= General intelligence factor
{title2=$g$}
The <general intelligence factor>, conventionally $g$, is a <latent variable> used to account for shared variation among performances on different cognitive tasks. <Charles Spearman>'s 1904 analysis distinguished a common component from task-specific contributions, helping establish the historical basis of this approach.<General Intelligence, Objectively Determined and Measured by Charles Spearman>[1904, chapter V, correlation and common versus specific components]{ref}
A simple one-factor <measurement model> writes a standardized test score as
$$
Z_j=\lambda_j g+\epsilon_j,
$$
where $\lambda_j$ is a <factor loading>. Suppose all variables have zero <mean>, $\operatorname{Var}(g)=1$, each $\epsilon_j$ has zero <covariance> with $g$, and distinct residuals have zero <covariance> with one another. Then for different tests,
$$
\operatorname{Cov}(Z_i,Z_j)=\lambda_i\lambda_j.
$$
Thus a single shared component can explain correlated scores even when the tests contain different content. The covariance equation follows from the stated assumptions; it does not identify a unique biological cause of the correlations.
<Factor analysis> estimates such models and evaluates their fit. The <general intelligence factor> should also be distinguished from a first component obtained by <principal component analysis>: one is a proposed latent structure with residual assumptions, while the other is a variance-maximizing summary of observed scores. Interpretation requires the tasks, population and measurement assumptions, rather than a number alone.
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