{"id":372,"date":"2022-02-21T18:01:56","date_gmt":"2022-02-21T18:01:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/?post_type=chapter&#038;p=372"},"modified":"2022-05-20T16:38:43","modified_gmt":"2022-05-20T16:38:43","slug":"z-score-and-the-empirical-rule-forming-connections","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/z-score-and-the-empirical-rule-forming-connections\/","title":{"raw":"Z-Score and the Empirical Rule: Forming Connections","rendered":"Z-Score and the Empirical Rule: Forming Connections"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>objectives for this activity<\/h3>\r\nDuring this activity, you will:\r\n<ul>\r\n \t<li><a href=\"#useEmp\">Utilize standardized scores and the Empirical Rule to determine if an observation is unusual.<\/a><\/li>\r\n \t<li><a href=\"#CompUseZ\">Compare two observations by calculating and comparing z-scores.<\/a><\/li>\r\n<\/ul>\r\nClick on a skill above to jump to its location in this activity.\r\n\r\n<\/div>\r\n<h2>What Is Unusual?<\/h2>\r\nIn a medical study, many observations are made in an effort to obtain a data sample representative of the population from which it was taken.\u00a0In this activity, you'll see how standardized scores and the Empirical Rule can be used to determine if an observation is usual or unusual.\r\n\r\nAround the world, pharmaceutical companies conduct clinical trials to evaluate the safety and efficacy of their drugs. Clinical trials are research studies performed on people and are aimed at evaluating if a new drug is safe and effective. People who participate in clinical trials are volunteers.\r\n\r\n<strong><img class=\"aligncenter wp-image-1034\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11224131\/Picture60-300x201.jpg\" alt=\"Two people in a room with a doctor who is writing something on a clipboard. Everyone is wearing face masks.\" width=\"523\" height=\"350\" \/><\/strong>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 1<\/h3>\r\n[ohm_question hide_question_numbers=1]241229[\/ohm_question]\r\n\r\n[reveal-answer q=\"253038\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"253038\"]What do <em>you<\/em> think?[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>video placement<\/h3>\r\n<span style=\"background-color: #e6daf7;\">[Intro video: In this activity, we'll use the landscape of a medical study to learn how the Empirical Rule can help us identify unusual observations of a quantitative variable. We'll also compare two observations by calculating and comparing their standardized scores. Data collected during medical studies tends to form a bell-shaped, unimodal distribution, with a large number of observations located near the mean and equal numbers of values further from the mean falling off to either side. [voice over the empirical rule image from What to Know 4E]. Under these conditions, we know that almost all the observations in the distribution fall within three standard deviations from the mean. This lets us set an exact threshold for how far away from the mean an observation can be located for us to consider it unusual. But human clinical trials require people willing to participate. They are costly and take a great deal of time to gather the appropriate data, even decades to study the effects of a drug over an entire lifespan. Can you think of a good alternative to having human volunteers participate in clinical trials? How about using mice instead? In this activity, we'll explore a medical study involving mice as we learn to apply the Empirical Rule and standardized scores.]<\/span>\r\n\r\n<\/div>\r\nMice are often used in medical studies to evaluate the effects of chemicals and pharmaceuticals. One reason for this is that scientists know a lot about the <span style=\"background-color: #ffff99;\">genome<\/span> of a mouse. They are bred in labs to be identical, so the only thing different between them is the treatment. Mice also have short lifespans, which allows scientists to model the effects of a drug over their entire lifespan (about\u00a0[latex]800[\/latex] days). It is much more difficult to understand the effects of a drug over the lifetime of a human.\r\n\r\nConsider a study concerned with learning how a drug or a treatment affects the body. The <span style=\"background-color: #ffff99;\">toxicity<\/span> of a chemical and its impact on vital organs is of interest when assessing the effects of a chemical treatment. A standard method used to measure the level of toxicity in an organ is to <span style=\"background-color: #ffff99;\">use the organ\u2019s weight<\/span>.[footnote] Sellers, R. S., Mortan, D., Michael, B., Bindhu, M., Roome, N., Johnson, J. K., Yano, B. L., Perry, R., &amp;\u00a0Schafer, K. (2007). Society of toxicologic pathology position paper: Organ weight recommendations for\u00a0toxicology studies. <em>Toxicologic Pathology.<\/em> 35(5), 751-755. https:\/\/doi.org\/10.1080\/01926230701595300[\/footnote]\r\n\r\n[reveal-answer q=\"283496\"]What is a genome?[footnote]<em>What is a genome?<\/em> (2017, January 6). Yourgenome. Retrieved from\u00a0<a href=\"https:\/\/www.yourgenome.org\/facts\/what-is-a-genome\">https:\/\/www.yourgenome.org\/facts\/what-is-a-genome<\/a>[\/footnote][\/reveal-answer]\r\n[hidden-answer a=\"283496\"]A genome is an organism\u2019s complete set of genetic instructions. Each genome contains all of the information needed to build that organism and allow it to grow and develop. Our bodies are made up of millions of cells, each with their own complete set of instructions for making us, like a recipe book for the body. This set of instructions is known as our genome and is made up of DNA.\u00a0 For more information on genomes, visit <a href=\"https:\/\/www.yourgenome.org\/facts\/what-is-a-genome\">https:\/\/www.yourgenome.org\/facts\/what-is-a-genome<\/a>.\r\n\r\n<hr \/>\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"233331\"]What is toxicity?[footnote]<em>Definition of toxicity<\/em>. (2021, March 29). RxList. Retrieved from\r\n<a href=\"https:\/\/www.rxlist.com\/toxicity\/definition.htm\">https:\/\/www.rxlist.com\/toxicity\/definition.htm<\/a>[\/footnote][\/reveal-answer]\r\n[hidden-answer a=\"233331\"]The degree to which a substance (a toxin or poison) can harm humans or animals.\r\n\r\n<hr \/>\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"128123\"]Why is organ weight used to measure toxicity?[footnote]Lazic, S. E., Semenova, E., &amp; Williams, D. P. (2020, April 20). <em>Determining organ weight toxicity with Bayesian causal models: Improving on the analysis of relative organ weights<\/em>. National Center for Biotechnology Information. <a href=\"https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC7170916\/\">https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC7170916\/<\/a>[\/footnote][\/reveal-answer]\r\n[hidden-answer a=\"128123\"]Organ weight changes are indicators of chemically-induced organ damage. Hence, changes in organ weight are often associated with treatment-related effects.\r\n\r\n<hr \/>\r\n\r\n[\/hidden-answer]\r\n<h3 id=\"useEmp\">The Empirical Rule<\/h3>\r\n<div class=\"textbox examples\">\r\n<h3>recall<\/h3>\r\nBefore beginning the activity below, recall the definition of the Empirical Rule. What does it state?\r\n\r\nCore skill: [reveal-answer q=\"725555\"]Explain the Empirical Rule[\/reveal-answer]\r\n[hidden-answer a=\"725555\"]\r\n\r\nThe Empirical Rule states that,\u00a0in a bell-shaped, unimodal distribution, almost all the observed data values, [latex]x[\/latex], lie within three standard deviations, [latex]\\sigma[\/latex], to either side of the mean, [latex]\\mu[\/latex]. Specifically,\r\n<p style=\"padding-left: 30px;\">[latex]68[\/latex]% of the observations lie within one standard deviation of the mean [latex]\\left(\\mu\\pm\\sigma\\right)[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]95[\/latex]% of the observations lie within two standard deviations of the mean [latex]\\left(\\mu\\pm2\\sigma\\right)[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]99.7[\/latex]% of the observations lie within three standard deviations of the mean\u00a0[latex]\\left(\\mu\\pm3\\sigma\\right)[\/latex]<\/p>\r\nFor this reason, the Empirical Rule is sometimes called the\u00a0[latex]68-95-99.7[\/latex] rule.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nConsider the weights of the livers and spleens in\u00a0[latex]26[\/latex]-week old female C57BL\/6J laboratory mice. The mean liver weight is\u00a0[latex]0.999[\/latex] grams (g) with a standard deviation of\u00a0[latex]0.087[\/latex] g, and the mean spleen weight is\u00a0[latex]0.086[\/latex] g with a standard deviation of\u00a0[latex]0.007[\/latex] g. Use this information along with the Empirical Rule to answer Questions 2 and 3 below. Round your answers to three decimal places.\r\n\r\n[reveal-answer q=\"782173\"]Click here to review the graph of the Empirical Rule if needed.[\/reveal-answer]\r\n[hidden-answer a=\"782173\"]\u00a0 <img class=\"alignnone size-medium 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=\"300\" height=\"295\" \/>\u00a0 [\/hidden-answer]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 2<\/h3>\r\n<span style=\"font-size: 1rem; text-align: initial;\">[ohm_question hide_question_numbers=1]241231[\/ohm_question]<\/span>\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">[reveal-answer q=\"19326\"]Hint[\/reveal-answer]<\/span>\r\n\r\n[hidden-answer a=\"19326\"]Calculate the values <strong>below <\/strong>and<strong> above<\/strong>\u00a0the mean, rounding to the nearest thousandth. The mean and standard deviation are given in the text above. Use the Empirical Rule (sometimes called the\u00a0[latex]68-95-99.7[\/latex] rule)[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 3<\/h3>\r\n<span style=\"font-size: 1rem; text-align: initial;\">[ohm_question hide_question_numbers=1]241232[\/ohm_question][reveal-answer q=\"803282\"]Hint[\/reveal-answer]<\/span>\r\n\r\n[hidden-answer a=\"803282\"]Calculate the values <strong>below <\/strong>and<strong> above<\/strong>\u00a0the mean, rounding to the nearest thousandth. The mean and standard deviation are given in the text above. Use the Empirical Rule (sometimes called the\u00a0[latex]68-95-99.7[\/latex] rule)[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>video placement<\/h3>\r\n<span style=\"background-color: #e6daf7;\">\u00a0[insert sub-summary video: \"In these questions, you calculated specific values for the liver and spleen weights of the mice that marked locations in the data exactly one, two, and three standard deviations below the mean (to the left on the graph) and above the mean (to the right)). [this is voice over the graph of the Empirical rule again.] So, [pointing to the horizontal axis] what values are associated with 68% of the liver weights? That's right, liver weights between 0.912 g and 1.086 g make up 68% of all the liver weights because these are all within one standard deviation of the mean. So, what do you think you'd consider an unusual liver weight, either unusually high or unusually low? In statistics, we oftentimes consider an observation unusual if it is at least two standard deviations away from the mean of a data set. What percentage of this data is within two standard deviations? That's right, 95% percent. In this context, between which two values of spleen and liver weights are 95% of the data located? I'll let you figure that one out for yourself and use it to answer the following question.\"]<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 4<\/h3>\r\n[ohm_question hide_question_numbers=1]241234[\/ohm_question]\r\n\r\n[reveal-answer q=\"861361\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"861361\"]What do <em>you<\/em> think? How extreme should an observation be for it to be an outlier: two standard deviations away from the mean? Three standard deviations away?[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3 id=\"CompUseZ\">Z-Scores<\/h3>\r\nA higher organ weight is an indicator of higher toxicity. Suppose a mouse has a liver weight of\u00a0[latex]1.07[\/latex] g and a spleen weight of\u00a0[latex]0.104[\/latex] g. Is either of these values extreme?\u00a0 How many standard deviations from the mean do these values lie, and in what direction? We can use the z-score for each of these values to help us answer these questions. In the following questions, calculate the z-score for these weights then interpret that score. Remember, the z-score is a\u00a0<em>number of standard deviations<\/em>, and has no units associated with it. It only gives relative proximity (distance and direction) from the mean of a quantitative variable.\r\n<div class=\"textbox examples\">\r\n<h3>recall<\/h3>\r\nIn the following questions, you'll need to calculate and interpret z-scores. Take a moment to refresh the formula if needed.\r\n\r\nCore skill: [reveal-answer q=\"785224\"]Express the formula obtaining a z-score[\/reveal-answer]\r\n[hidden-answer a=\"785224\"][latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x[\/latex] is an observation value, [latex]\\mu[\/latex] is the population mean, and [latex]\\sigma[\/latex] is the standard deviation[\/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]241235[\/ohm_question]\r\n\r\n[reveal-answer q=\"360167\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"360167\"]Use the formula for calculating z-scores and round your answer to the nearest thousandth. How many standard deviations from the mean is this weight?[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 6<\/h3>\r\n[ohm_question hide_question_numbers=1]241238[\/ohm_question]\r\n\r\n[reveal-answer q=\"674426\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"674426\"]Use the formula for calculating z-scores and round your answer to the nearest thousandth. How many standard deviations from the mean is this weight?[\/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]241241[\/ohm_question]\r\n\r\n[reveal-answer q=\"391619\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"391619\"]Which organ is relatively heavier than a typical weight? Put this in the context of the Empirical Rule.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>video placement<\/h3>\r\n<span style=\"background-color: #e6daf7;\">[wrap-up video: In the final question of this activity, you compared two organ weights, one liver and one spleen, to determine which had a higher level of toxicity. But the distribution for liver and spleen weights didn't have the same mean, so simply comparing one weight to the other wouldn't help. Mouse spleens are naturally much lighter than mouse livers. You needed to compare their \"unusualness\" instead. To do so, you calculated z-scores for each weight. This let you determine which of the two was further from the mean weight for all such mouse organs [voice over the Empirical graph again here], which let you know which of the two was relatively heavier for it's type. Remember that by calculating the z-score, you are calculating a distance in the distribution, not a weight in grams. Z-scores have no units associated with them. You found that the spleen showed a higher level of toxicity because the weight of the spleen was unusual, at 2.571 standard deviations above the mean. The weight of the liver, by contrast, was only 0.816 standard deviations away, within the middle 68% of all mouse liver weights.\"]<\/span>\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>objectives for this activity<\/h3>\n<p>During this activity, you will:<\/p>\n<ul>\n<li><a href=\"#useEmp\">Utilize standardized scores and the Empirical Rule to determine if an observation is unusual.<\/a><\/li>\n<li><a href=\"#CompUseZ\">Compare two observations by calculating and comparing z-scores.<\/a><\/li>\n<\/ul>\n<p>Click on a skill above to jump to its location in this activity.<\/p>\n<\/div>\n<h2>What Is Unusual?<\/h2>\n<p>In a medical study, many observations are made in an effort to obtain a data sample representative of the population from which it was taken.\u00a0In this activity, you&#8217;ll see how standardized scores and the Empirical Rule can be used to determine if an observation is usual or unusual.<\/p>\n<p>Around the world, pharmaceutical companies conduct clinical trials to evaluate the safety and efficacy of their drugs. Clinical trials are research studies performed on people and are aimed at evaluating if a new drug is safe and effective. People who participate in clinical trials are volunteers.<\/p>\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1034\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11224131\/Picture60-300x201.jpg\" alt=\"Two people in a room with a doctor who is writing something on a clipboard. Everyone is wearing face masks.\" width=\"523\" height=\"350\" \/><\/strong><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 1<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241229\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241229&theme=oea&iframe_resize_id=ohm241229\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q253038\">Hint<\/span><\/p>\n<div id=\"q253038\" class=\"hidden-answer\" style=\"display: none\">What do <em>you<\/em> think?<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>video placement<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[Intro video: In this activity, we&#8217;ll use the landscape of a medical study to learn how the Empirical Rule can help us identify unusual observations of a quantitative variable. We&#8217;ll also compare two observations by calculating and comparing their standardized scores. Data collected during medical studies tends to form a bell-shaped, unimodal distribution, with a large number of observations located near the mean and equal numbers of values further from the mean falling off to either side. [voice over the empirical rule image from What to Know 4E]. Under these conditions, we know that almost all the observations in the distribution fall within three standard deviations from the mean. This lets us set an exact threshold for how far away from the mean an observation can be located for us to consider it unusual. But human clinical trials require people willing to participate. They are costly and take a great deal of time to gather the appropriate data, even decades to study the effects of a drug over an entire lifespan. Can you think of a good alternative to having human volunteers participate in clinical trials? How about using mice instead? In this activity, we&#8217;ll explore a medical study involving mice as we learn to apply the Empirical Rule and standardized scores.]<\/span><\/p>\n<\/div>\n<p>Mice are often used in medical studies to evaluate the effects of chemicals and pharmaceuticals. One reason for this is that scientists know a lot about the <span style=\"background-color: #ffff99;\">genome<\/span> of a mouse. They are bred in labs to be identical, so the only thing different between them is the treatment. Mice also have short lifespans, which allows scientists to model the effects of a drug over their entire lifespan (about\u00a0[latex]800[\/latex] days). It is much more difficult to understand the effects of a drug over the lifetime of a human.<\/p>\n<p>Consider a study concerned with learning how a drug or a treatment affects the body. The <span style=\"background-color: #ffff99;\">toxicity<\/span> of a chemical and its impact on vital organs is of interest when assessing the effects of a chemical treatment. A standard method used to measure the level of toxicity in an organ is to <span style=\"background-color: #ffff99;\">use the organ\u2019s weight<\/span>.<a class=\"footnote\" title=\"Sellers, R. S., Mortan, D., Michael, B., Bindhu, M., Roome, N., Johnson, J. K., Yano, B. L., Perry, R., &amp;\u00a0Schafer, K. (2007). Society of toxicologic pathology position paper: Organ weight recommendations for\u00a0toxicology studies. Toxicologic Pathology. 35(5), 751-755. https:\/\/doi.org\/10.1080\/01926230701595300\" id=\"return-footnote-372-1\" href=\"#footnote-372-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283496\">What is a genome?<a class=\"footnote\" title=\"What is a genome? (2017, January 6). Yourgenome. Retrieved from\u00a0https:\/\/www.yourgenome.org\/facts\/what-is-a-genome\" id=\"return-footnote-372-2\" href=\"#footnote-372-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/span><\/p>\n<div id=\"q283496\" class=\"hidden-answer\" style=\"display: none\">A genome is an organism\u2019s complete set of genetic instructions. Each genome contains all of the information needed to build that organism and allow it to grow and develop. Our bodies are made up of millions of cells, each with their own complete set of instructions for making us, like a recipe book for the body. This set of instructions is known as our genome and is made up of DNA.\u00a0 For more information on genomes, visit <a href=\"https:\/\/www.yourgenome.org\/facts\/what-is-a-genome\">https:\/\/www.yourgenome.org\/facts\/what-is-a-genome<\/a>.<\/p>\n<hr \/>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q233331\">What is toxicity?<a class=\"footnote\" title=\"Definition of toxicity. (2021, March 29). RxList. Retrieved from\nhttps:\/\/www.rxlist.com\/toxicity\/definition.htm\" id=\"return-footnote-372-3\" href=\"#footnote-372-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a><\/span><\/p>\n<div id=\"q233331\" class=\"hidden-answer\" style=\"display: none\">The degree to which a substance (a toxin or poison) can harm humans or animals.<\/p>\n<hr \/>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q128123\">Why is organ weight used to measure toxicity?<a class=\"footnote\" title=\"Lazic, S. E., Semenova, E., &amp; Williams, D. P. (2020, April 20). Determining organ weight toxicity with Bayesian causal models: Improving on the analysis of relative organ weights. National Center for Biotechnology Information. https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC7170916\/\" id=\"return-footnote-372-4\" href=\"#footnote-372-4\" aria-label=\"Footnote 4\"><sup class=\"footnote\">[4]<\/sup><\/a><\/span><\/p>\n<div id=\"q128123\" class=\"hidden-answer\" style=\"display: none\">Organ weight changes are indicators of chemically-induced organ damage. Hence, changes in organ weight are often associated with treatment-related effects.<\/p>\n<hr \/>\n<\/div>\n<\/div>\n<h3 id=\"useEmp\">The Empirical Rule<\/h3>\n<div class=\"textbox examples\">\n<h3>recall<\/h3>\n<p>Before beginning the activity below, recall the definition of the Empirical Rule. What does it state?<\/p>\n<p>Core skill: <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q725555\">Explain the Empirical Rule<\/span><\/p>\n<div id=\"q725555\" class=\"hidden-answer\" style=\"display: none\">\n<p>The Empirical Rule states that,\u00a0in a bell-shaped, unimodal distribution, almost all the observed data values, [latex]x[\/latex], lie within three standard deviations, [latex]\\sigma[\/latex], to either side of the mean, [latex]\\mu[\/latex]. Specifically,<\/p>\n<p style=\"padding-left: 30px;\">[latex]68[\/latex]% of the observations lie within one standard deviation of the mean [latex]\\left(\\mu\\pm\\sigma\\right)[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]95[\/latex]% of the observations lie within two standard deviations of the mean [latex]\\left(\\mu\\pm2\\sigma\\right)[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]99.7[\/latex]% of the observations lie within three standard deviations of the mean\u00a0[latex]\\left(\\mu\\pm3\\sigma\\right)[\/latex]<\/p>\n<p>For this reason, the Empirical Rule is sometimes called the\u00a0[latex]68-95-99.7[\/latex] rule.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Consider the weights of the livers and spleens in\u00a0[latex]26[\/latex]-week old female C57BL\/6J laboratory mice. The mean liver weight is\u00a0[latex]0.999[\/latex] grams (g) with a standard deviation of\u00a0[latex]0.087[\/latex] g, and the mean spleen weight is\u00a0[latex]0.086[\/latex] g with a standard deviation of\u00a0[latex]0.007[\/latex] g. Use this information along with the Empirical Rule to answer Questions 2 and 3 below. Round your answers to three decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q782173\">Click here to review the graph of the Empirical Rule if needed.<\/span><\/p>\n<div id=\"q782173\" class=\"hidden-answer\" style=\"display: none\">\u00a0 <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium 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=\"300\" height=\"295\" \/>\u00a0 <\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 2<\/h3>\n<p><span style=\"font-size: 1rem; text-align: initial;\"><iframe loading=\"lazy\" id=\"ohm241231\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241231&theme=oea&iframe_resize_id=ohm241231\" width=\"100%\" height=\"150\"><\/iframe><\/span><\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\"><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q19326\">Hint<\/span><\/span><\/p>\n<div id=\"q19326\" class=\"hidden-answer\" style=\"display: none\">Calculate the values <strong>below <\/strong>and<strong> above<\/strong>\u00a0the mean, rounding to the nearest thousandth. The mean and standard deviation are given in the text above. Use the Empirical Rule (sometimes called the\u00a0[latex]68-95-99.7[\/latex] rule)<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 3<\/h3>\n<p><span style=\"font-size: 1rem; text-align: initial;\"><iframe loading=\"lazy\" id=\"ohm241232\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241232&theme=oea&iframe_resize_id=ohm241232\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q803282\">Hint<\/span><\/span><\/p>\n<div id=\"q803282\" class=\"hidden-answer\" style=\"display: none\">Calculate the values <strong>below <\/strong>and<strong> above<\/strong>\u00a0the mean, rounding to the nearest thousandth. The mean and standard deviation are given in the text above. Use the Empirical Rule (sometimes called the\u00a0[latex]68-95-99.7[\/latex] rule)<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>video placement<\/h3>\n<p><span style=\"background-color: #e6daf7;\">\u00a0[insert sub-summary video: &#8220;In these questions, you calculated specific values for the liver and spleen weights of the mice that marked locations in the data exactly one, two, and three standard deviations below the mean (to the left on the graph) and above the mean (to the right)). [this is voice over the graph of the Empirical rule again.] So, [pointing to the horizontal axis] what values are associated with 68% of the liver weights? That&#8217;s right, liver weights between 0.912 g and 1.086 g make up 68% of all the liver weights because these are all within one standard deviation of the mean. So, what do you think you&#8217;d consider an unusual liver weight, either unusually high or unusually low? In statistics, we oftentimes consider an observation unusual if it is at least two standard deviations away from the mean of a data set. What percentage of this data is within two standard deviations? That&#8217;s right, 95% percent. In this context, between which two values of spleen and liver weights are 95% of the data located? I&#8217;ll let you figure that one out for yourself and use it to answer the following question.&#8221;]<\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 4<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241234\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241234&theme=oea&iframe_resize_id=ohm241234\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q861361\">Hint<\/span><\/p>\n<div id=\"q861361\" class=\"hidden-answer\" style=\"display: none\">What do <em>you<\/em> think? How extreme should an observation be for it to be an outlier: two standard deviations away from the mean? Three standard deviations away?<\/div>\n<\/div>\n<\/div>\n<h3 id=\"CompUseZ\">Z-Scores<\/h3>\n<p>A higher organ weight is an indicator of higher toxicity. Suppose a mouse has a liver weight of\u00a0[latex]1.07[\/latex] g and a spleen weight of\u00a0[latex]0.104[\/latex] g. Is either of these values extreme?\u00a0 How many standard deviations from the mean do these values lie, and in what direction? We can use the z-score for each of these values to help us answer these questions. In the following questions, calculate the z-score for these weights then interpret that score. Remember, the z-score is a\u00a0<em>number of standard deviations<\/em>, and has no units associated with it. It only gives relative proximity (distance and direction) from the mean of a quantitative variable.<\/p>\n<div class=\"textbox examples\">\n<h3>recall<\/h3>\n<p>In the following questions, you&#8217;ll need to calculate and interpret z-scores. Take a moment to refresh the formula if needed.<\/p>\n<p>Core skill: <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q785224\">Express the formula obtaining a z-score<\/span><\/p>\n<div id=\"q785224\" class=\"hidden-answer\" style=\"display: none\">[latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x[\/latex] is an observation value, [latex]\\mu[\/latex] is the population mean, and [latex]\\sigma[\/latex] is the standard deviation<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 5<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241235\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241235&theme=oea&iframe_resize_id=ohm241235\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q360167\">Hint<\/span><\/p>\n<div id=\"q360167\" class=\"hidden-answer\" style=\"display: none\">Use the formula for calculating z-scores and round your answer to the nearest thousandth. How many standard deviations from the mean is this weight?<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 6<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241238\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241238&theme=oea&iframe_resize_id=ohm241238\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q674426\">Hint<\/span><\/p>\n<div id=\"q674426\" class=\"hidden-answer\" style=\"display: none\">Use the formula for calculating z-scores and round your answer to the nearest thousandth. How many standard deviations from the mean is this weight?<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 7<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241241\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241241&theme=oea&iframe_resize_id=ohm241241\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q391619\">Hint<\/span><\/p>\n<div id=\"q391619\" class=\"hidden-answer\" style=\"display: none\">Which organ is relatively heavier than a typical weight? Put this in the context of the Empirical Rule.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>video placement<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[wrap-up video: In the final question of this activity, you compared two organ weights, one liver and one spleen, to determine which had a higher level of toxicity. But the distribution for liver and spleen weights didn&#8217;t have the same mean, so simply comparing one weight to the other wouldn&#8217;t help. Mouse spleens are naturally much lighter than mouse livers. You needed to compare their &#8220;unusualness&#8221; instead. To do so, you calculated z-scores for each weight. This let you determine which of the two was further from the mean weight for all such mouse organs [voice over the Empirical graph again here], which let you know which of the two was relatively heavier for it&#8217;s type. Remember that by calculating the z-score, you are calculating a distance in the distribution, not a weight in grams. Z-scores have no units associated with them. You found that the spleen showed a higher level of toxicity because the weight of the spleen was unusual, at 2.571 standard deviations above the mean. The weight of the liver, by contrast, was only 0.816 standard deviations away, within the middle 68% of all mouse liver weights.&#8221;]<\/span><\/p>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-372-1\"> Sellers, R. S., Mortan, D., Michael, B., Bindhu, M., Roome, N., Johnson, J. K., Yano, B. L., Perry, R., &amp;\u00a0Schafer, K. (2007). Society of toxicologic pathology position paper: Organ weight recommendations for\u00a0toxicology studies. <em>Toxicologic Pathology.<\/em> 35(5), 751-755. https:\/\/doi.org\/10.1080\/01926230701595300 <a href=\"#return-footnote-372-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-372-2\"><em>What is a genome?<\/em> (2017, January 6). Yourgenome. Retrieved from\u00a0<a href=\"https:\/\/www.yourgenome.org\/facts\/what-is-a-genome\">https:\/\/www.yourgenome.org\/facts\/what-is-a-genome<\/a> <a href=\"#return-footnote-372-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-372-3\"><em>Definition of toxicity<\/em>. (2021, March 29). RxList. Retrieved from\r\n<a href=\"https:\/\/www.rxlist.com\/toxicity\/definition.htm\">https:\/\/www.rxlist.com\/toxicity\/definition.htm<\/a> <a href=\"#return-footnote-372-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-372-4\">Lazic, S. E., Semenova, E., &amp; Williams, D. P. (2020, April 20). <em>Determining organ weight toxicity with Bayesian causal models: Improving on the analysis of relative organ weights<\/em>. National Center for Biotechnology Information. <a href=\"https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC7170916\/\">https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC7170916\/<\/a> <a href=\"#return-footnote-372-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":175116,"menu_order":58,"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-372","chapter","type-chapter","status-publish","hentry"],"part":1252,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/372","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":9,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/372\/revisions"}],"predecessor-version":[{"id":1254,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/372\/revisions\/1254"}],"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\/372\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/media?parent=372"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapter-type?post=372"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/contributor?post=372"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/license?post=372"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}