{"id":263,"date":"2024-02-16T17:40:46","date_gmt":"2024-02-16T22:40:46","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=part&#038;p=263"},"modified":"2025-02-28T10:01:31","modified_gmt":"2025-02-28T15:01:31","slug":"confidence-intervals","status":"publish","type":"part","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/part\/confidence-intervals\/","title":{"raw":"Normal Distributions","rendered":"Normal Distributions"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUnderstand the shape, statistical properties and formulas for Normal Distributions.\r\n\r\n<\/div>\r\n<\/div>\r\n<h2>Properties of Normal Distributions<\/h2>\r\nA <a href=\"https:\/\/www.wolframalpha.com\/input?i=normal+distribution\">normal distribution<\/a> is:\r\n<ul>\r\n \t<li>The most common type of distribution<\/li>\r\n \t<li>It is continuous and has a \u201cbell\u201d shape.<\/li>\r\n \t<li>It is 'symmetric' about the mean (\u00b5) (see more in the 'SYMMETRIC' section)<\/li>\r\n \t<li>The total area under the normal curve is 1.<\/li>\r\n \t<li>Ie: the probability of being anywhere on the distribution=1.<\/li>\r\n<\/ul>\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributions.jpg\"><img class=\"alignnone wp-image-1917 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributions.jpg\" alt=\"Image with multiple bell shaped curves. Two of the curves have a population mean at zero and the standard deviation varies. The curve with the smaller standard deviation is less spread out and more concentrated around the middle. The curve with the mean at one is displaced to the right by one unit. It also has a larger standard deviation and is, therefore, more spread out.\" width=\"1388\" height=\"532\" \/><\/a>\r\n<h1>Calculating Probabilities<\/h1>\r\nIt would require a calculate technique called <a href=\"https:\/\/www.mathsisfun.com\/calculus\/integration-by-parts.html\">Integration by Parts<\/a> to calculate the probabilities by hand for the Normal Distribution. For this reason, we will only use Excel's <a href=\"https:\/\/support.microsoft.com\/en-gb\/office\/norm-dist-function-edb1cc14-a21c-4e53-839d-8082074c9f8d\">NORM.DIST()<\/a> function to calculate probabilities:\r\n<ul>\r\n \t<li>[latex]P[\/latex](at most or less than) =NORM.DIST([latex]x[\/latex], \u00b5, \u03c3, TRUE)<\/li>\r\n \t<li>[latex]P[\/latex](at least or more than) =1\u2212NORM.DIST([latex]x[\/latex], \u00b5, \u03c3, TRUE)<\/li>\r\n \t<li>Where \u00b5 (mu) is the mean of the distribution and \u03c3 (sigma) is the standard deviation.<\/li>\r\n<\/ul>\r\n<h1>Calculating X-Values<\/h1>\r\nIf we are looking to solve for the [latex]x[\/latex]-value instead of the probability, this is called an '<a href=\"https:\/\/www.radfordmathematics.com\/probabilities-and-statistics\/normal-distributions\/inverse-normal-distributions.html\">inverse<\/a>' problem and we use Excel's <a href=\"https:\/\/support.microsoft.com\/en-au\/office\/norm-inv-function-54b30935-fee7-493c-bedb-2278a9db7e13#:~:text=INV%20function,-Excel%20for%20Microsoft&amp;text=Returns%20the%20inverse%20of%20the,specified%20mean%20and%20standard%20deviation.\">NORM.INV()<\/a> function:\r\n<ul>\r\n \t<li>[latex]x[\/latex] = NORM.INV(Area to left of [latex]x[\/latex], \u00b5, \u03c3)<\/li>\r\n \t<li>[latex]x[\/latex] = NORM.INV(1\u2212 Area to right of [latex]x[\/latex], \u00b5, \u03c3)<\/li>\r\n<\/ul>\r\n<h1>Z-Scores<\/h1>\r\nA <a href=\"https:\/\/www.investopedia.com\/terms\/z\/zscore.asp\">z-score<\/a> is:\r\n<ul>\r\n \t<li>\"A statistical measurement that describes a value's relationship to the mean of a group of values.\"<\/li>\r\n \t<li>\"Measured in terms of standard deviations from the mean.\"<\/li>\r\n \t<li>\"A measure of an instrument's variability and can be used by traders to help determine volatility.\"<\/li>\r\n<\/ul>\r\nIt can be calculated using a formula if the [latex]x[\/latex]-value, \u00b5 (mu) and, \u03c3 (sigma) are given:\r\n\\[z = \\frac{x-\\mu}{\\sigma}\\]\r\n\r\n<span style=\"text-align: initial\">It can be calculated using Excel's <\/span><a style=\"text-align: initial\" href=\"https:\/\/support.microsoft.com\/en-us\/office\/norm-s-inv-function-d6d556b4-ab7f-49cd-b526-5a20918452b1#:~:text=Returns%20the%20inverse%20of%20the%20standard%20normal%20cumulative%20distribution.\">NORM.S.INV<\/a><span style=\"text-align: initial\"> function if the area\/probability is given: <\/span>\\[ z = \\text{NORM.S.INV}(\\text{Area to left of z}) = \\text{NORM.S.INV}(1-\\text{Area to right of z})\\]\r\n<h1>Symmetric Property<\/h1>\r\n<ul>\r\n \t<li>It is symmetric (or identical) on either side of the mean.<\/li>\r\n \t<li>The mean and median are equal.<\/li>\r\n \t<li>The data in this distribution is neither skewed left nor skewed right.<\/li>\r\n<\/ul>\r\n<h1>Statistical Properties<\/h1>\r\nThe following metrics apply to Normal Distributions:\r\n<ul>\r\n \t<li>population mean = \u00b5<\/li>\r\n \t<li>sample mean = x\u0304<\/li>\r\n \t<li>population standard deviation = \u03c3<\/li>\r\n \t<li>sample standard deviation = s<\/li>\r\n \t<li>mode = \u00b5 or x\u0304 (depending if population or sample given)<\/li>\r\n \t<li>variance = \u03c3<sup>2<\/sup> or s<sup>2<\/sup> (depending if population or sample given)<\/li>\r\n \t<li>symmetric (not skewed) and the skewness = 0<\/li>\r\n<\/ul>\r\n<h1>Video &amp; Resources Explaining Normal Distributions<\/h1>\r\n[embed]https:\/\/youtu.be\/pX-ldE3JOEs[\/embed]\r\n\r\n<span style=\"color: #003366\"><strong>Additional Resources<\/strong><\/span>:\r\n<ul>\r\n \t<li><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributions.pptx\">Click here<\/a> to download the Powerpoint slides that accompany the video.<\/li>\r\n \t<li><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributionSolutions.xlsx\">Click here<\/a> to download the Excel solutions for the Normal Distribution section.<\/li>\r\n<\/ul>\r\n<h1>Key Takeaways (EXERCISE)<\/h1>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Takeaways: Normal Distributions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDrag the words into the correct boxes for each section below:\r\n\r\n[h5p id=\"111\"]\r\n\r\n[h5p id=\"116\"]\r\n\r\n[h5p id=\"112\"]\r\n\r\n[h5p id=\"113\"]\r\n\r\n[h5p id=\"115\"]\r\n\r\n[h5p id=\"114\"]\r\n\r\nClick the sections below to reveal the solutions to the above exercises\r\n\r\n[h5p id=\"117\"]\r\n\r\n<\/div>\r\n<\/div>\r\n<h1>Your Own Notes (EXERCISE)<\/h1>\r\n<ul>\r\n \t<li>Are there any notes you want to take from this section? Is there anything you'd like to copy and paste below?<\/li>\r\n \t<li>These notes are for you only (they will not be stored anywhere)<\/li>\r\n \t<li>Make sure to download them at the end to use as a reference<\/li>\r\n<\/ul>\r\n[h5p id=\"16\"]","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Understand the shape, statistical properties and formulas for Normal Distributions.<\/p>\n<\/div>\n<\/div>\n<h2>Properties of Normal Distributions<\/h2>\n<p>A <a href=\"https:\/\/www.wolframalpha.com\/input?i=normal+distribution\">normal distribution<\/a> is:<\/p>\n<ul>\n<li>The most common type of distribution<\/li>\n<li>It is continuous and has a \u201cbell\u201d shape.<\/li>\n<li>It is &#8216;symmetric&#8217; about the mean (\u00b5) (see more in the &#8216;SYMMETRIC&#8217; section)<\/li>\n<li>The total area under the normal curve is 1.<\/li>\n<li>Ie: the probability of being anywhere on the distribution=1.<\/li>\n<\/ul>\n<p><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributions.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1917 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributions.jpg\" alt=\"Image with multiple bell shaped curves. Two of the curves have a population mean at zero and the standard deviation varies. The curve with the smaller standard deviation is less spread out and more concentrated around the middle. The curve with the mean at one is displaced to the right by one unit. It also has a larger standard deviation and is, therefore, more spread out.\" width=\"1388\" height=\"532\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributions.jpg 1388w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributions-300x115.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributions-1024x392.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributions-768x294.jpg 768w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributions-65x25.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributions-225x86.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributions-350x134.jpg 350w\" sizes=\"auto, (max-width: 1388px) 100vw, 1388px\" \/><\/a><\/p>\n<h1>Calculating Probabilities<\/h1>\n<p>It would require a calculate technique called <a href=\"https:\/\/www.mathsisfun.com\/calculus\/integration-by-parts.html\">Integration by Parts<\/a> to calculate the probabilities by hand for the Normal Distribution. For this reason, we will only use Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-gb\/office\/norm-dist-function-edb1cc14-a21c-4e53-839d-8082074c9f8d\">NORM.DIST()<\/a> function to calculate probabilities:<\/p>\n<ul>\n<li>[latex]P[\/latex](at most or less than) =NORM.DIST([latex]x[\/latex], \u00b5, \u03c3, TRUE)<\/li>\n<li>[latex]P[\/latex](at least or more than) =1\u2212NORM.DIST([latex]x[\/latex], \u00b5, \u03c3, TRUE)<\/li>\n<li>Where \u00b5 (mu) is the mean of the distribution and \u03c3 (sigma) is the standard deviation.<\/li>\n<\/ul>\n<h1>Calculating X-Values<\/h1>\n<p>If we are looking to solve for the [latex]x[\/latex]-value instead of the probability, this is called an &#8216;<a href=\"https:\/\/www.radfordmathematics.com\/probabilities-and-statistics\/normal-distributions\/inverse-normal-distributions.html\">inverse<\/a>&#8216; problem and we use Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-au\/office\/norm-inv-function-54b30935-fee7-493c-bedb-2278a9db7e13#:~:text=INV%20function,-Excel%20for%20Microsoft&amp;text=Returns%20the%20inverse%20of%20the,specified%20mean%20and%20standard%20deviation.\">NORM.INV()<\/a> function:<\/p>\n<ul>\n<li>[latex]x[\/latex] = NORM.INV(Area to left of [latex]x[\/latex], \u00b5, \u03c3)<\/li>\n<li>[latex]x[\/latex] = NORM.INV(1\u2212 Area to right of [latex]x[\/latex], \u00b5, \u03c3)<\/li>\n<\/ul>\n<h1>Z-Scores<\/h1>\n<p>A <a href=\"https:\/\/www.investopedia.com\/terms\/z\/zscore.asp\">z-score<\/a> is:<\/p>\n<ul>\n<li>&#8220;A statistical measurement that describes a value&#8217;s relationship to the mean of a group of values.&#8221;<\/li>\n<li>&#8220;Measured in terms of standard deviations from the mean.&#8221;<\/li>\n<li>&#8220;A measure of an instrument&#8217;s variability and can be used by traders to help determine volatility.&#8221;<\/li>\n<\/ul>\n<p>It can be calculated using a formula if the [latex]x[\/latex]-value, \u00b5 (mu) and, \u03c3 (sigma) are given:<br \/>\n\\[z = \\frac{x-\\mu}{\\sigma}\\]<\/p>\n<p><span style=\"text-align: initial\">It can be calculated using Excel&#8217;s <\/span><a style=\"text-align: initial\" href=\"https:\/\/support.microsoft.com\/en-us\/office\/norm-s-inv-function-d6d556b4-ab7f-49cd-b526-5a20918452b1#:~:text=Returns%20the%20inverse%20of%20the%20standard%20normal%20cumulative%20distribution.\">NORM.S.INV<\/a><span style=\"text-align: initial\"> function if the area\/probability is given: <\/span>\\[ z = \\text{NORM.S.INV}(\\text{Area to left of z}) = \\text{NORM.S.INV}(1-\\text{Area to right of z})\\]<\/p>\n<h1>Symmetric Property<\/h1>\n<ul>\n<li>It is symmetric (or identical) on either side of the mean.<\/li>\n<li>The mean and median are equal.<\/li>\n<li>The data in this distribution is neither skewed left nor skewed right.<\/li>\n<\/ul>\n<h1>Statistical Properties<\/h1>\n<p>The following metrics apply to Normal Distributions:<\/p>\n<ul>\n<li>population mean = \u00b5<\/li>\n<li>sample mean = x\u0304<\/li>\n<li>population standard deviation = \u03c3<\/li>\n<li>sample standard deviation = s<\/li>\n<li>mode = \u00b5 or x\u0304 (depending if population or sample given)<\/li>\n<li>variance = \u03c3<sup>2<\/sup> or s<sup>2<\/sup> (depending if population or sample given)<\/li>\n<li>symmetric (not skewed) and the skewness = 0<\/li>\n<\/ul>\n<h1>Video &amp; Resources Explaining Normal Distributions<\/h1>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"An introduction to Normal Distributions\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/pX-ldE3JOEs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><span style=\"color: #003366\"><strong>Additional Resources<\/strong><\/span>:<\/p>\n<ul>\n<li><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributions.pptx\">Click here<\/a> to download the Powerpoint slides that accompany the video.<\/li>\n<li><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributionSolutions.xlsx\">Click here<\/a> to download the Excel solutions for the Normal Distribution section.<\/li>\n<\/ul>\n<h1>Key Takeaways (EXERCISE)<\/h1>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways: Normal Distributions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Drag the words into the correct boxes for each section below:<\/p>\n<div id=\"h5p-111\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-111\" class=\"h5p-iframe\" data-content-id=\"111\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"An Introduction to Normal Distributions - Properties Key Takeaways\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-116\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-116\" class=\"h5p-iframe\" data-content-id=\"116\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"An Introduction to Normal Distributions - Calculating Probabilities Key Takeaways\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-112\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-112\" class=\"h5p-iframe\" data-content-id=\"112\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"An Introduction to Normal Distributions - Calculating X-Values Key Takeaways\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-113\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-113\" class=\"h5p-iframe\" data-content-id=\"113\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"An Introduction to Normal Distributions - Z-Scores Key Takeaways\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-115\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-115\" class=\"h5p-iframe\" data-content-id=\"115\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"An Introduction to Normal Distributions - Symmetric Property Key Takeaways\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-114\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-114\" class=\"h5p-iframe\" data-content-id=\"114\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"An Introduction to Normal Distributions - Statistical Properties Key Takeaways\"><\/iframe><\/div>\n<\/div>\n<p>Click the sections below to reveal the solutions to the above exercises<\/p>\n<div id=\"h5p-117\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-117\" class=\"h5p-iframe\" data-content-id=\"117\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"An Introduction to Normal Distributions - Properties Key Solutions\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Your Own Notes (EXERCISE)<\/h1>\n<ul>\n<li>Are there any notes you want to take from this section? Is there anything you&#8217;d like to copy and paste below?<\/li>\n<li>These notes are for you only (they will not be stored anywhere)<\/li>\n<li>Make sure to download them at the end to use as a reference<\/li>\n<\/ul>\n<div id=\"h5p-16\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-16\" class=\"h5p-iframe\" data-content-id=\"16\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key takeaways, notes and comments from this section document tool.\"><\/iframe><\/div>\n<\/div>\n","protected":false},"parent":0,"menu_order":9,"template":"","meta":{"pb_part_invisible":false,"pb_part_invisible_string":""},"contributor":[],"license":[],"class_list":["post-263","part","type-part","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/263","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/types\/part"}],"version-history":[{"count":25,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/263\/revisions"}],"predecessor-version":[{"id":2989,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/263\/revisions\/2989"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=263"}],"wp:term":[{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=263"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=263"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}