{"id":1046,"date":"2023-06-22T01:45:20","date_gmt":"2023-06-22T01:45:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/normal-distribution-learn-it-3\/"},"modified":"2025-05-16T02:24:30","modified_gmt":"2025-05-16T02:24:30","slug":"normal-distribution-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/normal-distribution-learn-it-3\/","title":{"raw":"Normal Distribution: Learn It 3","rendered":"Normal Distribution: Learn It 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li class=\"li1\">Understand the properties, characteristics, and importance of a normal distribution in statistical analysis.<\/li>\r\n\t<li class=\"li1\">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>Normal Curve<\/h2>\r\n<p>There are many normal curves. Even though all normal curves have the same bell shape, they vary in their center and spread.<\/p>\r\n<section class=\"textbox proTip\">The shape of a normal curve will depend on the <strong>standard deviation<\/strong>.\r\n\r\n<p>The standard deviation, [latex]\\sigma[\/latex] (pronounced \u201csigma\u201d), is often referred to as the <strong>scale parameter<\/strong>.<\/p>\r\n<\/section>\r\n<p>The shape of a normal curve will still be bell-shaped and unimodal, but the standard deviation will change how spread out or flat the curve appears.<\/p>\r\n<section class=\"textbox example\">In the following figure, all three curves have different heights and widths, but they are all still normal distributions with the same mean. As the standard deviation increases, the curve gets flatter. The standard deviation of the red curve (the tall curve) is [latex]1[\/latex], the standard deviation of the blue curve (the middle curve) is [latex]2[\/latex], and the standard deviation of the purple curve (the shorter curve) is [latex]3[\/latex].<img class=\"aligncenter wp-image-5832 size-large\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22014518\/Picture1-5.png\" alt=\"Three curves that get increasingly flatter as the standard deviation increases.\" width=\"1024\" height=\"444\" \/><\/section>\r\n<p>Let's explore the normal curve by changing its mean\u00a0<span style=\"font-size: 1rem; text-align: initial;\">([latex]\\mu[\/latex]) and its standard deviation ([latex]\\sigma[\/latex]).<\/span><\/p>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/normaldist\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\r\n<p><br \/>\r\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/normaldist\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]902[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1518[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1519[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1015[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li class=\"li1\">Understand the properties, characteristics, and importance of a normal distribution in statistical analysis.<\/li>\n<li class=\"li1\">Explain how changing the mean and standard deviation will change the characteristics of a normal curve.<\/li>\n<\/ul>\n<\/section>\n<h2>Normal Curve<\/h2>\n<p>There are many normal curves. Even though all normal curves have the same bell shape, they vary in their center and spread.<\/p>\n<section class=\"textbox proTip\">The shape of a normal curve will depend on the <strong>standard deviation<\/strong>.<\/p>\n<p>The standard deviation, [latex]\\sigma[\/latex] (pronounced \u201csigma\u201d), is often referred to as the <strong>scale parameter<\/strong>.<\/p>\n<\/section>\n<p>The shape of a normal curve will still be bell-shaped and unimodal, but the standard deviation will change how spread out or flat the curve appears.<\/p>\n<section class=\"textbox example\">In the following figure, all three curves have different heights and widths, but they are all still normal distributions with the same mean. As the standard deviation increases, the curve gets flatter. The standard deviation of the red curve (the tall curve) is [latex]1[\/latex], the standard deviation of the blue curve (the middle curve) is [latex]2[\/latex], and the standard deviation of the purple curve (the shorter curve) is [latex]3[\/latex].<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5832 size-large\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22014518\/Picture1-5.png\" alt=\"Three curves that get increasingly flatter as the standard deviation increases.\" width=\"1024\" height=\"444\" \/><\/section>\n<p>Let&#8217;s explore the normal curve by changing its mean\u00a0<span style=\"font-size: 1rem; text-align: initial;\">([latex]\\mu[\/latex]) and its standard deviation ([latex]\\sigma[\/latex]).<\/span><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/normaldist\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\n<p>\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/normaldist\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm902\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=902&theme=lumen&iframe_resize_id=ohm902&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1518\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1518&theme=lumen&iframe_resize_id=ohm1518&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1519\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1519&theme=lumen&iframe_resize_id=ohm1519&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1015\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1015&theme=lumen&iframe_resize_id=ohm1015&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":14,"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\/1046"}],"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":11,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1046\/revisions"}],"predecessor-version":[{"id":6689,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1046\/revisions\/6689"}],"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\/1046\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1046"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1046"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1046"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1046"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}