{"id":1275,"date":"2021-08-23T18:51:16","date_gmt":"2021-08-23T18:51:16","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1275"},"modified":"2023-12-05T09:08:35","modified_gmt":"2023-12-05T09:08:35","slug":"summary-mean-or-expected-value-and-standard-deviation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-mean-or-expected-value-and-standard-deviation\/","title":{"raw":"Summary: Mean or Expected Value and Standard Deviation","rendered":"Summary: Mean or Expected Value and Standard Deviation"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The mean (expected value) of a discrete random variable is found by multiplying each value of the random variable by its associated probability and summing the results.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The standard deviation of a discrete random variable describes the typical distance a value is from the mean.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">As the number of trials in a probability experiment increases, the mean approaches the theoretical expected value. This is called the law of large numbers.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>expected value:<\/strong> expected arithmetic average when an experiment is repeated many times; also called the mean. Notations: [latex]\u03bc[\/latex]. For a discrete random variable [latex](RV)[\/latex] with probability distribution function [latex]P(x)[\/latex], the definition can also be written in the form [latex]\\mu = \\sum x \\cdot P(x).[\/latex]\r\n\r\n<strong>mean:<\/strong> a number that measures the central tendency of a distribution; a common name for mean is \"average.\" For a probability distribution, the mean is written as [latex]\\mu[\/latex] and it is the long-term average of many trials.\r\n\r\n<strong>standard deviation of a probability distribution:\u00a0<\/strong>a number that measures how far the outcomes of a statistical experiment are from the mean of the distribution [latex]\\sigma = \\sqrt{\\sum [(x-M)^{2} \\cdot P(x)]}[\/latex].\r\n\r\n<strong>the Law of Large Numbers: <\/strong>as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency probability approaches zero","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The mean (expected value) of a discrete random variable is found by multiplying each value of the random variable by its associated probability and summing the results.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The standard deviation of a discrete random variable describes the typical distance a value is from the mean.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">As the number of trials in a probability experiment increases, the mean approaches the theoretical expected value. This is called the law of large numbers.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>expected value:<\/strong> expected arithmetic average when an experiment is repeated many times; also called the mean. Notations: [latex]\u03bc[\/latex]. For a discrete random variable [latex](RV)[\/latex] with probability distribution function [latex]P(x)[\/latex], the definition can also be written in the form [latex]\\mu = \\sum x \\cdot P(x).[\/latex]<\/p>\n<p><strong>mean:<\/strong> a number that measures the central tendency of a distribution; a common name for mean is &#8220;average.&#8221; For a probability distribution, the mean is written as [latex]\\mu[\/latex] and it is the long-term average of many trials.<\/p>\n<p><strong>standard deviation of a probability distribution:\u00a0<\/strong>a number that measures how far the outcomes of a statistical experiment are from the mean of the distribution [latex]\\sigma = \\sqrt{\\sum [(x-M)^{2} \\cdot P(x)]}[\/latex].<\/p>\n<p><strong>the Law of Large Numbers: <\/strong>as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency probability approaches zero<\/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-1275\">\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><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\/4-key-terms\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/4-key-terms<\/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":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/4-key-terms\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at 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