Spearman's Law of Diminishing Returns: Does g Weaken at High Ability?

Two people take the same battery. One scores 85 on a mean-100, standard-deviation-15 scale; the other scores 135. Both walk away with a single number that is supposed to describe how they did across every part of the test. There is a century-old argument in psychometrics that those two numbers are not doing equally good jobs — that the higher one is a worse summary of the person it describes. The claim is called Spearman's Law of Diminishing Returns, and it is one of the few ideas in intelligence research where the popular version and the evidence have drifted noticeably apart.

The observation Spearman made

Charles Spearman is the reason anyone talks about a general factor at all. His central finding, first set out in 1904 and developed through the 1920s, was that people who do well on one kind of mental test tend to do well on the others — vocabulary, arithmetic, spatial puzzles, memory span. Every pair of tests correlates positively, a pattern known as the positive manifold, and the shared variance is what the letter g refers to. Our explainer on what the g factor actually is covers that ground in full.

In his 1927 book The Abilities of Man, Spearman added a qualification that is far less often quoted. The strength of those correlations, he noted, was not constant across the range of ability. Among people who did poorly, the tests hung together tightly. Among people who did well, they came apart. He described the general factor as behaving like a resource that mattered enormously when it was scarce and mattered progressively less once there was plenty of it — hence the name later attached to the idea.

What "diminishing returns" means in practice

A correlation between two subtests describes how well one predicts the other. If verbal comprehension and matrix reasoning correlate at 0.55 in a group, knowing one score tells you a fair amount about the other. If they correlate at 0.30, it tells you much less. Spearman's claim is that the first figure describes low scorers and the second describes high scorers — and if that is right, several things follow:

• The general factor explains a smaller share of the differences between able people than between less able people.
• A full-scale score, which is essentially a weighted average of the parts, therefore loses more information at the top of the range than in the middle.
• Two people with the same high full-scale figure may have arrived there by genuinely different routes — one on verbal strength, another on spatial — more often than two people with the same average figure.
• Selecting on a single cut score does less to make a group homogeneous the higher the cut is set.

The evidence, which is genuinely mixed

The modern literature starts with a 1989 study by Douglas Detterman and Mark Daniel, who split standardisation samples by ability and compared the subtest correlation matrices. In the lowest-ability groups the average intercorrelation was in the region of twice what it was in the highest-ability groups. That result is the one usually cited, and it is a large effect — large enough that if it were the whole story, test manuals would need to say so on the first page.

It is not the whole story. Splitting a sample by ability and then measuring correlations inside each slice introduces a well-known statistical artefact: restricting the range of a variable reduces its correlations mechanically, whether or not anything about the underlying structure has changed. Much of the work since has been an argument about how to separate a real change in structure from that artefact. Approaches that model the change continuously rather than by cutting the sample into groups — moderated factor analysis, and methods that estimate the factor structure locally along the ability range — generally do find the effect in Spearman's direction, but smaller than the 1989 figures imply. A 2017 meta-analysis by Blum and Holling in Intelligence concluded that support for the law exists across a substantial body of studies while being moderated by sample and method. Other analyses, including work on how abilities differentiate across the lifespan, have found age effects considerably clearer than ability effects.

Why it matters if you have a high score

The practical reading is narrower than the idea's reputation. Nobody serious claims g stops existing at high ability, or that a 135 is uninformative. The claim is about how much of the story one number tells, and it bears on a specific set of decisions:

• Above roughly the 98th percentile, the index scores underneath the full-scale figure deserve more weight than they do in the middle of the range — a point that stands on its own regardless of how the SLODR argument resolves.
• A large gap between two index scores is more common at high ability and is not automatically a sign of a problem.
• Selection at a single high cut score produces a group that is more internally varied than the shared number suggests, which matters for gifted programmes designed as though the admitted students were alike.
• Comparing two high scores from different batteries is even less safe than comparing two average ones, because the profiles behind them can differ more.

None of that changes what a score is. It remains a position in a reference sample, carrying a published measurement error, and our note on why a reported IQ is a range rather than a point applies at 135 exactly as it does at 100. What the law adds, if it holds at the smaller size the better studies suggest, is a second reason to read the parts: not only is the total imprecise, it is summarising components that have started to go their own ways. The mechanics of that breakdown are set out in our piece on full-scale scores versus index scores.

What the ceiling looks like on a real report

The abstract argument has a concrete counterpart on the page of any score report, and it is worth knowing where to look. On the Wechsler batteries each subtest is reported as a scaled score with a mean of 10 and a standard deviation of 3, running from 1 to 19. Nineteen is not a description of performance; it is the top of the scale. A person who answers every item correctly and a person who answers nearly every item correctly both receive it.

That matters because a full-scale figure built from several subtests sitting at or near their ceiling is averaging numbers that have stopped discriminating. It is why extended norms exist for gifted assessment, allowing scaled scores above 19 to be derived where the standard table has run out, and why an evaluator working at the top of the range will often reach for a battery with more difficult items rather than trusting the total from a general-population instrument. If part of the reason correlations shrink at high ability is that the test has run out of room, then the fix is a harder test rather than a new theory — and separating those two explanations is exactly what makes this literature difficult.

How to read a high result properly

Three habits, in order. Convert the figure to a percentile against the scale it was measured on, because a bare number carries no meaning without its mean and standard deviation — our IQ percentile calculator does that conversion. Then look at the index scores rather than only the total, and note where the profile is uneven. Then treat both the total and each index as bands rather than points. A person described as "135" is more usefully described as someone in roughly the top 1 per cent overall, with particular strengths that the single figure has averaged away. A properly structured assessment reports those parts alongside the total, which is the form worth reading.

Spearman's qualification has aged better as a caution than as a law. The strong version — that g fades away at the top — is not what the careful studies support. The weak version is both supported and useful: the higher a score sits, the more work a single number is being asked to do, and the more the interesting information has moved into the shape underneath it. That is a good reason to ask what a test measured rather than only what it totalled, which is a reasonable thing to ask of any score at any level.

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