{"id":1044,"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-2\/"},"modified":"2024-03-14T19:09:19","modified_gmt":"2024-03-14T19:09:19","slug":"normal-distribution-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/normal-distribution-learn-it-2\/","title":{"raw":"Normal Distribution: Learn It 2","rendered":"Normal Distribution: Learn It 2"},"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 Distribution<\/h2>\r\n<p><strong>Normal distribution<\/strong> is one of the most common types of continuous distributions used in statistics. The model we saw about\u00a0the sodium content in milligrams (mg) of [latex]20[\/latex] different cereals is a normal distribution.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>normal distribution<\/h3>\r\n<p>A <strong>normal distribution<\/strong> is a mathematical model with a smooth bell-shaped curve to describe the bell-shaped data distributions.<\/p>\r\n<p>A normal distribution has the following characteristics:<\/p>\r\n<ol>\r\n\t<li>[latex]x[\/latex] is a continuous random variable.<\/li>\r\n\t<li>Symmetrical around the mean, [latex]\\mu[\/latex] (pronounced \u201cmu\u201d), the left side is a mirror image of the right side centered at the mean.<\/li>\r\n\t<li>There is one peak at the mean of bell-shaped distributions.<\/li>\r\n<\/ol>\r\n<p>The following graph displays a normal distribution.<\/p>\r\n<p style=\"text-align: left;\"><img class=\"aligncenter wp-image-5825 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/03231908\/picture-1.png\" alt=\"A normal distribution curve with a mean of 0 and a standard deviation of 1.\" width=\"742\" height=\"400\" \/><\/p>\r\n<\/section>\r\n<p><span style=\"font-size: 1rem; text-align: initial;\">Recall the Empirical Rule we learned earlier. We know that [latex]99.7\\%[\/latex] of the data fall within three standard deviations of the mean, so for normal distributions, we will be generally concerned with looking at values within [latex]\\pm3[\/latex] standard deviations ([latex]\\sigma[\/latex]) of the mean ([latex]\\mu[\/latex]). If we know that data is normally distributed and we know the mean and standard deviation, we can draw the graph for the normal distribution.<\/span><\/p>\r\n<section class=\"textbox proTip\">The normal distribution is <strong>centered at the mean ([latex]\\mu[\/latex])<\/strong>.\r\n\r\n<p>Sometimes the mean is called the \u201c<strong>location parameter.<\/strong>\u201d<\/p>\r\n<\/section>\r\n<p>The value of the mean gives the location of the distribution on the [latex]x[\/latex]-axis. Looking at the following graphs, the mean of the red graph (on the left) is [latex]0[\/latex], the mean of the blue graph (in the middle) is [latex]5[\/latex], and the mean of the green graph (on the right) is [latex]10[\/latex]. As the mean increases, the graphs shift to the right on the [latex]x[\/latex]-axis.<\/p>\r\n<p><img class=\"aligncenter wp-image-5827 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/03232108\/Picture1-3.png\" alt=\"Three normal curve comparisons, showing a mean of 0, 5, and 10.\" width=\"554\" height=\"440\" \/><\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1517[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1014[\/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 Distribution<\/h2>\n<p><strong>Normal distribution<\/strong> is one of the most common types of continuous distributions used in statistics. The model we saw about\u00a0the sodium content in milligrams (mg) of [latex]20[\/latex] different cereals is a normal distribution.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>normal distribution<\/h3>\n<p>A <strong>normal distribution<\/strong> is a mathematical model with a smooth bell-shaped curve to describe the bell-shaped data distributions.<\/p>\n<p>A normal distribution has the following characteristics:<\/p>\n<ol>\n<li>[latex]x[\/latex] is a continuous random variable.<\/li>\n<li>Symmetrical around the mean, [latex]\\mu[\/latex] (pronounced \u201cmu\u201d), the left side is a mirror image of the right side centered at the mean.<\/li>\n<li>There is one peak at the mean of bell-shaped distributions.<\/li>\n<\/ol>\n<p>The following graph displays a normal distribution.<\/p>\n<p style=\"text-align: left;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5825 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/03231908\/picture-1.png\" alt=\"A normal distribution curve with a mean of 0 and a standard deviation of 1.\" width=\"742\" height=\"400\" \/><\/p>\n<\/section>\n<p><span style=\"font-size: 1rem; text-align: initial;\">Recall the Empirical Rule we learned earlier. We know that [latex]99.7\\%[\/latex] of the data fall within three standard deviations of the mean, so for normal distributions, we will be generally concerned with looking at values within [latex]\\pm3[\/latex] standard deviations ([latex]\\sigma[\/latex]) of the mean ([latex]\\mu[\/latex]). If we know that data is normally distributed and we know the mean and standard deviation, we can draw the graph for the normal distribution.<\/span><\/p>\n<section class=\"textbox proTip\">The normal distribution is <strong>centered at the mean ([latex]\\mu[\/latex])<\/strong>.<\/p>\n<p>Sometimes the mean is called the \u201c<strong>location parameter.<\/strong>\u201d<\/p>\n<\/section>\n<p>The value of the mean gives the location of the distribution on the [latex]x[\/latex]-axis. Looking at the following graphs, the mean of the red graph (on the left) is [latex]0[\/latex], the mean of the blue graph (in the middle) is [latex]5[\/latex], and the mean of the green graph (on the right) is [latex]10[\/latex]. As the mean increases, the graphs shift to the right on the [latex]x[\/latex]-axis.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5827 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/03232108\/Picture1-3.png\" alt=\"Three normal curve comparisons, showing a mean of 0, 5, and 10.\" width=\"554\" height=\"440\" \/><\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1517\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1517&theme=lumen&iframe_resize_id=ohm1517&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1014\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1014&theme=lumen&iframe_resize_id=ohm1014&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":13,"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\/1044"}],"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":10,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1044\/revisions"}],"predecessor-version":[{"id":5951,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1044\/revisions\/5951"}],"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\/1044\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1044"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1044"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1044"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1044"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}