Central limit theorem: the cornerstone of modern statistics
According to the central limit theorem, the means of a random sample of size, , from a population with mean, µ, and variance, σ , distribute normally with mean, µ, and variance, [Formula: see text]. Using the central limit theorem, a variety of parametric tests have been developed under assumptions...
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| Published in | Korean journal of anesthesiology Vol. 70; no. 2; pp. 144 - 156 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Korea (South)
The Korean Society of Anesthesiologists
01.04.2017
Korean Society of Anesthesiologists 대한마취통증의학회 |
| Subjects | |
| Online Access | Get full text |
| ISSN | 2005-6419 2005-7563 |
| DOI | 10.4097/kjae.2017.70.2.144 |
Cover
| Abstract | According to the central limit theorem, the means of a random sample of size,
, from a population with mean, µ, and variance, σ
, distribute normally with mean, µ, and variance, [Formula: see text]. Using the central limit theorem, a variety of parametric tests have been developed under assumptions about the parameters that determine the population probability distribution. Compared to non-parametric tests, which do not require any assumptions about the population probability distribution, parametric tests produce more accurate and precise estimates with higher statistical powers. However, many medical researchers use parametric tests to present their data without knowledge of the contribution of the central limit theorem to the development of such tests. Thus, this review presents the basic concepts of the central limit theorem and its role in binomial distributions and the Student's t-test, and provides an example of the sampling distributions of small populations. A proof of the central limit theorem is also described with the mathematical concepts required for its near-complete understanding. |
|---|---|
| AbstractList | According to the central limit theorem, the means of a random sample of size, n, from a population with mean, µ, and variance, σ2, distribute normally with mean, µ, and variance, σ2n. Using the central limit theorem, a variety of parametric tests have been developed under assumptions about the parameters that determine the population probability distribution. Compared to non-parametric tests, which do not require any assumptions about the population probability distribution, parametric tests produce more accurate and precise estimates with higher statistical powers. However, many medical researchers use parametric tests to present their data without knowledge of the contribution of the central limit theorem to the development of such tests. Thus, this review presents the basic concepts of the central limit theorem and its role in binomial distributions and the Student's t-test, and provides an example of the sampling distributions of small populations. A proof of the central limit theorem is also described with the mathematical concepts required for its near-complete understanding. According to the central limit theorem, the means of a random sample of size, n, from a population with mean, µ, and variance, σ2, distribute normally with mean, µ, and variance, [Formula: see text]. Using the central limit theorem, a variety of parametric tests have been developed under assumptions about the parameters that determine the population probability distribution. Compared to non-parametric tests, which do not require any assumptions about the population probability distribution, parametric tests produce more accurate and precise estimates with higher statistical powers. However, many medical researchers use parametric tests to present their data without knowledge of the contribution of the central limit theorem to the development of such tests. Thus, this review presents the basic concepts of the central limit theorem and its role in binomial distributions and the Student's t-test, and provides an example of the sampling distributions of small populations. A proof of the central limit theorem is also described with the mathematical concepts required for its near-complete understanding.According to the central limit theorem, the means of a random sample of size, n, from a population with mean, µ, and variance, σ2, distribute normally with mean, µ, and variance, [Formula: see text]. Using the central limit theorem, a variety of parametric tests have been developed under assumptions about the parameters that determine the population probability distribution. Compared to non-parametric tests, which do not require any assumptions about the population probability distribution, parametric tests produce more accurate and precise estimates with higher statistical powers. However, many medical researchers use parametric tests to present their data without knowledge of the contribution of the central limit theorem to the development of such tests. Thus, this review presents the basic concepts of the central limit theorem and its role in binomial distributions and the Student's t-test, and provides an example of the sampling distributions of small populations. A proof of the central limit theorem is also described with the mathematical concepts required for its near-complete understanding. According to the central limit theorem, the means of a random sample of size, n, from a population with mean, μ, and variance, σ2, distribute normally with mean, μ, and variance, σ2 n . Using the central limit theorem, a variety of parametric tests have been developed under assumptions about the parameters that determine the population probability distribution. Compared to non-parametric tests, which do not require any assumptions about the population probability distribution, parametric tests produce more accurate and precise estimates with higher statistical powers. However, many medical researchers use parametric tests to present their data without knowledge of the contribution of the central limit theorem to the development of such tests. Thus, this review presents the basic concepts of the central limit theorem and its role in binomial distributions and the Student’s t-test, and provides an example of the sampling distributions of small populations. A proof of the central limit theorem is also described with the mathematical concepts required for its nearcomplete understanding. KCI Citation Count: 2 According to the central limit theorem, the means of a random sample of size, , from a population with mean, µ, and variance, σ , distribute normally with mean, µ, and variance, [Formula: see text]. Using the central limit theorem, a variety of parametric tests have been developed under assumptions about the parameters that determine the population probability distribution. Compared to non-parametric tests, which do not require any assumptions about the population probability distribution, parametric tests produce more accurate and precise estimates with higher statistical powers. However, many medical researchers use parametric tests to present their data without knowledge of the contribution of the central limit theorem to the development of such tests. Thus, this review presents the basic concepts of the central limit theorem and its role in binomial distributions and the Student's t-test, and provides an example of the sampling distributions of small populations. A proof of the central limit theorem is also described with the mathematical concepts required for its near-complete understanding. According to the central limit theorem, the means of a random sample of size, n , from a population with mean, µ, and variance, σ 2 , distribute normally with mean, µ, and variance, σ 2 n . Using the central limit theorem, a variety of parametric tests have been developed under assumptions about the parameters that determine the population probability distribution. Compared to non-parametric tests, which do not require any assumptions about the population probability distribution, parametric tests produce more accurate and precise estimates with higher statistical powers. However, many medical researchers use parametric tests to present their data without knowledge of the contribution of the central limit theorem to the development of such tests. Thus, this review presents the basic concepts of the central limit theorem and its role in binomial distributions and the Student's t-test, and provides an example of the sampling distributions of small populations. A proof of the central limit theorem is also described with the mathematical concepts required for its near-complete understanding. |
| Author | Kim, Jong Hae Kwak, Sang Gyu |
| AuthorAffiliation | 1 Department of Medical Statistics, School of Medicine, Catholic University of Daegu, Daegu, Korea 2 Department of Anesthesiology and Pain Medicine, School of Medicine, Catholic University of Daegu, Daegu, Korea |
| AuthorAffiliation_xml | – name: 2 Department of Anesthesiology and Pain Medicine, School of Medicine, Catholic University of Daegu, Daegu, Korea – name: 1 Department of Medical Statistics, School of Medicine, Catholic University of Daegu, Daegu, Korea |
| Author_xml | – sequence: 1 givenname: Sang Gyu orcidid: 0000-0003-0398-5514 surname: Kwak fullname: Kwak, Sang Gyu organization: Department of Medical Statistics, School of Medicine, Catholic University of Daegu, Daegu, Korea – sequence: 2 givenname: Jong Hae orcidid: 0000-0003-1222-0054 surname: Kim fullname: Kim, Jong Hae organization: Department of Anesthesiology and Pain Medicine, School of Medicine, Catholic University of Daegu, Daegu, Korea |
| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/28367284$$D View this record in MEDLINE/PubMed https://www.kci.go.kr/kciportal/ci/sereArticleSearch/ciSereArtiView.kci?sereArticleSearchBean.artiId=ART002211182$$DAccess content in National Research Foundation of Korea (NRF) |
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| Keywords | Probability Normal distribution Statistics Statistical distributions |
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| Snippet | According to the central limit theorem, the means of a random sample of size,
, from a population with mean, µ, and variance, σ
, distribute normally with... According to the central limit theorem, the means of a random sample of size, n, from a population with mean, µ, and variance, σ2, distribute normally with... According to the central limit theorem, the means of a random sample of size, n , from a population with mean, µ, and variance, σ 2 , distribute normally with... According to the central limit theorem, the means of a random sample of size, n, from a population with mean, μ, and variance, σ2, distribute normally with... |
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| Title | Central limit theorem: the cornerstone of modern statistics |
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