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 I.Q. tests.
When it comes to the question whether or not there is a difference in mean I.Q. between males and females, Jensen basically says no, after having considered a large amount of evidence (more precisely, Jensen is referring to the difference in g rather than in I.Q.). 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 5 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.
For better understanding, one should know that I.Q. tests measure not only g (the common factor shared by all tests for mental abilities), but that about one third of the variance in I.Q. is due to group factors (e.g. verbal, numerical, spatial, memory) common to some but not all tests, and to test specificity. Therefore a sex difference in mean I.Q. does not imply a sex difference in g, as some or all of the difference my lie in non-g factors.
I myself can not observe a mean difference directly as I only deal with high-range tests. There is a statistical report on sex differences on high range I.Q. tests, which is highly relevant regarding this matter.
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 concludes from this and from various other facts that the cause of the greater male variance in I.Q. lies mainly or entirely in non-g factors rather than in 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, 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 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 the onset of 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 at least some of the sex differences in mental abilities are likely caused by hormonal differences (oestrogen/testosterone balance), which work partly prenatally and partly after the onset of puberty, but are absent in childhood.
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 straightforward arithmetic (not at more complex math). On short-term memory the difference is greater; they score .3 σ (standard deviation) higher than males.
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 (the notion of females being better at verbal ability may be related to the popular confusion of talkativeness with verbal ability, and of "verbal" with "oral"; for clarity, "verbal", in psychometrics, simply means "relating to or in the form of words", and not "spoken rather than written", which is a secondary meaning of the word).
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 feature in the "performance" sections of some individual tests.
The largest single 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. When spatial and numerical ability (especially math) are combined, the mean sex difference becomes much greater still than are the separate differences on spatial and numerical ability.
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 dilemma 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.