{"id":474,"date":"2021-12-20T14:37:38","date_gmt":"2021-12-20T14:37:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=474"},"modified":"2022-02-17T20:13:54","modified_gmt":"2022-02-17T20:13:54","slug":"what-to-know-about-4e","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/what-to-know-about-4e\/","title":{"raw":"What to Know About Z-Score and the Empirical Rule: 4E - 29","rendered":"What to Know About Z-Score and the Empirical Rule: 4E &#8211; 29"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>goals for this section<\/h3>\r\nAfter completing this section, you should feel comfortable performing these skills.\r\n<ul>\r\n \t<li><a href=\"#defineZscore\">Define the standardized value, or z-score.<\/a><\/li>\r\n \t<li><a href=\"#convert\">Use technology to convert values into standardized scores.<\/a><\/li>\r\n \t<li><a href=\"#identNumStdDev\">Use a dotplot and histogram to identify the number of standard deviations from the mean of certain observations.<\/a><\/li>\r\n \t<li><a href=\"#calcZscore\">Calculate a value's standardized score by hand to determine its location relative to the mean.<\/a><\/li>\r\n \t<li><a href=\"#defineEmp\">Define the Empirical Rule.<\/a><\/li>\r\n<\/ul>\r\nClick on a skill above to jump to its location in this section.\r\n\r\n<\/div>\r\nIn the next activity, you will need to be able to convert values into standardized scores and use a value\u2019s standardized score to determine whether the value is above, below, or equal to the mean. You will also need to be able to explain the Empirical Rule. In this section, we'll use a dataset to explore how to perform necessary calculations by hand and using technology.\r\n<h2>Standardized Values<\/h2>\r\n<h3 id=\"defineZscore\">Definition of Standardized Value (Z-Score)<\/h3>\r\nA <strong>standardized value<\/strong>, or <strong>z-score<\/strong>, is the number of standard deviations an observation is away from the mean.\r\n\r\nTo learn how to calculate standardized values, we will analyze runtimes (in minutes) of G-rated movies. Within this context, the standardized value, or z-score, is the number of standard deviations a particular movie runtime is from the mean.\r\n\r\nIt is important to note that this distance is not measured in minutes; rather it is measured in standard deviations. Thus, a z-score of -2.3 is an observation that is 2.3 standard deviations <em>below<\/em> the mean, and a z-score of 2.3 is an observation that is 2.3 standard deviations <em>above<\/em> the mean. It is important to note that z-scores do not have units associated with them.\r\n<div class=\"textbox tryit\">\r\n<h3>standard deviation as a unit of distance<\/h3>\r\n<span style=\"background-color: #99cc00;\">[Perspective video -- a 3-instructor video showing how to think about standard deviation as a unit of distance -- i.e., illustrating values so many standard deviations above and below the mean of a bell-shaped, unimodal, symmetric distribution. Show how adding or subtracting std devs can obtain a new value at that location.]<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Interactive example<\/h3>\r\nA particular dataset has a mean of 61 and a standard deviation of 3.7. Calculate the following values.\r\n<ol>\r\n \t<li>A value that is one standard deviation <strong>above<\/strong> the mean.<\/li>\r\n \t<li>A value that is one standard deviation <strong>below<\/strong> the mean.<\/li>\r\n \t<li>A value that is one-and-a-half standard deviations\u00a0<strong>below<\/strong> the mean.<\/li>\r\n \t<li>A value that is two standard deviations\u00a0<strong>above\u00a0<\/strong>the mean.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"600825\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"600825\"]\r\n<ol>\r\n \t<li>61\u00a0 + 3.7\u00a0 = 64.7<\/li>\r\n \t<li>61\u00a0- 3.7\u00a0= 57.3<\/li>\r\n \t<li>61\u00a0- (1.5)(3.7) = 55.45<\/li>\r\n \t<li>61+ (2)(3.7) = 68.4<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3 id=\"convert\">Use Technology to Convert Values into Standardized Scores<\/h3>\r\nThe value of an observation is <strong>standardized<\/strong> using the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x[\/latex] represents the value of the observation, [latex]\\mu[\/latex] represents the population mean, [latex]\\sigma[\/latex] represents the population standard deviation, and [latex]z[\/latex] represents the standardized value, or z-score.\r\n\r\nWe'll use technology to calculate the standard deviation of a dataset in order to convert values to standardized scores.\r\n<div class=\"textbox\">\r\n\r\nGo to the Describing and Exploring Quantitative Variables tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\" target=\"_blank\" rel=\"noopener\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a>.\r\n<p style=\"text-align: left; padding-left: 30px;\">Step 1) Select the<strong> Single Group<\/strong> tab.<\/p>\r\n<p style=\"text-align: left; padding-left: 30px;\">Step 2) Locate the dropdown under <strong>Enter Data<\/strong> and select <strong>From Textbook<\/strong>.<\/p>\r\n<p style=\"text-align: left; padding-left: 30px;\">Step 3) Locate the drop-down menu under <strong>Dataset<\/strong> and select <strong>Movie Runtime (G Rated 1990-2016)<\/strong>.<\/p>\r\n<p style=\"text-align: left; padding-left: 30px;\">Step 4) Under <strong>Choose Type of Plot<\/strong>, select the options to create a <strong>Histogram<\/strong> and a <strong>Dotplot<\/strong> of runtime (in minutes).<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 1<\/h3>\r\nDescribe the shape of the dataset using the histogram and dotplot you created.\r\n\r\n[reveal-answer q=\"279127\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"279127\"]Recall the characteristics used to describe the shape of a distribution.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 2<\/h3>\r\nWhat is the mean runtime for this dataset?\r\n\r\n[reveal-answer q=\"256092\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"256092\"]Look under Descriptive Statistics in the tool[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 3<\/h3>\r\nWhat is the standard deviation for this dataset?\r\n\r\n[reveal-answer q=\"733914\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"733914\"]Look under Descriptive Statistics in the tool[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3 id=\"identNumStdDev\">Identify the Number of Standard Deviations an Observation is From the Mean<\/h3>\r\nUse the mean and the standard deviation you entered in Questions 2 and 3 to answer the following questions.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 4<\/h3>\r\nWhat movie runtime is one standard deviation <em>above<\/em> the mean? What movie runtime is one standard deviation <em>below<\/em> the mean? Round to the nearest tenth.\r\n[reveal-answer q=\"577093\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"577093\"]Use the process in the interactive example above to calculate these values.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 5<\/h3>\r\nWhat movie runtime is two standard deviations <em>above<\/em> the mean? What movie runtime is two standard deviations <em>below<\/em> the mean? Round to the nearest tenth.\r\n\r\n[reveal-answer q=\"147907\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"147907\"]Use the process in the interactive example above to calculate these values.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3 id=\"calcZscore\">Calculate a Z-Score to Determine a Value's Location Relative to the Mean<\/h3>\r\nNow that you have obtained the standard deviation of the dataset Runtimes using technology, you can calculate any observation's z-score to locate it in the dataset relative to the mean.\r\n<div class=\"textbox tryit\">\r\n<h3>calculating z-scores<\/h3>\r\n<span style=\"background-color: #99cc00;\">[Worked example video - a 3-instructor video that works through an example like questions 6 - 9]<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Interactive Example<\/h3>\r\nRecall, to calculate a z-score given an observation, use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x[\/latex] represents the value of the observation, [latex]\\mu[\/latex] represents the population mean, [latex]\\sigma[\/latex] represents the population standard deviation, and [latex]z[\/latex] represents the standardized value, or z-score.\r\n\r\nFor a dataset with a mean of 132 and standard deviation of 9.8, calculate the z-scores of the following observations, [latex]x[\/latex], and indicate if the given value is above or below the mean. Round answers to 2 decimal places.\r\n<ol>\r\n \t<li>[latex]x=112[\/latex]<\/li>\r\n \t<li>[latex]x=141[\/latex]<\/li>\r\n \t<li>[latex]x=158[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"172851\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"172851\"]\r\n<ol>\r\n \t<li>[latex]z=\\dfrac{112-132}{9.8}\\approx -2.04[\/latex]. This value is 2.04 standard deviations\u00a0<strong>below<\/strong> the mean.<\/li>\r\n \t<li>[latex]z=\\dfrac{141-132}{9.8}\\approx 0.92[\/latex]. This value is 0.92 standard deviations\u00a0<strong>above\u00a0<\/strong>the mean.<\/li>\r\n \t<li>[latex]z=\\dfrac{158-132}{9.8}\\approx 2.65[\/latex]. This value is 2.65 standard deviations\u00a0<strong>above<\/strong> the mean.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nUse the mean and the standard deviation you calculated in Questions 2 and 3 to answer Questions 6 - 9.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 6<\/h3>\r\nCalculate the z-score for the movie <em>101 Dalmatians<\/em>, which has a length of 103 minutes. How many standard deviations from the mean is this movie runtime? Make sure to indicate if this movie runtime is above or below the mean. Round your answer to 2 decimal places.\r\n\r\n[reveal-answer q=\"563409\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"563409\"]Use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x= 103[\/latex], [latex]\\mu=91[\/latex], and [latex]\\sigma=11.8[\/latex], and then use the order of operations.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 7<\/h3>\r\nCalculate the z-score for the movie <em>The Rookie<\/em>, which has a length of 127 minutes. How many standard deviations from the mean is this movie runtime? Make sure to indicate if this movie runtime is above or below the mean. Round your answer to 2 decimal places.\r\n\r\n[reveal-answer q=\"914105\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"914105\"]Use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x= 127[\/latex], [latex]\\mu=91[\/latex], and [latex]\\sigma=11.8[\/latex], and then use the order of operations.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 8<\/h3>\r\nCalculate the z-score for the movie <em>The Adventures of Elmo in Grouchland<\/em>, which has a length of 73 minutes. How many standard deviations from the mean is this movie runtime? Make sure to indicate if this movie runtime is above or below the mean. Round your answer to 2 decimal places.\r\n\r\n[reveal-answer q=\"84618\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"84618\"]Use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x=73[\/latex], [latex]\\mu=91[\/latex], and [latex]\\sigma=11.8[\/latex], and then use the order of operations.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 9<\/h3>\r\nCalculate the z-score for the movie <em>The Hunchback of Notre Dame<\/em>, which has a length of 91 minutes. How many standard deviations from the mean is this movie runtime? Make sure to indicate if this movie runtime is above or below the mean. Round your answer to 2 decimal places.\r\n\r\n[reveal-answer q=\"503665\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"503665\"]Use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x=91[\/latex], [latex]\\mu=91[\/latex], and [latex]\\sigma=11.8[\/latex], and then use the order of operations.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>The Empirical Rule<\/h2>\r\nIf a distribution is bell shaped, unimodal, and symmetric, then we can estimate how many observations are within a certain number of standard deviations. The <strong>Empirical Rule<\/strong> (also known as the 68-95-99.7 rule) is a guideline that predicts the percentage of observations within a certain number of standard deviations.\r\n<div class=\"textbox tryit\">\r\n<h3>the empirical rule<\/h3>\r\n<span style=\"background-color: #ffff00;\">[insert a video describing (but not using) the Empirical Rule]--&gt;this video could be good, but she refers back to other lessons and writes on the diagram in a way that could be confusing (calculating half of 68% and not others, uses x bar and s instead of mu and sigma, writes 99.7% on the outside of the bell while the others are clearly written inside). She begins an example at 4:41.<\/span>\r\n\r\n[embed]https:\/\/www.youtube.com\/watch?v=1AmZjyXKveM[\/embed]\r\n\r\n<\/div>\r\n<h3 id=\"defineEmp\">Define the Empirical Rule<\/h3>\r\nThe Empirical Rule states that:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">about 68% of observations in a dataset will be within one standard deviation of the mean.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">about 95% of the observations in a dataset will be within two standard deviations of the mean.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">about 99.7% of the observations in a dataset will be within three standard deviations of the mean.<\/li>\r\n<\/ul>\r\nGraphically, the Empirical Rule can be expressed like this:\r\n\r\n<strong><img class=\"alignnone wp-image-1033\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11224126\/Picture59-300x295.jpg\" alt=\"A bar graph with the highest bars in the middle and lower bars to either side. In the center, the x-axis is labeled &quot;mu.&quot; Three bars to the left, it is labeled &quot;mu - sigma,&quot; three more bars to the left it is labeled &quot;mu - 2 sigma,&quot; and three more to the left, it's labeled &quot;mu - 3 sigma.&quot; Three to the right of the center, it is labeled &quot;mu + sigma.&quot; Three more to the right and it is labeled &quot;mu + 2 sigma&quot; and three more to the right, it is labeled &quot;mu + 3 sigma.&quot; The center six bars are all green and labeled as 68&amp; collectively. The three leftmost center bars are labeled 34.1%, and the three rightmost center bars are also labeled 34.1%. The next three bars out on either side of the center six are each labeled 13.6% and the center 12 are all labeled 95% collectively. Lastly, the next three out on either side of the center twelve are each labeled 2.1% and all 18 are collectively labeled 99.7% \u2248 100%\" width=\"551\" height=\"541\" \/><\/strong>\r\n\r\nFill in the blank for each of\u00a0Questions 10 - 12\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 10<\/h3>\r\nIf a distribution is bell shaped, unimodal, and symmetric, then approximately 68% of the observations are between ____ standard deviation above the mean and ___ standard deviation below the mean.\r\n\r\n[reveal-answer q=\"495115\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"495115\"]Use the image and definition above. [\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 11<\/h3>\r\nIf a distribution is bell shaped, unimodal, and symmetric, then approximately 95% of the observations are between ____ standard deviations above the mean and ___ standard deviations below the mean.\r\n\r\n[reveal-answer q=\"532094\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"532094\"]Use the image and definition above.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 12<\/h3>\r\nIf a distribution is bell shaped, unimodal, and symmetric, then approximately 99.7% of the observations are between ____ standard deviations above the mean and ___ standard deviations below the mean.\r\n\r\n[reveal-answer q=\"820605\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"820605\"]Use the image and definition above.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nIn this section, you've seen how to convert observed values into standardized scores (z-scores) and that the value of the z-score gives meaningful information about the location of the observation with respect to the mean of a dataset. You also seen how to explain what the Empirical Rule is. Let's summarize where these skills showed up in the material.\r\n<ul>\r\n \t<li>In question 1 - 5, you converted values into standardized scores.<\/li>\r\n \t<li>In Questions 6 - 9, you used a value's standardized score to determine whether the value is above, below, or equal to the mean.<\/li>\r\n \t<li>In Questions 10 - 12, you explained the Empirical Rule.<\/li>\r\n<\/ul>\r\nBeing able to calculate z-scores and understanding the Empirical Rule will be necessary for completing the next activity. If you feel comfortable with these skills, it's time to move on!","rendered":"<div class=\"textbox learning-objectives\">\n<h3>goals for this section<\/h3>\n<p>After completing this section, you should feel comfortable performing these skills.<\/p>\n<ul>\n<li><a href=\"#defineZscore\">Define the standardized value, or z-score.<\/a><\/li>\n<li><a href=\"#convert\">Use technology to convert values into standardized scores.<\/a><\/li>\n<li><a href=\"#identNumStdDev\">Use a dotplot and histogram to identify the number of standard deviations from the mean of certain observations.<\/a><\/li>\n<li><a href=\"#calcZscore\">Calculate a value&#8217;s standardized score by hand to determine its location relative to the mean.<\/a><\/li>\n<li><a href=\"#defineEmp\">Define the Empirical Rule.<\/a><\/li>\n<\/ul>\n<p>Click on a skill above to jump to its location in this section.<\/p>\n<\/div>\n<p>In the next activity, you will need to be able to convert values into standardized scores and use a value\u2019s standardized score to determine whether the value is above, below, or equal to the mean. You will also need to be able to explain the Empirical Rule. In this section, we&#8217;ll use a dataset to explore how to perform necessary calculations by hand and using technology.<\/p>\n<h2>Standardized Values<\/h2>\n<h3 id=\"defineZscore\">Definition of Standardized Value (Z-Score)<\/h3>\n<p>A <strong>standardized value<\/strong>, or <strong>z-score<\/strong>, is the number of standard deviations an observation is away from the mean.<\/p>\n<p>To learn how to calculate standardized values, we will analyze runtimes (in minutes) of G-rated movies. Within this context, the standardized value, or z-score, is the number of standard deviations a particular movie runtime is from the mean.<\/p>\n<p>It is important to note that this distance is not measured in minutes; rather it is measured in standard deviations. Thus, a z-score of -2.3 is an observation that is 2.3 standard deviations <em>below<\/em> the mean, and a z-score of 2.3 is an observation that is 2.3 standard deviations <em>above<\/em> the mean. It is important to note that z-scores do not have units associated with them.<\/p>\n<div class=\"textbox tryit\">\n<h3>standard deviation as a unit of distance<\/h3>\n<p><span style=\"background-color: #99cc00;\">[Perspective video &#8212; a 3-instructor video showing how to think about standard deviation as a unit of distance &#8212; i.e., illustrating values so many standard deviations above and below the mean of a bell-shaped, unimodal, symmetric distribution. Show how adding or subtracting std devs can obtain a new value at that location.]<\/span><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Interactive example<\/h3>\n<p>A particular dataset has a mean of 61 and a standard deviation of 3.7. Calculate the following values.<\/p>\n<ol>\n<li>A value that is one standard deviation <strong>above<\/strong> the mean.<\/li>\n<li>A value that is one standard deviation <strong>below<\/strong> the mean.<\/li>\n<li>A value that is one-and-a-half standard deviations\u00a0<strong>below<\/strong> the mean.<\/li>\n<li>A value that is two standard deviations\u00a0<strong>above\u00a0<\/strong>the mean.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q600825\">Show Answer<\/span><\/p>\n<div id=\"q600825\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>61\u00a0 + 3.7\u00a0 = 64.7<\/li>\n<li>61\u00a0&#8211; 3.7\u00a0= 57.3<\/li>\n<li>61\u00a0&#8211; (1.5)(3.7) = 55.45<\/li>\n<li>61+ (2)(3.7) = 68.4<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h3 id=\"convert\">Use Technology to Convert Values into Standardized Scores<\/h3>\n<p>The value of an observation is <strong>standardized<\/strong> using the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x[\/latex] represents the value of the observation, [latex]\\mu[\/latex] represents the population mean, [latex]\\sigma[\/latex] represents the population standard deviation, and [latex]z[\/latex] represents the standardized value, or z-score.<\/p>\n<p>We&#8217;ll use technology to calculate the standard deviation of a dataset in order to convert values to standardized scores.<\/p>\n<div class=\"textbox\">\n<p>Go to the Describing and Exploring Quantitative Variables tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\" target=\"_blank\" rel=\"noopener\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a>.<\/p>\n<p style=\"text-align: left; padding-left: 30px;\">Step 1) Select the<strong> Single Group<\/strong> tab.<\/p>\n<p style=\"text-align: left; padding-left: 30px;\">Step 2) Locate the dropdown under <strong>Enter Data<\/strong> and select <strong>From Textbook<\/strong>.<\/p>\n<p style=\"text-align: left; padding-left: 30px;\">Step 3) Locate the drop-down menu under <strong>Dataset<\/strong> and select <strong>Movie Runtime (G Rated 1990-2016)<\/strong>.<\/p>\n<p style=\"text-align: left; padding-left: 30px;\">Step 4) Under <strong>Choose Type of Plot<\/strong>, select the options to create a <strong>Histogram<\/strong> and a <strong>Dotplot<\/strong> of runtime (in minutes).<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 1<\/h3>\n<p>Describe the shape of the dataset using the histogram and dotplot you created.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q279127\">Hint<\/span><\/p>\n<div id=\"q279127\" class=\"hidden-answer\" style=\"display: none\">Recall the characteristics used to describe the shape of a distribution.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 2<\/h3>\n<p>What is the mean runtime for this dataset?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q256092\">Hint<\/span><\/p>\n<div id=\"q256092\" class=\"hidden-answer\" style=\"display: none\">Look under Descriptive Statistics in the tool<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 3<\/h3>\n<p>What is the standard deviation for this dataset?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q733914\">Hint<\/span><\/p>\n<div id=\"q733914\" class=\"hidden-answer\" style=\"display: none\">Look under Descriptive Statistics in the tool<\/div>\n<\/div>\n<\/div>\n<h3 id=\"identNumStdDev\">Identify the Number of Standard Deviations an Observation is From the Mean<\/h3>\n<p>Use the mean and the standard deviation you entered in Questions 2 and 3 to answer the following questions.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 4<\/h3>\n<p>What movie runtime is one standard deviation <em>above<\/em> the mean? What movie runtime is one standard deviation <em>below<\/em> the mean? Round to the nearest tenth.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q577093\">Hint<\/span><\/p>\n<div id=\"q577093\" class=\"hidden-answer\" style=\"display: none\">Use the process in the interactive example above to calculate these values.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 5<\/h3>\n<p>What movie runtime is two standard deviations <em>above<\/em> the mean? What movie runtime is two standard deviations <em>below<\/em> the mean? Round to the nearest tenth.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q147907\">Hint<\/span><\/p>\n<div id=\"q147907\" class=\"hidden-answer\" style=\"display: none\">Use the process in the interactive example above to calculate these values.<\/div>\n<\/div>\n<\/div>\n<h3 id=\"calcZscore\">Calculate a Z-Score to Determine a Value&#8217;s Location Relative to the Mean<\/h3>\n<p>Now that you have obtained the standard deviation of the dataset Runtimes using technology, you can calculate any observation&#8217;s z-score to locate it in the dataset relative to the mean.<\/p>\n<div class=\"textbox tryit\">\n<h3>calculating z-scores<\/h3>\n<p><span style=\"background-color: #99cc00;\">[Worked example video &#8211; a 3-instructor video that works through an example like questions 6 &#8211; 9]<\/span><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Interactive Example<\/h3>\n<p>Recall, to calculate a z-score given an observation, use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x[\/latex] represents the value of the observation, [latex]\\mu[\/latex] represents the population mean, [latex]\\sigma[\/latex] represents the population standard deviation, and [latex]z[\/latex] represents the standardized value, or z-score.<\/p>\n<p>For a dataset with a mean of 132 and standard deviation of 9.8, calculate the z-scores of the following observations, [latex]x[\/latex], and indicate if the given value is above or below the mean. Round answers to 2 decimal places.<\/p>\n<ol>\n<li>[latex]x=112[\/latex]<\/li>\n<li>[latex]x=141[\/latex]<\/li>\n<li>[latex]x=158[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q172851\">Show Answer<\/span><\/p>\n<div id=\"q172851\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]z=\\dfrac{112-132}{9.8}\\approx -2.04[\/latex]. This value is 2.04 standard deviations\u00a0<strong>below<\/strong> the mean.<\/li>\n<li>[latex]z=\\dfrac{141-132}{9.8}\\approx 0.92[\/latex]. This value is 0.92 standard deviations\u00a0<strong>above\u00a0<\/strong>the mean.<\/li>\n<li>[latex]z=\\dfrac{158-132}{9.8}\\approx 2.65[\/latex]. This value is 2.65 standard deviations\u00a0<strong>above<\/strong> the mean.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Use the mean and the standard deviation you calculated in Questions 2 and 3 to answer Questions 6 &#8211; 9.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 6<\/h3>\n<p>Calculate the z-score for the movie <em>101 Dalmatians<\/em>, which has a length of 103 minutes. How many standard deviations from the mean is this movie runtime? Make sure to indicate if this movie runtime is above or below the mean. Round your answer to 2 decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q563409\">Hint<\/span><\/p>\n<div id=\"q563409\" class=\"hidden-answer\" style=\"display: none\">Use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x= 103[\/latex], [latex]\\mu=91[\/latex], and [latex]\\sigma=11.8[\/latex], and then use the order of operations.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 7<\/h3>\n<p>Calculate the z-score for the movie <em>The Rookie<\/em>, which has a length of 127 minutes. How many standard deviations from the mean is this movie runtime? Make sure to indicate if this movie runtime is above or below the mean. Round your answer to 2 decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q914105\">Hint<\/span><\/p>\n<div id=\"q914105\" class=\"hidden-answer\" style=\"display: none\">Use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x= 127[\/latex], [latex]\\mu=91[\/latex], and [latex]\\sigma=11.8[\/latex], and then use the order of operations.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 8<\/h3>\n<p>Calculate the z-score for the movie <em>The Adventures of Elmo in Grouchland<\/em>, which has a length of 73 minutes. How many standard deviations from the mean is this movie runtime? Make sure to indicate if this movie runtime is above or below the mean. Round your answer to 2 decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q84618\">Hint<\/span><\/p>\n<div id=\"q84618\" class=\"hidden-answer\" style=\"display: none\">Use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x=73[\/latex], [latex]\\mu=91[\/latex], and [latex]\\sigma=11.8[\/latex], and then use the order of operations.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 9<\/h3>\n<p>Calculate the z-score for the movie <em>The Hunchback of Notre Dame<\/em>, which has a length of 91 minutes. How many standard deviations from the mean is this movie runtime? Make sure to indicate if this movie runtime is above or below the mean. Round your answer to 2 decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q503665\">Hint<\/span><\/p>\n<div id=\"q503665\" class=\"hidden-answer\" style=\"display: none\">Use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x=91[\/latex], [latex]\\mu=91[\/latex], and [latex]\\sigma=11.8[\/latex], and then use the order of operations.<\/div>\n<\/div>\n<\/div>\n<h2>The Empirical Rule<\/h2>\n<p>If a distribution is bell shaped, unimodal, and symmetric, then we can estimate how many observations are within a certain number of standard deviations. The <strong>Empirical Rule<\/strong> (also known as the 68-95-99.7 rule) is a guideline that predicts the percentage of observations within a certain number of standard deviations.<\/p>\n<div class=\"textbox tryit\">\n<h3>the empirical rule<\/h3>\n<p><span style=\"background-color: #ffff00;\">[insert a video describing (but not using) the Empirical Rule]&#8211;&gt;this video could be good, but she refers back to other lessons and writes on the diagram in a way that could be confusing (calculating half of 68% and not others, uses x bar and s instead of mu and sigma, writes 99.7% on the outside of the bell while the others are clearly written inside). She begins an example at 4:41.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Empirical Rule Explained\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/1AmZjyXKveM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<h3 id=\"defineEmp\">Define the Empirical Rule<\/h3>\n<p>The Empirical Rule states that:<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">about 68% of observations in a dataset will be within one standard deviation of the mean.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">about 95% of the observations in a dataset will be within two standard deviations of the mean.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">about 99.7% of the observations in a dataset will be within three standard deviations of the mean.<\/li>\n<\/ul>\n<p>Graphically, the Empirical Rule can be expressed like this:<\/p>\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1033\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11224126\/Picture59-300x295.jpg\" alt=\"A bar graph with the highest bars in the middle and lower bars to either side. In the center, the x-axis is labeled &quot;mu.&quot; Three bars to the left, it is labeled &quot;mu - sigma,&quot; three more bars to the left it is labeled &quot;mu - 2 sigma,&quot; and three more to the left, it's labeled &quot;mu - 3 sigma.&quot; Three to the right of the center, it is labeled &quot;mu + sigma.&quot; Three more to the right and it is labeled &quot;mu + 2 sigma&quot; and three more to the right, it is labeled &quot;mu + 3 sigma.&quot; The center six bars are all green and labeled as 68&amp; collectively. The three leftmost center bars are labeled 34.1%, and the three rightmost center bars are also labeled 34.1%. The next three bars out on either side of the center six are each labeled 13.6% and the center 12 are all labeled 95% collectively. Lastly, the next three out on either side of the center twelve are each labeled 2.1% and all 18 are collectively labeled 99.7% \u2248 100%\" width=\"551\" height=\"541\" \/><\/strong><\/p>\n<p>Fill in the blank for each of\u00a0Questions 10 &#8211; 12<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 10<\/h3>\n<p>If a distribution is bell shaped, unimodal, and symmetric, then approximately 68% of the observations are between ____ standard deviation above the mean and ___ standard deviation below the mean.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q495115\">Hint<\/span><\/p>\n<div id=\"q495115\" class=\"hidden-answer\" style=\"display: none\">Use the image and definition above. <\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 11<\/h3>\n<p>If a distribution is bell shaped, unimodal, and symmetric, then approximately 95% of the observations are between ____ standard deviations above the mean and ___ standard deviations below the mean.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q532094\">Hint<\/span><\/p>\n<div id=\"q532094\" class=\"hidden-answer\" style=\"display: none\">Use the image and definition above.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 12<\/h3>\n<p>If a distribution is bell shaped, unimodal, and symmetric, then approximately 99.7% of the observations are between ____ standard deviations above the mean and ___ standard deviations below the mean.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q820605\">Hint<\/span><\/p>\n<div id=\"q820605\" class=\"hidden-answer\" style=\"display: none\">Use the image and definition above.<\/div>\n<\/div>\n<\/div>\n<h2>Summary<\/h2>\n<p>In this section, you&#8217;ve seen how to convert observed values into standardized scores (z-scores) and that the value of the z-score gives meaningful information about the location of the observation with respect to the mean of a dataset. You also seen how to explain what the Empirical Rule is. Let&#8217;s summarize where these skills showed up in the material.<\/p>\n<ul>\n<li>In question 1 &#8211; 5, you converted values into standardized scores.<\/li>\n<li>In Questions 6 &#8211; 9, you used a value&#8217;s standardized score to determine whether the value is above, below, or equal to the mean.<\/li>\n<li>In Questions 10 &#8211; 12, you explained the Empirical Rule.<\/li>\n<\/ul>\n<p>Being able to calculate z-scores and understanding the Empirical Rule will be necessary for completing the next activity. If you feel comfortable with these skills, it&#8217;s time to move on!<\/p>\n","protected":false},"author":25777,"menu_order":31,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-474","chapter","type-chapter","status-publish","hentry"],"part":621,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/474","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":38,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/474\/revisions"}],"predecessor-version":[{"id":3326,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/474\/revisions\/3326"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/621"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/474\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=474"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=474"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=474"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=474"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}