Editing G factor (psychometrics) (section)

==Cognitive ability testing==
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|+ Spearman's correlation matrix for six measures of school performance. All the correlations are positive, the ''positive manifold'' phenomenon. The bottom row shows the ''g'' loadings of each performance measure.<ref>Adapted from Jensen 1998, 24. The correlation matrix was originally published in Spearman 1904, and it is based on the school performance of a sample of English children. While this analysis is historically important and has been highly influential, it does not meet modern technical standards. See Mackintosh 2011, 44''ff''. and Horn & McArdle 2007 for discussion of Spearman's methods.</ref>
|-
! scope="col" | 
! scope="col" | Classics
! scope="col" | French
! scope="col" | English
! scope="col" | Math
! scope="col" | Pitch
! scope="col" | Music
|-
! scope="row" | Classics
| – || || || || ||
|-
! scope="row" | French
| .83 || – || || || ||
|-
! scope="row" | English
| .78 || .67 || – || || ||
|-
! scope="row" | Math
| .70 || .67 || .64|| – || ||
|-
! scope="row" | Pitch discrimination
| .66 || .65 || .54 || .45 || – ||
|-
! scope="row" | Music
| .63 || .57 || .51 || .51 || .40 || –
|- style="background:lightblue"
! scope="row" | ''g''
| .958 || .882 || .803 || .750 || .673 || .646
|}
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{| class="wikitable" style="text-align:center; width:200px; float:right; margin:0 2em 2em;"
|+ Subtest intercorrelations in a sample of Scottish subjects who completed the [[WAIS-R]] battery. The subtests are Vocabulary, Similarities, Information, Comprehension, Picture arrangement, Block design, Arithmetic, Picture completion, Digit span, Object assembly, and Digit symbol. The bottom row shows the ''g'' loadings of each subtest.
<ref>Adapted from Chabris 2007, Table 19.1.</ref>
|-
! scope="col" | 
! scope="col" | V
! scope="col" | S
! scope="col" | I
! scope="col" | C
! scope="col" | PA
! scope="col" | BD
! scope="col" | A
! scope="col" | PC
! scope="col" | DSp
! scope="col" | OA
! scope="col" | DS
|-
! scope="row" | V
| – || || || || || || || || || || 
|-
! scope="row" | S
|  .67||- || || || || || || || || ||
|-
! scope="row" | I
|  .72 ||.59||- || || || || || || || ||
|-
! scope="row" | C
|  .70 ||.58 ||.59|| -|| || || || || || ||
|-
! scope="row" | PA
|  .51 ||.53 ||.50 ||.42|| -|| || || || || ||
|-
! scope="row" | BD
|  .45 ||.46 ||.45 ||.39 ||.43|| -|| || || || ||
|-
! scope="row" | A
|  .48|| .43|| .55|| .45 ||.41 ||.44|| – || || || ||
|-
! scope="row" | PC
|  .49 ||.52|| .52 ||.46 ||.48 ||.45 ||.30|| -|| || ||
|-
! scope="row" | DSp
|  .46 ||.40 ||.36|| .36 ||.31 ||.32 ||.47|| .23|| -|| ||
|-
! scope="row" | OA
|  .32 ||.40 ||.32 ||.29|| .36|| .58 ||.33 ||.41 ||.14||- ||
|-
! scope="row" | DS
|  .32 ||.33 ||.26|| .30|| .28 ||.36 ||.28 ||.26|| .27 ||.25||-
|- style="background:lightblue"
! scope="row" | ''g''
| .83 || .80 || .80 || .75 || .70 || .70 || .68 || .68 || .56 || .56 || .48
|}

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[[File:Correlations between mental tests.svg|alt=Heat-map of mental test from the above given data|thumb|Correlations between mental tests|550x550px]]
[[Cognitive ability]] tests are designed to measure different aspects of cognition. Specific domains assessed by tests include mathematical skill, verbal fluency, [[spatial visualization ability|spatial visualization]], and memory, among others. However, individuals who excel at one type of test tend to excel at other kinds of tests, too, while those who do poorly on one test tend to do so on all tests, regardless of the tests' contents.<ref>Gottfredson 1998</ref> The English psychologist Charles Spearman was the first to describe this phenomenon.<ref>{{Cite book|last=Deary|first=I. J.|title=Intelligence. A Very Short Introduction|publisher=Oxford University Press|year=2001|isbn=9780192893215|pages=12}}</ref> In a famous research paper published in 1904,<ref>Spearman 1904</ref> he observed that children's performance measures across seemingly unrelated school subjects were positively correlated. This finding has since been replicated numerous times. The consistent finding of universally positive [[correlation matrix|correlation matrices]] of mental test results (or the "positive manifold"), despite large differences in tests' contents, has been described as "arguably the most replicated result in all psychology".<ref>Deary 2000, 6</ref> Zero or negative correlations between tests suggest the presence of [[sampling error]] or restriction of the range of ability in the sample studied.<ref name="jensen1992">Jensen 1992</ref>

Using [[factor analysis]] or related statistical methods, it is possible to identify a single common factor that can be regarded as a summary variable characterizing the correlations between all the different tests in a test battery. Spearman referred to this common factor as the ''general factor'', or simply ''g''. (By convention, ''g'' is always printed as a lower case italic.) Mathematically, the ''g'' factor is ''a source of variance among individuals'', which means that one cannot meaningfully speak of any one individual's mental abilities consisting of ''g'' or other factors to any specified degree. One can only speak of an individual's standing on ''g'' (or other factors) compared to other individuals in a relevant population.<ref name="jensen1992"/><ref>Jensen 1998, 28</ref><ref name="maas">van deer Maas et al. 2006</ref>

Different tests in a test battery may correlate with (or "load onto") the ''g'' factor of the battery to different degrees. These correlations are known as ''g'' loadings. An individual test taker's ''g'' factor score, representing their relative standing on the ''g'' factor in the total group of individuals, can be estimated using the ''g'' loadings. Full-scale IQ scores from a test battery will usually be highly correlated with ''g'' factor scores, and they are often regarded as estimates of ''g''. For example, the correlations between ''g'' factor scores and full-scale IQ scores from [[David Wechsler]]'s tests have been found to be greater than .95.<ref name="kamphaus2005"/><ref name="jensen1992"/><ref>Jensen 1998, 26, 36–39</ref> The terms IQ, general intelligence, general cognitive ability, general mental ability, or simply intelligence are frequently used interchangeably to refer to the common core shared by cognitive tests.<ref name="deary2010"/>

The ''g'' loadings of mental tests are always positive and usually range between .10 and .90, with a mean of about .60 and a standard deviation of about .15. [[Raven's Progressive Matrices]] is among the tests with the highest ''g'' loadings, around .80. Tests of vocabulary and general information are also typically found to have high ''g'' loadings.<ref>Jensen 1998, 26, 36–39, 89–90</ref><ref name="Jensen 2002">Jensen 2002</ref> However, the ''g'' loading of the same test may vary somewhat depending on the composition of the test battery.<ref name="floyd2009">Floyd et al. 2009</ref>

The complexity of tests and the demands they place on mental manipulation are related to the tests' ''g'' loadings. For example, in the forward digit span test the subject is asked to repeat a sequence of digits in the order of their presentation after hearing them once at a rate of one digit per second. The backward digit span test is otherwise the same except that the subject is asked to repeat the digits in the reverse order to that in which they were presented. The backward digit span test is more complex than the forward digit span test, and it has a significantly higher ''g'' loading. Similarly, the ''g'' loadings of arithmetic computation, spelling, and word reading tests are lower than those of arithmetic problem solving, text composition, and reading comprehension tests, respectively.<ref name="jensen1992"/><ref name="Jensen 1980, 213">Jensen 1980, 213</ref>

Test difficulty and ''g'' loadings are distinct concepts that may or may not be empirically related in any specific situation. Tests that have the same difficulty level, as indexed by the proportion of test items that are failed by test takers, may exhibit a wide range of ''g'' loadings. For example, tests of [[rote memory]] have been shown to have the same level of difficulty but considerably lower ''g'' loadings than many tests that involve reasoning.<ref name="Jensen 1980, 213"/><ref>Jensen 1998, 94</ref>