{"id":370,"date":"2022-02-21T18:01:23","date_gmt":"2022-02-21T18:01:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/?post_type=chapter&#038;p=370"},"modified":"2022-05-20T16:38:54","modified_gmt":"2022-05-20T16:38:54","slug":"z-score-and-the-empirical-rule-what-to-know","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/z-score-and-the-empirical-rule-what-to-know\/","title":{"raw":"Z-Score and the Empirical Rule: What to Know","rendered":"Z-Score and the Empirical Rule: What to Know"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Goals<\/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 values (also called standardized scores or z-scores) and use a value\u2019s standardized value 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 data set to explore how to perform necessary calculations by hand and using technology.\r\n<h2>Standardized Values<\/h2>\r\nYou learned in\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/comparing-variability-of-data sets-what-to-know\/\"><em>Comparing Variability of Data Sets: What to Know<\/em><\/a> that a standard deviation is a measure for how spread out observations are from the mean.\r\n\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\nFor example, in this section we will analyze runtimes (in minutes) of G-rated movies to learn how to calculate standardized values. 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 the distance of a particular movie runtime from the mean is not measured in minutes; rather it is measured in standard deviations. Thus, a z-score of\u00a0[latex]-2.3[\/latex] is an observation that is\u00a0[latex]2.3[\/latex] standard deviations <em>below<\/em> the mean, and a z-score of\u00a0[latex]2.3[\/latex] is an observation that is\u00a0[latex]2.3[\/latex] standard deviations <em>above<\/em> the mean. It is important to note that z-scores do not have units associated with them.\r\n<p style=\"text-align: left;\"><strong>Z-Score Formula\u00a0\u00a0<\/strong>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>\r\nBefore we use the formula to convert values into standardized values, let's recap our understanding of standard deviation. In <a href=\"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/comparing-variability-of-data sets-what-to-know\/\"><em>Comparing Variability of Data Sets: What to Know<\/em><\/a>, you learned to understand standard deviation as a measure of variability in a data set. You looked at the statistical components that went into the formulas for standard deviation and variance and saw that larger standard deviations could represent more variability, and vice-versa. We'd like to shift that perspective now and look at a unit of standard deviation as a distance from the mean of a data set in a distribution.\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 in a distribution -- 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 certain value at that location in the distribution. Show that a value's z-score (negative or positive) is that many std deviations away from the mean in that direction.]<\/span>\r\n\r\n<\/div>\r\nSee the example below for a demonstration, then try it out using the Movie Runtimes database to answer the questions below.\r\n<div class=\"textbox exercises\">\r\n<h3>inTeractive example<\/h3>\r\nLet's return again to the data set Sleep Study: Average Sleep, which we used in\u00a0<em><a href=\"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/comparing-variability-of-data sets-what-to-know\/\">Comparing Variability of Data Sets: What to Know<\/a><\/em> to learn about standard deviation as a measure of the variability of a data set.\r\n\r\nOpen the tool at\u00a0<a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a>\u00a0and select the Sleep Study: Average Sleep data set.\u00a0Display a histogram and dotplot and make a note of the mean and standard deviation in the descriptive statistics. Round your final answers to the questions below to 3 decimal places, as needed.\r\n<ol>\r\n \t<li>Describe the shape of the data set using the histogram and dotplot. For practice, display a boxplot as well and note the visual clues that you can use to determine the shape of the distribution from the boxplot.<\/li>\r\n \t<li>How does the relationship between the mean and median (given in descriptive statistics) help to support your analysis?<\/li>\r\n \t<li>What are the mean and standard deviation of the data set?<\/li>\r\n \t<li>What number of sleep hours lies one standard deviation above the mean? What value lies one standard deviation below?<\/li>\r\n \t<li>What number of sleep hours lie two standard deviations above and below the mean?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"599366\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"599366\"]\r\n<ol>\r\n \t<li>The distribution is unimodal and approximately symmetric. A few outliers lie to either side of the distribution but do so evenly.<\/li>\r\n \t<li>The mean and median are approximately equal, which supports that the outliers are approximately evenly distributed.<\/li>\r\n \t<li><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">[latex]\\bar{x}=7.97[\/latex] and [latex]s=0.965[\/latex]<\/span><\/li>\r\n \t<li>The standard deviation is 0.965. We can add that to the mean to determine the value exactly one standard deviation above the mean. Likewise, we can subtract that from the mean to find the value exactly one standard deviation below the mean.\r\n<ul>\r\n \t<li>[latex]7.97 + 0.965 = 8.935[\/latex]:\r\n<ul>\r\n \t<li>[latex]8.935[\/latex] hours of sleep lies <strong>one<\/strong> standard deviation <strong>above<\/strong> the mean.<\/li>\r\n \t<li>That is, the standardized value for the observation [latex]8.935[\/latex] hours is [latex]1[\/latex[] Its z-score is <strong>[latex]1[\/latex]<\/strong>.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>[latex]7.97 - 0.965 = 7.005[\/latex]:\r\n<ul>\r\n \t<li>[latex]7.005[\/latex] hours of sleep lies <strong>one<\/strong> standard deviation <strong>below<\/strong> the mean.<\/li>\r\n \t<li>That is, the standardized value for the observation [latex]7.005[\/latex] hours is [latex]-1[\/latex]. Its z-score is <strong>[latex]-1[\/latex]<\/strong><\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The standard deviation is [latex]0.965[\/latex]. We can add twice that to the mean to determine the value exactly two standard deviations above the mean. Likewise, we can subtract [latex]2*0.965[\/latex] from the mean to find the value exactly one standard deviation below the mean.\r\n<ul>\r\n \t<li>[latex]7.97 + 2*0.965 = 9.9[\/latex]:\r\n<ul>\r\n \t<li>[latex]9.9[\/latex] hours of sleep lies <strong>two<\/strong> standard deviations\u00a0<strong>above<\/strong> the mean.<\/li>\r\n \t<li>That is, the standardized value for the observation [latex]9.9[\/latex] hours is [latex]2[\/latex[] Its z-score is <strong>[latex]2[\/latex]<\/strong>.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>[latex]7.97 - 2*0.965 = 6.04[\/latex]:\r\n<ul>\r\n \t<li>[latex]6.04[\/latex] hours of sleep lies <strong>two<\/strong> standard deviations\u00a0<strong>below<\/strong> the mean.<\/li>\r\n \t<li>That is, the standardized value for the observation [latex]6.04[\/latex] hours is [latex]-2[\/latex]. Its z-score is\u00a0<strong>[latex]-2[\/latex].<\/strong><\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow it's your turn to try. Let's load the Movie Runtime data set in the technology to calculate the standard deviation of a data set and then convert values to standardized scores by hand.\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>Data Set<\/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\n[ohm_question hide_question_numbers=1]241207[\/ohm_question]\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\n[ohm_question hide_question_numbers=1]241208[\/ohm_question]\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\n[ohm_question hide_question_numbers=1]241209[\/ohm_question]\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\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\n[ohm_question hide_question_numbers=1]241210[\/ohm_question]\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\n[ohm_question hide_question_numbers=1]241211[\/ohm_question]\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=\"convert\">Standardizing a Score<\/h3>\r\n<span style=\"background-color: #ffff00;\">At this point, students will be presented with two datasets. They will be able to choose which one they would like to use to answer example questions.<\/span>\r\n\r\nNow that you have obtained the standard deviation of the data set Runtimes using technology, you can calculate any observation's z-score to locate it in the data set 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\nWe'll use data set Sleep Study: Average Sleep, you saw in\u00a0\u00a0<em><a href=\"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/comparing-variability-of-data sets-what-to-know\/\">Comparing Variability of Data Sets: What to Know<\/a><\/em>\u00a0to demonstrate how to calculate z-scores for individual observations in the data set. The mean [latex]\\mu[\/latex] and standard deviation [latex]\\sigma[\/latex] in the formula represent the population the sample came from. Since we don't know these, we'll use the sample mean and standard deviation in our calculations.\r\n\r\nThe mean of the data is 7.97 hours with a standard deviation of 0.965. Calculate the z-scores for each of the following observations\u00a0and indicate if the given value lies above or below the mean. Round your calculations to two decimal places.\r\n<ol>\r\n \t<li>[latex]6.93\\text{ hours}[\/latex]<\/li>\r\n \t<li>[latex]9.87\\text{ hours}[\/latex]<\/li>\r\n \t<li>[latex]7.97\\text{ hours}[\/latex]<\/li>\r\n \t<li>[latex]4.95\\text{ hours}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"172851\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"172851\"]\r\n<ol>\r\n \t<li>[latex]z=\\dfrac{6.93-7.97}{0.965}\\approx -1.08[\/latex]. This value is [latex]1.08[\/latex] standard deviations\u00a0<strong>below<\/strong> the mean.<\/li>\r\n \t<li>[latex]z=\\dfrac{9.87-7.97}{0.965}\\approx 1.97[\/latex]. This value is\u00a0[latex]1.97[\/latex] standard deviations\u00a0<strong>above\u00a0<\/strong>the mean.<\/li>\r\n \t<li>[latex]z=\\dfrac{7.97-7.97}{0.965}=0[\/latex]. This value is equal to the\u00a0mean.<\/li>\r\n \t<li>[latex]z=\\dfrac{4.95-7.97}{0.965}\\approx -3.13[\/latex]. This value is [latex]-3.13[\/latex] standard deviations\u00a0<strong>below<\/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\n[ohm_question hide_question_numbers=1]241215[\/ohm_question]\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\n[ohm_question hide_question_numbers=1]241216[\/ohm_question]\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\n[ohm_question hide_question_numbers=1]241217[\/ohm_question]\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\n[ohm_question hide_question_numbers=1]241218[\/ohm_question]\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\u00a0[latex]68-95-99.7[\/latex] 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\nThe Empirical Rule states that:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">about\u00a0[latex]68[\/latex]% of observations in a data set will be within one standard deviation of the mean.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">about\u00a0[latex]95[\/latex]% of the observations in a data set will be within two standard deviations of the mean.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">about\u00a0[latex]99.7[\/latex]% of the observations in a data set 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\n[ohm_question hide_question_numbers=1]241220[\/ohm_question]\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\n[ohm_question hide_question_numbers=1]241225[\/ohm_question]\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\n[ohm_question hide_question_numbers=1]241227[\/ohm_question]\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 data set. 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>Learning Goals<\/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 values (also called standardized scores or z-scores) and use a value\u2019s standardized value 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 data set to explore how to perform necessary calculations by hand and using technology.<\/p>\n<h2>Standardized Values<\/h2>\n<p>You learned in\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/comparing-variability-of-data sets-what-to-know\/\"><em>Comparing Variability of Data Sets: What to Know<\/em><\/a> that a standard deviation is a measure for how spread out observations are from the mean.<\/p>\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>For example, in this section we will analyze runtimes (in minutes) of G-rated movies to learn how to calculate standardized values. 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 the distance of a particular movie runtime from the mean is not measured in minutes; rather it is measured in standard deviations. Thus, a z-score of\u00a0[latex]-2.3[\/latex] is an observation that is\u00a0[latex]2.3[\/latex] standard deviations <em>below<\/em> the mean, and a z-score of\u00a0[latex]2.3[\/latex] is an observation that is\u00a0[latex]2.3[\/latex] standard deviations <em>above<\/em> the mean. It is important to note that z-scores do not have units associated with them.<\/p>\n<p style=\"text-align: left;\"><strong>Z-Score Formula\u00a0\u00a0<\/strong>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>Before we use the formula to convert values into standardized values, let&#8217;s recap our understanding of standard deviation. In <a href=\"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/comparing-variability-of-data sets-what-to-know\/\"><em>Comparing Variability of Data Sets: What to Know<\/em><\/a>, you learned to understand standard deviation as a measure of variability in a data set. You looked at the statistical components that went into the formulas for standard deviation and variance and saw that larger standard deviations could represent more variability, and vice-versa. We&#8217;d like to shift that perspective now and look at a unit of standard deviation as a distance from the mean of a data set in a distribution.<\/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 in a distribution &#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 certain value at that location in the distribution. Show that a value&#8217;s z-score (negative or positive) is that many std deviations away from the mean in that direction.]<\/span><\/p>\n<\/div>\n<p>See the example below for a demonstration, then try it out using the Movie Runtimes database to answer the questions below.<\/p>\n<div class=\"textbox exercises\">\n<h3>inTeractive example<\/h3>\n<p>Let&#8217;s return again to the data set Sleep Study: Average Sleep, which we used in\u00a0<em><a href=\"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/comparing-variability-of-data sets-what-to-know\/\">Comparing Variability of Data Sets: What to Know<\/a><\/em> to learn about standard deviation as a measure of the variability of a data set.<\/p>\n<p>Open the tool at\u00a0<a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a>\u00a0and select the Sleep Study: Average Sleep data set.\u00a0Display a histogram and dotplot and make a note of the mean and standard deviation in the descriptive statistics. Round your final answers to the questions below to 3 decimal places, as needed.<\/p>\n<ol>\n<li>Describe the shape of the data set using the histogram and dotplot. For practice, display a boxplot as well and note the visual clues that you can use to determine the shape of the distribution from the boxplot.<\/li>\n<li>How does the relationship between the mean and median (given in descriptive statistics) help to support your analysis?<\/li>\n<li>What are the mean and standard deviation of the data set?<\/li>\n<li>What number of sleep hours lies one standard deviation above the mean? What value lies one standard deviation below?<\/li>\n<li>What number of sleep hours lie two standard deviations above and below the mean?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q599366\">Show Answer<\/span><\/p>\n<div id=\"q599366\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The distribution is unimodal and approximately symmetric. A few outliers lie to either side of the distribution but do so evenly.<\/li>\n<li>The mean and median are approximately equal, which supports that the outliers are approximately evenly distributed.<\/li>\n<li><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">[latex]\\bar{x}=7.97[\/latex] and [latex]s=0.965[\/latex]<\/span><\/li>\n<li>The standard deviation is 0.965. We can add that to the mean to determine the value exactly one standard deviation above the mean. Likewise, we can subtract that from the mean to find the value exactly one standard deviation below the mean.\n<ul>\n<li>[latex]7.97 + 0.965 = 8.935[\/latex]:\n<ul>\n<li>[latex]8.935[\/latex] hours of sleep lies <strong>one<\/strong> standard deviation <strong>above<\/strong> the mean.<\/li>\n<li>That is, the standardized value for the observation [latex]8.935[\/latex] hours is [latex]1[\/latex[] Its z-score is <strong>[latex]1[\/latex]<\/strong>.<\/li>\n<\/ul>\n<\/li>\n<li>[latex]7.97 - 0.965 = 7.005[\/latex]:\n<ul>\n<li>[latex]7.005[\/latex] hours of sleep lies <strong>one<\/strong> standard deviation <strong>below<\/strong> the mean.<\/li>\n<li>That is, the standardized value for the observation [latex]7.005[\/latex] hours is [latex]-1[\/latex]. Its z-score is <strong>[latex]-1[\/latex]<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>The standard deviation is [latex]0.965[\/latex]. We can add twice that to the mean to determine the value exactly two standard deviations above the mean. Likewise, we can subtract [latex]2*0.965[\/latex] from the mean to find the value exactly one standard deviation below the mean.\n<ul>\n<li>[latex]7.97 + 2*0.965 = 9.9[\/latex]:\n<ul>\n<li>[latex]9.9[\/latex] hours of sleep lies <strong>two<\/strong> standard deviations\u00a0<strong>above<\/strong> the mean.<\/li>\n<li>That is, the standardized value for the observation [latex]9.9[\/latex] hours is [latex]2[\/latex[] Its z-score is <strong>[latex]2[\/latex]<\/strong>.<\/li>\n<\/ul>\n<\/li>\n<li>[latex]7.97 - 2*0.965 = 6.04[\/latex]:\n<ul>\n<li>[latex]6.04[\/latex] hours of sleep lies <strong>two<\/strong> standard deviations\u00a0<strong>below<\/strong> the mean.<\/li>\n<li>That is, the standardized value for the observation [latex]6.04[\/latex] hours is [latex]-2[\/latex]. Its z-score is\u00a0<strong>[latex]-2[\/latex].<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Now it's your turn to try. Let's load the Movie Runtime data set in the technology to calculate the standard deviation of a data set and then convert values to standardized scores by hand.<\/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>Data Set<\/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><iframe loading=\"lazy\" id=\"ohm241207\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241207&theme=oea&iframe_resize_id=ohm241207\" width=\"100%\" height=\"150\"><\/iframe><\/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><iframe loading=\"lazy\" id=\"ohm241208\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241208&theme=oea&iframe_resize_id=ohm241208\" width=\"100%\" height=\"150\"><\/iframe><\/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><iframe loading=\"lazy\" id=\"ohm241209\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241209&theme=oea&iframe_resize_id=ohm241209\" width=\"100%\" height=\"150\"><\/iframe><\/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<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><iframe loading=\"lazy\" id=\"ohm241210\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241210&theme=oea&iframe_resize_id=ohm241210\" width=\"100%\" height=\"150\"><\/iframe><\/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><iframe loading=\"lazy\" id=\"ohm241211\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241211&theme=oea&iframe_resize_id=ohm241211\" width=\"100%\" height=\"150\"><\/iframe><\/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=\"convert\">Standardizing a Score<\/h3>\n<p><span style=\"background-color: #ffff00;\">At this point, students will be presented with two datasets. They will be able to choose which one they would like to use to answer example questions.<\/span><\/p>\n<p>Now that you have obtained the standard deviation of the data set Runtimes using technology, you can calculate any observation's z-score to locate it in the data set 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 - a 3-instructor video that works through an example like questions 6 - 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>We'll use data set Sleep Study: Average Sleep, you saw in\u00a0\u00a0<em><a href=\"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/comparing-variability-of-data sets-what-to-know\/\">Comparing Variability of Data Sets: What to Know<\/a><\/em>\u00a0to demonstrate how to calculate z-scores for individual observations in the data set. The mean [latex]\\mu[\/latex] and standard deviation [latex]\\sigma[\/latex] in the formula represent the population the sample came from. Since we don't know these, we'll use the sample mean and standard deviation in our calculations.<\/p>\n<p>The mean of the data is 7.97 hours with a standard deviation of 0.965. Calculate the z-scores for each of the following observations\u00a0and indicate if the given value lies above or below the mean. Round your calculations to two decimal places.<\/p>\n<ol>\n<li>[latex]6.93\\text{ hours}[\/latex]<\/li>\n<li>[latex]9.87\\text{ hours}[\/latex]<\/li>\n<li>[latex]7.97\\text{ hours}[\/latex]<\/li>\n<li>[latex]4.95\\text{ hours}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q172851\">Show Solution<\/span><\/p>\n<div id=\"q172851\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]z=\\dfrac{6.93-7.97}{0.965}\\approx -1.08[\/latex]. This value is [latex]1.08[\/latex] standard deviations\u00a0<strong>below<\/strong> the mean.<\/li>\n<li>[latex]z=\\dfrac{9.87-7.97}{0.965}\\approx 1.97[\/latex]. This value is\u00a0[latex]1.97[\/latex] standard deviations\u00a0<strong>above\u00a0<\/strong>the mean.<\/li>\n<li>[latex]z=\\dfrac{7.97-7.97}{0.965}=0[\/latex]. This value is equal to the\u00a0mean.<\/li>\n<li>[latex]z=\\dfrac{4.95-7.97}{0.965}\\approx -3.13[\/latex]. This value is [latex]-3.13[\/latex] standard deviations\u00a0<strong>below<\/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 - 9.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 6<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241215\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241215&theme=oea&iframe_resize_id=ohm241215\" width=\"100%\" height=\"150\"><\/iframe><\/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><iframe loading=\"lazy\" id=\"ohm241216\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241216&theme=oea&iframe_resize_id=ohm241216\" width=\"100%\" height=\"150\"><\/iframe><\/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><iframe loading=\"lazy\" id=\"ohm241217\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241217&theme=oea&iframe_resize_id=ohm241217\" width=\"100%\" height=\"150\"><\/iframe><\/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><iframe loading=\"lazy\" id=\"ohm241218\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241218&theme=oea&iframe_resize_id=ohm241218\" width=\"100%\" height=\"150\"><\/iframe><\/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\u00a0[latex]68-95-99.7[\/latex] 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]--&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<p>The Empirical Rule states that:<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">about\u00a0[latex]68[\/latex]% of observations in a data set will be within one standard deviation of the mean.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">about\u00a0[latex]95[\/latex]% of the observations in a data set will be within two standard deviations of the mean.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">about\u00a0[latex]99.7[\/latex]% of the observations in a data set 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 - 12<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 10<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241220\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241220&theme=oea&iframe_resize_id=ohm241220\" width=\"100%\" height=\"150\"><\/iframe><\/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><iframe loading=\"lazy\" id=\"ohm241225\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241225&theme=oea&iframe_resize_id=ohm241225\" width=\"100%\" height=\"150\"><\/iframe><\/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><iframe loading=\"lazy\" id=\"ohm241227\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241227&theme=oea&iframe_resize_id=ohm241227\" width=\"100%\" height=\"150\"><\/iframe><\/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'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 data set. You also seen how to explain what the Empirical Rule is. Let's summarize where these skills showed up in the material.<\/p>\n<ul>\n<li>In question 1 - 5, you converted values into standardized scores.<\/li>\n<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>\n<li>In Questions 10 - 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's time to move on!<\/p>\n","protected":false},"author":175116,"menu_order":57,"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-370","chapter","type-chapter","status-publish","hentry"],"part":1252,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/370","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/users\/175116"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/370\/revisions"}],"predecessor-version":[{"id":1250,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/370\/revisions\/1250"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/parts\/1252"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/370\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/media?parent=370"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapter-type?post=370"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/contributor?post=370"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/license?post=370"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}