{"id":645,"date":"2020-07-20T22:09:11","date_gmt":"2020-07-21T02:09:11","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&#038;p=645"},"modified":"2025-06-19T17:02:06","modified_gmt":"2025-06-19T21:02:06","slug":"5-3-standard-normal-density-curve","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/5-3-standard-normal-density-curve\/","title":{"raw":"5.3 Standard Normal Density Curve","rendered":"5.3 Standard Normal Density Curve"},"content":{"raw":"The standardized variable (z-score) has a mean of 0 and a standard deviation of 1. We can also standardize the normal variable [latex]X \\sim N(\\mu, \\sigma)[\/latex] by [latex]Z=\\frac{X - \\mu}{\\sigma}[\/latex] and [latex]Z[\/latex] follows a standard normal distribution with mean 0 and standard deviation 1, i.e., [latex]Z \\sim N(0,1)[\/latex]. <a id=\"retfig5.4\"><\/a>\r\n\r\n[caption id=\"attachment_2762\" align=\"aligncenter\" width=\"1008\"]<img class=\"wp-image-2762 size-full\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2022\/04\/standardize-1.png\" alt=\"Two normal distributions are shown. The one on the right is the standard z-score. The grey area demonstrates how standardising works. Image description available.\" width=\"1008\" height=\"435\" \/> <strong>Figure 5.4<\/strong>: Mapping from Normal N(\u00b5, \u03c3) to Standard Normal N(0, 1). [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig5.4\">Image Description (See Appendix D Figure 5.4)<\/a>][\/caption]The standard normal density curve has the following properties:\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Facts: Properties of a Standard Normal Density Curve<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li>Total area under the curve is 1.<\/li>\r\n \t<li>The curve extends from [latex]- \\infty[\/latex]\u00a0to [latex]+ \\infty[\/latex].<\/li>\r\n \t<li>Symmetric at 0, which means the area to the <strong>right<\/strong> of a positive number [latex]a[\/latex]\u00a0is equal to the area to the <strong>left<\/strong> of [latex]-a[\/latex]. For example, [latex]P(Z \\ge 2) = P(Z \\le -2).[\/latex]<\/li>\r\n \t<li>Empirical rule:\r\n<ol>\r\n \t<li>68.26% of the observations are within the interval [-1, 1]. The area under the curve between -1 and 1 is 0.6826.<\/li>\r\n \t<li>95.44% of the observations are within the interval [-2, 2]. The area under the curve between -2 and 2 is 0.9544.<\/li>\r\n \t<li>99.74% of the observations are within the interval [-3, 3]. The area under the curve between -3 and 3 is 0.9974.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<span style=\"text-align: initial; font-size: 1em;\">Standardization converts all normal distributions to a single one\u2014the standard normal distribution. Therefore, we can calculate the probabilities of events relative to any normal distribution using only the standard normal density curve.<a id=\"retfig5.5\"><\/a><\/span>\r\n\r\n[caption id=\"attachment_3040\" align=\"aligncenter\" width=\"720\"]<img class=\"wp-image-3040 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/normal_standardization_one.png\" alt=\"Three normal density curves are shown over a horizontal axis. Arrows indicate how each of them can be converted to a standard normal curve. Image description available.\" width=\"720\" height=\"501\" \/> <strong>Figure 5.5<\/strong>: Converting Normal Distributions to Standard Normal. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig5.5\">Image Description (See Appendix D Figure 5.5)<\/a>][\/caption]\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example: Density Curve and Probability<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSuppose the grades\u00a0in a Statistics class are approximately normally distributed with mean [latex]\\mu=70[\/latex]\u00a0and standard deviation [latex]\\sigma=10[\/latex].\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>The percentage (proportion, to be more precise) of students with a grade <strong>below<\/strong> 60 equals the area under the standard normal curve to the <strong>left<\/strong> of <strong>-1<\/strong>. The [latex]z[\/latex]-score is calculated by [latex]z = \\frac{x- \\mu}{\\sigma} = \\frac{60 - 70}{10} = -1[\/latex].<\/li>\r\n \t<li>The percentage (proportion) of students with a grade <strong>above<\/strong> 90 equals the area under the standard normal curve to the <strong>right<\/strong> of <strong>2<\/strong>. The [latex]z[\/latex]-score is calculated by\u00a0[latex]z = \\frac{x- \\mu}{\\sigma} = \\frac{90 - 70}{10} = 2[\/latex].<\/li>\r\n \t<li>The percentage (proportion) of students with a grade between 60 and 90 equals the area under the standard normal curve between <strong>-1<\/strong> and <strong>2<\/strong>. Figure 5.4 gives a graphical presentation of this question with [latex]\\mu=70, \\sigma=10, a=60[\/latex] and [latex]b=90[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>","rendered":"<p>The standardized variable (z-score) has a mean of 0 and a standard deviation of 1. We can also standardize the normal variable [latex]X \\sim N(\\mu, \\sigma)[\/latex] by [latex]Z=\\frac{X - \\mu}{\\sigma}[\/latex] and [latex]Z[\/latex] follows a standard normal distribution with mean 0 and standard deviation 1, i.e., [latex]Z \\sim N(0,1)[\/latex]. <a id=\"retfig5.4\"><\/a><\/p>\n<figure id=\"attachment_2762\" aria-describedby=\"caption-attachment-2762\" style=\"width: 1008px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2762 size-full\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2022\/04\/standardize-1.png\" alt=\"Two normal distributions are shown. The one on the right is the standard z-score. The grey area demonstrates how standardising works. Image description available.\" width=\"1008\" height=\"435\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standardize-1.png 1008w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standardize-1-300x129.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standardize-1-768x331.png 768w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standardize-1-65x28.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standardize-1-225x97.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standardize-1-350x151.png 350w\" sizes=\"auto, (max-width: 1008px) 100vw, 1008px\" \/><figcaption id=\"caption-attachment-2762\" class=\"wp-caption-text\"><strong>Figure 5.4<\/strong>: Mapping from Normal N(\u00b5, \u03c3) to Standard Normal N(0, 1). [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig5.4\">Image Description (See Appendix D Figure 5.4)<\/a>]<\/figcaption><\/figure>\n<p>The standard normal density curve has the following properties:<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Facts: Properties of a Standard Normal Density Curve<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li>Total area under the curve is 1.<\/li>\n<li>The curve extends from [latex]- \\infty[\/latex]\u00a0to [latex]+ \\infty[\/latex].<\/li>\n<li>Symmetric at 0, which means the area to the <strong>right<\/strong> of a positive number [latex]a[\/latex]\u00a0is equal to the area to the <strong>left<\/strong> of [latex]-a[\/latex]. For example, [latex]P(Z \\ge 2) = P(Z \\le -2).[\/latex]<\/li>\n<li>Empirical rule:\n<ol>\n<li>68.26% of the observations are within the interval [-1, 1]. The area under the curve between -1 and 1 is 0.6826.<\/li>\n<li>95.44% of the observations are within the interval [-2, 2]. The area under the curve between -2 and 2 is 0.9544.<\/li>\n<li>99.74% of the observations are within the interval [-3, 3]. The area under the curve between -3 and 3 is 0.9974.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p><span style=\"text-align: initial; font-size: 1em;\">Standardization converts all normal distributions to a single one\u2014the standard normal distribution. Therefore, we can calculate the probabilities of events relative to any normal distribution using only the standard normal density curve.<a id=\"retfig5.5\"><\/a><\/span><\/p>\n<figure id=\"attachment_3040\" aria-describedby=\"caption-attachment-3040\" style=\"width: 720px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3040 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/normal_standardization_one.png\" alt=\"Three normal density curves are shown over a horizontal axis. Arrows indicate how each of them can be converted to a standard normal curve. Image description available.\" width=\"720\" height=\"501\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/normal_standardization_one.png 720w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/normal_standardization_one-300x209.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/normal_standardization_one-65x45.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/normal_standardization_one-225x157.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/normal_standardization_one-350x244.png 350w\" sizes=\"auto, (max-width: 720px) 100vw, 720px\" \/><figcaption id=\"caption-attachment-3040\" class=\"wp-caption-text\"><strong>Figure 5.5<\/strong>: Converting Normal Distributions to Standard Normal. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig5.5\">Image Description (See Appendix D Figure 5.5)<\/a>]<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example: Density Curve and Probability<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Suppose the grades\u00a0in a Statistics class are approximately normally distributed with mean [latex]\\mu=70[\/latex]\u00a0and standard deviation [latex]\\sigma=10[\/latex].<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>The percentage (proportion, to be more precise) of students with a grade <strong>below<\/strong> 60 equals the area under the standard normal curve to the <strong>left<\/strong> of <strong>-1<\/strong>. The [latex]z[\/latex]-score is calculated by [latex]z = \\frac{x- \\mu}{\\sigma} = \\frac{60 - 70}{10} = -1[\/latex].<\/li>\n<li>The percentage (proportion) of students with a grade <strong>above<\/strong> 90 equals the area under the standard normal curve to the <strong>right<\/strong> of <strong>2<\/strong>. The [latex]z[\/latex]-score is calculated by\u00a0[latex]z = \\frac{x- \\mu}{\\sigma} = \\frac{90 - 70}{10} = 2[\/latex].<\/li>\n<li>The percentage (proportion) of students with a grade between 60 and 90 equals the area under the standard normal curve between <strong>-1<\/strong> and <strong>2<\/strong>. Figure 5.4 gives a graphical presentation of this question with [latex]\\mu=70, \\sigma=10, a=60[\/latex] and [latex]b=90[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n","protected":false},"author":19,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-645","chapter","type-chapter","status-publish","hentry"],"part":588,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/645","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/users\/19"}],"version-history":[{"count":28,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/645\/revisions"}],"predecessor-version":[{"id":5536,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/645\/revisions\/5536"}],"part":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/parts\/588"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/645\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=645"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=645"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=645"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=645"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}