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I am a layperson currently reading "The Bell Curve" and its various rebuttals such as "The Intelligence and How to get it". I do not want to go into controversies of the topics per se and those are irrelevant to my question.

To quote from Wikipedia:

"Heritability" is defined as the proportion of variance in a trait which is attributable to genetic variation within a defined population in a specific environment.1 Heritability takes a value ranging from 0 to 1; a heritability of 1 indicates that all variation in the trait in question is genetic in origin and a heritability of 0 indicates that none of the variation is genetic.

If for the sake of this question, if I were to assume heritability is 0.5, then is it right to understand that there is a 50% chance that I could have inherited my intelligence (as measured using standardised IQ test) and another 50% chance that my intelligence is not inherited? I also read that genes set the upper limit of intelligence, in which case is it right to assume that if a person did not inherit her intelligence because she falls into the latter group, are her limits of intelligence determined entirely by environment and education (or do genes still play a smaller role?).

I appreciate that the extent of heritability is scientifically debated among experts, but I am interested to know what is the debate really about and how to interpret the arguments related to nature vs. nurture. To explain my confusion further - a person who is only 170 cm likely inherited his height from his parents who are only 165 cm. A bit of extra height may have come from better nutrition (than a generation ago) which is an environmental factor. What would be a similar example when it comes to IQ?

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As noted, heritability is calculated at a population level. It says nothing about inheritance at an individual level. For example, given parents with certain known IQs, can we use heritability to calculate the range of possible IQs of their children or how much of their children's IQ is attributable to genetics? No. As Wikipedia puts it:

Heritability measures the proportion of variation in a trait that can be attributed to genes, and not the proportion of a trait caused by genes.

It seems to me that this value is practically useless for anything other than general comparisons between other heritability values! (See my question on Cross Validated (stats.SE) asking to confirm this.)

Put simply: A trait with high heritability indicates that on average, children closely resemble their parents (and twins closely resemble each other) in that trait; conversely, a trait with low heritability suggests that children can have a wide variety of outcomes in that trait. For example, if the heritability of height is high, then I would expect most children of tall parents to be relatively tall; whereas if the heritability of height were low, then tall parents could have children of virtually any height. Stated technically, a heritability of 50% means that half of the variance seen within a population can be accounted for by corresponding variation in genes:

It measures how much of the variation of a trait can be attributed to variation of genetic factors, as opposed to variation of environmental factors.

Environmental factors known to affect IQ include nutrition, toxins, prenatal, perinatal, and family environments.

The reason that the question about the heritability of IQ remains open, is because as long as there are both genetic and environmental differences between populations, then it is a challenge to convincingly tease out which factors (including measurement error and random chance) cause the differences observed between those populations.

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