{"id":1983,"date":"2021-09-17T14:25:26","date_gmt":"2021-09-17T14:25:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1983"},"modified":"2022-02-18T05:28:54","modified_gmt":"2022-02-18T05:28:54","slug":"descriptive-statistics-formula-review","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/descriptive-statistics-formula-review\/","title":{"raw":"Descriptive Statistics Formula Review","rendered":"Descriptive Statistics Formula Review"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Calculate the population mean, variance and standard deviation<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Calculate the sample mean, variance and standard deviation<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn Module 2 you learned formulas for numerical characteristics of populations and samples. The table below summarizes the formulas and notation for measures of center and spread.\r\n<div align=\"left\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Population\u00a0<\/strong><\/td>\r\n<td><strong>Sample<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Population size [latex]N[\/latex]<\/td>\r\n<td>Sample size [latex]n[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Population mean [latex]\\mu = \\frac{\\sum{x}}{N}[\/latex]<\/td>\r\n<td>Sample mean [latex]\\overline{x}=\\frac{\\sum{x}}{n}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Population variance [latex]\\sigma ^2. = \\frac{\\sum{(x- \\mu)^2}}{N}[\/latex]<\/td>\r\n<td>Sample variance [latex]s^2=\\frac{\\sum{(x-\\overline{x})^2}}{n-1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Population standard deviation [latex]\\sigma = \\sqrt{\\frac{\\sum{(x-\\mu)^2}}{N}}[\/latex]<\/td>\r\n<td>Sample standard deviation [latex]s=\\sqrt{\\frac{\\sum{(x-\\overline{x})^2}}{n-1}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe mean measures the \"center\" of a distribution. If you had to guess the value of a variable using a single number, your best guess for the value which comes closest to a randomly selected value might be the mean if the distribution is symmetric.\r\n\r\nThe variance and standard deviation measure the \"spread\" of values about the mean. The larger the variance and standard deviation are, the more spread out the values will be. Note that the standard deviation is always just the square root of the variance.\r\n\r\nThe dot plots below give calorie contents for children\u2019s and adult\u2019s cereal. We can see that the variance and standard deviation are greater for adult cereals since the observations are more spread out.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/images\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation2of4_image27.png\" alt=\"Dotplot showing calorie content of adult and children's cereals\" width=\"383\" height=\"250\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<strong>Calculating a Population Mean, Variance, and Standard Deviation<\/strong>\r\n\r\nThanushka took five exams in his statistics class last semester. His scores are shown below.\r\n<p style=\"text-align: center;\">[latex]89, 96, 97, 95, 98[\/latex]<\/p>\r\nCalculate the mean, variance, and standard deviation.\r\n\r\n[reveal-answer q=\"867857\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"867857\"]\r\n\r\nThe data values represent all of Thanushka\u2019s exam scores, so these statistics refer to population parameters.\r\n\r\nThe mean is [latex]\\mu = \\frac{89+96+97+95+98}{5} = \\frac{475}{5} = 95[\/latex].\r\n\r\nThe variance is [latex]\\sigma ^2 = \\frac{(89-95)^2 + (96-95)^2 + (97-95)^2 + (95-95)^2 + (98-95)^2}{5} = \\frac{50}{5} = 10[\/latex].\r\n\r\nThe standard deviation is [latex]\\sigma = \\sqrt{10} \\approx 3.1623[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<strong>Calculating a Sample Mean, Variance, and Standard Deviation<\/strong>\r\n\r\nA statistics professor took a random sample of five final exams from her statistics class last semester. The scores are shown below.\r\n<p style=\"text-align: center;\">[latex]89, 96, 97, 95, 98[\/latex]<\/p>\r\nCalculate the mean, variance, and standard deviation.\r\n\r\n[reveal-answer q=\"660614\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"660614\"]\r\n\r\nThe data values represent a sample of exam scores, so these statistics refer to sample statistics.\r\n\r\nThe mean is [latex]\\overline{x} = \\frac{89+96+97+95+98}{5} = \\frac{475}{5} = 95[\/latex].\r\n\r\nThe variance is [latex]s^2 = \\frac{(89-95)^2 + (96-95)^2 + (97-95)^2 + (95-95)^2 + (98-95)^2}{5-1} = \\frac{50}{4} = 12.5[\/latex].\r\n\r\nThe standard deviation is [latex]s = \\sqrt{12.5} \\approx 3.5355[\/latex].\r\n\r\nWe usually use technology to calculate the mean, variance and standard deviation.\r\n\r\n<strong>To find these using a TI-83, 83+, or 84+ calculator:<\/strong>\r\n\r\nFirst, press the STAT key and select 1:Edit.\r\n\r\nEnter the data into L1.\r\n\r\nSelect STAT, CALC, and 1: 1-Var Stats.\r\n\r\nPress ENTER.\r\n\r\nYou will see displayed both a population standard deviation, [latex]\\sigma _x[\/latex], and the sample standard deviation, [latex]s_x[\/latex]. The population mean, [latex]\\mu[\/latex], will be the same as a sample mean, [latex]\\overline{x}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]7025[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]7063[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Calculate the population mean, variance and standard deviation<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Calculate the sample mean, variance and standard deviation<\/li>\n<\/ul>\n<\/div>\n<p>In Module 2 you learned formulas for numerical characteristics of populations and samples. The table below summarizes the formulas and notation for measures of center and spread.<\/p>\n<div style=\"text-align: left;\">\n<table>\n<tbody>\n<tr>\n<td><strong>Population\u00a0<\/strong><\/td>\n<td><strong>Sample<\/strong><\/td>\n<\/tr>\n<tr>\n<td>Population size [latex]N[\/latex]<\/td>\n<td>Sample size [latex]n[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Population mean [latex]\\mu = \\frac{\\sum{x}}{N}[\/latex]<\/td>\n<td>Sample mean [latex]\\overline{x}=\\frac{\\sum{x}}{n}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Population variance [latex]\\sigma ^2. = \\frac{\\sum{(x- \\mu)^2}}{N}[\/latex]<\/td>\n<td>Sample variance [latex]s^2=\\frac{\\sum{(x-\\overline{x})^2}}{n-1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Population standard deviation [latex]\\sigma = \\sqrt{\\frac{\\sum{(x-\\mu)^2}}{N}}[\/latex]<\/td>\n<td>Sample standard deviation [latex]s=\\sqrt{\\frac{\\sum{(x-\\overline{x})^2}}{n-1}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The mean measures the &#8220;center&#8221; of a distribution. If you had to guess the value of a variable using a single number, your best guess for the value which comes closest to a randomly selected value might be the mean if the distribution is symmetric.<\/p>\n<p>The variance and standard deviation measure the &#8220;spread&#8221; of values about the mean. The larger the variance and standard deviation are, the more spread out the values will be. Note that the standard deviation is always just the square root of the variance.<\/p>\n<p>The dot plots below give calorie contents for children\u2019s and adult\u2019s cereal. We can see that the variance and standard deviation are greater for adult cereals since the observations are more spread out.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/images\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation2of4_image27.png\" alt=\"Dotplot showing calorie content of adult and children's cereals\" width=\"383\" height=\"250\" \/><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p><strong>Calculating a Population Mean, Variance, and Standard Deviation<\/strong><\/p>\n<p>Thanushka took five exams in his statistics class last semester. His scores are shown below.<\/p>\n<p style=\"text-align: center;\">[latex]89, 96, 97, 95, 98[\/latex]<\/p>\n<p>Calculate the mean, variance, and standard deviation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q867857\">Show Answer<\/span><\/p>\n<div id=\"q867857\" class=\"hidden-answer\" style=\"display: none\">\n<p>The data values represent all of Thanushka\u2019s exam scores, so these statistics refer to population parameters.<\/p>\n<p>The mean is [latex]\\mu = \\frac{89+96+97+95+98}{5} = \\frac{475}{5} = 95[\/latex].<\/p>\n<p>The variance is [latex]\\sigma ^2 = \\frac{(89-95)^2 + (96-95)^2 + (97-95)^2 + (95-95)^2 + (98-95)^2}{5} = \\frac{50}{5} = 10[\/latex].<\/p>\n<p>The standard deviation is [latex]\\sigma = \\sqrt{10} \\approx 3.1623[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p><strong>Calculating a Sample Mean, Variance, and Standard Deviation<\/strong><\/p>\n<p>A statistics professor took a random sample of five final exams from her statistics class last semester. The scores are shown below.<\/p>\n<p style=\"text-align: center;\">[latex]89, 96, 97, 95, 98[\/latex]<\/p>\n<p>Calculate the mean, variance, and standard deviation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q660614\">Show Answer<\/span><\/p>\n<div id=\"q660614\" class=\"hidden-answer\" style=\"display: none\">\n<p>The data values represent a sample of exam scores, so these statistics refer to sample statistics.<\/p>\n<p>The mean is [latex]\\overline{x} = \\frac{89+96+97+95+98}{5} = \\frac{475}{5} = 95[\/latex].<\/p>\n<p>The variance is [latex]s^2 = \\frac{(89-95)^2 + (96-95)^2 + (97-95)^2 + (95-95)^2 + (98-95)^2}{5-1} = \\frac{50}{4} = 12.5[\/latex].<\/p>\n<p>The standard deviation is [latex]s = \\sqrt{12.5} \\approx 3.5355[\/latex].<\/p>\n<p>We usually use technology to calculate the mean, variance and standard deviation.<\/p>\n<p><strong>To find these using a TI-83, 83+, or 84+ calculator:<\/strong><\/p>\n<p>First, press the STAT key and select 1:Edit.<\/p>\n<p>Enter the data into L1.<\/p>\n<p>Select STAT, CALC, and 1: 1-Var Stats.<\/p>\n<p>Press ENTER.<\/p>\n<p>You will see displayed both a population standard deviation, [latex]\\sigma _x[\/latex], and the sample standard deviation, [latex]s_x[\/latex]. The population mean, [latex]\\mu[\/latex], will be the same as a sample mean, [latex]\\overline{x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm7025\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=7025&theme=oea&iframe_resize_id=ohm7025&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm7063\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=7063&theme=oea&iframe_resize_id=ohm7063&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-1983\">\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><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>Concepts in Statistics. <strong>Provided by<\/strong>: Open Learning Initiative. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/oli.cmu.edu\">http:\/\/oli.cmu.edu<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>QID 7025, 7063. <strong>Authored by<\/strong>: Lippman, D. <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>: IMathAS Community License CC-BY + GPL<\/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":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Concepts in Statistics\",\"author\":\"\",\"organization\":\"Open Learning Initiative\",\"url\":\"http:\/\/oli.cmu.edu\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"QID 7025, 7063\",\"author\":\"Lippman, D\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1983","chapter","type-chapter","status-publish","hentry"],"part":269,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1983","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1983\/revisions"}],"predecessor-version":[{"id":3751,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1983\/revisions\/3751"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/269"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1983\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1983"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1983"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1983"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1983"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}