{"id":877,"date":"2023-03-20T19:17:51","date_gmt":"2023-03-20T19:17:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-variability-of-data-sets-learn-it-1\/"},"modified":"2025-05-08T03:09:31","modified_gmt":"2025-05-08T03:09:31","slug":"comparing-variability-of-data-sets-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-variability-of-data-sets-learn-it-1\/","title":{"raw":"Measures of Variability: Learn It 1","rendered":"Measures of Variability: Learn It 1"},"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>We have learned that the shape and measures of center are important characteristics to describe a data set.<\/p>\r\n<p>Another important characteristic of any data set is the variation within the data. In some data sets, the data values are concentrated closely near the center; in other data sets, the data values are more widely spread out from the center. So, another way to describe data numerically is to find and use the measures of <strong>spread<\/strong>.<\/p>\r\n\r\n[caption id=\"attachment_5623\" align=\"aligncenter\" width=\"475\"]<img class=\"wp-image-5623\" style=\"font-size: 14pt;\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/30003013\/4.3.L.Diagram-3.png\" alt=\"A flow chart beginning with Graph the distribution of a quantitative variable. Describe the following: with one arrow pointing to Overall pattern and another arrow pointing to Deviations from the pattern. The overall pattern box points to shape, center, and spread, with the latter being highlighted. The deviations from the pattern box points to outliers.\" width=\"475\" height=\"406\" \/> Figure 1. When analyzing a graph, describe the overall pattern (shape, center, spread) and look for deviations, or outliers, that don\u2019t follow the pattern.[\/caption]\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>variability<\/h3>\r\n<p><strong>Variability<\/strong> in statistics refers to a measure of how spread out, or dispersed, the data set is.<\/p>\r\n<p>Standard deviation, variance, and range are all calculated measures of variability.<\/p>\r\n<\/section>\r\n<h2>Comparing Variability<\/h2>\r\n<p>We can visually assess variability using graphical displays such as histograms and dotplots. When looking at a histogram or a dotplot, consider whether the data appears to be more spread out from the center (greater variability), or more clustered toward the center (less variability). These visual clues help us\u00a0recognize distributions that have more or less variability than others.<\/p>\r\n<section class=\"textbox tryIt\">Histograms displaying the distribution of two quantitative variables with different amounts of variability are shown below. Which do you think has less variability than the other? Explain your reasoning. For example: What visual clue could help you decide if your answers were correct?\r\n[caption id=\"attachment_1039\" align=\"aligncenter\" width=\"1196\"]<img class=\"wp-image-1039 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5772\/2022\/02\/12025120\/LessMore_Question.jpg\" alt=\"Two histograms are shown. The one on the left appears tightly clustered. The one on the right appears more widely dispersed.\" width=\"1196\" height=\"250\" \/> Figure 2. These two histograms show distributions with different amounts of variability, illustrating how data sets can differ in how spread out their values are.[\/caption]\r\n[reveal-answer q=\"510090\"]Show solution[\/reveal-answer][hidden-answer a=\"510090\"]\u00a0 The data in the distribution on the left varies less than that of the one on the right. Note how the data in the first histogram is tightly clustered around the center and tails off quickly to either side. The data in the second histogram is widely dispersed across the graph.\r\n[caption id=\"attachment_1037\" align=\"aligncenter\" width=\"1156\"]<img class=\"wp-image-1037 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5772\/2022\/02\/12024114\/LessMoreVariabiliy.jpg\" alt=\"Two histograms side by side. The one on the left has data tightly clustered toward the center while the one on the right shows data spread out widely and evenly across the graph.\" width=\"1156\" height=\"290\" \/> Figure 3. The left histogram shows data with less variability\u2014most values are close to the center. The right histogram shows more variability, with data spread out more widely across different values.[\/caption]\r\nThe visual indication of variability is how tightly clustered or widely dispersed the data appears in the display. [\/hidden-answer]<\/section>\r\n<p>It can be easier to visualize variability using a dotplot instead of a histogram because the individual data points (or observations) are visible in the dotplot.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1 hide_question_numbers=1]2121[\/ohm2_question]<\/section>\r\n<h2>Range<\/h2>\r\n<p><strong>Range<\/strong> is a value that can describe the spread of the data set. When the range is larger, it indicates more variability in the data. However, range only utilizes two observations in the entire data set to measure variability, so it is not an ideal measure of spread when used alone.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>range<\/h3>\r\n<p>Range = maximum value \u2013 minimum value\u00a0 =\u00a0 largest value \u2013 smallest value<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">The following dotplots show the potassium content in 76 cereals. Compare children\u2019s cereals to adult cereals.\r\n[caption id=\"\" align=\"alignnone\" width=\"622\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031645\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image10.png\" alt=\"Dotplots showing potassium content of 76 children\u2019s and adult cereals. \" width=\"622\" height=\"216\" \/> Figure 4. A dotplot of the potassium content in 76 cereals, sorted by adult and children's cereals.[\/caption]\r\n<br \/>\r\n[reveal-answer q=\"180951\"]Which type of cereal has more variability in potassium content?[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"180951\"]<br \/>\r\nWe see that there is more variability in the potassium content of the adult cereals than in the children\u2019s cereals.<br \/>\r\nWe can also measure this spread using range.\r\n\r\n<p style=\"padding-left: 40px;\">The range of potassium content is larger for the adult cereals than for the children\u2019s cereals. The children\u2019s cereal set has a range of [latex]90[\/latex] (because [latex]110 \u2212 20 = 90[\/latex]), whereas the adult cereal set has a range of [latex]315[\/latex] (because [latex]330 \u2212 15 = 315[\/latex]).<\/p>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"671\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031647\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image11.png\" alt=\"Overall range of potassium content for both adult and children's of cereals\" width=\"671\" height=\"357\" \/> Figure 5. Dot plots comparing potassium content in children\u2019s and adult cereals, with adult cereals showing a wider range of values than children\u2019s cereals.[\/caption]\r\n\r\n<p>[\/hidden-answer]<\/p>\r\n<\/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>We have learned that the shape and measures of center are important characteristics to describe a data set.<\/p>\n<p>Another important characteristic of any data set is the variation within the data. In some data sets, the data values are concentrated closely near the center; in other data sets, the data values are more widely spread out from the center. So, another way to describe data numerically is to find and use the measures of <strong>spread<\/strong>.<\/p>\n<figure id=\"attachment_5623\" aria-describedby=\"caption-attachment-5623\" style=\"width: 475px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5623\" style=\"font-size: 14pt;\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/30003013\/4.3.L.Diagram-3.png\" alt=\"A flow chart beginning with Graph the distribution of a quantitative variable. Describe the following: with one arrow pointing to Overall pattern and another arrow pointing to Deviations from the pattern. The overall pattern box points to shape, center, and spread, with the latter being highlighted. The deviations from the pattern box points to outliers.\" width=\"475\" height=\"406\" \/><figcaption id=\"caption-attachment-5623\" class=\"wp-caption-text\">Figure 1. When analyzing a graph, describe the overall pattern (shape, center, spread) and look for deviations, or outliers, that don\u2019t follow the pattern.<\/figcaption><\/figure>\n<section class=\"textbox keyTakeaway\">\n<h3>variability<\/h3>\n<p><strong>Variability<\/strong> in statistics refers to a measure of how spread out, or dispersed, the data set is.<\/p>\n<p>Standard deviation, variance, and range are all calculated measures of variability.<\/p>\n<\/section>\n<h2>Comparing Variability<\/h2>\n<p>We can visually assess variability using graphical displays such as histograms and dotplots. When looking at a histogram or a dotplot, consider whether the data appears to be more spread out from the center (greater variability), or more clustered toward the center (less variability). These visual clues help us\u00a0recognize distributions that have more or less variability than others.<\/p>\n<section class=\"textbox tryIt\">Histograms displaying the distribution of two quantitative variables with different amounts of variability are shown below. Which do you think has less variability than the other? Explain your reasoning. For example: What visual clue could help you decide if your answers were correct?<\/p>\n<figure id=\"attachment_1039\" aria-describedby=\"caption-attachment-1039\" style=\"width: 1196px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1039 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5772\/2022\/02\/12025120\/LessMore_Question.jpg\" alt=\"Two histograms are shown. The one on the left appears tightly clustered. The one on the right appears more widely dispersed.\" width=\"1196\" height=\"250\" \/><figcaption id=\"caption-attachment-1039\" class=\"wp-caption-text\">Figure 2. These two histograms show distributions with different amounts of variability, illustrating how data sets can differ in how spread out their values are.<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q510090\">Show solution<\/button><\/p>\n<div id=\"q510090\" class=\"hidden-answer\" style=\"display: none\">\u00a0 The data in the distribution on the left varies less than that of the one on the right. Note how the data in the first histogram is tightly clustered around the center and tails off quickly to either side. The data in the second histogram is widely dispersed across the graph.<\/p>\n<figure id=\"attachment_1037\" aria-describedby=\"caption-attachment-1037\" style=\"width: 1156px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1037 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5772\/2022\/02\/12024114\/LessMoreVariabiliy.jpg\" alt=\"Two histograms side by side. The one on the left has data tightly clustered toward the center while the one on the right shows data spread out widely and evenly across the graph.\" width=\"1156\" height=\"290\" \/><figcaption id=\"caption-attachment-1037\" class=\"wp-caption-text\">Figure 3. The left histogram shows data with less variability\u2014most values are close to the center. The right histogram shows more variability, with data spread out more widely across different values.<\/figcaption><\/figure>\n<p>The visual indication of variability is how tightly clustered or widely dispersed the data appears in the display. <\/p><\/div>\n<\/div>\n<\/section>\n<p>It can be easier to visualize variability using a dotplot instead of a histogram because the individual data points (or observations) are visible in the dotplot.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2121\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2121&theme=lumen&iframe_resize_id=ohm2121&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Range<\/h2>\n<p><strong>Range<\/strong> is a value that can describe the spread of the data set. When the range is larger, it indicates more variability in the data. However, range only utilizes two observations in the entire data set to measure variability, so it is not an ideal measure of spread when used alone.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>range<\/h3>\n<p>Range = maximum value \u2013 minimum value\u00a0 =\u00a0 largest value \u2013 smallest value<\/p>\n<\/section>\n<section class=\"textbox example\">The following dotplots show the potassium content in 76 cereals. Compare children\u2019s cereals to adult cereals.<\/p>\n<figure style=\"width: 622px\" 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\/15031645\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image10.png\" alt=\"Dotplots showing potassium content of 76 children\u2019s and adult cereals.\" width=\"622\" height=\"216\" \/><figcaption class=\"wp-caption-text\">Figure 4. A dotplot of the potassium content in 76 cereals, sorted by adult and children&#8217;s cereals.<\/figcaption><\/figure>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q180951\">Which type of cereal has more variability in potassium content?<\/button><\/p>\n<div id=\"q180951\" class=\"hidden-answer\" style=\"display: none\">\nWe see that there is more variability in the potassium content of the adult cereals than in the children\u2019s cereals.<br \/>\nWe can also measure this spread using range.<\/p>\n<p style=\"padding-left: 40px;\">The range of potassium content is larger for the adult cereals than for the children\u2019s cereals. The children\u2019s cereal set has a range of [latex]90[\/latex] (because [latex]110 \u2212 20 = 90[\/latex]), whereas the adult cereal set has a range of [latex]315[\/latex] (because [latex]330 \u2212 15 = 315[\/latex]).<\/p>\n<figure style=\"width: 671px\" 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\/15031647\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image11.png\" alt=\"Overall range of potassium content for both adult and children's of cereals\" width=\"671\" height=\"357\" \/><figcaption class=\"wp-caption-text\">Figure 5. Dot plots comparing potassium content in children\u2019s and adult cereals, with adult cereals showing a wider range of values than children\u2019s cereals.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":13,"menu_order":29,"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\/877"}],"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":12,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/877\/revisions"}],"predecessor-version":[{"id":6473,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/877\/revisions\/6473"}],"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\/877\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=877"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=877"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=877"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=877"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}