{"id":1043,"date":"2023-06-22T01:45:17","date_gmt":"2023-06-22T01:45:17","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/normal-distribution-learn-it-1\/"},"modified":"2025-05-16T02:23:52","modified_gmt":"2025-05-16T02:23:52","slug":"normal-distribution-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/normal-distribution-learn-it-1\/","title":{"raw":"Normal Distribution: Learn It 1","rendered":"Normal Distribution: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Understand the properties, characteristics, and importance of a normal distribution in statistical analysis.<\/li>\r\n\t<li>Explain how changing the mean and standard deviation will change the characteristics of a normal curve.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Continuous Random Variables<\/h2>\r\n<p>Previously, we studied discrete (listable) random variables and their distributions. In this section, we will explore <strong>continuous<\/strong> (decimal-valued) <strong>random variables<\/strong> that can take on values anywhere in an interval. For example, a person\u2019s exact weight without rounding is a continuous random variable. If rounded to the nearest pound, weight is a discrete random variable. Decimal-valued numbers appear frequently in real life, often in measuring things such as weight or length. To best study real-life data, we need to build a solid foundation in continuous probability distributions.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1514[\/ohm2_question]<\/section>\r\n<h3>Probability Density Curve<\/h3>\r\n<p>In a continuous probability distribution, probabilities are represented as areas under a curve.<\/p>\r\n<section class=\"textbox example\">You can see below a histogram of the sodium content in milligrams (mg) of [latex]20[\/latex] different cereals modeled by a curve. Note that models are not perfect representations of the data.<img class=\"aligncenter wp-image-1571\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/10\/11162913\/Picture15.png\" alt=\"A histogram of the sodium content in milligrams (mg) of 20 different cereals, with a normal curve overlayed over the histogram.\" width=\"474\" height=\"379\" \/><br \/>\r\nThe figure above has a similar shape as the histogram, with a few differences. The y-axis is now labeled \u201c<strong>density<\/strong>\u201d.<br \/>\r\nTo understand density, let\u2019s focus on the area in the first bar that represents the cereals with sodium contents between [latex]0[\/latex] and [latex]50[\/latex] mg. The percentage of cereals that have less than [latex]50[\/latex] mg of sodium is [latex]\\dfrac{1}{20}[\/latex], or [latex]0.05 (5\\%)[\/latex].\r\n\r\n<p>Thus, the shaded area of the density plot that is less than [latex]50[\/latex] mg of sodium is [latex]50*0.001 = 0.05[\/latex], where [latex]0.001[\/latex] is the height of the rectangle and [latex]50[\/latex] is the width.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1515[\/ohm2_question]<\/section>\r\n<section class=\"textbox proTip\">For all\u00a0continuous random variables, the probability distribution can be approximated by a smooth curve called a\u00a0<strong>probability density curve<\/strong>.<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand the properties, characteristics, and importance of a normal distribution in statistical analysis.<\/li>\n<li>Explain how changing the mean and standard deviation will change the characteristics of a normal curve.<\/li>\n<\/ul>\n<\/section>\n<h2>Continuous Random Variables<\/h2>\n<p>Previously, we studied discrete (listable) random variables and their distributions. In this section, we will explore <strong>continuous<\/strong> (decimal-valued) <strong>random variables<\/strong> that can take on values anywhere in an interval. For example, a person\u2019s exact weight without rounding is a continuous random variable. If rounded to the nearest pound, weight is a discrete random variable. Decimal-valued numbers appear frequently in real life, often in measuring things such as weight or length. To best study real-life data, we need to build a solid foundation in continuous probability distributions.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1514\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1514&theme=lumen&iframe_resize_id=ohm1514&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Probability Density Curve<\/h3>\n<p>In a continuous probability distribution, probabilities are represented as areas under a curve.<\/p>\n<section class=\"textbox example\">You can see below a histogram of the sodium content in milligrams (mg) of [latex]20[\/latex] different cereals modeled by a curve. Note that models are not perfect representations of the data.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1571\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/10\/11162913\/Picture15.png\" alt=\"A histogram of the sodium content in milligrams (mg) of 20 different cereals, with a normal curve overlayed over the histogram.\" width=\"474\" height=\"379\" \/><br \/>\nThe figure above has a similar shape as the histogram, with a few differences. The y-axis is now labeled \u201c<strong>density<\/strong>\u201d.<br \/>\nTo understand density, let\u2019s focus on the area in the first bar that represents the cereals with sodium contents between [latex]0[\/latex] and [latex]50[\/latex] mg. The percentage of cereals that have less than [latex]50[\/latex] mg of sodium is [latex]\\dfrac{1}{20}[\/latex], or [latex]0.05 (5\\%)[\/latex].<\/p>\n<p>Thus, the shaded area of the density plot that is less than [latex]50[\/latex] mg of sodium is [latex]50*0.001 = 0.05[\/latex], where [latex]0.001[\/latex] is the height of the rectangle and [latex]50[\/latex] is the width.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1515\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1515&theme=lumen&iframe_resize_id=ohm1515&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox proTip\">For all\u00a0continuous random variables, the probability distribution can be approximated by a smooth curve called a\u00a0<strong>probability density curve<\/strong>.<\/section>\n","protected":false},"author":8,"menu_order":12,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":3053,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1043"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":12,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1043\/revisions"}],"predecessor-version":[{"id":6688,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1043\/revisions\/6688"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/3053"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1043\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1043"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1043"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1043"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1043"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}