Spearman's law of diminishing returns, a pattern still in dispute

0 The short answer

Spearman's law of diminishing returns is a well documented pattern with a disputed cause: in many samples, tests correlate less and the general factor explains less of the score differences at higher ability, but skewed scores and disturbance can reproduce part of the pattern, and several large samples do not show it. A meta-analysis of 106 articles and 408 studies found that mean correlations and g loadings fall as ability rises, by about 0.10 per 30 IQ points across all studies with IQ data and about 0.15 in the better quality half (the 0.10 is our arithmetic on the authors' table). The decisive distinction is between structure and accuracy: the law concerns how much of the variation a single factor explains, not whether a high score is correct, and it never turns a high score into a false one. For a reader with a high or uneven profile, wide spread across indices is an expected finding, not a defect of the test or of the person.

1 What Did Spearman Claim, and What Does the Law Not Say?

Spearman's claim is narrow: the shared factor behind ability tests accounts for a smaller share of the differences between people as ability rises, so the tests become less interchangeable among high scorers. The starting point is the observation that scores on different ability tests correlate positively, a pattern called the positive manifold. Spearman explained it with a general factor, g, and the extent to which a test reflects that factor is its g loading. The page on the g factor covers the definition in full. The law adds a second claim about the factor's strength. In the words of the 2017 meta-analysis, the cognitive ability differentiation hypothesis, also called Spearman's law of diminishing returns, proposes that ability tests are less correlated and less g loaded in higher ability populations (Blum and Holling, 2017, Intelligence).

The evidence Spearman offered in 1927 was a comparison of groups. As McGill summarizes it, he compared correlation matrices for 78 children of ordinary ability and 22 children of below average ability and found a mean correlation between tasks of .47 in the first group and .78 in the second (McGill, 2015). Blum and Holling report that Spearman explained the difference by saying that higher skilled groups had more mental "energy" and so benefited less from further increments of it. He compared this to a principle from economics, the law of diminishing returns, which holds that when one input to a process keeps rising and the others stay fixed, the growth of the output eventually slows. The name of the law is therefore an analogy, not a measured quantity.

Blum and Holling also point out what Spearman left open. He did not say whether the relation between ability and g saturation is linear, curved or logarithmic, and he did not restrict the claim to an age or an ability range. That vagueness explains why a century of studies reached different verdicts: each study had to decide for itself what the law predicts and how to detect it.

Three misreadings are worth removing at the start.

• The law does not say that high scorers have less intelligence or "less g" as a personal possession. A g loading is a property of a test within a group of people. The claim is about how strongly the tests move together in that group.
• The law does not say that g disappears at the top. The reported shifts are gradual, and the meta-analysis found a linear decline, not a cliff (details below).
• The law does not say that a high score is unreliable or inflated. Whether a score is accurate is a question of measurement error and norms, covered in the page on reliability and validity. The law concerns structure: how the scores of the different tests relate to each other.

A related idea is often confused with this one. The ability differentiation hypothesis concerns people at different ability levels. The age differentiation hypothesis concerns people at different ages, and the two are tangled because ability changes with age in childhood. The section on age below separates them. The page on what IQ measures and the page on the history of IQ testing give the wider background for both.

2 How Is the Law Tested, and Why Does the Method Matter?

Every test of the law compares something across ability levels, and the way the comparison is built decides what a result can show. The oldest method sorts people into groups by score, then compares how strongly the tests correlate within each group. The measure can be the mean correlation among tests or the share of variance captured by the first unrotated factor, and Blum and Holling describe the average correlation as equivalent to the total variance explained by the first unrotated factor, which is its eigenvalue divided by the number of tests.

Three points follow from the table. First, a result obtained by splitting at a cut score depends on where the cut is. In their two Danish samples, Hartmann and Teasdale found that the outcome depended on the cut points and the tests used (details in the next section). Second, the newer designs were created to avoid the arbitrary split. Reynolds, Keith and Beretvas describe the limitation they wanted to remove as the "a priori division of respondents into separate ability or IQ groups", and they used factor mixture modeling, in which the model finds the classes (Reynolds, Keith and Beretvas, 2010). Third, the contemporary criteria for the law are specific. In the simulation work discussed below, three are examined: a negatively skewed latent g, a positive correlation between g and the log of the residual variance, and nonlinear loadings that bend along the g scale (Sorjonen and Melin, 2020).

Why does it matter that a mean correlation is the usual summary? Because it is easy to compute from published tables, which is why a meta-analysis is possible at all. The cost is that a mean correlation also falls when scores are noisier and when the range of scores is restricted, and sorting people by score restricts the range inside each group. Skew and disturbance, the two artifacts examined by the simulation studies below, work through the same summary. The page on how IQ scores are normed explains the standardization samples that many of these studies reused, and the page on the standard deviation of 15 explains why a restricted range changes the numbers.

3 What Did the Early Studies Find, and Where Did They Stop Agreeing?

The modern literature began with a rediscovery in 1989 and split within fifteen years into studies that confirmed the pattern and large samples that could not. Reynolds and colleagues write that Detterman and Daniel were credited with rediscovering the law after Spearman's original observation. Detterman and Daniel reported two studies with a combined 4,261 participants. In both, low IQ groups showed higher correlations than high IQ groups, and the same direction appeared in three kinds of correlation: among IQ subtests, among measures of cognitive abilities, and between IQ and those measures (Detterman and Daniel, 1989, Intelligence). McGill adds that they divided samples given the Wechsler scales into five ability groups and found that subtest intercorrelations decreased from the low to the high groups.

The first large failures to replicate came from military data. Hartmann and Teasdale tested two samples of Danish military draftees (33,833 and 25,020, with 95 percent aged 18 or 19) using four subtests and two cut points. They did not confirm the law. The data instead showed higher g saturation in the high ability group, and the result depended on the cut points and the tests (Hartmann and Teasdale, 2004, Intelligence). Hartmann and Reuter then used 6,980 children aged 12 to 16 from the 1997 cohort of the National Longitudinal Survey of Youth, tested on 12 computer adaptive subtests of the Armed Services Vocational Aptitude Battery. They split the group either by total score or by a single subtest, and could not confirm the law under either method (Hartmann and Reuter, 2006, Intelligence).

Between those poles sits a study that found the predicted direction with very small effects. te Nijenhuis and Hartmann used five samples of Dutch and immigrant participants, in which large mean score differences created the ability groups. Variance, reliability and factor structure were highly comparable across groups, and the effect sizes were in the predicted direction but small: from negative 0.012 to 0.097, averaging 0.02 across an ability difference of 1 to 1.5 standard deviations (te Nijenhuis and Hartmann, 2006, Intelligence). The abstract does not make the unit of the effect size explicit, so the numbers are best read as a statement of order of magnitude: small.

Why did capable researchers using large samples reach opposite conclusions? The studies differed in age (children, adolescents, young adults), in test batteries (Wechsler and Kaufman scales against military aptitude batteries), in how groups were formed and in what was compared. A military draft sample is a near complete cohort of one age, which is a different population from a norming sample spanning many ages. None of these differences is a flaw, but together they mean that the law cannot be tested once and settled, and they are the reason a meta-analysis was needed.

4 What Does the 2017 Meta-Analysis Show?

Blum and Holling found that the mean correlation among ability tests falls along a straight line as the mean IQ of a study sample rises, with a slope of about 0.0033 per IQ point across all studies and about 0.005 per point in the better quality studies. Their search covered a century of publications and ended with 106 articles describing 408 studies and 188,489 individuals. Of the 408 studies, 98 reported data that could be tied to IQ and 394 reported the age of the respondents, so the IQ question rests on a quarter of the studies. The effect size was the average correlation among the tests in each correlation matrix, and a three level multilevel model handled the fact that several studies came from the same article (Blum and Holling, 2017).

Without any moderator, the overall mean correlation was .3374 (standard error .0118), which corresponds to about 41 percent of variance for the first factor, and the spread between studies was large (a standard deviation of .1104 for the effect). The funnel plot gave no hint of publication bias, and the three potential outliers did not change the results. With the mean IQ of the study group as the predictor, the regression was .6493 minus .0033 times the mean IQ, significant at the .001 level. Adding quadratic and cubic terms did not help, so the authors treated the relation as linear. Dropping the single study whose mean IQ exceeded 140, which was an influential case, gave .6730 minus .0036 times the mean IQ.

Their Table 2 turns the regressions into predicted mean correlations at fixed group IQs. The first column below is for all 98 studies with IQ data, the second for the half of studies with higher methodological quality scores, and the third for the best quartile.

Reading the table needs three cautions. First, the drop from a group mean of 100 to a group mean of 130 is .3193 minus .2203, or about 0.10, in the full set, and .3165 minus .1635, or about 0.15, in the better quality half (our arithmetic on the authors' table; the authors themselves state the figure of approximately .15 for a difference of two standard deviations, 30 IQ points, in the better half). The full set shows a relative drop of about 31 percent from 100 to 130, our arithmetic again. Second, the lowest quality subgroup, the studies scoring below the median on the authors' quality scale, showed no significant slope (.1007 minus .0023 times the mean IQ). The pattern is therefore strongest in the studies the authors graded better, which supports the result, but the authors also note that the quality scale had a mean of only 1.25 on a possible maximum of 4 and that no study reached 3. Third, only one study had a mean IQ above 140, so the 145 row is an extension of the line toward very little data.

Two further results matter for interpretation. The year of publication had no significant effect, so the relation did not appear to change across the century of data. And the mean IQ of a study and its mean age were almost uncorrelated across studies (r of negative .05), so the two predictors carry different information. The authors conclude that the meta-analysis supports the differentiation hypothesis and that "we cannot assume an invariant cognitive structure for different ability and age levels."

That conclusion is more modest than it sounds. It is a statement about group means in published correlation matrices. It does not tell any single person how their own tests will relate, and it inherits the design of the studies it pools, many of which sorted people into groups by score, the approach the artifact critiques target. The authors acknowledge that it remains uncertain how much extra factorial complexity a decline of .15 would produce in practice. The page on reliability and validity explains why a correlation among tests is a statement about a group and not about a person.

5 Which Later Studies Agree, and Which Do Not?

Studies that avoided cut scores or used very large samples are split: factor mixture models and two large applicant samples agree with the pattern, a twin study finds only a weak trend in children, and a working memory study finds none. The table summarizes the studies this page relies on, with the sample and the main result as each paper reports it.

Several details deserve a closer look. In the twin study, the pooled first component also rose with age, from 48.45 percent at age 7 to 54.31 percent at age 10 (Arden and Plomin, 2007, Personality and Individual Differences). The authors report F ratios and not standardized effect sizes, and they describe the consistent advantage of the low ability group as a trend, not a robust effect.

The KABC-II study is the cleanest model based test because the classes were found by the model instead of by a cut score. Its abstract summarizes the result as a smaller g variance in classes with higher g means. The same paper breaks the variance down. In the first subsample the common factors explained .54 of the average subtest's variance in the low class and .39 in the high class, and the share explained by broad abilities hardly moved (.16 against .18). The loss was therefore a loss in the general factor's share, while the remainder, which holds subtest specific variance and measurement error, grew from .46 to .61. The difference in latent g means between classes was large, d = 0.77 (Reynolds, Keith and Beretvas, 2010).

The DAS-II study matters for readers of index scores because it examined composites, not single subtests, across four age groups. Reynolds concludes that specific ability patterns deserve more attention at higher levels of g (Reynolds, 2013, School Psychology Quarterly, as summarized in its ERIC record).

The Russian samples address the artifact critique directly. Sugonyaev and Radchenko split each sample by total score and then transformed the scores in each half sample to normalized stens separately, to equate variability and minimize skewness, in answer to the criticism of Murray and colleagues. They report that the findings were consistent with the law and that the result held in a second sample, and they conclude that it refutes the suggestion of an artifactual origin (Sugonyaev and Radchenko, 2018). That is the authors' own conclusion, and the next section shows why not everyone accepts that this closes the question.

Finally, the working memory result shows that the law, if real, need not hold for every kind of ability. Kroczek and colleagues tested the relation between fluid intelligence and working memory in three kinds of tasks, tapping short term storage, attention control and relational integration, and found no significant effect in any. Each task shared virtually the same amount of variance with fluid intelligence in low and high ability people, which the authors take to suggest that the two rely on the same mechanism (Kroczek, Ociepka and Chuderski, 2016, Polish Psychological Bulletin). The page on working memory tests for adults covers what those tasks measure, and the page on fluid reasoning tests covers the fluid side (Gf in the CHC model).

6 Is the Pattern a Statistical Artifact?

Two simulation studies show that ordinary features of test data, skewed score distributions and disturbances that affect some test takers more than others, can produce the pattern without any change in g, so the law's support depends on methods that survive those artifacts. The first critique concerns skew. Murray, Dixon and Johnson separated traditional methods of testing the law, from which most of its supporting evidence comes, from contemporary methods. They showed that skewed distributions of observed scores can produce effects that look like the law, and that negative skew produces the reverse. The skew involved was modest, within values usually considered acceptable for parametric analysis. When contemporary methods were applied to the simulated data, they found heteroscedastic residuals but no nonlinear g loadings and no skewed g. Part of the skew effect was reproduced in real data from the Minnesota Study of Twins Reared Apart. The authors conclude that traditional methods cannot separate skew artifacts from true effects, which calls the empirical support for the law into question (Murray, Dixon and Johnson, 2013, Intelligence, as recorded in the University of Edinburgh research portal).

The second critique concerns disturbance. Sorjonen and Melin simulated 10,000 people in each run, with 15 test scores driven by a true g drawn from a normal distribution, and multiplied the scores by a disturbance factor between 0 and 1 that varied between people. Their examples of disturbance are low motivation, illness and linguistic confusion. With the traditional method, the explained variance fell as the selection mean rose in every disturbed condition, which is the pattern the law predicts, so disturbance alone sufficed. With the contemporary criteria (1,000 simulations per condition), a negatively skewed disturbance, in which a minority of people perform well below their true ability, satisfied every criterion in 81 percent of runs, while no disturbance satisfied them in any run. Normal and positively skewed disturbance never met the full set, because g came out positively skewed (Sorjonen and Melin, 2020, PeerJ).

The authors are careful about what this means. They note that a negative skew of g offers some protection against false positives, and that a negatively skewed disturbance is nonetheless realistic. They do not claim that all supporting findings are artifacts, and they say that the law may still reflect a real phenomenon even if disturbance contributes. They recommend that researchers try to eliminate disturbing factors, while acknowledging that true ability is never measured perfectly.

Where does that leave the debate? Three statements are safe. The pattern of lower correlations at higher ability appears in a large share of published studies, in the meta-analysis, and in at least some designs that avoid cut scores. The traditional methods that produced most of the early evidence are vulnerable to skew, and a negatively skewed disturbance can imitate even the contemporary criteria. And no source examined here says that the law is either proved or refuted. The practical consequence for a reader is the same in both cases. Whatever its cause, the observed pattern is that scores on different tests are somewhat less tightly linked among high scorers, and a profile should be read with that in mind. The page on whether IQ is real sets this debate beside the broader critiques of the construct, and the page on what the g factor does not mean covers the common overstatements.

7 Does the Law Change How Well a Test Predicts Outcomes?

At the level of prediction the evidence is thinner and mixed: one study of a standardization sample found that the general score explained less achievement variance in an above average group, while a very large study of aptitude tests and grades found that linear effects dominated. The structural claim of the law is about how tests relate to each other. A practical reader cares more about a second claim, that a g loaded score predicts outcomes less well at higher ability. The two are linked in theory, because a test that reflects g less should carry less of g's predictive power, but they are tested with different data.

McGill examined this directly in the standardization sample of the Woodcock-Johnson III (N = 4,722), using hierarchical regression to ask how much achievement variance the general ability score explained in below average, average and above average classes. The general score explained significantly less in the above average group (R squared of .11 to .20) than in the below average group (.24 to .32). The broad ability clusters added more prediction at the higher level for reading and written language, which is the pattern the law predicts, but the mathematics models showed a discordant pattern in which the cluster contribution fell. The author lists limits plainly: an archived standardization sample, arbitrary ability classes that restrict the range within each class, small subgroups and a sample of children and adolescents. He also states that more research is needed to know whether the results are an artifact or a real challenge to validity estimates (McGill, 2015).

A different design looked for the loss of predictive power with a different tool. Coyle compared linear and quadratic relations among g (estimated from the Armed Services Vocational Aptitude Battery), aptitude tests and college grade point averages in large survey and admissions datasets, and the paper reports, in its own title, that linear effects dominate (Coyle, 2015, Intelligence). Only the title and bibliographic record of that paper were available to this review, so its sample sizes and effect sizes are not repeated here. A relation that stays close to a straight line is what one would expect if scores keep their predictive power as ability rises, which is the opposite of what the law would suggest for grades (our reading of the title, not a quotation). The two studies used different designs, so they are not a direct contradiction. McGill split a standardization sample into classes and compared the share of achievement variance explained, while Coyle looked for bends in the relation between continuous scores and grades, and neither is a test of the other.

What should a reader take from this? The prediction level evidence does not support the statement that high ability stops mattering, and it does not support the statement that a general score is as informative about outcomes at every level. It is mixed, from different designs, and it does not settle either. The pages on IQ and academic achievement and on IQ and job performance give the meta-analytic evidence on prediction at the population level, where the question is how strongly scores relate to outcomes across everyone, not what happens within one ability class.

8 Is This About Ability Level or About Age?

Ability differentiation and age differentiation are different hypotheses, and the studies that tested both do not point the same way. The age version says that the structure of abilities changes across the life span, from a fairly unified ability to a loosely organized set of abilities. Blum and Holling describe how the two ideas intersect: a younger age is usually marked by a lower ability level, so a comparison of children with adults mixes both. Spearman himself compared children and adults but proposed a decline of g saturation with ability and not with age.

Their meta-analysis addressed the two predictors separately. With the mean age of the study group as the predictor, the mean correlation among tests rose by about .0012 per year of age, which means that for samples whose mean age is 10 years higher the expected mean correlation is .01 higher. That is a linear increase across the life span, the opposite direction from the ability result. The authors discuss two readings. For later life, falling ability could explain the rise in g saturation, which would fit the ability version. For childhood, where ability rises quickly, a rising correlation contradicts what the age hypothesis predicts, which is a decrease. They also note two limits for the childhood data: many studies reported the school grade and not the age, so age had to be estimated, and group means can hide ability differences within a group (Blum and Holling, 2017).

The twin study points the same way in a narrower setting. The first principal component explained more variance as the children grew older, from 48.45 percent at age 7 to 54.31 percent at age 10, which runs opposite to a decline in g saturation with age in childhood. The practical lesson is that a finding labeled "differentiation" has to be read with its variable in mind: ability level, age, or both mixed together. The page on how IQ changes with age explains why scores and structure both shift across the life span.

9 What Does the Law Imply for Reading a High Score or an Uneven Profile?

If the law holds even in part, wider spread across indices is expected at higher levels, so an uneven profile in a high scorer is a normal finding to read with its confidence intervals, and not a sign that the Full Scale score is wrong. The step from the evidence to a profile needs one piece of arithmetic, and the arithmetic is ours, not the authors'. For two scores with the same standard deviation, the standard deviation of their difference is that standard deviation times the square root of two times one minus the correlation. With a standard deviation of 15, a correlation of .7 gives a spread of differences of about 11.6 points, a correlation of .5 gives 15.0 and a correlation of .3 gives about 17.7. Lower correlations among abilities therefore mean wider gaps between them. The meta-analysis reports correlations among tests, not among index scores, and high scoring groups also have a narrower spread of scores, which works in the other direction, so the net effect on gap size is an inference and not a published result. The page on index score differences works through critical values and base rates for gaps in detail.

There is direct evidence at the level of composite scores. In the DAS-II study, g loadings of five composites decreased as the level of g rose, and the author suggested that specific ability patterns deserve more attention at higher levels of g (Reynolds, 2013). McGill reached a similar practical conclusion from the Woodcock-Johnson data, that more interpretive weight for broad abilities may be warranted for people with higher general ability. In the KABC-II mixture study, the share of subtest variance carried by the broad abilities stayed about the same across classes while g's share fell, so the broad abilities carry relatively more of the story in the higher class.

None of this makes a high Full Scale score less meaningful or less accurate. The studies describe a smaller share of variance carried by g, not zero, and the overall score still summarizes standing across abilities. The page on the Full Scale IQ explains what the composite summarizes, and the page on the General Ability Index covers a composite built from a subset of subtests. A reader who sees a wide spread between a verbal and a quantitative score can also compare the two directly; the page on verbal versus quantitative ability looks at that contrast.

A practical way to read a spread-out profile, in order:

1. Compare each index with its own interval. Each index score is an estimate with error around it, and a gap smaller than the combined error is not evidence of a real difference.
2. Describe gaps in IQ points, not percentile points. Equal steps in percentile are unequal steps in IQ, a point made in the page on IQ scores and percentiles.
3. Look for a pattern before an outlier. Two indices that move together tell a different story from one low index among five similar ones.
4. Expect some scatter. In a high scoring profile, a gap that would look striking in a profile near the mean can be ordinary.
5. Keep the question separate from a diagnosis. A group statistic about scatter is never a statement about why one person's gap appears, and nothing on this page is a clinical statement.

The page on gifted adults discusses uneven profiles in that group, and the page on twice exceptional people covers the narrower, clinical situation where a strong ability and a disability coexist, which this page does not address.

10 What Does the Law Not Imply?

The law does not support the common stronger readings, such as that g vanishes at the top, that a high score is inflated, that high ability stops mattering or that the pattern has been shown to be an artifact. The table sets each reading against what the studies in this page report.

The last row deserves emphasis. Every study here works with groups: classes, halves, study means. A statement such as "tests correlate less in high ability groups" cannot be turned into "your own tests will correlate less". The page on the g factor explains why a statistical factor is a feature of a population and not a mechanism inside one person's head, and the page on common myths about IQ tests lists other readings that the evidence does not support.

11 What Does an ACIS Report Show, and What Should You Not Conclude From It?

An ACIS report shows a Full Scale IQ and six indices, each with a percentile and a 95 percent confidence interval, which gives a reader the information needed to see spread across indices, but it does not test the law and it cannot say why a profile is uneven. The assessment has 20 subtests in six cognitive domains drawn from the Cattell-Horn-Carroll model: verbal comprehension (VCI), fluid reasoning (FRI), quantitative reasoning (QRI), visual spatial processing (VSI), working memory (WMI) and processing speed (PSI). Indices and the Full Scale IQ are on the standard scale with a mean of 100 and a standard deviation of 15, and subtest results are scaled scores from 1 to 19 with a mean of 10 and a standard deviation of 3. The page on the six cognitive domains and the page on the CHC model explain the structure, and the page on what each subtest measures covers the 20 tasks.

ACIS sells three forms, each a one time payment with no subscription, with prices read on October 20, 2026 and subject to change: Quick at 15 dollars (6 subtests in 3 domains, about 45 minutes), Optimized at 30 dollars (13 subtests in 5 domains, about 110 minutes) and Full Scale at 50 dollars (all 20 subtests in six domains, about 175 minutes). Breaks are allowed, a free trial needs no card, there is a 5 day quality guarantee and a test must be completed within 30 days. This bears on the topic directly: a spread across six indices can only be seen with the Full Scale form, because the shorter forms cover fewer domains.

The limits matter as much as the features. The test is online and unsupervised. It is not a clinical or diagnostic instrument, and it is not for hiring, school accommodations or admission to high IQ societies. It is available in English only. This page uses no ACIS data and makes no statement about how the law behaves in ACIS scores. Readers who want documentation of the scoring and norms can read the technical manual. A reader who sees an uneven profile in a report should read it as the section above describes: with its intervals, in IQ points, as a pattern, and without a diagnosis.

12 What Do the Standards and the APA Guidelines Say About Reading Scores Across Ability Levels?

The Standards for Educational and Psychological Testing (AERA, APA and NCME, 2014) and the American Psychological Association guidelines both ask that an interpretation of a score or a profile be backed by evidence for that specific interpretation, which is the right test for any claim built on the law. The Standards make two demands that bear on the topic. Standard 1.13 says that when the rationale for an interpretation depends on premises about the relationships among parts of a test, evidence about the internal structure should be provided, and its comment adds that when a test provides more than one score, the interrelationships of those scores should be shown to be consistent with the construct being assessed. A composite score rests on exactly such a premise, and the law is a reminder that the strength of those relationships may not be the same at every ability level. Standard 1.14 says that when interpretation of subscores, score differences or profiles is suggested, the rationale and relevant evidence in support of the interpretation should be provided. Its comment states that evidence for the validity of two or more separate scores would not necessarily justify an interpretation of the difference between them (AERA, APA and NCME, 2014).

For a reader, the two standards translate into one practical rule: a gap between two indices needs its own evidence before it is called meaningful, and a high or uneven profile is not exempt from that rule or condemned by it. The meta-analysis and the model based studies are evidence about structure at group level. They are not evidence about the validity of one person's difference between two indices.

The APA Guidelines for Psychological Assessment and Evaluation, approved by the APA Council of Representatives in March 2020, speak to professionals, but their statements are useful to anyone reading a report. On interpretation, they say that psychologists strive to understand the conceptual meaning of scores and the technical range of interpretation of any given set of scores, and that accurate interpretation depends on integrating multiple sources of data. They also name overinterpretation, inconsistent interpretation and selective interpretation as common errors in reporting results (APA, 2020). A single online score, however carefully made, is one source of data. For any decision that matters, such as an educational placement or a clinical question, the appropriate step is a conversation with a qualified professional, who can combine the score with history, observation and other tests.

13 Sources Behind This Page

Every figure above is traceable to one of the following, and each is linked at the point where it is used. Where this page states a figure that is not in a source, it says so and labels it as our arithmetic. For several articles only the abstract was read, in library, repository or search records, and the full text was not opened; the figures taken from them are limited to what the abstract states.

• American Educational Research Association, American Psychological Association and National Council on Measurement in Education. Standards for Educational and Psychological Testing. AERA, 2014, Standards 1.13 and 1.14 and their comments.
• American Psychological Association, APA Task Force on Psychological Assessment and Evaluation Guidelines. APA Guidelines for Psychological Assessment and Evaluation, approved by the APA Council of Representatives in March 2020, the passage on interpretation.
• Arden R, Plomin R. Scant evidence for Spearman's law of diminishing returns in middle childhood. Personality and Individual Differences, 2007, volume 42, issue 4, pages 743 to 753.
• Blum D, Holling H. Spearman's law of diminishing returns. A meta-analysis. Intelligence, 2017, volume 65, pages 60 to 66.
• Coyle TR. Relations among general intelligence (g), aptitude tests, and GPA: Linear effects dominate. Intelligence, 2015, volume 53, pages 16 to 22.
• Detterman DK, Daniel MH. Correlations of mental tests with each other and with cognitive variables are highest for low IQ groups. Intelligence, 1989, volume 13, issue 4, pages 349 to 359.
• Hartmann P, Reuter M. Spearman's "Law of Diminishing Returns" tested with two methods. Intelligence, 2006, volume 34, issue 1, pages 47 to 62.
• Hartmann P, Teasdale TW. A test of Spearman's "Law of Diminishing Returns" in two large samples of Danish military draftees. Intelligence, 2004, volume 32, issue 5, pages 499 to 508.
• Kroczek B, Ociepka M, Chuderski A. No Spearman's Law of Diminishing Returns for the working memory and intelligence relationship. Polish Psychological Bulletin, 2016, volume 47, issue 1, pages 73 to 80.
• McGill RJ. Spearman's Law of Diminishing Returns (SLODR): Examining effects at the level of prediction. Journal of Psychology and Behavioral Science, 2015, volume 3, issue 1, pages 24 to 36.
• Murray AL, Dixon H, Johnson W. Spearman's law of diminishing returns: A statistical artifact? Intelligence, 2013, volume 41, issue 5, pages 439 to 451.
• Reynolds MR. Interpreting the g loadings of intelligence test composite scores in light of Spearman's law of diminishing returns. School Psychology Quarterly, 2013, volume 28, issue 1, pages 63 to 76.
• Reynolds MR, Keith TZ, Beretvas SN. Use of factor mixture modeling to capture Spearman's law of diminishing returns. Intelligence, 2010, volume 38, issue 2, pages 231 to 241.
• Sorjonen K, Melin B. Diminishing returns as a function of disturbance. PeerJ, 2020, volume 8, article e9490.
• Sugonyaev K, Radchenko Y. Spearman's Law of Diminishing Returns: Investigation on large-scales Russian samples. Bulletin of the South Ural State University, series Psychology, 2018, volume 11, issue 1, pages 5 to 21.
• te Nijenhuis J, Hartmann P. Spearman's "Law of Diminishing Returns" in samples of Dutch and immigrant children and adults. Intelligence, 2006, volume 34, issue 5, pages 437 to 447.

14 Frequently Asked Questions