In the book Good science, bad science, and bogus (1989) famous pseudoscience debunker Martin Gardner also mentions Cyril Burt’s twin experiments as a scientific forgery, as was the “truth” at that time. Today Sir Cyril Burt is vindicated: he did not falsify or invent data, he did not invent non-existing assistants, and the IQ correlation he found for monozygotic twins raised apart (0.77) has been verified by later measurements. We can trust this correlation figure. It supports the hereditarian position of intelligence. Yet, it is not in any means in conflict with the data showing that the environment also has a role in IQ. Let me demonstrate this by calculating an estimate to the IQ of a parent based on the IQ of a child.
The child inherits half of his/her genes from each parent. A son does inherit a bit more IQ-related genes from their mother than from their father as the son’s X-chromosome is from the mother while the Y-chromosome mainly contains stuff to produce sperm. But the effect of IQ related genes in the X-chromosome is very important only in genes that cause low IQ: even a single bad gene in the X-chromosome can cause serious mental deterioration in boys, while good IQ related genes in the X-chromosome have only slight effects, similar as good IQ related genes in all chromosomes. Thus, one can say that boys do not inherit their intelligence form the mother, but they may inherit their stupidity from the mother, even from a completely intelligent mother, if they get the wrong gene from mother’s X-chromosome. Let us be positive and assume that the mother has two good X-chromosomes. Then the genetic contribution from both parents is about the same.
Let us assume that the standard deviation of IQ is 15 for the population. The Pearson correlation coefficient for IQ of monozygotic twins raised apart is r=0.77, thus the shared genes contribute the standard deviation 15*√0.77=13 IQ points. The rest of the standard deviation, √(152-132)=7 points, is caused by the environment. Next we divide this standard deviation 13 between the two parents. Assuming that both parents contribute equally, one parent contributes 13/√2=9.2 points.
We can check if this result is correct. If the result is correct, the correlation of IQs between the father and the son should be (9.2/15)2=0.376.
For monozygotic twins raised together the IQ correlation is 0.86, i.e., the standard deviation of 15*√0.86 IQ points. The contribution of the shared environment is thus √(0.86*152-132)=√24.5 IQ points. Siblings share as many genes as parents with their children. The IQ correlation between siblings raised together is 0.47, i.e., 15*√0.47 points. Subtracting the shared environment of √24.5 points, we get the genetic correlation between siblings as √(0.47*152-24.5)=9 points, which means the correlation of (9/15)2=0.36. This agrees well with the 0.376 that we calculated, thus the model is reasonably correct.
A parent and a child share 9.2 points of the standard deviation for genetic reasons and the rest is determined by two random variables (one for the child, one for the parent) with the standard deviation √(152-9.22)=12. If the parent and the child do not share an environment, these random numbers are independent and have the standard deviation 12. Each random variable is within two 2 standard deviations (±24) with the probability of 95.6%. The probability that both the parent and the child are within 2 standard deviations in their individual distributions is 91.4%=0.9562. Thus, with the probability of 91.4% the IQs of the child and the parent do not differ by more than 48 points.
This prediction does not much differ from the one for two unrelated people. It does constraint something, but almost nothing that can happen in the real life, like if the child has the IQ of over 180, then most probably the parents have an IQ over 132. It does not work in the low IQ direction: very intelligent parents can have a child where a gene causing mental retardation is expressed.
Assuming that both the child’s and the parent’s IQs are in the one standard deviation range, they differ at most by 12+12=24 IQ points. However, for this to happen the parent’s IQ should be in the 1 SD range (68.2% of cases) and the child’s IQ also in the 1 SD range (again 68.2% of the cases). This range of at most 24 points IQ difference is already rather tight, but the probability of this to happen is only 46.5%.
If we assume that the parent and the child share an environment, like live in the same home, then the correlation is about 0.47. Now we get two components: 15*√0.47=10.3 and 15*√(1-0.47)=10.9 as the standard deviations of inherited and independent random numbers. As 10.9 is a bit smaller than 12, we have 91.4% chance that the IQs of the parent and the child differ less than 43.6 points. With the probability of 46.5% the IQ scores are within 22 points. Again, this does not give much information of the IQ of a parent if the IQ of the child is known. Clever parents can have a stupid child, or a clever child can have stupid parents or siblings.
It is still interesting to know how much the parents may have suffered from lack of education. From the twin studies we concluded that 7 points in the standard deviation is from the environment. Thus, 68.2% of the parents may have dropped (or gained) up to 7 points just because they did not have education. But this was so in England and not in the war time. There are much worse environments. The parent maybe had only eight years of primary school. The child may have gone to the university and got 20 years of education. Each year of education gives some 1-1.5 points to IQ, so the child should have 12-18 IQ points more. This agrees well with the estimation that the parent lost 7 points and the child gained 7 points because of education.
As seen from these calculations, the fact that intelligence is largely determined by genes, it does not follow that the IQ scores of parents and children, or siblings, should be very similar. High similarity is guaranteed only with monozygotic twins. More important is the average of the population and the standard deviation.
For simplicity, I did not include regression to the mean in the calculations. It should be included if studying large deviations from the population mean.