{"id":1047,"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-4\/"},"modified":"2025-05-16T02:25:26","modified_gmt":"2025-05-16T02:25:26","slug":"normal-distribution-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/normal-distribution-learn-it-4\/","title":{"raw":"Normal Distribution: Learn It 4","rendered":"Normal Distribution: Learn It 4"},"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>Standard Normal Distribution ([latex]z[\/latex] Distribution)<\/h2>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>[latex]z[\/latex] distribution<\/h3>\r\n<p>A normal distribution with a mean ([latex]\\mu[\/latex]) = 0 and a standard deviation ([latex]\\sigma[\/latex]) = 1 is called the <strong>standard normal distribution<\/strong>.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1520[\/ohm2_question]<\/section>\r\n<h3><strong>Let\u2019s Summarize<\/strong><\/h3>\r\n<ul>\r\n\t<li>A continuous random variable is not limited to distinct values. We cannot display the probability distribution for a continuous random variable with a table or histogram. We use a density curve to assign probabilities to intervals of [latex]x[\/latex]-values. We use the area under the density curve to find probabilities.<\/li>\r\n\t<li>We use a normal density curve to model the probability distribution for many variables, such as weight, shoe sizes, foot lengths, and other human physical characteristics. Normal curves are mathematical models. We use <strong>[latex]\\mu[\/latex] <\/strong>to represent the mean of a normal curve and <strong>[latex]\\sigma[\/latex]<\/strong> to represent the standard deviation of a normal curve. We use Greek letters to remind us that the normal curve is not a distribution of real data. It is a mathematical model based on a mathematical equation. We use this mathematical model to represent the perfect bell-shaped distribution.<\/li>\r\n\t<li>Recall: For a normal curve, the empirical rule for normal curves tells us that [latex]68\\%[\/latex] of the observations fall within [latex]1[\/latex] standard deviation of the mean, [latex]95\\%[\/latex] within [latex]2[\/latex] standard deviations of the mean, and [latex]99.7\\%[\/latex] within [latex]3[\/latex] standard deviations of the mean.<\/li>\r\n\t<li>To compare [latex]x[\/latex]-values from different distributions, we can standardize the values into a standard normal distribution. If we convert the [latex]x[\/latex]-values into [latex]z[\/latex]-scores, the distribution of [latex]z[\/latex]-scores is also a normal density curve with a mean of [latex]0[\/latex] and a standard deviation of [latex]1[\/latex]. This curve is called the <strong>standard normal distribution<\/strong>. We can then use the standard normal curve to find probabilities for any normal distribution.<\/li>\r\n<\/ul>","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>Standard Normal Distribution ([latex]z[\/latex] Distribution)<\/h2>\n<section class=\"textbox keyTakeaway\">\n<h3>[latex]z[\/latex] distribution<\/h3>\n<p>A normal distribution with a mean ([latex]\\mu[\/latex]) = 0 and a standard deviation ([latex]\\sigma[\/latex]) = 1 is called the <strong>standard normal distribution<\/strong>.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1520\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1520&theme=lumen&iframe_resize_id=ohm1520&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3><strong>Let\u2019s Summarize<\/strong><\/h3>\n<ul>\n<li>A continuous random variable is not limited to distinct values. We cannot display the probability distribution for a continuous random variable with a table or histogram. We use a density curve to assign probabilities to intervals of [latex]x[\/latex]-values. We use the area under the density curve to find probabilities.<\/li>\n<li>We use a normal density curve to model the probability distribution for many variables, such as weight, shoe sizes, foot lengths, and other human physical characteristics. Normal curves are mathematical models. We use <strong>[latex]\\mu[\/latex] <\/strong>to represent the mean of a normal curve and <strong>[latex]\\sigma[\/latex]<\/strong> to represent the standard deviation of a normal curve. We use Greek letters to remind us that the normal curve is not a distribution of real data. It is a mathematical model based on a mathematical equation. We use this mathematical model to represent the perfect bell-shaped distribution.<\/li>\n<li>Recall: For a normal curve, the empirical rule for normal curves tells us that [latex]68\\%[\/latex] of the observations fall within [latex]1[\/latex] standard deviation of the mean, [latex]95\\%[\/latex] within [latex]2[\/latex] standard deviations of the mean, and [latex]99.7\\%[\/latex] within [latex]3[\/latex] standard deviations of the mean.<\/li>\n<li>To compare [latex]x[\/latex]-values from different distributions, we can standardize the values into a standard normal distribution. If we convert the [latex]x[\/latex]-values into [latex]z[\/latex]-scores, the distribution of [latex]z[\/latex]-scores is also a normal density curve with a mean of [latex]0[\/latex] and a standard deviation of [latex]1[\/latex]. This curve is called the <strong>standard normal distribution<\/strong>. We can then use the standard normal curve to find probabilities for any normal distribution.<\/li>\n<\/ul>\n","protected":false},"author":8,"menu_order":15,"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\/1047"}],"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":7,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1047\/revisions"}],"predecessor-version":[{"id":6690,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1047\/revisions\/6690"}],"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\/1047\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1047"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1047"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1047"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1047"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}