As a mathematician, I have a deep contempt of the whole world of modern "quantitative" psychology. This is, at best, self-deception, futile attempt to measure non-measurable "quantities", which really are not quantities at all. Many times in my professional life I was invited "to do math" for some psychological study, and in final analysis it inevitably happened that quality of the data made impossible unambiguous interpretation. The very concepts of this "science" are so elastic and even fluid, that they positively defy any math rigor. By very slight manipulation, which is unavoidable for any formalization, you can shape your conclusions to any desired outcome. Does it mean that psychology is not a science? Certainly not in the sense in which physics, chemistry or even biology are. This is “humanitarian science”, that is more philosophy and art, and this is hypocrisy to pretend that in this field any “hard” facts independent on world-view are possible. This is another illustration and interpretation of Orwell”s statement that “any humanitarian is hypocrite”.
I can add some recent experience. Two or three month ago in “Science” there was an article comparing men’s and women’s cognitive skills by comparing their respective IQ on a vast, ostensibly representative sample. Authors get two nice bell-shaped curves, so close to each other, that superimposed on the single graph they virtually coincided. But after cutting through three pages of dull technicalities, I burst with laugh. The whole trick was the definition of measured quantity. Such thing as intellect can not, of course, be quantified by single number. That is, when the procedure of measurement is defined, you get one of myriad of possible definition of what intellect is, depending on what type of task included into tests. Some deal with logic reasoning, some with pattern recognition, some with common sense and linguistic skills, and all are relative to “scores” that other people got on the same set of tasks. Every block contain many repetitive variation on the same theme. No wonder, that in such tests on any large sample a normal Gaussian distribution is obtained. The mean value is normalized to be 100 points; but really on different types of tasks different people get different results. Some are better in logic, some in geometry, some in arithmetic; proportion of different type of tasks is arbitrary, just as amount of time for every block. This arbitrariness means that any concrete “recipe” of the test is a definition of the measured quantity. Recipe by definition is adjusted so that the results were independent on sexual composition of the sample. If they are not, your recipe is turned down as “gender biased”. This means that in reality, the definition of intellectual capabilities by IQ is arbitrary chosen so it can not differentiate between the two sexes, and completely non-fit to study any real differences if they exist. But the next blunder in this study is even more egregious. Authors normalized experimental curves for male and female sub-samples by their respective standard deviations — for males by standard deviation for males, and for females by standard deviation for females! As every statistician know, Gaussian distribution has only two free parameters — mean and standard deviation, so when you normalize it by both you have the only one possible result — The Normalized Normal Distribution! It simply can not be different for different samples, irrespective of the nature of the set under study, and this has not anything to do with psychology, men and women, and with any physical reality at all. This is statistical artifact — Central Limit Theorem. It neither proves nor disproves anything, except complete mathematical illiteracy (innumeracy?) of the authors and ideological blindness of editorial board of this, in the old days reputable journal (“Science”), which, after its hijacking by Leftist Donald Kennedy, became a laughing-stock and megaphone of Leftist Agitprop.