{"id":1898,"date":"2021-09-15T12:16:47","date_gmt":"2021-09-15T12:16:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1898"},"modified":"2023-12-05T09:22:19","modified_gmt":"2023-12-05T09:22:19","slug":"summary-the-central-limit-theorem-for-sample-means-averages","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-the-central-limit-theorem-for-sample-means-averages\/","title":{"raw":"Summary: The Central Limit Theorem for Sample Means (Averages)","rendered":"Summary: The Central Limit Theorem for Sample Means (Averages)"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The central limit theorem (for means) states that even if a population distribution is non-normal or the shape is unknown, the shape of the sampling distribution of the sample mean will be approximately normal if the sample size is large enough.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The mean of the sampling distribution of the sample means is equal to the mean of the population.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The standard deviation of the sampling distribution of the sample mean is the population mean divided by the square root of the sample size.\u00a0 This is also called the standard error of the mean.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>central limit theorem (for means): <\/strong>given a random variable (RV) with known mean \u03bc and known standard deviation, \u03c3, if\u00a0the size <em>(n)<\/em> of the sample is sufficiently large, then [latex]\\overline{X} \\sim N (M, \\frac{\\sigma}{\\sqrt{n}})[\/latex].\u00a0If the size (n) of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The standard deviation of the distribution of the sample means, [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex], is called the standard error of the mean.\r\n\r\n<strong>sampling distribution: <\/strong>given simple random samples of size n from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution\r\n\r\n<strong>standard error of the mean:\u00a0<\/strong>the standard deviation of the distribution of the sample means, or [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex]","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The central limit theorem (for means) states that even if a population distribution is non-normal or the shape is unknown, the shape of the sampling distribution of the sample mean will be approximately normal if the sample size is large enough.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The mean of the sampling distribution of the sample means is equal to the mean of the population.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The standard deviation of the sampling distribution of the sample mean is the population mean divided by the square root of the sample size.\u00a0 This is also called the standard error of the mean.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>central limit theorem (for means): <\/strong>given a random variable (RV) with known mean \u03bc and known standard deviation, \u03c3, if\u00a0the size <em>(n)<\/em> of the sample is sufficiently large, then [latex]\\overline{X} \\sim N (M, \\frac{\\sigma}{\\sqrt{n}})[\/latex].\u00a0If the size (n) of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The standard deviation of the distribution of the sample means, [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex], is called the standard error of the mean.<\/p>\n<p><strong>sampling distribution: <\/strong>given simple random samples of size n from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution<\/p>\n<p><strong>standard error of the mean:\u00a0<\/strong>the standard deviation of the distribution of the sample means, or [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex]<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1898\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1898","chapter","type-chapter","status-publish","hentry"],"part":262,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1898","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1898\/revisions"}],"predecessor-version":[{"id":3714,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1898\/revisions\/3714"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/262"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1898\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1898"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1898"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1898"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1898"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}