{"id":26824,"date":"2021-11-05T13:00:10","date_gmt":"2021-11-05T13:00:10","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=26824"},"modified":"2024-05-22T07:34:50","modified_gmt":"2024-05-22T07:34:50","slug":"how-to-calculate-the-normal-distribution-using-the-table","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/how-to-calculate-the-normal-distribution-using-the-table\/","title":{"rendered":"How to Calculate the Normal Distribution Using the Table?"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n  <center><\/p>\n<h1>How to Calculate the Normal Distribution Using the Table?<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\">\n      <strong>Summary<\/strong><br \/>\n      <em>In this class, we will address the topic of the normal distribution, one of the most common continuous probability distributions. We will analyze how a random variable X with parameters \u03bc and \u03c3 can follow a normal distribution and how this distribution can be standardized through a substitution in the integral. However, we will note that despite the simplification through standardization, the analytical calculation of the integral remains a challenge, as there are no algebraic expressions to express its result. To overcome this problem, we will present a standard normal distribution table built in Excel, offering numerical approximations to the function&#8217;s values. We will explain in detail how to use the table, including the interpretation of cumulative probabilities as areas under the curve. Finally, the class concludes with a series of practical exercises focused on calculating probabilities and estimates using the normal distribution table and the concept of the standard normal distribution.<\/em>\n    <\/p>\n<p>  <\/center><br \/>\n  <\/p>\n<p style=\"text-align:center;\">\n    <strong>LEARNING OBJECTIVES:<\/strong><br \/>\n    By the end of this class, the student will be able to:\n  <\/p>\n<ol>\n<li><strong>Demonstrate<\/strong> the relationship between the standard normal distribution and the normal distribution with parameters <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma<\/span><\/span> established through an integral substitution.<\/li>\n<li><strong>Solve<\/strong> practical problems using the standard normal distribution table.<\/li>\n<li><strong>Build<\/strong> and use a standard normal distribution table using Excel.<\/li>\n<\/ol>\n<p>  <center><br \/>\n    <strong><u>TABLE OF CONTENTS<\/u>:<\/strong><br \/>\n    <a href=\"#1\">The Problem and the Solution of the Normal Distribution<\/a><br \/>\n    <a href=\"#2\">How Do I Use This Table?<\/a><br \/>\n    <a href=\"#3\">Exercises<\/a><br \/>\n  <\/center><br \/>\n  <center><br \/>\n    <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/GdoFK696k3s\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><br \/>\n  <\/center>\n<\/div>\n<p><a name=\"1\"><\/a><\/p>\n<h2>The Problem and the Solution of the Normal Distribution<\/h2>\n<p style=\"text-align:justify;\">One of the most widely used <a href=\"http:\/\/toposuranos.com\/material\/es\/distribuciones-continuas-de-probabilidad\/\" rel=\"noopener\" target=\"_blank\">continuous probability distributions<\/a> is the normal distribution. A random variable <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> follows a normal distribution with parameters <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu,\\sigma<\/span><\/span> if it satisfies that<\/p>\n<p class=\"eq\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X\\leq x) =\\Phi_{\\mu,\\sigma}(x) = \\int_{-\\infty}^x \\frac{1}{\\sqrt{2\\pi}\\sigma} e^{-\\frac{1}{2} \\left(\\frac{t-\\mu}{\\sigma}\\right)^2}dt <\/span><\/span><\/p>\n<p>and we have also <a href=\"https:\/\/www.youtube.com\/watch?v=kdxgrB1h98g&amp;t=1188s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">seen<\/span><\/strong><\/a> that this distribution can be \u00abstandardized\u00bb if we make a substitution in the integral of the form <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle z= \\frac{t-\\mu}{\\sigma},<\/span><\/span> obtaining:<\/p>\n<p class=\"eq\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\Phi_{\\mu,\\sigma}(x) = \\int_{-\\infty}^{\\frac{x-\\mu}{\\sigma}} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{t^2}{2} }dt = \\Phi_{0,1}\\left(\\frac{x-\\mu}{\\sigma} \\right) <\/span><\/span><\/p>\n<p style=\"text-align:justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=GdoFK696k3s&amp;t=270s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>Although this standardization process<\/strong><\/span><\/a> simplifies the integral, it is still not enough to allow us to calculate a single probability for this distribution. It happens that evaluating this integral analytically is not something we can perform; in fact, there are no algebraic expressions to express the result of these integrals, and that puts us in trouble. However, if we have numerical approximations for some values of the standard normal distribution function, we can start making some estimates. These values are summarized in this <strong>standard normal distribution table<\/strong> that I built in Excel for you &gt;:D<\/p>\n<p><center><br \/>\n  <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/tabla-distribucion-normalb.png\" alt=\"\" width=\"784\" height=\"774\" class=\"aligncenter size-full wp-image-26819 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/tabla-distribucion-normalb.png\" alt=\"\" width=\"784\" height=\"774\" class=\"aligncenter size-full wp-image-26819 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/tabla-distribucion-normalb.png 784w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/tabla-distribucion-normalb-300x296.png 300w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/tabla-distribucion-normalb-768x758.png 768w\" sizes=\"(max-width: 784px) 100vw, 784px\" \/><\/noscript><br \/>\n  <em>You can download this table in Excel format at the following <a href=\"https:\/\/www.dropbox.com\/s\/9yff75b70wd75xn\/tabla%20-%20distribucion%20normal%20estandar.xlsx?dl=0\" rel=\"noopener\" target=\"_blank\">link.<\/a> I recommend reviewing that file because you will learn how to build your own table in case you cannot find one \ud83d\ude42<\/em><br \/>\n<\/center><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>How Do I Use This Table?<\/h2>\n<p style=\"text-align:justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=GdoFK696k3s&amp;t=622s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>To use this table easily<\/strong><\/span><\/a> you must remember that, in addition to indicating a cumulative probability, it represents an area under the curve (of the density function). This can be seen represented in the following figure:<\/p>\n<p><center><br \/>\n  <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/comousarlatablab.png\" alt=\"\" width=\"784\" height=\"774\" class=\"aligncenter size-full wp-image-26820 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/comousarlatablab.png\" alt=\"\" width=\"784\" height=\"774\" class=\"aligncenter size-full wp-image-26820 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/comousarlatablab.png 784w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/comousarlatablab-300x296.png 300w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/comousarlatablab-768x758.png 768w\" sizes=\"(max-width: 784px) 100vw, 784px\" \/><\/noscript><br \/>\n<\/center><\/p>\n<p style=\"text-align:justify;\">Having understood this, the next step is to explain the actual use of the table. What we see here is that the value <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">z<\/span><\/span> at which we will evaluate the function <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(z)<\/span><\/span> is separated into two parts: in the column, you will find its expansion up to the first decimal digit, and in the row, you will find the second decimal digit; so, for example: the number <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">z=1.72<\/span><\/span> can be found as if it were formed by \u00abcoordinates\u00bb: the vertical is <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1.7,<\/span><\/span> and the horizontal <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0.02.<\/span><\/span> Finally, the numerical approximation of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(1.72)<\/span><\/span> will appear in the block of coordinates \u00ab1.7\u00bb and \u00ab0.02\u00bb, which shows a value of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0.957284.<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Exercises<\/h2>\n<ol>\n<li>Estimate the value of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(2.93)<\/span><\/span><\/li>\n<li>Estimate the value of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{\\mu=12,\\sigma=3}(11.5)<\/span><\/span><\/li>\n<li>A random variable <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> follows a normal distribution <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">N(\\mu=37,\\sigma=9).<\/span><\/span> Calculate <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(35 \\lt X \\lt 43).<\/span><\/span><\/li>\n<li>The number of tomatoes that rot daily at a vendor&#8217;s stall has a mean <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu=50<\/span><\/span> with a standard deviation <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma=15.<\/span><\/span> If the vendor brings tomatoes 3 times a week, estimate the number of days in a month when more than 60 tomatoes will rot.<\/li>\n<li>Assuming that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X \\sim N(\\mu,\\sigma)<\/span><\/span> calculate the following probabilities:\n<ol>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(\\mu - \\sigma \\leq X \\leq \\mu + \\sigma)<\/span><\/span><\/li>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X \\leq \\mu - \\sigma \\vee \\mu + \\sigma \\leq X)<\/span><\/span><\/li>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X \\leq \\mu - n\\sigma \\vee \\mu + n\\sigma \\leq X)<\/span><\/span><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><center><br \/>\n  <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/6tEAj-_dd_c\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><br \/>\n<\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>How to Calculate the Normal Distribution Using the Table? Summary In this class, we will address the topic of the normal distribution, one of the most common continuous probability distributions. We will analyze how a random variable X with parameters \u03bc and \u03c3 can follow a normal distribution and how this distribution can be standardized [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":26821,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":4,"footnotes":""},"categories":[567,670],"tags":[],"class_list":["post-26824","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","category-probabilities-and-statistics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>How to Calculate the Normal Distribution Using the Table? - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Learn to use the standardized normal distribution table to calculate probabilities and solve practical problems.\" \/>\n<meta name=\"robots\" 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Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. 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