San José State University
Department of Economics |
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Rodolfo Alejo
Gonzalez' Theory of the Origin of Gender Differences in the Variation in the Distributions of Intellectual Abilities |
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The purpose of this page is to present a hypothesis of Rodolfo A. Gonzalez, a professor at San José State University, to the effect that significant elements of human intelligence are a result of genes carried only on the X-chromosome. A male gets his X-chromosome only from his mother but a female gets one X-chromosome from her mother and one from her father. Under the Gonzalez hypothesis this means that a female's ability due to these factors is a function of two variables, the genes inherited from both her parents, whereas a male's abilities are a function of the genes inherited from his mother. Thus there should be a lower variation in female abilities than in male abilities. In effect, females have a more diversified portfolio of abilities compared to the undiversified portfolio for males. Gonzalez' hypothesis has some interesting implications. If a male has some outstanding intellectual ability (associated with the X-chromosome) he is likely to be disappointed in the abilities of his sons because that ability can only be passed on to his daughters. He would be especially likely to be disappointed in his sons if he chose their mother on the basis of her beauty rather than her intellectual capabilities.
A male looking for a male descendant who has inherited an X-chromosome-related trait would have to look to his grandsons from his daughters. But only half of them are likely to have the trait he is looking for.
In contrast a female with some special intellectual trait is likely to find it in both her sons and daughters. Her sons could have it to a greater degree than she does.
Rodolfo Gonzalez' hypothesis may not be the whole truth but it seems to capture some element of the truth and is well worth considering. I believe, from my personal observations, that not all intellectual abilities are solely carried on the X-chromosome but there may be some abilities for which this is true. What makes Gonzalez' hypothesis especially intriguing is that it suggest some quantitative levels for gender differences in the standard deviations of abilities.
There is a distribution in the level of human abilities ranging from high values to low values with most people having abilities near the average. The shape of the distribution function is usually a normal, bell-shaped curve. Such normal distributions are characterized by their mean or average and their spread or dispersion. The spread or dispersion of a normal distribution is measured by its standard deviation.
Even if two population groups have distributions with the same mean values a difference in standard deviations can result in substantially different proportions in the numbers of people with above some specified levels. For example, in the diagram shown below the means of the two distributions are the same but one distribution has a higher standard deviation than the other. Proportion of the population above some level, such as the one shown, will be significantly higher for the population with the higher variation.
For purposes of the later presentation let us rephrase the above point. A population group with a smaller variation in the distribution of some ability will have a significantly smaller proportion of the population which have an ability level above a specified value than a population group with a higher variation. The table below shows the proportions above specified level as a function of the ratio of standard deviations compared to a base distribution.
Ratio of Standard Deviation | Proportions Beyond a Number of Standard Deviation Units Above the Mean | |||||
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0 | 1 | 2 | 3 | 4 | ||
1.00 | 0.500 | 0.15866 | 0.02275 | 0.00135 | 0.00003 | |
0.90 | 0.500 | 0.13326 | 0.01313 | 0.00043 | 0.00000 | |
0.80 | 0.500 | 0.10565 | 0.00621 | 0.00009 | 0.00000 | |
0.70 | 0.500 | 0.07656 | 0.00214 | 0.00001 | 0.00000 | |
0.60 | 0.500 | 0.04779 | 0.00043 | 0.00000 | 0.00000 |
Note that the numbers listed as 0.00000 are not truly zero but just less than 0.000005.
The interesting statistics is the ratio of the numbers in the various categories compared to the distribution with the higher standard deviation. These values are shown in the table below:
Ratio of Standard Deviation | Relative Proportions Beyond a Number of Standard Deviation Units Above the Mean | |||||
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0 | 1 | 2 | 3 | 4 | ||
1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | |
0.90 | 1.00 | 1.19 | 1.73 | 3.14 | -- | |
0.80 | 1.00 | 1.50 | 3.66 | 15.00 | -- | |
0.70 | 1.00 | 2.07 | 10.63 | 135 | -- | |
0.60 | 1.00 | 3.32 | 52.91 | -- | -- |
What the above table indicates, for example, is that if it requires a level of ability that is three standard deviations above the mean for the higher variation group to be classed as notable then the number of notable individuals from the high variation group will be 3.14 times as many as notables from a group that has a standard deviation that is 90 percent of the value for the high variation group. This is even when the means for the two groups are exactly the same. It also means there will be 3.14 times as many individuals in the high variation group which as three standard deviations below average.
If the low variation group has a standard deviation 80 percent of that of the high variation group then the relative numbers in the notable category rises to 15 to one. And for a standard deviation 70 percent of the high variation group the relative number of notables jumps to 135 to one.
Thus even without a difference in means the relative numbers in some category that is significantly above the mean will depend very strongly upon the relative values of the standard deviations of the distributions.
IQ test scores are standardized to have a mean of 100 and a standard deviation of 15. Dr. Gina Losasso estimates the standard deviation of female IQs to be 13.4. In order for the male-female combined population to have a standard deviation of 15 the standard deviation of males would have to be 16.4. The ratio of the standard deviation of females to the standard deviation of males would then be 0.83. Based upon this figure of 0.83 the ratio of males with IQs of 145 and over to females with IQs of 145 and over should be about 6.5 to one.
Suppose that a female's abilities are based upon an average of levels inherited from her mother and father; i.e., x = (x_{mother} + x_{father})/2. This would imply that the standard deviation of abilities for females should be 0.707 times the standard deviation for males. The 0.707 arises as the square root of one half.
If there is a positive correlation between the abilities female inherit from their mothers and fathers then the ratio of the standard deviations of females to males should be greater than the 0.707 mentioned above. The formula for the ratio is
where ρ is the correlation coefficient between the X-chromosome genes of the mother and father. If there was perfect correlation (ρ=1) between the genes inherited by a female from her mother and father then then there would be no difference in the variation between females and males (R=1). On the other hand if the correlation coefficient was 0.5 then the ratio of variations would be R=0.866.
The above computations are based upon the supposition that a female's X-chromosome-related abilities are a simple average of the values based upon the genes inherited from her mother and father.
The functional relation may well be more complicated than a simple average. For example, for dominant-recessive genes the ratio of two gene variation to single gene variation is 0.866. This figure is based upon a distribution of 50% in in two levels for the single source case and a distribution of 25% in one level and 75% in the other level for the double source case in which one gene is dominant. In any case the X-chromosome-related ability level of a female is a function of two variables whereas that of a male is a function of one variable. The variance of a binary function of two identically distributed random variables is never greater than the variance of the corresponding unary function.
Whatever the explanation it is a statistical fact that the variance of the distribution of abilities among males is greater than the variance of the distribution for females. In a sense this difference can be interpreted as more natural genetic experimentation with the males of the species than the females.
The higher proportion of geniuses among males is counterbalanced with a higher proportion of idiots. Nature recognizes that the females of the species are the foundation of its survival whereas there is a great surplus of males in any species and it does not matter that a significant proportion is wasted in experiments that fail. The successful experiments are predominantly male but so are the unsuccessful experiments. Nature does waste females in experiments. Nature is amazingly wise not to waste females in that way.
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