The Conspiracy of Silence


Nobel Prizes and the Stock Market
    In her book, "Beyond Terman - contemporary longitudinal studies of giftedness and talent", Rena Subotnik discusses the prior environments that predispose toward Nobel Prizes. I won't say anything about "old boy" networks.
    There's an article in this month's issue of Smart Money that confirms what I had deduced from the calculations I ran a few weeks ago: that the stock market has reached the highest valuations in his history by a factor of, at its peak, two-to-one. The author says,
    "As I wrote here on May 10, the stock market, in December 1996, launched into a prolonged period of consistently high valuation. Ever since then, the prices that investors have been willing to pay for stocks have been far greater in relation to average expected earnings than ever before in postwar history. Ironically, this period of high valuation began almost immediately after Federal Reserve Chairman Alan Greenspan warned for the first time of "irrational exuberance." Only during the panics of October 1998 and September 2001 did the markets return and then only briefly to anything like what Greenspan would consider "rational." Those brief moments proved to be bottoms. We might be approaching such a moment right now."
    The author concludes that the stock market is going to turn up again in due course, and we're going to have business as usual. I certainly can't gainsay him, but I'm still concerned. Stock valuations in terms of the Federal Reserve's yield formula may not be out-of-line, but the formula itself is of recent vintage, and may possibly be something that won't guide investments twenty years from today. One of its vulnerabilities seems to me to be that of putting all one's eggs in one basket. If price-to-earnings ratios are the defining measure of value in  pricing stocks, then price-to-earnings ratios become the sole target for spin and window dressing, and in fact, earnings scandals seem to be the concern du jour that's holding back the stock market right now.

Agreement of the Ln-Normal Model with WISC Manual
    John Scoville has found a table in the administration manual for the Wechsler Intelligence-Scale-for-Children (WISC) that converts Stanford-Binet IQs to Wechsler IQs. He writes that it, "
basically agrees with the Sare data.
    I've ordered a copy of the WISC user's guide, and should receive it within a day or two. But that's a very important data set in establishing the validity of the ln-normal model. 
    One important fact about this ln-normal relationship is that it would seem to me to be all-or-nothing. There aren't any parameters to jiggle to make it fit except the standard deviation of the logarithms (which John has determined is very close to, if, possibly, slightly below 0.15). 
    The first step will be to buttress the link between children's Stanford-Binet IQs and the Wechsler IQ scale. (The Wechsler IQ scale is a deviation-"IQ" scale with a standard deviation of 15.) Then it will be time to look at how far this can be pushed. 
In Defense of Deviation "IQs"
    One of the motivations for cleaving to a deviation-"IQ" is that in the mental-age/chronological-age formula, chronological age can't be as easily defined as it is for children. The rate of rise of intelligence slows down markedly after puberty, as it does with height, with "g" peaking in the twenties, and crystallized intelligence continuing in some cases to rise into midlife. Then it slowly declines, as shown in this age curve. Since it's based upon percentiles, the adoption of deviation IQs allowed the test-takers to assign the same IQ to someone who was 75-years old that they would to that same individual at 25, even though, on average, the 75-year-old would score only 90% as high as she did at 25. (Of course, the same approach could be taken with Stanford-Binet IQs by upping the chronological age divisor somewhat between 16 and 25, and then lowering it gradually with rising age until, by age 75, the chronological-age divisor were 90% of what it had been at 25.)
Deviation-IQs are Relative; Ratio-IQs are Absolute
    Mental age gives us an absolute meter-stick against which to measure IQs (although not as meaningful a scale as I would like), whereas deviation-IQs are relative, defined by percentile rankings. I could assign percentiles to a population whose IQs lay between 100 and 101, and then go on to assign deviation-IQs from 0 to 200 to the people whose IQs lie between 100 and 101. Now, there are claims that deviation "IQs" are the only true IQs. After all, after correcting for the difference in standard deviations (15 vs. 16), Wechsler IQs are only a point or two lower than S-B IQs over the range from 70 to 130, and that's the range for which David Wechsler certified his test. The growing difference above an IQ of 130 between IQs delivered by the two tests may simply be caused by uneven rates of maturation. Maybe the brighter the child, the sooner the child stops growing mentally. Maybe adult intelligence covers a considerably smaller range of mental capacities than childhood intelligence. With deviation IQs, there's no way to tell. And who cares what happens above IQ 130, anyway? We've shown that above an IQ of about 120, there are no significant differences in life outcomes. 
    Or so it might be argued.

Patrick Wahl's Ln-Normal Distribution Simulators
First, from Eric Weisstein's Math World pages:

Second, from a Swiss institute I don't know: 
The last link is a Java applet ... like the familiar simulation, where falling balls bounce randomly
through a regular array of pins and pile up into a heaped normal distribution, except that these parameters are multiplicative, and the resulting heap is log-normal. Fun to watch, if you pick the higher speeds. (You can't re-set once it starts, you have to refresh the screen and then enter new parameters.)"

   When Patrick sent his email, for some reason, I wasn't able to get these to work, but now they're working fine. Thanks again, Patrick.

Are Geniuses Both Born and Made?
   Between 1912 and 1926, the Lutheran High School in Budapest produced four of the leading physicists and one of the leading mathematicians of the twentieth century: Leo Szilard, Eugene Wigner, John von Neumann, Edward Teller and Paul Erdos. The proximate cause? An inspiring science teacher by the name of László Rátz. In the meantime, the physics department at the University of Rome was producing the "Italian school" of geniuses, Enrico Fermi, Bruno Rossi, Bruno Pontecorvo, and Emilio Segre. The secret? An inspiring department head (who was also a senator).
   These individuals were undoubtedly extraordinarily gifted by birth, but they also were inspired by a mentor at an age when they were making lifetime career decisions. Students of this process observe that to transition from being a prodigy to a trailblazing adult begins with family support, then mentoring by a nurturing mentor, and finally, mentoring under an exacting taskmaster (or taskmistress) who can teach the burgeoning genius her/his craft.
   I have published this before, but it needs to go into the intelligence site map, as does this.

The Army General Classification Test (AGCT) Scores
    I had planned to present the Army General Classification Test data, and to compare it with a ln-normal distribution, but upon closer examination, I discovered that it gives Wechsler "IQs", only it uses a standard deviation of 20 instead of 15(!?) In other words, it converts the frequencies of occurrence to z-scores, multiplied by 20. Consequently, it can't be used to compare ratio-IQs with a ln-normal distribution.

    In touting the implications of the thesis that the natural logarithms of IQs are normally distributed, I want to be careful that I don't get carried away, and present this as gospel. I'm enthusiastic so far about the fit that seems to be occurring between the logarithms of IQs and a normal distribution. At the same time, to the best of my knowledge, this hasn't had the benefit of any review by experts in the field of psychometrics, or if so, it hasn't been at my behest. For all I know, ln-normal models of I distributions may have been explored and abandoned 50 years ago. Also, there are some hard questions that may be posed if we assume that IQ distributions are ln-normal. For example, Terman, et al, observed that the Concept Mastery Test rose by the equivalent of 5 or 6 points of IQ between the administration of the CMT-A form of the test in 1940 and the CMT-T version in 1950. Both these tests are tests of crystallized intelligence rather than fluid intelligence, so these findings aren't at variance with the current idea that vocabulary can rise into one's 50's and 60's. But how do we dovetail this into a ln-normal model of IQs? The disparity between fluid intelligence, which rises at 6 points per decade, and crystallized intelligence, which, for a given-aged test subject, has risen little or no enhancement over the decades, seems to me to force a split between these two aspects of intelligence even though factor analysis reveals only a single general factor, "g".
Does Fluid Intelligence Decline with Age, or Is It Overtaken By Rapidly Rising Norms?
    Certain components of intelligence seem to change with age. On the other hand, this might be simply a manifestation of the rapid, 6-point-per-decade rise of fluid intelligence, coupled with little or no change in vocabulary, general information. or arithmetic. This could make it appear as though people experience decreasing fluid intelligence over the decades, when, in fact, they're as bright as they ever were, but slower and slower compared with younger folk, who are boosting the norms with seven-league strides at 6 points per decade. 
Does Cognitive Decline with Aging Vary Markedly from One Individual to the Next?
    I suspect that when it comes to the effects of aging on cognitive powers, a lot will depend upon general health and mental activity, and may vary markedly from one person to the next. Arteriosclerosis and prodromal manifestations of Alzheimer's Disease probably also enter in. You wonder if brain size and reserve capacity play a role, also.

    Last night's link to the table converting Wechsler "IQs" to Stanford-Binet IQs went in the wrong direction. It's now been corrected.

Plotting Plots
    One interesting corollary to this investigation of IQ distributions is that the Wechsler "IQ" isn't an IQ at all but is the natural logarithm of the IQ! For example, a Wechsler IQ of 144 becomes 0.44 after subtracting 100 and dividing by 100 to convert it to a fractional deviation. Taking the exponential of 0.44 yields 1.5527, which, when multiplied by 100 to convert it to an IQ becomes an IQ of 155.27, or about 155.
    A precision table converting Wechsler "IQs" to Stanford-Binet (true) IQs may be found here.
    The table below reproduces the comparison among Dr. Terman's IQ distribution, a Gaussian IQ distribution, and a ln-normal (Sare) distribution. These tabular results are then plotted in the two figures below the table. 

IQ Range
"IQ"  Range

Gaussian Prediction*

Predicted by  Sare

Observed by Terman

% Found by Terman


133-136 637





137-140 264





141-143 100





144-146 34





147-149 11





150-152 3





153-155 0.77





156-158 0.18





159-161 0.04





162-164 0.0085





164-166 0.00125





167-169 0.00022





169+ 0.00003




* - This column shows the distribution of IQs that Dr. Terman must have expected to find, based upon a presumed Gaussian distribution of IQs in his subject population. I have used a standard deviation of 16 in calculating the Gaussian predictions because that's what I think Dr. Terman would have employed. 

    You can see in the Figure below how the Terman distribution of IQ scores (dark blue) differs from both the Gaussian and the ln-normal distributions. 

        The bar chart below affords a slightly different perspective on this data.

The Meaning of Regression to the Mean?
Regression to the mean takes place at or near the cutoff boundary

    In thinking about the test-retest regression to the mean that occurs, it penetrated my numb skull today that regression to the mean must occur as a form of downward diffusion of scores that occurs when we establish a threshold score on a first test and then give the test again. There will be a some test subjects who barely passed the threshold the first time we gave them the test who will fall just below the threshold the second time we give them the test. However, for the most part, test subjects who scored far above the threshold the first time they took the test will probably still be well above the threshold the second time the take the test. 
If we retested the upper half of the population (IQ = 101+), the regression would be away from the mean.
    One interesting case would be that in which we selected the entire upper half of the IQ distribution on our first test--everyone with IQ's of 101 or above. If we tested them again a few months later on the same test or on an equivalent form of the same test, some of their scores on the second test would have drifted below 101. In  this case, the regression would be not to the mean but away from the mean.
If we retest an entire population on a different version of the same test, there will be no regression to the mean.
    If we retest an entire population, there can be no regression to the mean. If we retest with an equivalent form of the same test, the population will have to exhibit the same distribution curve that it did with the first test (except for statistical fluctuations, which will be minimized if the size of the population is sufficiently large). Individuals within the population may be expected to have slightly different scores than they got the first time they took the test, but the overall envelope must remain unchanged.
Groups within a group that are selected by a lower-end cutoff will regress toward the mean on a retest
    Individual groups within a selected population who were selected by a lower-end cutoff, like all those who scored 170+ on a childhood Stanford-Binet test, will diffuse downward below 170 on a retest, while other children who scored in the 160's (and below) will diffuse upward. So we would expect the average score of the children who scored 170+ on their childhood S-B test to be less on a retest  than it was on their first test (although the overall number of children with IQ's of 170+ should stay the same. There will just be some different children in the 170+ group following the second testing. 
The higher scorers among the 260,000+ schoolchildren screened in 1921-22 should still be present when the group was retested in later life.
    This has important consequences for the Terman Study. Since I think that the Terman screening probably identified all or nearly all of the students at IQs above, perhaps, 160, it follows that all of them would still be present during the follow-up IQ-testing that occurred in 1940 and in 1950.       

Why I'm Finding It Difficult to Get Complete Data
    The Terman Group Test scores are cited in Dr. Terman's first volume in his series, "Genetic Studies of Genius", but with no translation table converting them to IQ scores, Consequently, I've been unable to arrive at numbers for his total set of Termites. I could only use the numbers for the "main group", which constitutes a little over 40% of them. I've come across a few other numbers, such as the fact that there were 77 children identified with IQs of 170 or above,  that 26 of the 77 were children with IQs above 180, and that the highest IQ was either 201, or in Ellen Winner's, "Gifted Children, Myths and Realities",  pg. 24, that the IQs in the group ranged from 135 to 196. 

Update, 6-18-2002:  Today, I found mention of the fact that the average IQ of the high school students who took the Terman Group Test in 1921-22 "was 142.6, but because of the low ceiling on this test we estimate that the mean "was spuriously reduced by at least 8 to 10 points. If so, the mean IQ of the total gifted group when selected was not far from 151 in terms of 1916 Stanford-Binet scores." (Lewis Terman, Genetic Studies of Genius, Vol. IV., pg. 135).
Dr. Terman Explains Later That He Really Wasn't Trying to Find All the Eligible Children
    In his 1940 follow-up volume, Dr. Terman doesn't claim to have made a clean sweep of the entire 260,000+ children that he screened. Instead, he says in 1940, he wanted the higher IQs among the top 1%, though that's not the way it's' presented in his first volume, published in 1925. In his first volume, Dr. Terman spends several pages discussing steps taken to insure that most of the children who were eligible for the study were included in it. He mentions that spot checks were made at a school in Los Angeles with 350 children and a school in San Francisco with 800 children, both of which which were resifted for eligible children. No new entries were found in the Los Angeles school and ten new children were picked up in the San Francisco retest. However, he concludes that overall, about 90% of the children with IQs above 140 were caught with their filter. (You wonder how he rationalized the fact that they found only 629 children with IQs of 140+ in the "main group" when a Gaussian distribution would have predicted 1,050.) 

Dr. Ellen Winner on the Terman Study
    Dr. Winner says (p. 24) that on the Stanford-Binet, "About two or three children out of a hundred have IQs of 130 or higher; only about one in a hundred has an IQ of 140 or higher. Only one in ten thousand to thirty thousand will score 160 or higher, and only one in a million will score above 180." As mentioned above, Dr. Terman found 26 children with IQs at or above 180 out of 260,000 schoolchildren. Using Dr. Winner's frequency of 1 child in 1,000,000  with an IQ of 180+, 26 children would exhaust the quota for 26,000,000 children, or all  of the schoolchildren in the United States in 1921! I can only guess at the number of children with IQs above 160, but I would guess, considering the way the numbers are running, that it would  be about 200. Using Dr. Winner's frequency measure of 1 child in 10,000 to 1 child in 30,000, that would exhaust a population of about 2,000,000 to 6,000,000 schoolchildren, or about 8 to 24 times the size of Dr. Terman's sample.
    Dr. Winner is one of our leading experts on giftedness.
   I don't mean to knock Dr. Terman or Dr. Winner for the anomalies I seem to be finding. I'm highly impressed with, and grateful for the research they've carried out. They've done such good and important work that their results are widely quoted, making it important that (in my opinion) what they say be reviewed.

Returning to the Scene of the Crime
    I keep returning to the Terman Study like a detective to the scene of a crime. The Terman Study is still pivotal in discussions of the intelligence and the highly gifted. There are strange and major irregularities in the published Terman results.

Why Didn't Dr. Terman Pursue or Proclaim His Major Discovery?
    As far back as 1921-22, Dr. Terman and his colleagues discovered that the IQs of his children didn't fit a Gaussian distribution even approximately. Now when a scientist finds something that deviates wildly from what theory would predict, it's usually a great discovery. Normally, he will follow up his anomalous findings with an investigation of the anomalies, or if he doesn't, someone else will. Dr. Terman wrote in 1925 that his data weren't fitting the expected normal distribution. And yet, in 1940, when Drs. Terman and McNemar are reviewing the average 152 childhood IQ of the Termites, they describe this 152 IQ as falling 3.15 standard deviations from the population norm. But their own data showed that it didn't. A 152 IQ lies about 2.8 S. D. from the mean IQ of 100. And why didn't someone over that 18-year period investigate the IQ distributions that must have been pouring in from all the state-by-state IQ testing that was taking place?

Why Didn't Anyone Question Dr. Terman's Strange Distribution of IQs?
    Why hasn't anyone said anything about the gross under-representation of Termites in the 140-to-155 IQ range? There are only 20%  more children in the 140-144 IQ range than there are in the 150-154 IQ range. Didn't anyone question that? Yet people run around blithely quoting the Terman Study and Dr. Terman's results without any question being raised about his strange data.

What Was the National Intelligence Test?
    Dr. Terman pre-selected his children with the National Intelligence Test. What was the National Intelligence Test? Was it some rehash of the Army Alpha? The year 1921 was only five years after the introduction of the first intelligence test, the 1916 Stanford-Binet, and only three years after the development and introduction of the Army Alpha intelligence test. I never heard about a "National Intelligence Test" in 1945.

What Was the Terman Group Test?
    Dr. Terman rated his high school "Termites" using the Terman Group Test (TGT). However, he doesn't provide any IQ scores for the high school students . .  only their raw scores on the TGT. What is the Terman Group Test? What are the IQ scores for the Terman Group Test? How was the Terman Group Test normed? What ever became of it? Like the "National Intelligence Test", I never heard of the Terman Group Test in 1945, although many other tests were mentioned in the literature of that day.

Who Else Has Tried to Correct for Ceiling Effects?
    Dr. Terman "corrected" the IQs of his oldest and/or brightest students for ceiling effects. No one corrects IQ scores for ceiling effects today. How was that justified?

No, Virginia, There Really Is No Grinch Who Stole Christmas
    No, I don't really think there was any conspiracy involved, except, perhaps, a "conspiracy" of 'If you don't mention it, maybe they won't notice it", or "We don't know what to do with it, so let's not bring it up" And yet, it's one of those situations when, even though no conspiracy was involved, it seems as though one might almost as well have been.