Spurious Correlations and Spearman's g
Spearman's g, of course, is IQ. When various tests of mental abilities (verbal, mathematical, and geometrical, for example) are given, it is found that scores tend to be positively correlated, so that better performance on one type of test is correlated with better performance on others. Factor analysis is a tool for analyzing such correlations. If we measure a couple of parameters that are strongly correlated, like human height and weight, for example, and display them on a graph, they will tend to cluster in a roughly elliptical region along a line. Factor analysis finds the line of best bit. For poorly correlated variables, like perhaps time of day and height, clustering will be less evident.
Factor analysis works in higher dimensions too. The essential idea is to transform the original measurement variables into linear combinations that resolve the highest amount of variance.
If one measures a large number of variables, or simulates a large number of random variables, chance will dictate that some of them will appear to be correlated. This fact has led astray numerous critics of IQ, including Stephen Jay Gould (in The Mismeasure of Man and now Arun G., a smart and well-educated guy whose anti-IQ zealotry seems to make him forget his math.
So how does one separate such spurious correlations from real ones? The test is durability. Purely random correlations disappear when more measurements are made. Moreover, their domain is narrow. Two independent measurements being randomly correlated can happen - three, ten or more, not so likely. The correlations of IQ exams have persisted over hundreds of different exams and millions of test takers. Moreover, they have been shown to correlate strongly with educational and other measures of successful performance.
Spearman thought that the correlation pointed to a single general ('g') factor that explained the correlations. We now know that this is a bit simplistic. Factor analysis can tease out several factors that exhibit significant correlations, but g has never disappeared nor has it ever been adequately explained.