{"id":673,"date":"2021-08-13T16:01:53","date_gmt":"2021-08-13T16:01:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=673"},"modified":"2023-12-05T08:58:08","modified_gmt":"2023-12-05T08:58:08","slug":"measures-of-the-spread-of-data-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/measures-of-the-spread-of-data-2\/","title":{"raw":"Measures of the Spread of Data","rendered":"Measures of the Spread of Data"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul id=\"list123523\">\r\n \t<li>Calculate and interpret <strong><em>z<\/em>-scores<\/strong><\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Comparing Values from Different Data Sets<\/h2>\r\nThe standard deviation is useful when comparing data values that come from different data sets. If the data sets have different means and standard deviations, then comparing the data values directly can be misleading.\r\n<ul>\r\n \t<li>For each data value, calculate how many standard deviations away from its mean the value is.<\/li>\r\n \t<li>Use the formula: value = mean + (#ofSTDEVs)(standard deviation); solve for #ofSTDEVs.<\/li>\r\n \t<li><em>#ofSTDEVs <\/em>= [latex]\\frac{\\mathrm{value} - \\mathrm{mean}}{\\mathrm{standard \\ deviation}} [\/latex]<\/li>\r\n \t<li>Compare the results of this calculation.<\/li>\r\n<\/ul>\r\n#ofSTDEVs is often called a \"[latex]z[\/latex]-score\"; we can use the symbol [latex]z[\/latex]. In symbols, the formulas become:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Sample<\/td>\r\n<td>[latex]x=\\overline{x}+zs[\/latex]<\/td>\r\n<td>[latex]z = \\frac{x - \\overline{x}}{s}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Population<\/td>\r\n<td>[latex]x = \u03bc + z\u03c3[\/latex]<\/td>\r\n<td>[latex]z = \\frac{x - \u03bc}{\u03c3}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nTwo students, John and Eric, from different high schools, wanted to find out who had the highest GPA when compared to his school. Which student had the highest GPA when compared to his school?\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Student<\/th>\r\n<th>GPA<\/th>\r\n<th>School Mean GPA<\/th>\r\n<th>School Standard Deviation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>John<\/td>\r\n<td>[latex]2.85[\/latex]<\/td>\r\n<td>[latex]3.0[\/latex]<\/td>\r\n<td>[latex]0.7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Eric<\/td>\r\n<td>[latex]77[\/latex]<\/td>\r\n<td>[latex]80[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"124079\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124079\"]\r\n\r\nFor each student, determine how many standard deviations (#ofSTDEVs) his GPA is away from the average, for his school. Pay careful attention to signs when comparing and interpreting the answer.\r\n\r\n[latex]z[\/latex] = # of STDEVs = [latex]\\frac{\\mathrm{value} - \\mathrm{mean}}{\\mathrm{standard \\ deviation}} [\/latex] =\u00a0 [latex]\\frac{x-\u03bc}{\u03c3}[\/latex]\r\n\r\nFor John, [latex]z[\/latex] = # ofSTDEVs = [latex]\\displaystyle\\frac{{2.85 - 3.00}}{{0.7}}=-{0.21}[\/latex]\r\n\r\nFor Eric, [latex]z[\/latex] = # ofSTDEVs = [latex]\\displaystyle\\frac{{77- 80}}{{10}}=\u22120.3[\/latex]\r\n\r\nJohn has the better GPA when compared to his school because his GPA is [latex]0.21[\/latex] standard deviations\u00a0<strong data-redactor-tag=\"strong\">below<\/strong> his school's mean while Eric GPA is [latex]0.3[\/latex] standard deviations <strong data-redactor-tag=\"strong\">below<\/strong> his school's mean.\r\n\r\nJohn's [latex]z[\/latex]-score of [latex]\u20130.21[\/latex] is higher than Eric [latex]z[\/latex]-score of [latex]\u20130.3[\/latex]. For GPA, higher values are better, so we conclude that Eric has the better GPA when compared to his school.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nTwo swimmers, Angie and Beth, from different teams, wanted to find out who had the fastest time for the 50 meter freestyle when compared to her team. Which swimmer had the fastest time when compared to her team?\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Swimmer<\/th>\r\n<th>Time (seconds)<\/th>\r\n<th>Team Mean Time<\/th>\r\n<th>Team Standard Deviation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Angie<\/td>\r\n<td>[latex]26.2[\/latex]<\/td>\r\n<td>[latex]27.2[\/latex]<\/td>\r\n<td>[latex]0.8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Beth<\/td>\r\n<td>[latex]27.3[\/latex]<\/td>\r\n<td>[latex]30.1[\/latex]<\/td>\r\n<td>[latex]1.4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"124080\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124080\"]\r\nFor Angie: [latex]z=\\frac{\\left(26.2-27.2\\right)}{0.8}=1.25[\/latex]\r\n\r\nFor Beth: [latex]z=\\frac{\\left(27.3-30.1\\right)}{1.4}=-2[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following lists give a few facts that provide a little more insight into what the standard deviation tells us about the distribution of the data.\r\n\r\n<strong data-redactor-tag=\"strong\">For ANY data set, no matter what the distribution of the data is:<\/strong>\r\n<ul>\r\n \t<li>At least [latex]75[\/latex]% of the data is within two standard deviations of the mean.<\/li>\r\n \t<li>At least [latex]89[\/latex]% of the data is within three standard deviations of the mean.<\/li>\r\n \t<li>At least [latex]95[\/latex]% of the data is within [latex]4.5[\/latex] standard deviations of the mean.<\/li>\r\n \t<li>This is known as Chebyshev's Rule.<\/li>\r\n<\/ul>\r\n<strong data-redactor-tag=\"strong\">For data having a distribution that is BELL-SHAPED and SYMMETRIC:<\/strong>\r\n<ul>\r\n \t<li>Approximately [latex]68[\/latex]% of the data is within one standard deviation of the mean.<\/li>\r\n \t<li>Approximately [latex]95[\/latex]% of the data is within two standard deviations of the mean.<\/li>\r\n \t<li>More than [latex]99[\/latex]% of the data is within three standard deviations of the mean.<\/li>\r\n \t<li>This is known as the Empirical Rule.<\/li>\r\n \t<li>It is important to note that this rule only applies when the shape of the distribution of the data is bell-shaped and symmetric. We will learn more about this when studying the \"Normal\" or \"Gaussian\" probability distribution in later chapters.<\/li>\r\n<\/ul>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul id=\"list123523\">\n<li>Calculate and interpret <strong><em>z<\/em>-scores<\/strong><\/li>\n<\/ul>\n<\/div>\n<h2>Comparing Values from Different Data Sets<\/h2>\n<p>The standard deviation is useful when comparing data values that come from different data sets. If the data sets have different means and standard deviations, then comparing the data values directly can be misleading.<\/p>\n<ul>\n<li>For each data value, calculate how many standard deviations away from its mean the value is.<\/li>\n<li>Use the formula: value = mean + (#ofSTDEVs)(standard deviation); solve for #ofSTDEVs.<\/li>\n<li><em>#ofSTDEVs <\/em>= [latex]\\frac{\\mathrm{value} - \\mathrm{mean}}{\\mathrm{standard \\ deviation}}[\/latex]<\/li>\n<li>Compare the results of this calculation.<\/li>\n<\/ul>\n<p>#ofSTDEVs is often called a &#8220;[latex]z[\/latex]-score&#8221;; we can use the symbol [latex]z[\/latex]. In symbols, the formulas become:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Sample<\/td>\n<td>[latex]x=\\overline{x}+zs[\/latex]<\/td>\n<td>[latex]z = \\frac{x - \\overline{x}}{s}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Population<\/td>\n<td>[latex]x = \u03bc + z\u03c3[\/latex]<\/td>\n<td>[latex]z = \\frac{x - \u03bc}{\u03c3}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Two students, John and Eric, from different high schools, wanted to find out who had the highest GPA when compared to his school. Which student had the highest GPA when compared to his school?<\/p>\n<table>\n<thead>\n<tr>\n<th>Student<\/th>\n<th>GPA<\/th>\n<th>School Mean GPA<\/th>\n<th>School Standard Deviation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>John<\/td>\n<td>[latex]2.85[\/latex]<\/td>\n<td>[latex]3.0[\/latex]<\/td>\n<td>[latex]0.7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Eric<\/td>\n<td>[latex]77[\/latex]<\/td>\n<td>[latex]80[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q124079\">Show Solution<\/span><\/p>\n<div id=\"q124079\" class=\"hidden-answer\" style=\"display: none\">\n<p>For each student, determine how many standard deviations (#ofSTDEVs) his GPA is away from the average, for his school. Pay careful attention to signs when comparing and interpreting the answer.<\/p>\n<p>[latex]z[\/latex] = # of STDEVs = [latex]\\frac{\\mathrm{value} - \\mathrm{mean}}{\\mathrm{standard \\ deviation}}[\/latex] =\u00a0 [latex]\\frac{x-\u03bc}{\u03c3}[\/latex]<\/p>\n<p>For John, [latex]z[\/latex] = # ofSTDEVs = [latex]\\displaystyle\\frac{{2.85 - 3.00}}{{0.7}}=-{0.21}[\/latex]<\/p>\n<p>For Eric, [latex]z[\/latex] = # ofSTDEVs = [latex]\\displaystyle\\frac{{77- 80}}{{10}}=\u22120.3[\/latex]<\/p>\n<p>John has the better GPA when compared to his school because his GPA is [latex]0.21[\/latex] standard deviations\u00a0<strong data-redactor-tag=\"strong\">below<\/strong> his school&#8217;s mean while Eric GPA is [latex]0.3[\/latex] standard deviations <strong data-redactor-tag=\"strong\">below<\/strong> his school&#8217;s mean.<\/p>\n<p>John&#8217;s [latex]z[\/latex]-score of [latex]\u20130.21[\/latex] is higher than Eric [latex]z[\/latex]-score of [latex]\u20130.3[\/latex]. For GPA, higher values are better, so we conclude that Eric has the better GPA when compared to his school.\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Two swimmers, Angie and Beth, from different teams, wanted to find out who had the fastest time for the 50 meter freestyle when compared to her team. Which swimmer had the fastest time when compared to her team?<\/p>\n<table>\n<thead>\n<tr>\n<th>Swimmer<\/th>\n<th>Time (seconds)<\/th>\n<th>Team Mean Time<\/th>\n<th>Team Standard Deviation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Angie<\/td>\n<td>[latex]26.2[\/latex]<\/td>\n<td>[latex]27.2[\/latex]<\/td>\n<td>[latex]0.8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Beth<\/td>\n<td>[latex]27.3[\/latex]<\/td>\n<td>[latex]30.1[\/latex]<\/td>\n<td>[latex]1.4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q124080\">Show Solution<\/span><\/p>\n<div id=\"q124080\" class=\"hidden-answer\" style=\"display: none\">\nFor Angie: [latex]z=\\frac{\\left(26.2-27.2\\right)}{0.8}=1.25[\/latex]<\/p>\n<p>For Beth: [latex]z=\\frac{\\left(27.3-30.1\\right)}{1.4}=-2[\/latex]\n<\/p><\/div>\n<\/div>\n<\/div>\n<p>The following lists give a few facts that provide a little more insight into what the standard deviation tells us about the distribution of the data.<\/p>\n<p><strong data-redactor-tag=\"strong\">For ANY data set, no matter what the distribution of the data is:<\/strong><\/p>\n<ul>\n<li>At least [latex]75[\/latex]% of the data is within two standard deviations of the mean.<\/li>\n<li>At least [latex]89[\/latex]% of the data is within three standard deviations of the mean.<\/li>\n<li>At least [latex]95[\/latex]% of the data is within [latex]4.5[\/latex] standard deviations of the mean.<\/li>\n<li>This is known as Chebyshev&#8217;s Rule.<\/li>\n<\/ul>\n<p><strong data-redactor-tag=\"strong\">For data having a distribution that is BELL-SHAPED and SYMMETRIC:<\/strong><\/p>\n<ul>\n<li>Approximately [latex]68[\/latex]% of the data is within one standard deviation of the mean.<\/li>\n<li>Approximately [latex]95[\/latex]% of the data is within two standard deviations of the mean.<\/li>\n<li>More than [latex]99[\/latex]% of the data is within three standard deviations of the mean.<\/li>\n<li>This is known as the Empirical Rule.<\/li>\n<li>It is important to note that this rule only applies when the shape of the distribution of the data is bell-shaped and symmetric. We will learn more about this when studying the &#8220;Normal&#8221; or &#8220;Gaussian&#8221; probability distribution in later chapters.<\/li>\n<\/ul>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-673\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>OpenStax Statistics. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/statistics\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/statistics\/pages\/1-introduction<\/li><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":35,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"OpenStax Statistics\",\"author\":\"\",\"organization\":\"\",\"url\":\"https:\/\/openstax.org\/books\/statistics\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/statistics\/pages\/1-introduction\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"Open Stax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at 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