Normal distribution

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Definition

Description

In probability theory, the normal (or Gaussian) distribution is a very commonly occurring continuous probability distribution—a function that tells the probability that an observation in some context will fall between any two real numbers. For example, the distribution of income measured on a log scale is normally distributed in some contexts, as is often the distribution of grades on a test administered to many people. Normal distributions are extremely important in statistics and are often used in the natural and social sciences for real-valued random variables whose distributions are not known.

The normal distribution is immensely useful because of the central limit theorem, which states that, under mild conditions, the mean of many random variables independently drawn from the same distribution is distributed approximately normally, irrespective of the form of the original distribution: physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have a distribution very close to the normal. Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.

The Gaussian distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped (such as Cauchy's, Student's, and logistic). The terms Gaussian function and Gaussian bell curve are also ambiguous because they sometimes refer to multiples of the normal distribution that cannot be directly interpreted in terms of probabilities.[1]

Bell Curve Book

The Bell Curve: Intelligence and Class Structure in American Life is a 1994 book by American psychologist Richard J. Herrnstein (who died before the book was released) and American political scientist Charles Murray. Its central argument is that human intelligence is substantially influenced by both inherited and environmental factors and is a better predictor of many personal dynamics, including financial income, job performance, chance of unwanted pregnancy, and involvement in crime than are an individual's parental socioeconomic status, or education level. The book also argues that those with high intelligence, the "cognitive elite", are becoming separated from those of average and below-average intelligence, and that this is a dangerous social trend with the United States moving toward a more divided society similar to that in Latin America.

The book was controversial, especially where the authors wrote about racial differences in intelligence and discussed the implications of those differences. The authors were reported throughout the popular press as arguing that these IQ differences are genetic. They wrote in chapter 13: "It seems highly likely to us that both genes and the environment have something to do with racial differences." The introduction to the chapter more cautiously states, "The debate about whether and how much genes and environment have to do with ethnic differences remains unresolved."

The book's title comes from the bell-shaped normal distribution of intelligence quotient (IQ) scores in a population.

Shortly after publication, many people rallied both in criticism and defense of the book. A number of critical texts were written in response to the book.