the distributions of mental capacity
vary in symmetry, the units of standard deviation change in significance
from one form of distribution to another. Minus 2 S. D. may exclude very
different portions of groups differently distributed, while it would
always exclude the same proportion if the distributions had the same
symmetry, or skewness.
Under conditions of variable symmetry there is a sense in which the same
relative physical score in units running from zero ability to the best
ability would always have an equivalent objective meaning, but this
might not express equivalent development conditions at different ages.
For example, with shifting forms of distribution, to say that a child of
six years had reached three-fifths of the best development for his age
on an objective scale might give no significant indication of how nearly
he was keeping pace with those three-fifths of the best ability of
another age. Neither would his position in units of the deviation of
ability at his age give this information without knowledge of the form
of the distribution of ability at his age. With varying forms of
distribution at different stages of development this would afford an
insurmountable difficulty.
[Illustration: FIG. 5. _Hypothetical Development Curves (Changing Forms
of Distribution)_]
With unknown or varying types of distribution it is desirable to utilize
percentiles as equivalent units for comparing individuals at different
stages of development. They differ somewhat from ranks in an order of
noticeable differences. With an indefinitely large group, such ranks
would mark off only those cases which were indistinguishable in merit.
These units would be numbered in order from the highest to the lowest in
ranks of just distinguishable merit, a different number of individuals
conceivably occurring at the single steps. Psychologically the
percentiles are somewhat less significant because they are not
conceivable in steps of just noticeable differences. Percentiles have
less value in _comparing abilities in the same distribution_, but have
decided advantages when _comparing corresponding abilities in different
distributions_. Except at points where merit is indistinguishable, they
signify that a certain proportion of a group is ahead in the struggle
for existence. They are thus units of relative rank. Moreover, they are
directly translatable into units of the deviation in case the form of
the distribution of ability has been determined. This is a special
advantage if the forms of distribution turn out to be normal or even
uniform.
In using percentiles it is to be remembered that equal differences
between percentiles _are not comparable in the same distribution_ except
in the sense of the same extra proportions of the group to be met in
competition. A change in the degree of ability from the lowest
percentile to the lowest 2 percentile would be very different from the
change in the degree represented by the 50 percentile to the next
percentile above. Differences in the ability of individuals ranking near
each other in the middle of the same percentile series would be
distinguished with difficulty while it would be easy to make such
discriminations at the extremes.
The special value of the percentile units in measurement of ability lies
in the comparison of individuals of corresponding position in
corresponding groups in which the ability may not be assumed to
distribute alike. The concept that 995 out of every 1000 randomly
selected individuals at his age are ahead of a particular individual in
the struggle for existence has very definite and significant meaning
which is quite comparable from one period of life to another regardless
of the form of the distribution. We shall return to this question of
equivalent units in distributions of unlike symmetry when we compare the
definitions of the borderlines of deficiency in terms of intelligence
quotient, coefficient of intelligence, standard deviation and
percentage. Corresponding percentages of corresponding groups have a
more useful definite significance of equivalence than any other units of
measurement of mental ability available when the forms of distribution
vary at different stages of development or are uncertain, as seems to be
true with tested abilities.
B. THE CURVES OF MENTAL DEVELOPMENT.
When we endeavor to make our ideas of mental development more definite,
we are assisted by thinking of the various stages in graphic form. This
is especially true when trying to think of the position of the deficient
individuals, relative to the average individuals and to genius.
In diagrammatically presenting these concepts in Fig. 3 and Fig. 5 we do
not wish to assume that all the principles on which the developmental
curves have been plotted have been decided. If they make clearer the
points still under discussion and direct the discussion to specific
features so that more data may be brought to bear upon the empirical
determination of their characteristics, they will serve a useful
purpose. For our present ends, we shall consider only certain features
which have a bearing upon the interpretation of developmental scales and
the quantitative definition of the borderline.
In the graphic presentation of the curves of development in Figures 3
and 5 the relative position at various ages has been suggested
hypothetically for those of the best ability and median, or middle
ability, as well as the borderline of the deficients.
It is evident that these graphs should represent equivalent ability at
each stage of development measured by as objective a scale of
measurement as possible. In the graphs this scale is assumed to be
composed of physical units with its zero at zero ability. The deficient
group is distinguished by the portion with a grated shading. The
distribution curves of individual ability we have already mentioned in
connection with scales of measurement. Fig. 3 is constructed on the
assumption of a normal distribution of ability at each age extending to
the same zero ability. Fig. 5 on the assumption of distributions of
varying form.
Otis has given a very able logical analysis of certain concepts
underlying the testing of mental development (_163_). His discussion
differs from the present in its aim to determine the proper mental age
for particular tests, a question which I have not considered. It also
supplements the present discussion by showing the changing value of the
same intelligence quotient with normal distributions of ability under
certain assumptions as to range of ability and decrease in the annual
increments of ability with age.
(a) THE SIGNIFICANCE OF AVERAGE CURVES OF DEVELOPMENT.
Some investigators are apparently inclined to question the significance
of any curve of mental development on account of the very different
forms of development which they have found in particular cases. A
quotation from Goddard will state this problem:
“It seems to me that there is considerable evidence that there are a
good many children that develop at a normal rate up to a certain age
and then slow down; some slowing down gradually and others rapidly.
This is possibly accounted for by accidental conditions. Dr. Healy's
case of traumatic feeble-mindedness is a good illustration of this.
We have quite a good many cases, not a large percentage as yet,
where it is pretty clear that they have developed very nearly
normally up to the age of seven, eight or nine, so that I am very
skeptical as to the possibility of formulating a rule for
determining the rate of development. Many cases are uniform in
slowness while others vary a great deal; some slow up more rapidly
than others as has already been stated....
“Morons are not usually discovered until twelve or fourteen years of
age. The picture to me of the development of the feeble-minded is
rather that these different types develop each in his own way very
much as the physical side develops. Different families have
different determiners of development. Just as it was determined
before I was born that I should be five feet, ten inches tall, I
developed that height and no further. In the same way, probably,
that determiner carries with it the determination of the rate of
development and the time. This carries with it the fact that I
should have been an average boy from birth. As a matter of fact I
was very much under-size until I was fifteen or sixteen years of
age. Then I shot up. Other cases are over-size. It may be a false
analogy, but it seems to me to illustrate the rate at which these
cases develop” (_111_).
This view raises clearly the question how far the curve of average
development represents a common tendency of different individuals in
development. Are the individual curves of development so varied in form
that an average curve does nothing but obscure their significance? The
study of individual curves of growth in height and weight by Baldwin
indicates that the bigger children tend to develop earlier, the smaller
later (_73_). The individual curves of mental development may be
analogous. If so, the average curves may not adequately represent the
common tendencies of development. Nevertheless, it is to be remembered
that with height and weight the average curves do retain a decided
usefulness, which nobody, I suppose, would seriously question.
An analogous problem arises when we consider the question of variations
in the maturity of different mental processes. Besides the question
whether the average curve is useful in view of the variation among
individuals in their rates of maturity for the same process, the
psychologists have a still more difficult problem about curves of
general ability. These curves are built by combining the results of
numerous psycho-physical tests which are very different in type. We need
to raise the question whether the type of process measured by memory for
digits, for example, matures at the same rate as those processes
measured by other memory tests: in general, how much a single test or
combination of tests represents a common process. Furthermore, we need
to inquire whether processes measured by memory tests mature like those
measured by tests emphasizing reasoning, imagination, motor ability and
other groups of activities. We thus have the problems of the different
rates of maturity of the different tested processes in the same
individual and of common tendencies among these specific processes.
In order more clearly to present this problem of the significance of
developmental curves for different processes, I have brought together
the age norms from 8 to 14 years for 40 tests as given by different
investigators. No norms were included which were not based on tests of
at least 25 individuals. After 14 years the data which have been
collected are open to the objection that the norms for the older ages
would be seriously affected by the fact that they were obtained upon
children remaining in school, usually in the elementary school, _i. e._,
upon groups, among which a large portion of those of better or of poorer
ability had been eliminated. The relative position of the norms for
older ages are, therefore, not comparable with those of children who are
of the ages of compulsory attendance. The results published are
inadequate below 8 years for most of the tests, so I have not extended
the curves to earlier ages. In 14 instances the data for boys and girls
were only given separately. In these I have used the norms for the boys.
A prepubertal break in a combined curve may, therefore, indicate a sex
difference. In most cases the norms were given for the sexes combined,
and the difference is unimportant for the points considered.
The variation in age norms with different tests is shown graphically in
Figures 6, 7 and 8. In order that the various tests may be plotted on
the same scale, so as to compare changes in development for the
different tested processes, I have used the average increase in ability
from 8 to 9 years of age for each test as a common measure and
arbitrarily plotted the slant of the curve between these ages at 45
degrees. The increase from 8 to 9 is represented by 10 units on the
objective scale to the left of the graphs. On this basis it is possible
roughly to compare changes in the absolute annual increase at different
ages for the same test and for different tests. It assumes that the
units in which each test is scored are equivalent for that test. An
average difference between the basal ages or between any two ages cannot
be assumed to be accompanied by the same distribution of increases.
Moreover, the 8-year norm is at different distances from zero for the
different tests so that the relative increase from 8 to 9 cannot be
regarded alike for the different tests. The method, however, is
sufficiently accurate for illustrating the very different forms of the
developmental curves which might be expected if they were measured by
absolute increases from year to year. Even the variation in the slant of
the lines at the different ages gives a graphic picture which will
assist in interpreting the significance of average curves of general
ability. As the curves stand, they show the norms for each age for any
test, as if placed on its own objective scale, and the various objective
scales have been harmonized on the assumption that the norms at 8 and 9
years are accurate. We thus have a simple representation of the absolute
changes in the abilities tested from age to age by the same tests
relative to a single objective scale. It will not give a seriously
erroneous picture for any tested ability so long as the units in which
the particular test is scored may be presumed to be objectively equal.
The tests on which Figures 6, 7, and 8 were based included
practically all which were reported in the researches used. They
were as follows: Norsworthy (_159_), perception of 100-gram
weight, cancelling A's (boys), ideas remembered from four simple
sentences, memory of related and of unrelated words, part-wholes,
genus-species, opposites and reverse of opposites given the next
day, “a-t” test. J. Allen Gilbert (_108_), taps in 5 seconds,
fatigue in tapping, visual reaction time, color-discrimination
reaction time, reproduction of 2-second interval. Smedley (_51_,
No. 3), strength of right-hand grip (boys), taps in 30 seconds
(boys), ergograph; visual, auditory, audio-visual, and
audio-visual-articulatory memory for digits. W. H. Pyle, Standards
of Mental Efficiency (J. of Educ. Psychol., 1913, IV., 61-70),
uncontrolled association, opposites, part-wholes, genus-species,
digit-symbol and symbol-digit substitution, memory for concrete
and for abstract words, memory of Marble Statue selection, (only
boys' norms used for each). Pyle and Anderson combined by Whipple
(_220_) two word-building tests (boys). Anderson as given by
Whipple memory for letter squares. D. F. Carpenter, Mental Age
Tests (J. of Educ. Psychol., 1913, IV., 538-544), substitution of
colors in forms and of numbers in forms, perception time in
marking A's, concentration, _i. e._, difference in time of last
test under distraction, memory of pictures of objects, all tests
devised by Carrie R. Squire. Stenquist (_54_), construction test.
Sylvester (_191_), form-board test.
[Illustration: FIG. 6. _Tests of the Development of Memory Processes.
Medians at Each Age of the Central Tendencies of the Tests._]
[Illustration: FIG. 7. _Different Types of Development. Medians at
Each Age of the Central Tendencies of the Tests._]
[Illustration: FIG. 8. _Forty Curves of Development. Distribution at
Each Age of the Central Tendencies of the Tests._]
In Fig. 6 curves A and B are Smedley's tests; curve C includes in
addition Norsworthy's unrelated words, Pyle's memory for concrete
and abstract terms, Anderson's letter-squares, Carpenter's memory
for pictures, and Gilbert's for the time interval; curve E includes
Pyle's two and Carpenter's two substitution tests; curve F includes
Pyle's Marble Statue and Norsworthy's memory for related words and
for sentences; curve S is Norsworthy's; curve D is the combination
of these 17 tests.
In Fig. 7 curve H includes Gilbert's visual reaction time,
Norsworthy's A and a-t tests, Carpenter's two A tests; curve I
includes Gilbert's and Smedley's tapping tests; curve J is the
median of the central tendencies of all 40 tests; curve K includes
Norsworthy's two opposites and her part-whole and genus-species
tests, the Pyle opposites, genus-species and part-whole tests; curve
L is the same as D, curve M includes Smedley's strength of grip and
ergograph tests and Gilbert's fatigue of tapping; curve N includes
Pyle and Anderson's word building tests and Pyle's uncontrolled word
association test.
In Fig. 8 curve P is Gilbert's visual reaction time test, curve S is
Norsworthy's test for memory of unrelated words, the other curves
are the median and quartiles for the central tendencies of all 40
tests after each was expressed at each age in terms of the gain from
8 to 9 years taken as a unit.
Several points are to be noted about the nature of the curves for
different tests. In Fig. 6 showing the curves for different forms of
memory tests, that for the memory of digits is very different in
character from that for memory of related material. The most extreme
differences in the time of maturity are shown by the test for memory for
digits presented orally and the substitution of color in forms, the
former continues to increase so rapidly relative to the absolute
increase from 8 to 9 years that it cannot be represented in the graph
reaching 539 units of the scale by 14 years of age, while improvement in
ability in the latter is not measured after 9 years. We cannot take time
to discuss how much of the differences between the various curves may be
due to the nature of the tests themselves, the form of scoring the
results, or the condition under which they were given, selection of
subjects, etc. The conclusion is safe, however, that when groups of
three or four tests of similar type show such marked differences as
those for memory of digits and memory for related material we may expect
similar differences in the rates of maturity of the corresponding
processes.
From Fig. 7 we may learn that tests emphasizing functions such as speed
of motor or perceptual motor reaction, curves H and I, are notably
different in their form from curves for tests of imaginative processes,
curve N. As we group tests together covering larger ranges of activity
we approach the median curve for general ability. Note the median curve
for 17 memory tests (curve L) compared with the median for the 40 tests
(curve J). By empirical studies we might pick out types of tests which
would most closely represent the maturity of average ability. For
example, the median for the substitution tests, curve E, resembles the
median for the memory tests, curve D, more closely than does that of the
4 digit tests, curve B. Curve K, for 7 association tests, resembles the
median for the 40 tests, curve J, much more closely than the curve for
the perceptual-motor speed tests, curve H. This difference can not be
explained by the use of 7 instead of 5 tests in calculating the central
tendency of the group. It probably means that the sort of
psycho-physical processes usually tested more closely represent on the
average the abilities shown in association tests than they do the
abilities shown by speed of motor reaction. The significance of this
sort of analysis for those constructing a scale for measuring
intellectual ability is obvious.
Fig. 8 shows the median and quartile range for the central tendencies of
the 40 tests and gives examples of two extremely different tests, visual
reaction time and memory for unrelated words. How closely these
particular tests represent fundamental differences in the maturity of
different processes, we cannot, of course, be sure without prolonged
research; but nobody would question that analogous differences would be
found in different processes. When we think of curves of general ability
we must, therefore, keep in mind the light which might be thrown on them
by an analysis of the various processes tested in the particular scale
used.
Another feature of all developmental curves which is apparent as soon as
the causes of development are considered, is that growth in an
individual is the result of several factors. These include the native
capacity, the rate at which that capacity manifests itself
instinctively, and the external stimuli which encourage or retard that
manifestation. To some extent these factors vary independently. Our
curves of development will never completely express all the facts until
they analyse out all these factors for each of the processes. In the
meantime we shall be able to think of general trends of development by
considering average curves. The fact that they represent combinations of
unanalyzed factors must, however, make us very cautious in interpreting
our norms.
(b) CHANGES IN THE RATE OF DEVELOPMENT.
There has been considerable discussion of the form of the curves of
mental development. The logical aspects of the curves on the assumption
of normal distribution of ability at each age and uniform age of
maturity have been treated by Otis (_163_) and the bearing of these
assumptions upon the Binet scale pointed out. Thorndike has plotted the
developmental curves for a dozen tests on the basis of the variability
at 12 years of age used as unit and gives a chapter in his Educational
Psychology to the changes with maturity (_198_, Chap. XI). Bobertag
suggests that the rates of development of normal and deficient children
are analogous to the upward progress of two projectiles fired from such
different heights that the force of gravity would retard the lower
projectile more than the upper (_81_). This analogy supposes that the
rate of maturity would continually decrease and that those who were
feebler mentally would be arrested in their developmental earlier.
Bobertag, Kuhlmann (_137_, _138_) and Otis give evidence from the
results of Binet testing that the rate of development decreases with
age. The percentages of older children passing certain positions on the
Binet scale or certain tests taken from it were found to change less at
year intervals for the older ages. This evidence is not conclusive
unless we know that the positions compared are at the same point in the
distributions of ability at the beginning of the periods of growth. The
same percentage change at a point farther away from the central tendency
would mean a larger growth than at the middle of the distribution, when
judged either in reference to a physical scale or to units of deviation.
While recognizing that the complete curve of mental development is
logarithmic in form Pearson contends that, when measured by Jaederholm's
adaptation of the Binet scale, development is adequately represented by
a straight line from 6 to 15 years of age (_164_). As this conclusion is
based upon the use, as equivalent units, of years of excess and
deficiency at all these ages the data lacks the cogency of a scale of
equal physical units.
With the Point Scale it is not known whether the units in different
parts of the scale are equivalent. Without assuming that they are equal
it is impossible to discover the form of curves of development from the
records of children at a series of ages. Yerkes and Wood publish a curve
of the increase of intellectual ability based upon point-scale
measurements, which resembles in form the hypothetical curves. They say:
“The point-scale method has the merit of indicating directly the
rate, or annual increments of intellectual growth. We do not claim
for our measurements a high degree of accuracy, especially in the
case of the early years of childhood. But even the roughly
determined curve of intellectual growth from four to eighteen years,
which we present below, has considerable interest for the genetic
psychologist and for the psychological examiner. We have ascertained
that whether measured by the ratio of the increment of increase,
year by year, to the norm for the appropriate year or by the ratio
of the extreme range of scores to appropriate year norms,
intellectual development rapidly diminishes in rate, at least from
the fifth year onward” (_169_, p. 603).
Waiving the question whether annual increases or the range of
measurements relative to the age norms would be satisfactory indications
of the change in the rate of growth, it seems to be fairly clear that
neither of these criteria would be adequate unless we first knew that
the units in which they were measured were equivalent at different
portions of the scale. To show that the point scale units are even
theoretically equivalent it would seem to be necessary to assume, on the
basis of normal distribution of ability, that each unit of the deviation
for each age distribution either equaled the same number of scale units
or the same proportion of the total distance from lowest to highest
ability at each age measured in the point-scale units. The originators
of the scale do not seem to have planned it with this in view. Moreover,
the difficulty of empirically demonstrating such equivalence of units on
a point scale or any form of the Binet scale prevents its use for
indicating curves of mental development, however serviceable it may be
for other purposes.
The simplest demonstration of the form of the development curves is
applying the same test, scored in equal physical units, to children of
different ages. In Figs. 6, 7, and 8 the evidence from tests was
assembled for ages 8 to 14 inclusive. It is probable, however, that the
form of these development curves, when the unit of measurement was
anything but time taken for the same task, has been affected by the
difference in the real value of units called by the same name, _e. g._,
giving the opposite of one word is not always equal to giving the
opposite of another.
The best developmental curves empirically determined are probably those
for the form board presented by Sylvester (_191_), Wallin (_212_) and
Young (_227_) since in each of these cases the same test was presented
at all ages and the scores were in equal physical units of seconds. It
can hardly be supposed, however, that the form board curves alone would
be typical of average mental development. To know something about the
general curve of mental development we need a combination of a number of
mental tests scored on scales of equal units. These may be either equal
physical units or units on scales for mental development similar to
those of Thorndike and others for measuring educational products,
handwriting, arithmetic, spelling, _etc._
That either a straight line or a simple curve would represent the
development of ability from birth to maturity is very doubtful. When we
consider the entire developmental curve from birth nobody doubts that
there is a change in the rate of development at the time of the arrest
of instinctive changes at adolescence. There are probably fluctuations
in the rate before this final arrest. Pintner and Paterson also assume a
complex curve of development (_44_). Whether the fluctuations should be
allowed for in the description of the borderline of deficiency is the
important question in our study. With measurements of bodily growth we
noted that changes in the rate of maturity are accompanied by a skewness
of distribution of ability at the ages affected. The same effect may be
expected with mental measurements. The percentage method of defining the
borderline of deficiency has an advantage when the form of distribution
at any age is uncertain (See Chap. XIV, d.). Since the changes in the
rate of development are most likely to be important at the prepubertal
and adolescent ages the description of the borderline in terms of
deviation or quotient may be expected to be most uncertain at this
period. Moreover, none of the quantitative definitions of the
borderline, except the percentage method, remain equivalent if rates of
development of normal and deficient children change relative to each
other, a question we shall now consider.
(c) THE QUESTION OF EARLIER ARREST OF DEFICIENT CHILDREN.
It has been assumed by Bobertag (_81_), Stern (_88_), Goddard (_117_)
and others that deficient children reach their maturity earlier than
normal children. If this were true the curves of mental development for
the average and for the deficient children should not be expected to
retain their same relative positions after the idiots had begun to show
arrested development. Moreover, unless this arrest were compensated by
some peculiar form of accelerated growth among those above normal
ability, we might expect that the distributions of ability would change
in form at the various ages after arrest had begun. A relative increase
in the distance of older deficients from the average as compared with
younger deficients may be interpreted as meaning either the earlier
cessation of growth of the deficients or a change in the relative rates
of growth of individuals of different mental capacity. When fully
considered the present evidence from the Binet tests fails, I believe,
to demonstrate the earlier arrest of the deficients, although it is
undoubtedly true that the Binet scale may not be fine enough to measure
the improvement of idiots. We shall take up certain investigations that
bear upon this point.
Goddard has reported tests upon the same group of 346 inmates in an
institution for the feeble-minded who were tested three years in
succession (_117_). The paper suggests that the idiots, as a group
increased less in absolute ability than those of higher mental age. The
average gain for 55 idiots who tested I or II mentally was about half a
test in the two years. In order to reach our present problem, however,
we must know that the idiots, for example, developed relatively less
mentally than did those of the higher grades of ability in the imbecile
and moron groups of _the same life-ages_. This question cannot be
answered from the paper. It probably cannot be adequately answered from
mental age results on account of the irregularity in the value of the
year units at different points on the Binet scales.
Bobertag summarizes Chotzen's data obtained by the examination of the
children in the Breslau Hilfsschulen with the Binet scale. He believes
that the position on an objective scale attained by the average of these
retarded children is progressively lower with advancing age relative to
the average position attained by normal children, assuming that the
quotient for normal children remained constant at each age. The average
intelligence quotients of all the children in the special schools
(exclusive of those testing III or less) was 0.79 for those 8 years of
age, 0.72 for those 9 years, 0.70 at 10, and 0.67 at 11-12 (_81_, p.
534).
Stern also compiled a table from Chotzen's results which shows this
decrease in intelligence quotients with life-age separately for each
group of those whom Chotzen by his expert diagnosis regarded as
imbeciles, morons, doubtful, and not feeble-minded although attending
the special schools (_188_, p. 80). This table is reproduced here as
Table XX. On the surface it suggests that the quotients of the extreme
groups are nearer together at the older ages, instead of being farther
apart. The objection to this evidence from the Binet scale is that the
norms are not equivalent for different ages on the scale used. Since the
objective norms on the Binet scale are more difficult to attain at the
older ages this variation would tend to make older children show lower
quotients than the same children would show at younger ages, so that
such tables are quite uncertain in significance.
TABLE XX.
AVERAGE INTELLIGENCE QUOTIENTS OF CHILDREN OF DIFFERENT ABILITY. (From
Chotzen's Tables X & XI.)
─────────────┬──────────────┬──────────────┬─────────────┬─────────────
LIFE-AGE │ NOT │ DOUBTFUL │ MORONS │ IMBECILES
│FEEBLE-MINDED │ DEFECT │ │
─────────────┼──────────────┼──────────────┼─────────────┼─────────────
8 │ 0.92 │ 0.84 │ 0.76 │ 0.71
9 │ 0.85 │ 0.81 │ 0.77 │ 0.67
10 │ (0.80) │ (0.80) │ 0.74 │ 0.62
11 │ (0.73) │ (0.68) │ 0.71 │ (0.64)
12 │ (0.75) │ (0.75) │ (0.73) │ (0.61)
13 │ │ (0.73) │ │
─────────────┴──────────────┴──────────────┴─────────────┴─────────────
The Jaederholm data with his form of the Binet scale, as treated by
Pearson, shows a straight regression line for the backward children
which falls below the normal development line on the average four months
of mental age for each additional year of life from 7-14 (_167_).
Accepting Pearson's interpretation that a year of excess or deficiency
and a year of growth is a constant unit, we find that the deficient
group from special classes was falling continually behind the normals
with increase of age a relatively greater distance from any rational
reference point. Pearson accounts for this change in the distance
between the two groups of normal and backward children, as I understand
his paper, by supposing that with increase in age more and more normal
children become deficient. It would seem that this data would be more
easily explained by supposing that the distributions became skewed
toward deficiency for the older ages, rather than that the distributions
remained normal and became flatter.
The best evidence as to the relative positions of the curves for
deficients and those for average ability would be provided by using
psychological tests that could be adequately scored in terms of equal
physical units for the same task. The position of various lower
percentiles relative to the average or to an assumed reference point
could then be compared on the same objective scale. I have reviewed
studies of this type in discussing skewed distributions in Chap. XIII,
A, c. I there reached the conclusion that the weight of the evidence was
that the distributions were slightly skewed in the direction of
deficiency, although the evidence was not conclusive. We are now raising
the further question whether this skewness increases with age.
On account of the difficulty of determining the points for zero ability
in terms of the physical scales used, let us see what conclusion might
be reached if we calculated the relative distance of median and low
ability of equivalent degree from the scores of the same higher degree
of ability assumed as a reference point at the various ages. There seems
to be no reason in the theory of measurement why the highest score
instead of the lowest score in random samples might not be used for a
reference point for comparing the distances between normal and deficient
children at different ages. Instead of using the highest single score,
it would be better to use the upper quartile or quintile since it would
be less affected by a chance error in giving the test.
Applying this method to determining the relative position of median and
retarded ability I have calculated the data for the form board test
cited previously from Sylvester (_191_) and from Young (_227_). This
affords the only adequate evidence of which I know, derived from tests
scored in equal physical units given to sufficiently large groups to
indicate whether or not the retarded group changes its relative position
from the normal group at different ages. The comparison is shown in Fig.
9. With Sylvester's data the distance of the lower quartile in ability
from the median is compared with the distance of the upper quartile from
the median, the latter distance being taken as a unit. With Young's data
for Witmer's form board the quintile is used instead of the quartile and
each sex is given separately. Since Young's table shows the scores for
half ages, it was necessary to take the average of the two scores, thus
giving the approximate score for the middle of the complete age group.
The graph discloses no pronounced tendency for the retarded group to
fall relatively farther behind the median with increase in age. There
are, however, notable fluctuations in the relative positions of the
groups so that at 7 years with Young's data for boys and at 13 years for
Sylvester's curve the retarded group is twice as far from the median
relative to the distance between the median and the corresponding better
group as it is at some other times. It is possible that the curves for
the older groups of those of poorer ability are too high since it is
likely that more of the actually deficient children tend to be dropped
from the public school classes with increase in age. Nevertheless, so
far as the evidence at present goes it is not sufficient to determine
whether the backward and the corresponding better group show a general
change in their relative distances from the median with approach to
maturity.
[Illustration: FIG. 9. _Relative Positions at Each Age of the Median and
of Corresponding Bright and Retarded Children with the Form Board
Test._]
On the other hand the curves indicate the tendency for the distributions
to be skewed toward deficiency and for the relative distances to
fluctuate as we should expect if the accelerations in growth occurred at
different ages for those of different ability. The data of Young suggest
that there may be sex differences in the age of acceleration, the
backward girls showing accelerations, relative to the upper group at
ages 7 and 12, a year or more before the boys. For Sylvester's data the
ratio of the distance between the median and the lower quartile divided
by the distance between the median and the upper quartile for each of
the age groups is as follows: 5 yrs. 1.8, 6 yrs. 2.4, 7 yrs. 3.0, 8 yrs.
2.0, 9 yrs. 2.2, 10 yrs. 2.4, 11 yrs. 2.0, 12 yrs. 1.8, 13 yrs. 3.0, 14
yrs. 2.1. For Young's data the corresponding ratios are—Boys: 6 yrs.
1.5, 7 yrs. 1.9, 8 yrs. 1.5, 9 yrs. 0.8, 10 yrs. 1.6, 11 yrs. 1.2, 12
yrs. 1.4, 13 yrs. 1.0, 14 yrs. 1.3. Girls: 6 yrs. 1.7, 7 yrs. 1.0, 8
yrs. 1.5, 9 yrs. 0.9, 10 yrs. 1.0, 11 yrs. 1.3, 12 yrs. 0.9, 13 yrs.
1.5, 14 yrs. 1.4. Changes in the rate of growth causing asymmetrical
distributions are to be expected throughout the periods of growth. A
fundamental skewness toward deficient mental capacity, therefore, would
be indicated only if it were found at maturity or at ages when the
average rate is decreasing, when the more capable individuals would
theoretically approach relatively nearer the deficients if the latter
accelerated later.
So far as physical growth is concerned Baldwin (_74_, _75_) has shown
with repeated annual measurements on the same group of children that the
period of adolescent acceleration shifts from 12½ years for the tallest
boy to 16 years for the shortest boy. For the tallest girl the maximum
height was attained at 14½, for the shortest at 17 years, 3 months.
Maturity may be reached at 11 years by a tall well nourished girl, while
with a short girl light in weight it may be delayed until 16. “Children
above medium height between the chronological ages of 6-18 grow in
stature and in physiological maturity in advance of those below the
medium height, and they may be physiologically from one to four or five
years older than those below the medium height. Those above the medium
height have their characteristic pubescent changes and accelerations
earlier than those below; there is a relative shifting of the
accelerated period according to the individuals' relative heights”
(_74_).
Doll presents evidence from the physical measurements of a large
feeble-minded group in institutions which he suggests shows that the
shorter among them cease growing earlier. When the height of these
feeble-minded is measured in relation to the Smedley percentiles of the
height of normal children of their corresponding ages, he finds a
correlation of -.20 between age and percentiles of height, the taller
relative to normals being younger. He says: “This confirms Goddard's
similar conclusion, but negatives for the feeble-minded at least, the
theory affirmed by some writers, that children who grow at a retarded
rate continue their growth to a later age” (_98_ p. 51). On the contrary
this minus correlation is more likely to mean only that the Smedley
norms on school children are too high for the older ages because of the
excess of taller children who remain for the high school work. This
would give the minus correlation without supposing that the taller
individuals continue their growth to a later age, as he thinks.
Moreover, a total longer period of physical growth for smaller, less
normal, children has been demonstrated. Boas (_80_) says: “Among the
poor the period of diminishing growth which precedes adolescence is
lengthened and the acceleration of adolescence sets in later; therefore,
the whole period of growth is lengthened but the total amount of growth
during the larger period is less than during the shorter period of the
well-to-do” (_80_). A reversal in growth tendency between brain capacity
and size of body, which is supposed when the mentally deficient are said
to arrest earlier, would be one of the most puzzling paradoxes in the
study of development. We should, therefore, be exceedingly cautious
before accepting the hypothesis of the earlier maturity of deficient
children.
A complicated situation is presented when we come to represent
graphically the effect on the distributions of these differences in
growth among those of different intellectual capacity. In the
hypothetical diagrams, Fig. 5, it is shown how arrest of development
might be presented graphically in relation to the distribution curves,
ability being measured on the same physical scale. The earlier
acceleration and earlier maturity of those of better ability are
indicated. The distributions are shown as skewed at all ages after
birth. Equivalent units of mental development at different ages can be
found only in corresponding percentages of the groups, not in the units
of the deviation or in development quotients relative to the averages at
different ages. In other words the lowest 0.5% continues to be an
equivalent unit while -3 S. D. measures different portions of the group
and different portions of the distance from lowest to highest ability.
Corresponding percentages retain one common significance, namely, that
the same proportion of the group is ahead in the struggle for survival,
regardless of the form of the distribution.
It is hoped that the discussion of the statistical problems connected
with the quantitative study of mental development has given more meaning
to the different attempts to devise scales for measuring mental ability.
It should be noted that the same relative development at different ages,
expressed relative to the distance from lowest to highest ability
measured in equal objective units, does not correspond to the same
relative development measured in percentages of the groups, as soon as
the forms of the distributions change. The theoretical considerations
show that we have available at once a perfectly definite and clear
method of stating relative development in terms of corresponding
percentages of corresponding groups. If the groups distribute normally
these units are translatable into units of the standard deviation of the
group. If the distributions change in symmetry the only equivalent units
of deficiency available are in terms of corresponding percentages
reading from either end of the group. On the other hand percentile units
are not equivalent in _amount_ of change for the same distribution, so
they are of most importance for comparing different age distributions of
uncertain forms.
Until we have a scale of equal objective units for mental ability, it is
not possible to obtain a measure of relative development which shall
take into account the _amount_ of relative change. We must be content to
measure the change in percentile rank (changes in serial position) of an
individual relative to those of his own age.
Having clarified our conceptions of mental development and brought them
into harmony with certain suppositions regarding the distribution of
ability and its change from year to year, we are in a better position to
evaluate in the following chapter the different objective methods of
defining the borderline of feeble-mindedness.