In this discussion of sex differences I rely mostly on chapter 13, "Sex differences in g", from Arthur Jensen's book "The g factor", and a little bit on chapter 4, "Conditions for excellence", from Hans Eysenck's book "Genius - The natural history of creativity". Also on Richard Lynn's home page, and on my own experience with high-range tests.
When it comes to the question whether or not there is difference in mean I.Q. between males and females, Jensen basically says no, after having considered a large amount of evidence. Eysenck is a little bit more skeptical and points out that the usual assumption of equal I.Q. of the sexes may be flawed. Based on data also mentioned by Jensen (R. Lynn, 1994, "Sex differences in intelligence and brain size: a paradox resolved"), Eysenck suggests 4 I.Q. points as a conservative estimate of the difference (favouring males). Lynn, on his home page, simple states in adults the difference is about 4 points.
Both Jensen and Eysenck indicate that the question is hard to answer, as I.Q. tests like Stanford-Binet and WAIS have traditionally been constructed to show no sex difference in total score, by leaving out or counterbalancing items that show sex differences. Such tests therefore are not capable of measuring a possible difference between the sexes.
I myself can not observe a mean difference directly as I only deal with high-range tests. I will return to this point further on with regard to the variance difference.
The male variance in I.Q. is greater than that for females; Jensen says this difference is greatest in math and spatial ability. In math the male variance is 1.1 to 1.3 times greater (He does not give the difference for I.Q. or g).
In the high range, my own observation to date is that at or above the 98th percentile there are about twice more males than females, while at or above the 99.9th percentile there are about 15 times more males. These estimates are based on the male/female ratios in certain high I.Q. societies and on analysis of male and female performance on my tests. Trying to make this fit in terms of standard deviation ("variance" is, incidentally, the square of the standard deviation), I find that when the male and female mean are both I.Q. 100, the male standard deviation must be about 33% greater than the female standard deviation. However, if a mean difference of 5 points in favour of males existed, the male standard deviation would only need to be about 11% greater. I do not yet know which is true (or if the truth lies in between). I must say though that a difference of 33% seems unlikely.
A remark sometimes made regarding the male/female ratio in I.Q. societies and among high-range candidates is that females may be so lowly represented because they simply do not like taking intelligence tests, or even because they attach less value to intelligence than males do as a result of having been raised and socialized to value other traits higher in women. In other words, that their representation is not proportional to their actual presence at high intelligence levels. The following facts however speak against this:
These facts make it almost inescapable to conclude that the male/female ratio among high-range candidates and in I.Q. societies is roughly representative of the actual male/female ratio at high intelligence levels. Sometimes one has to leave a prejudice (equality) behind and accept the facts, even if one would like them to be otherwise. That I say "roughly representative" is not because I believe that "liking tests" or social or socialization factors may still contribute, but to leave open the possibility that yet other personality features are at play next to intelligence, like associative horizon.
Girls mature earlier verbally, and after puberty boys catch up. The male advantage on spatial and numerical ability (discussed further on) is not yet present in young children, and develops slowly during childhood and puberty. Important to realize here is that the sex differences in mental abilities are caused by hormonal differences (estrogen/testosterone balance), which work partly prenatally and partly after puberty.
If there is a mean difference in I.Q. between the sexes, this will be fully expressed only in adults, and not yet in children. In any case, it seems that when testing children, e.g. for giftedness, one should be aware of these developments and differences, the risk being that one selects too many girls and too few boys as "gifted".
Females are slightly better than males at straight-forward arithmetic (not at more complex math). On short-term memory the difference is greater; they score .3 σ (standard deviation) higher than males, and one may suppose that this disposes them for multi-tasking.
A verbal ability type that consistently favours females is "fluency"; such tests require the candidate to name as many as possible words starting with a given letter within a limited time. Females are also better at reading, writing, grammar and spelling. The popular notion that females are better than males at verbal ability on the whole is not true; they are only better at these specific tasks, while there is no or as good as no sex difference in verbal ability on the whole.
Other tasks at which females outscore males are those involving perceptual speed (e.g. matching figures) and clerical checking, both speed and accuracy (e.g. underlining certain letters in a text, or digit/symbol coding). Their advantage on such tasks varies from .2 to .4 σ. Females are also better at motor coordination and finger and manual dexterity, but those are not mental abilities in a strict sense, although they do occur in the "performance" sections of some individual tests.
The largest difference is that in spatial ability; the mental manipulation of figures in two or more dimensions. The difference varies from .3 to .5 σ. Studies link this difference to prenatal testosterone levels. A sex difference in spatial ability is also found in some animals, which suggests it is a more general biological phenomenon.
Then there is a difference in numerical ability (except for simple arithmetic) of .1 to .25 σ And as already said, in both spatial and numerical ability there is also a large difference in variance, favouring males.
As for verbal ability, males are better at tests of general knowledge. In verbal reasoning there is as good as no difference.
Two vital points follow from the above:
A dillemma that comes forth from these facts: Should "giftedness" be defined within children, or within adults? I am inclined to say "Beyond doubt within adults" to this question.