{"id":879,"date":"2023-03-20T19:17:54","date_gmt":"2023-03-20T19:17:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-variability-of-data-sets-learn-it-2\/"},"modified":"2025-05-11T22:42:25","modified_gmt":"2025-05-11T22:42:25","slug":"comparing-variability-of-data-sets-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-variability-of-data-sets-learn-it-2\/","title":{"raw":"Measures of Variability: Learn It 2","rendered":"Measures of Variability: Learn It 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Describe the differences in variability in histograms and dotplots.<\/li>\r\n\t<li>Calculate and describe standard deviation.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>As stated previously, range only utilizes two observations in the entire data set to measure variability. It is not an ideal measure of spread when used alone.<\/p>\r\n<p>In constructing a measure of spread about the center (i.e., the average or mean), we want to compute how far a \u201ctypical\u201d number is away from the mean.<\/p>\r\n<h2>Calculating Deviation from the Mean<\/h2>\r\n<p>Let\u2019s consider the sample data set [latex]2, 2, 4, 5, 6, 7, 9[\/latex].<br \/>\r\nThe mean of this data set is [latex]\\stackrel{\u00af}{x}=\\frac{2&amp;plus;2&amp;plus;4&amp;plus;5&amp;plus;6&amp;plus;7&amp;plus;9\\text{}}{7}\\text{}=\\text{}\\frac{35}{7}=5[\/latex].<\/p>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"722\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031639\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image2.png\" alt=\"Dotplot of data set with the mean marked by vertical blue line\" width=\"722\" height=\"146\" \/> Figure 1. A dotplot with a mean of 5.[\/caption]\r\n\r\n<p>Here is a dotplot of this data set with the mean marked by the vertical blue line.<\/p>\r\n<p>We can see that some data is close to the mean, and some data is further from the mean.<\/p>\r\n<p>Since we want to see how the data points deviate from the mean, we determine how far each point is from the mean. We compute the difference between each of these values and the mean. These differences are called the <strong>deviations<\/strong>.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>deviation<\/h3>\r\n<p>In statistics, <strong>deviation<\/strong> is a measure of difference between the observed value [latex]x[\/latex] of a variable and the mean [latex]\\bar{x}[\/latex].<\/p>\r\n<p style=\"text-align: center;\">Deviation = [latex]x-\\bar{x}[\/latex]<\/p>\r\n<\/section>\r\n<table class=\"alignleft\" style=\"border-collapse: collapse; width: 340px; height: 113px;\" border=\"1\">\r\n<tbody>\r\n<tr class=\"oli_table\" style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 112.943px; text-align: center;\" align=\"center\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 145.078px; height: 15px; text-align: center;\"><strong>Deviation <\/strong><strong>[latex]=(x-\\bar{x})[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 112.943px; height: 14px; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 145.078px; height: 14px; text-align: center;\">[latex]2 \u2212 5 = \u22123[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 112.943px; height: 14px; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 145.078px; height: 14px; text-align: center;\">[latex]2 \u2212 5 = \u22123[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 112.943px; height: 14px; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 145.078px; height: 14px; text-align: center;\">[latex]4 \u2212 5 = \u22121[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 112.943px; height: 14px; text-align: center;\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 145.078px; height: 14px; text-align: center;\">[latex]5 \u2212 5 = 0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 112.943px; height: 14px; text-align: center;\">[latex]6[\/latex]<\/td>\r\n<td style=\"width: 145.078px; height: 14px; text-align: center;\">[latex]6 \u2212 5 = 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 112.943px; height: 14px; text-align: center;\">[latex]7[\/latex]<\/td>\r\n<td style=\"width: 145.078px; height: 14px; text-align: center;\">[latex]7 \u2212 5 = 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 112.943px; height: 14px; text-align: center;\">[latex]9[\/latex]<\/td>\r\n<td style=\"width: 145.078px; height: 14px; text-align: center;\">[latex]\u00a09 \u2212 5 = 4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>When visualized on a dotplot, these differences are viewed as distances between each point and the mean. A negative difference indicates that the data point is to the <em>left<\/em> of the mean (shown in blue on the graph below). A positive difference indicates that the data point is to the <em>right<\/em> of the mean (shown in green on the graph below).<\/p>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"537\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031641\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image3.png\" alt=\"Dotplot where negative differences are shown as data points to the left of the mean; positive differences are shown as data points to the right\" width=\"537\" height=\"161\" \/> Figure 2. Dot plot showing distances from the mean. Negative differences (left of the mean) are shown in blue, and positive differences (right of the mean) are shown in green.[\/caption]\r\n\r\n<p>Our goal is to develop a single measurement that summarizes a typical distance from the mean.<\/p>\r\n<p>Now, let\u2019s practice determining the distance of a single data point from the mean, a.k.a., the deviation from the mean: <strong>[latex](x-\\bar{x})[\/latex]<\/strong>.<\/p>\r\n\r\n[caption id=\"attachment_1149\" align=\"aligncenter\" width=\"650\"]<img class=\"wp-image-1149\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/03\/20191753\/pexels-kelly-4170461.jpg\" alt=\"Hurricane aftermath from an aerial camera showing a damaged neighborhood with destroyed houses.\" width=\"650\" height=\"365\" \/> Figure 3. By analyzing the cost of hurricanes like this one, we can measure how far each event deviates from the average, revealing just how severe some storms really are.[\/caption]\r\n\r\n<p>Hurricanes cause extensive amounts of damage. Let's consider the amount of damage in dollars of the [latex]30[\/latex] most expensive hurricanes to have hit the U.S. mainland between 1990 and 2010.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]259[\/ohm2_question]<\/section>\r\n<h3>Representations of large numbers<\/h3>\r\n<p><span id=\"LargeNumbers\">Take a moment to consider the units within our data set. In the table presented in the question above, we see hurricane damage in millions of dollars. Look at the last number in the table: [latex]11,227[\/latex]. Presumably, that means [latex]11,227[\/latex] million dollars.<\/span><\/p>\r\n<p style=\"text-align: center;\">[latex]$11,227[\/latex] millions [latex]=$11,227,000,000=$11.227[\/latex] billions<\/p>\r\n<p><span id=\"LargeNumbers\">The hurricanes contributing to this data were catastrophic, causing billions of dollars of damage.\u00a0<\/span><\/p>\r\n<h3>The sign (+ or -) of deviation from the mean<\/h3>\r\n<p>Also, consider the meanings behind the sign (+ or -) of the deviation values you found in the question above. The + and - are important when making inferences regarding the data values.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]261[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]262[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Describe the differences in variability in histograms and dotplots.<\/li>\n<li>Calculate and describe standard deviation.<\/li>\n<\/ul>\n<\/section>\n<p>As stated previously, range only utilizes two observations in the entire data set to measure variability. It is not an ideal measure of spread when used alone.<\/p>\n<p>In constructing a measure of spread about the center (i.e., the average or mean), we want to compute how far a \u201ctypical\u201d number is away from the mean.<\/p>\n<h2>Calculating Deviation from the Mean<\/h2>\n<p>Let\u2019s consider the sample data set [latex]2, 2, 4, 5, 6, 7, 9[\/latex].<br \/>\nThe mean of this data set is [latex]\\stackrel{\u00af}{x}=\\frac{2&plus;2&plus;4&plus;5&plus;6&plus;7&plus;9\\text{}}{7}\\text{}=\\text{}\\frac{35}{7}=5[\/latex].<\/p>\n<figure style=\"width: 722px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031639\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image2.png\" alt=\"Dotplot of data set with the mean marked by vertical blue line\" width=\"722\" height=\"146\" \/><figcaption class=\"wp-caption-text\">Figure 1. A dotplot with a mean of 5.<\/figcaption><\/figure>\n<p>Here is a dotplot of this data set with the mean marked by the vertical blue line.<\/p>\n<p>We can see that some data is close to the mean, and some data is further from the mean.<\/p>\n<p>Since we want to see how the data points deviate from the mean, we determine how far each point is from the mean. We compute the difference between each of these values and the mean. These differences are called the <strong>deviations<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>deviation<\/h3>\n<p>In statistics, <strong>deviation<\/strong> is a measure of difference between the observed value [latex]x[\/latex] of a variable and the mean [latex]\\bar{x}[\/latex].<\/p>\n<p style=\"text-align: center;\">Deviation = [latex]x-\\bar{x}[\/latex]<\/p>\n<\/section>\n<table class=\"alignleft\" style=\"border-collapse: collapse; width: 340px; height: 113px;\">\n<tbody>\n<tr class=\"oli_table\" style=\"height: 15px;\">\n<td style=\"height: 15px; width: 112.943px; text-align: center;\" align=\"center\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td style=\"width: 145.078px; height: 15px; text-align: center;\"><strong>Deviation <\/strong><strong>[latex]=(x-\\bar{x})[\/latex]<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 112.943px; height: 14px; text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 145.078px; height: 14px; text-align: center;\">[latex]2 \u2212 5 = \u22123[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 112.943px; height: 14px; text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 145.078px; height: 14px; text-align: center;\">[latex]2 \u2212 5 = \u22123[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 112.943px; height: 14px; text-align: center;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 145.078px; height: 14px; text-align: center;\">[latex]4 \u2212 5 = \u22121[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 112.943px; height: 14px; text-align: center;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 145.078px; height: 14px; text-align: center;\">[latex]5 \u2212 5 = 0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 112.943px; height: 14px; text-align: center;\">[latex]6[\/latex]<\/td>\n<td style=\"width: 145.078px; height: 14px; text-align: center;\">[latex]6 \u2212 5 = 1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 112.943px; height: 14px; text-align: center;\">[latex]7[\/latex]<\/td>\n<td style=\"width: 145.078px; height: 14px; text-align: center;\">[latex]7 \u2212 5 = 2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 112.943px; height: 14px; text-align: center;\">[latex]9[\/latex]<\/td>\n<td style=\"width: 145.078px; height: 14px; text-align: center;\">[latex]\u00a09 \u2212 5 = 4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When visualized on a dotplot, these differences are viewed as distances between each point and the mean. A negative difference indicates that the data point is to the <em>left<\/em> of the mean (shown in blue on the graph below). A positive difference indicates that the data point is to the <em>right<\/em> of the mean (shown in green on the graph below).<\/p>\n<figure style=\"width: 537px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031641\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image3.png\" alt=\"Dotplot where negative differences are shown as data points to the left of the mean; positive differences are shown as data points to the right\" width=\"537\" height=\"161\" \/><figcaption class=\"wp-caption-text\">Figure 2. Dot plot showing distances from the mean. Negative differences (left of the mean) are shown in blue, and positive differences (right of the mean) are shown in green.<\/figcaption><\/figure>\n<p>Our goal is to develop a single measurement that summarizes a typical distance from the mean.<\/p>\n<p>Now, let\u2019s practice determining the distance of a single data point from the mean, a.k.a., the deviation from the mean: <strong>[latex](x-\\bar{x})[\/latex]<\/strong>.<\/p>\n<figure id=\"attachment_1149\" aria-describedby=\"caption-attachment-1149\" style=\"width: 650px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1149\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/03\/20191753\/pexels-kelly-4170461.jpg\" alt=\"Hurricane aftermath from an aerial camera showing a damaged neighborhood with destroyed houses.\" width=\"650\" height=\"365\" \/><figcaption id=\"caption-attachment-1149\" class=\"wp-caption-text\">Figure 3. By analyzing the cost of hurricanes like this one, we can measure how far each event deviates from the average, revealing just how severe some storms really are.<\/figcaption><\/figure>\n<p>Hurricanes cause extensive amounts of damage. Let&#8217;s consider the amount of damage in dollars of the [latex]30[\/latex] most expensive hurricanes to have hit the U.S. mainland between 1990 and 2010.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm259\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=259&theme=lumen&iframe_resize_id=ohm259&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Representations of large numbers<\/h3>\n<p><span id=\"LargeNumbers\">Take a moment to consider the units within our data set. In the table presented in the question above, we see hurricane damage in millions of dollars. Look at the last number in the table: [latex]11,227[\/latex]. Presumably, that means [latex]11,227[\/latex] million dollars.<\/span><\/p>\n<p style=\"text-align: center;\">[latex]$11,227[\/latex] millions [latex]=$11,227,000,000=$11.227[\/latex] billions<\/p>\n<p><span>The hurricanes contributing to this data were catastrophic, causing billions of dollars of damage.\u00a0<\/span><\/p>\n<h3>The sign (+ or -) of deviation from the mean<\/h3>\n<p>Also, consider the meanings behind the sign (+ or -) of the deviation values you found in the question above. The + and &#8211; are important when making inferences regarding the data values.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm261\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=261&theme=lumen&iframe_resize_id=ohm261&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm262\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=262&theme=lumen&iframe_resize_id=ohm262&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":13,"menu_order":30,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":834,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/879"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":14,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/879\/revisions"}],"predecessor-version":[{"id":6634,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/879\/revisions\/6634"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/834"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/879\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=879"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=879"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=879"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=879"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}