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# Intelligence quotient

Intelligence quotient
Diagnostics

An example of one kind of IQ test item, modeled after items in the Raven's Progressive Matrices test.
ICD-9-CM94.01
MedlinePlus001912

(Redirected from Iq)
Intelligence quotient
Diagnostics

An example of one kind of IQ test item, modeled after items in the Raven's Progressive Matrices test.
ICD-9-CM94.01
MedlinePlus001912

An intelligence quotient, or IQ, is a score derived from one of several standardized tests designed to assess intelligence. The abbreviation "IQ" comes from the German term Intelligenz-Quotient, originally coined by psychologist William Stern. When modern IQ tests are devised, the mean (average) score within an age group is set to 100 and the standard deviation (SD) almost always to 15, although this was not always so historically.[1] Thus, the intention is that approximately 95% of the population scores within two SDs of the mean, i.e. has an IQ between 70 and 130.

IQ scores have been shown to be associated with such factors as morbidity and mortality,[2] parental social status,[3] and, to a substantial degree, biological parental IQ. While the heritability of IQ has been investigated for nearly a century, there is still debate about the significance of heritability estimates[4][5] and the mechanisms of inheritance.[6]

IQ scores are used as predictors of educational achievement, special needs, job performance and income. They are also used to study IQ distributions in populations and the correlations between IQ and other variables. The average IQ scores for many populations have been rising at an average rate of three points per decade since the early 20th century, a phenomenon called the Flynn effect. It is disputed whether these changes in scores reflect real changes in intellectual abilities.

## History

### Early history

The first large-scale mental test may have been the imperial examination system in China. Modern mental testing began in France in the 19th century. It contributed to separating mental retardation from mental illness and reducing the neglect, torture, and ridicule heaped on both groups.[7]

Englishman Francis Galton coined the terms psychometrics and eugenics, and developed a method for measuring intelligence based on nonverbal sensory-motor tests. It was initially popular, but was abandoned after the discovery that it had no relationship to outcomes such as college grades.[7][8]

French psychologist Alfred Binet, together with psychologists Victor Henri and Théodore Simon, after about 15 years of development, published the Binet-Simon test in 1905, which focused on verbal abilities. It was intended to identify mental retardation in school children.[7] The score on the Binet-Simon scale would reveal the child's mental age. For example, a six-year-old child who passed all the tasks usually passed by six-year-olds—but nothing beyond—would have a mental age that exactly matched his chronological age, 6.0. (Fancher, 1985). In Binet's view, there were limitations with the scale and he stressed what he saw as the remarkable diversity of intelligence and the subsequent need to study it using qualitative, as opposed to quantitative, measures (White, 2000). American psychologist Henry H. Goddard published a translation of it in 1910. The eugenics movement in the USA seized on it as a means to give them credibility in diagnosing mental retardation, and thousands of American women, most of them poor African Americans, were forcibly sterilized based on their scores on IQ tests, often without their consent or knowledge.[9] American psychologist Lewis Terman at Stanford University revised the Binet-Simon scale, which resulted in the Stanford-Binet Intelligence Scales (1916). It became the most popular test in the United States for decades.[7][10][11][12]

### General factor (g)

The many different kinds of IQ tests use a wide variety of methods. Some tests are visual, some are verbal, some tests only use abstract-reasoning problems, and some tests concentrate on arithmetic, spatial imagery, reading, vocabulary, memory or general knowledge. The psychologist Charles Spearman in 1904 made the first formal factor analysis of correlations between the tests. He found a single common factor explained the positive correlations among tests. This is an argument still accepted in principle by many psychometricians. Spearman named it g for "general factor" and labelled the smaller, specific factors or abilities for specific areas s. In any collection of IQ tests, by definition the test that best measures g is the one that has the highest correlations with all the others. Most of these g-loaded tests typically involve some form of abstract reasoning. Therefore, Spearman and others have regarded g as the (perhaps genetically determined) real essence of intelligence. This is still a common but not universally accepted view. Other factor analyses of the data, with different results, are possible. Some psychometricians regard g as a statistical artifact. One of the best measures of g is Raven's Progressive Matrices which is a test of visual reasoning.[1][7]

### The War Years

During World War I, a way was needed to evaluate and assign recruits. This led to the rapid development of several mental tests. The testing generated controversy and much public debate. Nonverbal or "performance" tests were developed for those who could not speak English or were suspected of malingering.[7] After the war, positive publicity on army psychological testing helped to make psychology a respected field.[13] Subsequently, there was an increase in jobs and funding in psychology.[14] Group intelligence tests were developed and became widely used in schools and industry.[15]

L.L. Thurstone argued for a model of intelligence that included seven unrelated factors (verbal comprehension, word fluency, number facility, spatial visualization, associative memory, perceptual speed, reasoning, and induction). While not widely used, it influenced later theories.[7]

David Wechsler produced the first version of his test in 1939. It gradually became more popular and overtook the Binet in the 1960s. It has been revised several times, as is common for IQ tests, to incorporate new research. One explanation is that psychologists and educators wanted more information than the single score from the Binet. Wechsler's 10+ subtests provided this. Another is Binet focused on verbal abilities, while the Wechsler also included nonverbal abilities. The Binet has also been revised several times and is now similar to the Wechsler in several aspects, but the Wechsler continues to be the most popular test in the United States.[7]

### Cattell–Horn–Carroll theory

Raymond Cattell (1941) proposed two types of cognitive abilities in a revision of Spearman's concept of general intelligence. Fluid intelligence (Gf) was hypothesized as the ability to solve novel problems by using reasoning, and crystallized intelligence (Gc) was hypothesized as a knowledge-based ability that was very dependent on education and experience. In addition, fluid intelligence was hypothesized to decline with age, while crystallized intelligence was largely resistant. The theory was almost forgotten, but was revived by his student John L. Horn (1966) who later argued Gf and Gc were only two among several factors, and he eventually identified 9 or 10 broad abilities. The theory continued to be called Gf-Gc theory.[7]

John B. Carroll (1993), after a comprehensive reanalysis of earlier data, proposed the Three Stratum theory, which is a hierarchical model with three levels. The bottom stratum consists of narrow abilities that are highly specialized (e.g., induction, spelling ability). The second stratum consists of broad abilities. Carroll identified eight second-stratum abilities. Carroll accepted Spearman's concept of general intelligence, for the most part, as a representation of the uppermost, third stratum.[16][17]

More recently (1999), a merging of the Gf-Gc theory of Cattell and Horn with Carroll's Three-Stratum theory has led to the Cattell–Horn–Carroll theory. It has greatly influenced many of the current broad IQ tests.[7]

It is argued that this reflects much of what is known about intelligence from research. A hierarchy of factors is used; g is at the top. Under it are 10 broad abilities that in turn are subdivided into 70 narrow abilities. The broad abilities are:[7]

• Fluid intelligence (Gf) includes the broad ability to reason, form concepts, and solve problems using unfamiliar information or novel procedures.
• Crystallized intelligence (Gc) includes the breadth and depth of a person's acquired knowledge, the ability to communicate one's knowledge, and the ability to reason using previously learned experiences or procedures.
• Quantitative reasoning (Gq) is the ability to comprehend quantitative concepts and relationships and to manipulate numerical symbols.
• Reading and writing ability (Grw) includes basic reading and writing skills.
• Short-term memory (Gsm) is the ability to apprehend and hold information in immediate awareness, and then use it within a few seconds.
• Long-term storage and retrieval (Glr) is the ability to store information and fluently retrieve it later in the process of thinking.
• Visual processing (Gv) is the ability to perceive, analyze, synthesize, and think with visual patterns, including the ability to store and recall visual representations.
• Auditory processing (Ga) is the ability to analyze, synthesize, and discriminate auditory stimuli, including the ability to process and discriminate speech sounds that may be presented under distorted conditions.
• Processing speed (Gs) is the ability to perform automatic cognitive tasks, particularly when measured under pressure to maintain focused attention.
• Decision/reaction time/speed (Gt)reflects the immediacy with which an individual can react to stimuli or a task (typically measured in seconds or fractions of seconds; it is not to be confused with Gs, which typically is measured in intervals of 2–3 minutes). See Mental chronometry.

Modern tests do not necessarily measure of all of these broad abilities. For example, Gq and Grw may be seen as measures of school achievement and not IQ.[7] Gt may be difficult to measure without special equipment.

g was earlier often subdivided into only Gf and Gc, which were thought to correspond to the nonverbal or performance subtests and verbal subtests in earlier versions of the popular Wechsler IQ test. More recent research has shown the situation to be more complex.[7]

Modern comprehensive IQ tests no longer give a single score. Although they still give an overall score, they now also give scores for many of these more restricted abilities, identifying particular strengths and weaknesses of an individual.[7]

### Other theories

J.P. Guilford's Structure of Intellect (1967) model used three dimensions which when combined yielded a total of 120 types of intelligence. It was popular in the 1970s and early 1980s, but faded due to both practical problems and theoretical criticisms.[7]

Alexander Luria's earlier work on neuropsychological processes lead to the PASS theory (1997). It argued that only looking at one general factor was inadequate for researchers and clinicians who worked with learning disabilities, attention disorders, mental retardation, and interventions for such disabilities. The PASS model covers four kinds of processes (planning process, attention/arousal process, simultaneous processing, and successive processing). The planning processes involve decision making, problem solving, and performing activities and requires goal setting and self-monitoring. The attention/arousal process involves selectively attending to a particular stimulus, ignoring distractions, and maintaining vigilance. Simultaneous processing involves the integration of stimuli into a group and requires the observation of relationships. Successive processing involves the integration of stimuli into serial order. The planning and attention/arousal components comes from structures located in the frontal lobe, and the simultaneous and successive processes come from structures located in the posterior region of the cortex.[18][19][20] It has influenced some recent IQ tests, and been seen as a complement to the Cattell-Horn-Carroll theory described above.[7]

## Modern tests

Approximately 95% of the population have scores within two standard deviations (SD) of the mean. If one SD is 15 points, as is common in almost all modern tests, then 95% of the population are within a range of 70 to 130, and 98% are below 131. Alternatively, two-thirds of the population have IQ scores within one SD of the mean, i.e. within the range 85-115.

IQ scales are ordinally scaled.[21][22][23][24] While one standard deviation is 15 points, and two SDs are 30 points, and so on, this does not imply that mental ability is linearly related to IQ, such that IQ 50 means half the cognitive ability of IQ 100. In particular, IQ points are not percentage points.

The correlation between IQ test results and achievement test results is about 0.7.[7][25]

## Mental age vs. modern method

The IQs of a large enough population are calculated so they conform[26] to a normal distribution with a mean of 100 and a standard deviation of 15.

German psychologist William Stern proposed a method of scoring children's intelligence tests in 1912. He calculated what he called a Intelligenz-Quotient score, or IQ, as the quotient of the 'mental age' (the age group which scored such a result on average) of the test-taker and the 'chronological age' of the test-taker, multiplied by 100. Terman used this system for the first version of the Stanford-Binet Intelligence Scales.[27] This method has several problems such as the fact that it cannot be used to score adults.

Wechsler introduced a different procedure for his test that is now used by almost all IQ tests. When an IQ test is constructed, a standardization sample representative of the general population takes the test. The median result is defined to be equivalent to 100 IQ points. In almost all modern tests, a standard deviation of the results is defined to be equivalent to 15 IQ points. When a subject takes an IQ test, the result is ranked compared to the results of the standardization sample and the subject is given an IQ score equal to those with the same test result in the standardization sample.

The values of 100 and 15 were chosen to get somewhat similar scores as in the older type of test. Likely as a part of the rivalry between the Binet and the Wechsler, the Binet until 2003 chose to have 16 for one SD, causing considerable confusion. Today, almost all tests use 15 for one SD. Modern scores are sometimes referred to as "deviation IQs," while older method age-specific scores are referred to as "ratio IQs."[7][28]

## Reliability and validity

Psychometricians generally regard IQ tests as having high statistical reliability.[citation needed] A high reliability implies that-- although test-takers may have varying scores when taking the same test on differing occasions, and they may have varying scores when taking different IQ tests at the same age-- the scores generally agree with one another and across time. A test-taker's score on any one IQ test is surrounded by an error band that shows, to a specified degree of confidence, what the test-taker's true score is likely to be. For modern tests, the standard error of measurement is about three points, or in other words, the odds are about two out of three that a person's true IQ is in range from three points above to three points below the test IQ. Another description is there is a 95% chance the true IQ is in range from four to five points above to four to five points below the test IQ, depending on the test in question. Clinical psychologists generally regard them as having sufficient statistical validity for many clinical purposes.[7][29][30]

IQ scores can differ to some degree for the same individual on different IQ tests (age 12–13 years). (IQ score table data and pupil pseudonyms adapted from description of KABC-II norming study cited in Kaufman 2009.[7])
PupilKABC-IIWISC-IIIWJ-III
Asher9095111
Brianna125110105
Colin10093101
Danica116127118
Elpha9310593
Fritz106105105
Georgi9510090
Hector112113103
Imelda1049697
Jose1019986
Keoku817875
Leo116124102

## Flynn effect

Since the early 20th century, raw scores on IQ tests have increased in most parts of the world.[31][32][33] When a new version of an IQ test is normed, the standard scoring is set so performance at the population median results in a score of IQ 100. The phenomenon of rising raw score performance means if test-takers are scored by a constant standard scoring rule, IQ test scores have been rising at an average rate of around three IQ points per decade. This phenomenon was named the Flynn effect in the book The Bell Curve after James R. Flynn, the author who did the most to bring this phenomenon to the attention of psychologists.[34][35]

Researchers have been exploring the issue of whether the Flynn effect is equally strong on performance of all kinds of IQ test items, whether the effect may have ended in some developed nations, whether or not there are social subgroup differences in the effect, and what possible causes of the effect might be.[36] Flynn's observations have prompted much new research in psychology and "demolish some long-cherished beliefs, and raise a number of other interesting issues along the way."[32]

## IQ and age

IQ can change to some degree over the course of childhood.[37] However, in one longitudinal study, the mean IQ scores of tests at ages 17 and 18 were correlated at r=.86 with the mean scores of tests at ages five, six and seven and at r=.96 with the mean scores of tests at ages 11, 12 and 13.[38]

IQ scores for children are relative to children of a similar age. That is, a child of a certain age does not do as well on the tests as an older child or an adult with the same IQ. But relative to persons of a similar age, or other adults in the case of adults, they do equally well if the IQ scores are the same.[38] To convert a child's IQ score into an adult score the following calculation should be made: $\mathrm{IQ_{child}} * \mathrm{age}/16 = \mathrm{IQ_{adult}}$. The number 16 is used to indicate the age at which supposedly the IQ reaches its peak.[39]

For decades, practitioners' handbooks and textbooks on IQ testing have reported IQ declines with age after the beginning of adulthood. However, later researchers pointed out this phenomenon is related to the Flynn effect and is in part a cohort effect rather than a true aging effect.

A variety of studies of IQ and aging have been conducted since the norming of the first Wechsler Intelligence Scale drew attention to IQ differences in different age groups of adults. Current consensus is that fluid intelligence generally declines with age after early adulthood, while crystallized intelligence remains intact. Both cohort effects (the birth year of the test-takers) and practice effects (test-takers taking the same form of IQ test more than once) must be controlled to gain accurate data. It is unclear whether any lifestyle intervention can preserve fluid intelligence into older ages.[40]

The peak of capacity for both fluid intelligence and crystallized intelligence occurs at age 26. This is followed by a slow decline.[41]

## Genetics and environment

Environmental and genetic factors play a role in determining IQ. Their relative importance has been the subject of much research and debate.

### Heritability

Heritability is defined as the proportion of variance in a trait which is attributable to genotype within a defined population in a specific environment. A number of points must be considered when interpreting heritability.[42] Heritability measures the proportion of 'variation' in a trait can be attributed to genes, and not the proportion of a trait caused by genes. The value of heritability can change if the impact of environment (or of genes) in the population is substantially altered. A high heritability of a trait does not mean environmental effects, such as learning, are not involved. Since heritability increases during childhood and adolescence, one should be cautious drawing conclusions regarding the role of genetics and environment from studies where the participants are not followed until they are adults.

Studies have found the heritability of IQ in adult twins to be 0.7 to 0.8 and in children twins 0.45 in the Western world.[38][43][44] It may seem reasonable to expect genetic influences on traits like IQ should become less important as one gains experiences with age. However, the opposite occurs. Heritability measures in infancy are as low as 0.2, around 0.4 in middle childhood, and as high as 0.8 in adulthood.[45] One proposed explanation is that people with different genes tend to reinforce the effects of those genes, for example by seeking out different environments.[38] Debate is ongoing about whether these heritability estimates are too high due to not adequately considering various factors, such as that the environment may be relatively more important in families with low socioeconomic status or the effect of the maternal (fetal) environment.

Recent research suggests that molecular genetics of psychology and social science requires approaches that go beyond the examination of candidate genes.[46]

### Shared family environment

Family members have aspects of environments in common (for example, characteristics of the home). This shared family environment accounts for 0.25–0.35 of the variation in IQ in childhood. By late adolescence, it is quite low (zero in some studies). The effect for several other psychological traits is similar. These studies have not looked at the effects of extreme environments, such as in abusive families.[38][47][48][49]

### Non-shared family environment and environment outside the family

Although parents treat their children differently, such differential treatment explains only a small amount of nonshared environmental influence. One suggestion is that children react differently to the same environment due to different genes. More likely influences may be the impact of peers and other experiences outside the family.[38][48]

### Individual genes

A very large proportion of the over 17,000 human genes are thought to have an impact on the development and functionality of the brain.[50].

While a number of individual genes have been reported to be associated with IQ. Examples include CHRM2, microcephalin, and ASPM. However, Deary and colleagues (2009) argued no evidence has been replicated.[51], a finding supported by Chabris et al (2012)[52]

Recently, FNBP1L polymorphisms, specifically the SNP rs236330 have been associated with normally varying intelligence differences in adults [53] and in children [54].

### Gene-environment interaction

David Rowe reported an interaction of genetic effects with Socioeconomic Status, such that the heritability was high in high-SES families, but much lower in low-SES families [55]. This has been replicated in infants [56], children [57] and adolescents [58] in the US, though not outside the US, for instance a reverse result was reported in the UK [59]

Dickens and Flynn (2001) have argued that genes for high IQ initiate environment shaping feedback, as genetic effects cause bright children to seek out more stimulating environments that further increase IQ. In their model, an environment effects decay over time (the model could be adapted to include possible factors, like nutrition in early childhood, that may cause permanent effects). The Flynn effect can be explained by a generally more stimulating environment for all people. The authors suggest that programs aiming to increase IQ would be most likely to produce long-term IQ gains if they caused children to persist in seeking out cognitively demanding experiences.[60][61]

## Interventions

In general, educational interventions, as those described below, have shown short-term effects on IQ, but long-term follow-up is often missing. For example, in the US very large intervention programs such as the Head Start Program have not produced lasting gains in IQ scores. More intensive, but much smaller, projects Abecedarian Project have reported lasting effects, often on Socioeconomic status variables, rather than IQ.[38].

A placebo controlled double-blind experiment found that vegetarians who took 5 grams of creatine per day for six weeks showed a significant improvement on two separate tests of fluid intelligence, Raven's Progressive Matrices, and the backward digit span test from the WAIS. The treatment group was able to repeat longer sequences of numbers from memory and had higher overall IQ scores than the control group. The researchers concluded that "supplementation with creatine significantly increased intelligence compared with placebo."[62] A subsequent study found that creatine supplements improved cognitive ability in the elderly.[63] However, a study on young adults (0.03 g/kg/day for six weeks, e.g., 2 g/day for 150-pound individual) failed to find any improvements.[64]

Recent studies have shown that training in using one's working memory may increase IQ. A study on young adults published in April 2008 by a team from the Universities of Michigan and Bern supports the possibility of the transfer of fluid intelligence from specifically designed working memory training.[65][66] Further research will be needed to determine nature, extent and duration of the proposed transfer. Among other questions, it remains to be seen whether the results extend to other kinds of fluid intelligence tests than the matrix test used in the study, and if so, whether, after training, fluid intelligence measures retain their correlation with educational and occupational achievement or if the value of fluid intelligence for predicting performance on other tasks changes. It is also unclear whether the training is durable of extended periods of time.[67]

## Music and IQ

Musical training in childhood has been found to correlate with higher than average IQ.[68] In a 2004 study indicated that 6 year old children who received musical training (voice or piano lessons) had an average increase in IQ of 7.0 points while children who received alternative training (i.e. drama) or no training had an average increase in IQ of only 4.3 points (which may be consequence of the children entering grade school) as indicated by full scale IQ. Children were tested using Wechsler Intelligence Scale for Children–Third Edition, Kaufman Test of Educational Achievement and Parent Rating Scale of the Behavioral Assessment System for Children.[68]

Listening to classical music was reported to increase IQ; specifically spatial ability. In 1994 Frances Rauscher and Gorden Shaw reported that college students who listened to 10 minutes of Mozart's Sonata for Two Pianos, showed an increase in IQ of 8 to 9 points on the spatial subtest on the Standford-Binet Intelligence Scale.[69] The phenomenon was coined the Mozart effect.

Multiple attempted replications (e.g. [70]) have shown that this is at best a short-term effect (lasting no longer than 10 to 15 minutes), and is not related to IQ-increase.[71]

### Music lessons

In 2004, Schellenberg devised an experiment to test his hypothesis that music lessons can enhance the IQ of children. He had 144 samples of 6 year old children which were put into 4 groups; keyboard lessons, vocal lessons, drama lessons or no lessons at all, for 36 weeks. The samples' IQ was measured both before and after the lessons had taken place using the Wechsler Intelligence Scale for Children–Third Edition, Kaufman Test of Educational Achievement and Parent Rating Scale of the Behavioral Assessment System for Children. All four groups had increases in IQ, most likely resulted by the entrance of grade school. The notable difference with the two music groups compared to the two controlled groups was a slightly higher increase in IQ. The children in the control groups on average had an increase in IQ of 4.3 points, while the increase in IQ of the music groups was 7.0 points. Though the increases in IQ were not dramatic, one can still conclude that musical lessons does have a positive effect for children, if taken at a young age. It is hypothesized that improvements in IQ occur after musical lessons because the music lessons encourage multiple experiences which generates progression in a wide range of abilities for the children. Testing this hypothesis however, has proven difficult.[72]

Another test also performed by Schellenberg tested the effects of musical training in adulthood. He had two groups of adults, one group whom were musically trained and another group who were not. He administered tests of intelligence quotient and emotional intelligence to the trained and non-trained groups and found that the trained participants had an advantage in IQ over the untrained subjects even with gender, age, environmental issues (e.g. income, parent's education) held constant. The two groups, however, score similarly in the emotional intelligence test. The test results (like the previous results) show that there is a positive correlation between musical training and IQ, but it is not evident that musical training has a positive effect on emotional intelligence.[73]

## IQ and brain anatomy

Several neurophysiological factors have been correlated with intelligence in humans, including the ratio of brain weight to body weight and the size, shape and activity level of different parts of the brain. Specific features that may affect IQ include the size and shape of the frontal lobes, the amount of blood and chemical activity in the frontal lobes, the total amount of gray matter in the brain, the overall thickness of the cortex and the glucose metabolic rate.

## Health and IQ

Health is important in understanding differences in IQ test scores and other measures of cognitive ability. Several factors can lead to significant cognitive impairment, particularly if they occur during pregnancy and childhood when the brain is growing and the blood–brain barrier is less effective. Such impairment may sometimes be permanent, sometimes be partially or wholly compensated for by later growth.[citation needed]

Developed nations have implemented several health policies regarding nutrients and toxins known to influence cognitive function. These include laws requiring fortification of certain food products and laws establishing safe levels of pollutants (e.g. lead, mercury, and organochlorides). Improvements in nutrition, and in public policy in general, have been implicated in worldwide IQ increases.[citation needed]

Cognitive epidemiology is a field of research that examines the associations between intelligence test scores and health. Researchers in the field argue that intelligence measured at an early age is an important predictor of later health and mortality differences.

## Social outcomes

Intelligence is a better predictor of educational and work success than any other single score. [74]

Some measures of educational SAT aptitude are essentially IQ tests; For instance Frey and Detterman (2004) reported a correlation of 0.82 between g (general intelligence factor) and SAT scores [75] another has found correlation of 0.81 between g and GCSE scores.[76]

Correlations between IQ scores (general cognitive ability) and achievement test scores are reported to be 0.81 by Deary and colleagues, with the explained variance ranging "from 58.6% in Mathematics and 48% in English to 18.1% in Art and Design".[76]

### School performance

The American Psychological Association's report "Intelligence: Knowns and Unknowns" states that wherever it has been studied, children with high scores on tests of intelligence tend to learn more of what is taught in school than their lower-scoring peers. The correlation between IQ scores and grades is about .50. This means that the explained variance is 25%. Achieving good grades depends on many factors other than IQ, such as "persistence, interest in school, and willingness to study" (p. 81).[38]

It has been found IQ correlation with school performance depends on the IQ measurement used. For undergraduate students, the Verbal IQ as measured by WAIS-R has been found to correlate significantly (0.53) with the GPA of the last 60 hours. In contrast, Performance IQ correlation with the same GPA was only 0.22 in the same study.[77]

### Job performance

According to Schmidt and Hunter, "for hiring employees without previous experience in the job the most valid predictor of future performance is general mental ability."[74] The validity of IQ as a predictor of job performance is above zero for all work studied to date, but varies with the type of job and across different studies, ranging from 0.2 to 0.6.[78] The correlations were higher when the unreliability of measurement methods was controlled for.[38] While IQ is more strongly correlated with reasoning and less so with motor function,[79] IQ-test scores predict performance ratings in all occupations.[74] That said, for highly qualified activities (research, management) low IQ scores are more likely to be a barrier to adequate performance, whereas for minimally-skilled activities, athletic strength (manual strength, speed, stamina, and coordination) are more likely to influence performance.[74] It is largely through the quicker acquisition of job-relevant knowledge that higher IQ mediates job performance.

In establishing a causal direction to the link between IQ and work performance, longitudinal studies by Watkins and others suggest that IQ exerts a causal influence on future academic achievement, whereas academic achievement does not substantially influence future IQ scores.[80] Treena Eileen Rohde and Lee Anne Thompson write that general cognitive ability, but not specific ability scores, predict academic achievement, with the exception that processing speed and spatial ability predict performance on the SAT math beyond the effect of general cognitive ability.[81]

The US military has minimum enlistment standards at about the IQ 85 level. There have been two experiments with lowering this to 80 but in both cases these men could not master soldiering well enough to justify their costs [82]

Some US police departments have set a maximum IQ score for new officers (for example: 125, in New London, CT), under the argument that those with overly-high IQs will become bored and exhibit high turnover in the job. This policy has been challenged as discriminatory, but upheld by at least one US District court.[83]

The American Psychological Association's report "Intelligence: Knowns and Unknowns" states that since the explained variance is 29%, other individual characteristics such as interpersonal skills, aspects of personality etc. are probably of equal or greater importance, but at this point there are no equally reliable instruments to measure them.[38]

### Income

While it has been suggested that "in economic terms it appears that the IQ score measures something with decreasing marginal value. It is important to have enough of it, but having lots and lots does not buy you that much."[84][85], large scale longitudinal studies indicate an increase in IQ translates into an increase in performance at all levels of IQ: i.e., that ability and job performance are monotonically linked at all IQ levels.[86] Charles Murray, coauthor of The Bell Curve, found that IQ has a substantial effect on income independently of family background.[87]

The link from IQ to wealth is much less strong that than from IQ to job performance. Some studies indicate that IQ is unrelated to net worth.[88][89]

The American Psychological Association's 1995 report Intelligence: Knowns and Unknowns stated that IQ scores accounted for (explained variance) about quarter of the social status variance and one-sixth of the income variance. Statistical controls for parental SES eliminate about a quarter of this predictive power. Psychometric intelligence appears as only one of a great many factors that influence social outcomes.[38]

Some studies claim that IQ only accounts for (explains) a sixth of the variation in income because many studies are based on young adults, many of whom have not yet reached their peak earning capacity, or even their education. On pg 568 of The g Factor, Arthur Jensen claims that although the correlation between IQ and income averages a moderate 0.4 (one sixth or 16% of the variance), the relationship increases with age, and peaks at middle age when people have reached their maximum career potential. In the book, A Question of Intelligence, Daniel Seligman cites an IQ income correlation of 0.5 (25% of the variance).

A 2002 study[90] further examined the impact of non-IQ factors on income and concluded that an individual's location, inherited wealth, race, and schooling are more important as factors in determining income than IQ.

### IQ and crime

The American Psychological Association's 1995 report Intelligence: Knowns and Unknowns stated that the correlation between IQ and crime was -0.2. It was -0.19 between IQ scores and number of juvenile offenses in a large Danish sample; with social class controlled, the correlation dropped to -0.17. A correlation of 0.20 means that the explained variance is less than 4%. It is important to realize that the causal links between psychometric ability and social outcomes may be indirect. Children with poor scholastic performance may feel alienated. Consequently, they may be more likely to engage in delinquent behavior, compared to other children who do well.[38]

In his book The g Factor (1998), Arthur Jensen cited data which showed that, regardless of race, people with IQs between 70 and 90 have higher crime rates than people with IQs below or above this range, with the peak range being between 80 and 90.

The 2009 Handbook of Crime Correlates stated that reviews have found that around eight IQ points, or 0.5 SD, separate criminals from the general population, especially for persistent serious offenders. It has been suggested that this simply reflects that "only dumb ones get caught" but there is similarly a negative relation between IQ and self-reported offending. That children with conduct disorder have lower IQ than their peers "strongly argues" for the theory.[91]

A study of the relationship between US county-level IQ and US county-level crime rates found that higher average IQs were associated with lower levels of property crime, burglary, larceny rate, motor vehicle theft, violent crime, robbery, and aggravated assault. These results were not "confounded by a measure of concentrated disadvantage that captures the effects of race, poverty, and other social disadvantages of the county."[92]

### Other correlations with IQ

In addition, IQ and its correlation to health, violent crime, gross state product, and government effectiveness are the subject of a 2006 paper in the publication Intelligence. The paper breaks down IQ averages by U.S. states using the federal government's National Assessment of Educational Progress math and reading test scores as a source.[93]

The American Psychological Association's 1995 report Intelligence: Knowns and Unknowns stated that the correlations for most "negative outcome" variables are typically smaller than 0.20, which means that the explained variance is less than 4%.[38]

Tambs et al.[94][better source needed] found that occupational status, educational attainment, and IQ are individually heritable; and further found that "genetic variance influencing educational attainment ... contributed approximately one-fourth of the genetic variance for occupational status and nearly half the genetic variance for IQ." In a sample of U.S. siblings, Rowe et al.[95] report that the inequality in education and income was predominantly due to genes, with shared environmental factors playing a subordinate role.

A recent USA study connecting political views and intelligence has shown that the mean adolescent intelligence of young adults who identify themselves as "very liberal" is 106.4, while that of those who identify themselves as "very conservative" is 94.8.[96] Two other studies conducted in the UK reached similar conclusions.[97][98]

There are also other correlations such as those between religiosity and intelligence and fertility and intelligence.

### Real-life accomplishments

Average adult combined IQs associated with real-life accomplishments by various tests[99][100]
AccomplishmentIQTest/studyYear
MDs, JDs, or PhDs125+WAIS-R1987
K-BIT1992
115WAIS-R
1–3 years of college104KAIT
K-BIT
105–110WAIS-R
Clerical and sales workers100–105
High school graduates, skilled workers (e.g., electricians, cabinetmakers)100KAIT
WAIS-R
97K-BIT
1–3 years of high school (completed 9–11 years of school)94KAIT
90K-BIT
95WAIS-R
Semi-skilled workers (e.g. truck drivers, factory workers)90–95
Elementary school dropouts (completed 0–7 years of school)80–85
Have 50/50 chance of reaching high school75
Average IQ of various occupational groups:[101]
AccomplishmentIQTest/studyYear
Professional and technical112
Clerical workers, sales workers, skilled workers, craftsmen, and foremen101
Semi-skilled workers (operatives, service workers, including private household)92
Unskilled workers87
Type of work that can be accomplished:[99]
AccomplishmentIQTest/studyYear
Adults can harvest vegetables, repair furniture60

There is considerable variation within and overlap between these categories. People with high IQs are found at all levels of education and occupational categories. The biggest difference occurs for low IQs with only an occasional college graduate or professional scoring below 90.[7]

## Group differences

Among the most controversial issues related to the study of intelligence is the observation that intelligence measures such as IQ scores vary between ethnic and racial groups and sexes. While there is little scholarly debate about the existence of some of these differences, their causes remain highly controversial both within academia and in the public sphere.

### Sex

Most IQ tests are constructed so that there are no overall score differences between females and males.[102] Because environmental factors affect brain activity and behavior, where differences are found, it can be difficult for researchers to assess whether or not the differences are innate. Areas where differences have been found include verbal and mathematical ability.

### Race

The 1996 Task Force investigation on Intelligence sponsored by the American Psychological Association concluded that there are significant variations in IQ across races.[38] The problem of determining the causes underlying this variation relates to the question of the contributions of "nature and nurture" to IQ. Psychologists such as Alan S. Kaufman[103] and Nathan Brody[104] and statisticians such as Bernie Devlin[105] argue that there are insufficient data to conclude that this is because of genetic influences. One of the most notable researchers arguing for a strong genetic influence on these average score differences is Arthur Jensen. In contrast, other researchers such as Richard Nisbett argue that environmental factors can explain all of the average group differences.[106]

## Public policy

In the United States, certain public policies and laws regarding military service,[107] [108] education, public benefits,[109] capital punishment,[110] and employment incorporate an individual's IQ into their decisions. However, in the case of Griggs v. Duke Power Co. in 1971, for the purpose of minimizing employment practices that disparately impacted racial minorities, the U.S. Supreme Court banned the use of IQ tests in employment, except when linked to job performance via a Job analysis. Internationally, certain public policies, such as improving nutrition and prohibiting neurotoxins, have as one of their goals raising, or preventing a decline in, intelligence.

A diagnosis of mental retardation is in part based on the results of IQ testing. Borderline intellectual functioning is a categorization where a person has below average cognitive ability (an IQ of 71–85), but the deficit is not as severe as mental retardation (70 or below).

In the United Kingdom, the eleven plus exam which incorporated an intelligence test has been used from 1945 to decide, at eleven years old, which type of school a child should go to. They have been much less used since the widespread introduction of comprehensive schools.

## Criticism and views

### Relation between IQ and intelligence

IQ is the most researched approach to intelligence and by far the most widely used in practical setting. However, although IQ attempts to measure some notion of intelligence, it may fail to act as an accurate measure of "intelligence" in its broadest sense. IQ tests only examine particular areas embodied by the broadest notion of "intelligence", failing to account for certain areas which are also associated with "intelligence" such as creativity or emotional intelligence.

There are critics such as Keith Stanovich who do not dispute the stability of IQ test scores or the fact that they predict certain forms of achievement rather effectively. They do argue, however, that to base a concept of intelligence on IQ test scores alone is to ignore many important aspects of mental ability.[3][111]

### Criticism of g

Some scientists dispute IQ entirely. In The Mismeasure of Man (1996), paleontologist Stephen Jay Gould criticized IQ tests and argued that that they were used for scientific racism. He argued that g was a mathematical artifact and criticized:

...the abstraction of intelligence as a single entity, its location within the brain, its quantification as one number for each individual, and the use of these numbers to rank people in a single series of worthiness, invariably to find that oppressed and disadvantaged groups—races, classes, or sexes—are innately inferior and deserve their status.(pp. 24–25)

Psychologist Peter Schönemann was also a persistent critic of IQ, calling it "the IQ myth". He argued that g is a flawed theory and that the high heritability estimates of IQ are based on false assumptions.[112][113]

Psychologist Arthur Jensen has rejected the criticism by Gould and also argued that even if g was replaced by a model with several intelligences this would change the situation less than expected. All tests of cognitive ability would continue to be highly correlated with one another and there would still be a black-white gap on cognitive tests.[114]

### Test bias

The American Psychological Association's report Intelligence: Knowns and Unknowns stated that in the United States IQ tests as predictors of social achievement are not biased against African Americans since they predict future performance, such as school achievement, similarly to the way they predict future performance for Caucasians.[38]

However, IQ tests may well be biased when used in other situations. A 2005 study stated that "differential validity in prediction suggests that the WAIS-R test may contain cultural influences that reduce the validity of the WAIS-R as a measure of cognitive ability for Mexican American students,"[115] indicating a weaker positive correlation relative to sampled white students. Other recent studies have questioned the culture-fairness of IQ tests when used in South Africa.[116][117] Standard intelligence tests, such as the Stanford-Binet, are often inappropriate for children with autism; the alternative of using developmental or adaptive skills measures are relatively poor measures of intelligence in autistic children, and may have resulted in incorrect claims that a majority of children with autism are mentally retarded.[118]

### Outdated methodology

A 2006 article stated that contemporary psychologic research often did not reflect substantial recent developments in psychometrics and "bears an uncanny resemblance to the psychometric state of the art as it existed in the 1950s."[119]

### "Intelligence: Knowns and Unknowns"

In response to the controversy surrounding The Bell Curve, the American Psychological Association's Board of Scientific Affairs established a task force in 1995 to write a report on the state of intelligence research which could be used by all sides as a basis for discussion, "Intelligence: Knowns and Unknowns". The full text of the report is available through several websites.[38][120]

In this paper the representatives of the association regret that IQ-related works are frequently written with a view to their political consequences: "research findings were often assessed not so much on their merits or their scientific standing as on their supposed political implications".

The task force concluded that IQ scores do have high predictive validity for individual differences in school achievement. They confirm the predictive validity of IQ for adult occupational status, even when variables such as education and family background have been statistically controlled. They stated that individual differences in intelligence are substantially influenced by both genetics and environment.

The report stated that a number of biological factors, including malnutrition, exposure to toxic substances, and various prenatal and perinatal stressors, result in lowered psychometric intelligence under at least some conditions. The task force agrees that large differences do exist between the average IQ scores of blacks and whites, saying:

The cause of that differential is not known; it is apparently not due to any simple form of bias in the content or administration of the tests themselves. The Flynn effect shows that environmental factors can produce differences of at least this magnitude, but that effect is mysterious in its own right. Several culturally based explanations of the Black/ White IQ differential have been proposed; some are plausible, but so far none has been conclusively supported. There is even less empirical support for a genetic interpretation. In short, no adequate explanation of the differential between the IQ means of Blacks and Whites is presently available.

The APA journal that published the statement, American Psychologist, subsequently published eleven critical responses in January 1997, several of them arguing that the report failed to examine adequately the evidence for partly genetic explanations.

### Dynamic assessment

Notable and increasingly influential[121][122] alternative to the wide range of standard IQ tests originated in the writings of psychologist Lev Vygotsky (1896-1934) of his most mature and highly productive period of 1932-1934. The notion of the zone of proximal development that he introduced in 1933, roughly a year before his death, served as the banner for his proposal to diagnose development as the level of actual development that can be measured by the child's independent problem solving and, at the same time, the level of proximal, or potential development that is measured in the situation of moderately assisted problem solving by the child.[123] The maximum level of complexity and difficulty of the problem that the child is capable to solve under some guidance indicates the level of potential development. Then, the difference between the higher level of potential and the lower level of actual development indicates the zone of proximal development. Combination of the two indexes—the level of actual and the zone of the proximal development—according to Vygotsky, provides a significantly more informative indicator of psychological development than the assessment of the level of actual development alone.[124][125]

The ideas on the zone of development were later developed in a number of psychological and educational theories and practices. Most notably, they were developed under the banner of dynamic assessment that focuses on the testing of learning and developmental potential[126][127][128] (for instance, in the work of Reuven Feuerstein and his associates,[129] who has criticized standard IQ testing for its putative assumption or acceptance of "fixed and immutable" characteristics of intelligence or cognitive functioning). Grounded in developmental theories of Vygotsky and Feuerstein, who recognized that human beings are not static entities but are always in states of transition and transactional relationships with the world, dynamic assessment received also considerable support in the recent revisions of cognitive developmental theory by Joseph Campione, Ann Brown, and John D. Bransford and in theories of multiple intelligences by Howard Gardner and Robert Sternberg.[130]

## High IQ societies

There are social organizations, some international, which limit membership to people who have scores as high as or higher than the 98th percentile on some IQ test or equivalent. Mensa International is perhaps the best known of these. There are other groups requiring a score above the 98th percentile.

## Reference charts

IQ reference charts are tables suggested by test publishers to divide intelligence ranges in various categories.

## References

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# A restaurant offers a $12 dinner special that has 7 choices for an appetizer, 12 choices for an entree, and 6 choices for a dessert. EunoR 2021-09-08 Answered A restaurant offers a$12 dinner special that has 7 choices for an appetizer, 12 choices for an entree, and 6 choices for a dessert. How many different meals are available when you select an appetizer, an entree,and a dessert?
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## Expert Answer

izboknil3
Answered 2021-09-09 Author has 99 answers
According to fundamental counting principal it is known that, if the first event occurs in m ways, and the second event occurs in n ways, then two events occur in as sequence of mn ways.
There are 7 choices of appetizer, 12 choices of entrée, and 6 choices of dessert.
Hence, total number of different meals is, $7×12×6=504.$
Thus, 504 different means are available to select an appetizer, an entrée, and a dessert.
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# How does the lack of information increase as temperature increases?

Suppose one knows nothing about the concept of entropy. How can we argue that the lack of information/ignorance about the system typically increases with the increase in the temperature using the formula of the canonical probability $p_i=e^{-\beta E_i}/Z$ where $Z$ is the canonical partition function? Assume that a system has a fixed volume and a fixed number of particles.

Here is the objective. If I can argue that lack of information typically increases with temperature, I can use that to argue that entropy typically increases with temperature by equating lack of information with entropy.

• How are you defining "lack of information" without basically just restating the definition of entropy? – probably_someone May 22 '18 at 3:45
• Entropy is just a name given to the quantity $-\sum p_i \ln p_i$. The probabilities $p_i$ can be a measure of information about the system. – mithusengupta123 May 22 '18 at 3:49
• In what specific way do you define this measure of information? – probably_someone May 22 '18 at 4:03
• Agree with @probably_someone. You need to first sharpen your question into a mathematically well-defined one. – Kite.Y Jul 20 '18 at 14:48
• You may find the following helpful/relevant: physics.stackexchange.com/q/137050/59023 or physics.stackexchange.com/a/177972/59023 . – honeste_vivere Jul 24 '18 at 14:00

At absolute 0, assuming the ground state is a crystal, there is no information encoded in the state, so there is no lack of information about the state.

When we heat up the system, the amount of information encoded in the state (in terms of the positions and motion of all the atoms in the material) increases. But we are learning barely any of this information. Therefore, our lack of information increases, not because we're forgetting anything but because there's more information in the system to lack.

Where is this extra information coming from? From the way we heat up the system. Suppose we shine microwaves on it. We don't know which atoms the microwaves are interacting with, so we don't know the resulting motion of the atoms.

So entropy increases as temperature increases.

$$p_i \propto e^{-\frac{E_i}{K_b T}}$$

At infinite temperature, all states have equal probability and you have "minimal information"

Intuitively: at zero temperature there is no thermal energy so only one state is accessible - the ground state (the bottom of the energy landscape). So you don't lack any information on what configuration the system could be in - you know for sure that it's in the ground state. As you add in thermal energy, more and more states become energetically accessible, so there are more possible configurations the system could be in. So the amount of information in the system that you don't know (just from knowing the temperature) increases. Instead of hanging around the bottom of the energy landscape, the system could also be found higher up, because of its internal thermal energy.

Let's imagine a system $\mathcal{S}$ coupled to a large heat bath $\mathcal{E}$, the environment.

As $\mathcal{S}$ interacts with $\mathcal{E}$, correlations between these subsystems build up. Information about the system's state is proliferated into $\mathcal{E}$ in form of these correlations. In order to retain this info we would have to measure large chunks of the environment.

Since we are unable to do this, our uncertainty about the system's state increases. Maybe one way to quantify this would indeed be to consider the thermal state for the environment and a pure state (perfect information). The general time evolution of the whole density matrix would then lead the emergence of a mixed state if you trace (average) out the environment states.

The rate at which this happens will depend on the bath-bath correlation time which in turn depends on the temperature.

My second answer, which probably isn't what you're looking for, but just in case it is:

I just realized that you might be looking for a mathematically rigorous proof that the entropy of the thermal distribution: $$H_\beta = - \sum_i p_i \log p_i \ \ \ \ \mathrm{where} \ \ \ \ p_i = e^{-\beta E_i}/Z$$ is increasing in temperature $T = \beta^{-1}$. Here's a sketch of one.

Lemma 1
For a given average energy $E_\mathrm{ave} = \sum_i p_i E_i$, the thermal distribution with energy $E_\mathrm{ave}$ maximizes the entropy over all distributions with that energy.

Proof sketch:
Use Lagrange multipliers.

Lemma 2
For the thermal distribution at any energy $E_\mathrm{ave}$ with $0 < \beta < \infty$, you can find a probability distribution with slightly higher average energy $E_\mathrm{ave}+\,\epsilon$ and greater entropy.

Proof sketch:
Find two energies $E_j < E_k$ and move some probability mass from $p_j$ to $p_k$.

Lemma 3
When we increase the temperature $\beta^{-1}$, we increase the average energy.

Proof sketch:
Show that for any two positive temperatures $\beta^{-1} < {\tilde{\beta}}^{-1}$, there is an energy $E_m$ such that in the thermal distributions, if $E_i < E_m$, then $p_i > \tilde{p}_i$ and if $E_i>E_m$, then $p_i < \tilde{p}_i$.

This can be shown by straightforward calculation.

Now we can prove the theorem. Start with average energy $E_1$. By Lemma 2, we can increase the average energy to $E_2 = E_1+\epsilon\,$ and find a distribution with higher energy. But the thermal distribution at average energy $E_2$ has higher entropy than this distribution by Lemma 1. Thus, we've increased both the energy and the energy of the thermal distribution. But by Lemma 3, the thermal distribution at average energy $E_2$ also has higher temperature.


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# [arch-releng] pacman.conf.{i686,x86_64} ignored by archiso?

Gerardo Exequiel Pozzi vmlinuz386 at yahoo.com.ar
Fri Apr 6 11:41:35 EDT 2012

On 04/06/2012 12:12 PM, Dieter Plaetinck wrote:
> On Fri, 06 Apr 2012 11:48:27 -0300
> Gerardo Exequiel Pozzi<vmlinuz386 at yahoo.com.ar>  wrote:
>
>> On 04/06/2012 11:24 AM, Dieter Plaetinck wrote:
>>> in the releng env
>>> (http://projects.archlinux.org/users/dieter/releng.git/tree/) it seems the
>>> pacman.conf.{i686,x86_64} files are now being ignored. (i enabled testing in there but it's not used)
>>> Frankly I don't even know/remember how this ever worked, I can't find a reference to these files anywhere.
>>> How does one control which pacman.conf archiso must use during building of the arch environment it puts on the isos?
>>>
>>> Dieter
>>>
>> I do not know about how these scripts works, the only thing that I can
>> build.sh (wrapper to mkarchiso) does not have a parameter to set a
>> pacman.conf, so mkarchiso by default will use /etc/pacman.conf.
> has this always been like this?
Yes, never used neither in the age of Makefile instead of build.sh. At
some point we used a local copy of pacman.conf in overlay (unmodified
from pacman package when mkarchroot does not support NOCOPY)
>
>> Can you point me where is supposed to be used these files in
>>
> exactly, that's what I said.  I don't know where these files are ever used. They should be used when building the iso's.
>
Weird. sure?

But think that these files are never used, because:

They are out-of-date (compared to newer pacman.conf) -> There are no
"SigLevel" -> Default "SigLevel" is used, that is "Optional TrustedOnly"
by pacman-4 -> without manual interaction pacman will not work. -> no ISOs.

But we have ISOs all times :) Or I am missing something?

--
Gerardo Exequiel Pozzi
\cos^2\alpha + \sin^2\alpha = 1


text:
Metal forming and materials processing

Beside casting and machining, forming is one of the main processes for manufacturing of components from metallic materials. Forming processes include sheet metal forming as well as bulk metal forging. Metal forming generally involves significant plastic deformations, elevated temperatures, high deformation velocities and an increased risk for initiation of cracks in the material. These macroscopic process conditions are intimately connected to the microstructure evolution inside the material.

In Hallberg et al. (2007) and Hallberg et al. (2010), deep-drawing of stainless steel is used as application examples for constitutive models where martensitic phase transformation is considered, see the illustration below. As the martensite phase is much harder than the parent austenitic phase, the material properties may change dramatically in the presence of this kind of diffusionless - and thus rapid - phase transformation.

The influence of deformation rate and material pre-processing in metal forging is studied in Hallberg et al. 2009. Different behavior of a 100Cr6 steel, due to previous tempering or annealing, was studied in high strain rate axisymmetric compression, experimentally as well as through numerical simulations.

The illustration below is taken from Hallberg et al. 2009. Note the development of a "shear cross" (white lines) in the tempered - and much harder - material. This localized deformation is absent in the annealed specimen.

Another example of metal forming is rolling, for example discussed in Hallberg (2013). A conventional rolling process can be made asymmetric by different methods in order to increase the deformation imposed onto the sheet.

The asymmetry of the process can be induced by having different radii $r$ of the rolls, by different roller velocities, i.e. $\omega_{1}\ne\omega_{2}$ or by different friction/lubrication conditions at each side of the sheet. The asymmetry increases the shear deformation of the rolled sheet and hence the total amount of effective plastic deformation. This is utilized in severe plastic deformation (SPD) processes for production of very fine-grained metals.


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Contents

Log loss metric explained

LogLoss is a classification metric based on probabilities. It measures the performance of a classification model where the prediction input is a probability value between 0 and 1. For any given problem, a smaller LogLoss value means better predictions.

Formula

In order to calculate LogLoss the classifier must assign a probability to each class rather than simply yielding the most likely class (class with the largest probability). LogLoss formula is as follows:

Let’s find out what every symbol in this formula means.

• N - is a number of samples (or instances)
• M - is a number of possible labels
• y_ij - takes value 1 if label j is the correct classification for instance i, and 0 otherwise
• p_ij - is the model probability of assigning label j to instance i

Example calculation

Let’s calculate LogLoss in 2 ways:

• employ excellent sklearn library
• calculate with pure Python without any libraries

Data

Let’s assume that we predict car maker. We have only three possible labels: `audi`, `bmw`, `tesla`. And we have 8 rows of data with known labels:

``````data = ['audi', 'tesla', 'tesla', 'bmw', 'audi', 'bmw', 'audi', 'tesla']
``````

And we have a classifier that for given data gives following probabilities:

``````probs = [
[0.6, 0.3, 0.1], [0.45, 0.45, 0.1], [0.50, 0.00, 0.50], [1.00, 0.00, 0.00],
[0.2, 0.6, 0.2], [0.10, 0.10, 0.8], [0.33, 0.33, 0.34], [0.30, 0.40, 0.30]
]
``````

Please notice that lengh of `probs` array is equal to length of `data` array, and each array in `probs` array has size 3 (that is equal to the number of all possible labels). Internal arrays in `probs` array are probabilities of labels: 1st item is for `audi`, 2nd item is for `bmw`, 3rd item is for `tesla`. E.g. consider first array: `[0.6, 0.3, 0.1]`. Here `0.6` is probability of `audi`, `0.3` - for `bmw` and `0.1` for `tesla`. Also notice, that labels are sorted alphabetically, and so are probabilities for them (I will rely on this fact when implementing our own function that calculates LogLoss).

Now let’s calculate LogLoss for this classifier.

LogLoss in sklearn

Here is how to use sklearn.metrics.log_loss for this:

``````from sklearn.metrics import log_loss
data = ['audi', 'tesla', 'tesla', 'bmw', 'audi', 'bmw', 'audi', 'tesla']
probs = [
[0.6, 0.3, 0.1], [0.45, 0.45, 0.1], [0.50, 0.00, 0.50], [1.00, 0.00, 0.00],
[0.2, 0.6, 0.2], [0.10, 0.10, 0.8], [0.33, 0.33, 0.34], [0.30, 0.40, 0.30]
]
ll = log_loss(data, probs)
print(ll)
``````

The result is `5.53374909081`.

Handy LogLoss calculation

Now let’s calculate LogLoss with pure Python, not using any fancy library. Here is calculation for each data instance, summing all it up finally:

``````from math import log
ll = [None]*8
EPS = 1e-15
ll[0] = 1*log(0.6)+0*log(0.3)+0*log(0.1)
ll[1] = 0*log(0.45)+0*log(0.45)+1*log(0.1)
ll[2] = 0*log(0.5)+0*log(EPS)+1*log(0.5)
ll[3] = 0*log(1.0)+1*log(EPS)+0*log(EPS)
ll[4] = 1*log(0.2)+0*log(0.6)+0*log(0.2)
ll[5] = 0*log(0.1)+1*log(0.1)+0*log(0.8)
ll[6] = 1*log(0.33)+0*log(0.33)+0*log(0.34)
ll[7] = 0*log(0.3)+0*log(0.4)+1*log(0.3)
logloss = -(1/8)*sum(ll)
print(logloss)
``````

The result will we the same as with `sklearn`: `5.53374909081`.

Let’s consider this calculation. You see that instead of zero I am using here a small value: `EPS`, equal to `1e-15`. This is because logarythm of `0.0` is minus infinity. To workaround this I am using reasonably small value: `1e-15`.

Now let’s write our own function to calculate LogLoss. Let’s split it to 2 fundtions:

• one is validating input data: `logloss_validate`
• another one is doing actual calculations, only if validation was successful: `logloss`
``````def logloss_validate(a_true, a_probs, eps):
# 1. Validate input variables
if len(a_true) != len(a_probs):
raise ValueError('Length of a_true does not match length of a_probs')
unique_labels = set(a_true)
# 2. Validate number of unique labels
if len(unique_labels) < 2:
raise ValueError('Should have at least 2 unique labels')
# 3. Validate eps value
if eps <= 0.0:
raise ValueError('Eps should be very small (near zero) positive value')
# 4. Validate probabilities
for item in a_probs:
# 4.1. Check that item in a_probs is iterable
try:
itemiter = iter(item)
except TypeError as te:
raise ValueError('Item in a_probs should be an iterable')
# 4.2. Check that length of item in a_probs is equal to the number of unique labels
if len(item) != len(unique_labels):
raise ValueError('Size of item in a_probs does not match number of unique labels')
for i in item:
if i < 0.0:
raise ValueError('Some items of a_probs have negative values')

def logloss(a_true, a_probs, eps=1e-15):
logloss_validate(a_true, a_probs, eps)
from math import log

result = 0.0
# uniqalize labels and sort them alphabetically
unique_sorted_labels = sorted(set(a_true))
label_map = {}
label_idx = 0
# put index of each unique sorted label into a map
# I will use this map to get this label probability by label value itself
for label in unique_sorted_labels:
label_map[label] = label_idx
label_idx = label_idx + 1
for true_label, probs in zip(a_true, a_probs):
true_label_idx = label_map[true_label]
# Here I rely on the fact that label probabilities are sorted alphabetically
prob = probs[true_label_idx]
if (prob < eps):
prob = eps
result = result + log(prob)
return -1/len(a_true)*result;
``````

Then, if we call `logloss(data, probs)` we will get the same result for LogLoss metric: `5.53374909081`.


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Perceptrons got a lot of folks excited back in the day. However, there was a bit of a problem. Say we are trying to classify digits correctly using a network of perceptrons. Our model is given an image of the digit "1" and incorrectly classifies it as a "2". Why don't we just go ahead and tweak the weights and bias to improve our model's classification? Well we can, but by doing so we get unpredictable results. For example, say we do tune the parameters and correctly predict "1" in the next training run, but now our model is performing poorly on other digits. We've essentially kicked the can down the road, and our model begins to juggle between different weights and biases with no significant gains in accuracy across all digits.

This sensitivity to change quickly becomes unwieldy when tackling machine learning tasks, even the slightest change to the parameters will produce a completely different output. Instead, we need a model that gives us greater control of the output so we are able to fine tune the model to achieve better performance. In other words, for any small change in the weights and bias, we only want a small change in the output.

With this simple idea, the sigmoid neuron was born. The math can be kinda scary at first but some generous folks have broken it down into edible chunks for us. I'll summarise for you. The sigmoid has similar qualities to a perceptron but a few key differences. Like it's sibling, the sigmoid neuron still takes inputs $x_1, x_2, ..., x_j$, however $x_j$ can be any value between $0$ and $1$ (e.g. 0.7456). Weights are still assigned to each input $w_1, w_2, ..., w_3$, and the overall bias remains. However, the output is no longer a 0 or 1, instead it's any number between 0 and 1 depending on the magnitude of $w.x + b$. This is because of the $\sigma$ being applied to get $\sigma(w.x +b)$.

So what does the sigmoid part of it actually do? It does this: $$\sigma(z) \equiv \frac{1}{1+e^{-z}}$$

Now let's go another step further to make things more explicit. The output of a sigmoid neuron with the inputs, weights and bias detailed above is: $$\frac{1}{1 + \exp(-\sum_jw_jx_j-b)}$$

Wowza. It's starting to look pretty funky. Despite our function looking more complex, the sigmoid neuron still possesses similarities to the perceptron. For instance, say $z$, or $w.x+b$ is a big number - $x^{-z}=2.718281828^{-100}=0$. By finishing the calculation, we produce 1 as the output - feel free to pause here and calculate this yourself. This demonstrates that when $w.x + b$ is large and positive, the output will be 1. On the other hand, if the input is large and negative, the output will be 0. This is awfully similar to the perceptron, right? Where the sigmoid differentiates itself, is when z lands in the between these two extremes, with subtle tweaks to the weights and biases producing small corresponding updates to the output.

This post was developed with the help of:

Michael Nielsen, Neural Networks and Deep Learning link