{"id":41,"date":"2022-05-20T16:59:04","date_gmt":"2022-05-20T16:59:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/interpreting-the-mean-and-median-of-a-dataset-what-to-know\/"},"modified":"2022-08-08T15:59:06","modified_gmt":"2022-08-08T15:59:06","slug":"interpreting-the-mean-and-median-of-a-dataset-what-to-know","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/interpreting-the-mean-and-median-of-a-dataset-what-to-know\/","title":{"raw":"Interpreting the Mean and Median of a Data Set: Learn It 1","rendered":"Interpreting the Mean and Median of a Data Set: Learn It 1"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Goals<\/h3>\r\nAfter completing this section, you should feel comfortable performing these skills.\r\n<ul>\r\n \t<li><a href=\"#IntMeanMedian\">Interpret the median of a data set.<\/a><\/li>\r\n \t<li><a href=\"#IntMeanMedian\">Interpret the mean of a data set.<\/a><\/li>\r\n \t<li><a href=\"#IdentSkew\">Identify whether a data set is left-skewed, symmetric, or right-skewed.<\/a><\/li>\r\n \t<li><a href=\"#IdentSkew\">Identify in which data set the mean is greater than, less than, or approximately equal to the median.<\/a><\/li>\r\n \t<li><a href=\"#resistant\">Identify which of the mean or median is resistant to skew.<\/a><\/li>\r\n<\/ul>\r\nClick on a skill above to jump to its location in this section.\r\n\r\n<\/div>\r\nWhen examining the distribution of a quantitative variable using a histogram or a dotplot, we often find that the distribution follows a bell shape with a mound of observances in the middle of the distribution and even amounts of data falling to the right and left. But sometimes a distribution's values are bunched up to one side or the other, with a few observations stretching way out to the other side. You may recall from <em>What to Know About Applications of Histograms: 3D <\/em>that there are specialized statistical terms we use for these different distribution shapes: skewness and symmetry. In this section, you'll learn that there are certain ways the mean of the data relates to the median under these different shapes.\r\n\r\n<img class=\"aligncenter size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11184151\/Picture131.png\" alt=\"An image of three histograms: left skewed, in which the data is bunched up to the right with a long tail of data to the left; symmetric, in which the data is mounded in the center and falls away evenly to either side; and right-skewed, in which the data is bunched up to the left with a tail of data falling away to the right.\" width=\"1576\" height=\"608\" \/>\r\n<h2>Skewness<\/h2>\r\nRecall that we say a quantitative variable has a <strong>right-skewed<\/strong> distribution or a <strong>positive skew<\/strong> if there is a \"tail\" of infrequent values on the right (upper) end of the distribution. We say a data set has an approximately <strong>symmetric<\/strong> distribution if values are similarly distributed on either side of the mean\/median. We say a data set has a <strong>left-skewed<\/strong> distribution or a <strong>negative skew<\/strong> if there is a \"tail\" of infrequent values on the left (lower) end of the distribution.\r\nhttps:\/\/www.geogebra.org\/m\/fvdqmz2z\r\nRefresh your memory for how to describe the shape of a histogram by trying the question in the interactive example below.\r\n<div class=\"textbox exercises\">\r\n<h3>interactive example<\/h3>\r\nSeveral histograms are displayed below. Provide a description of the shape of each.\r\n\r\n<img class=\"aligncenter size-full wp-image-1024\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5772\/2022\/02\/11210038\/Shape_Hist.jpg\" alt=\"A group of four histograms. The first is mounded in the middle and tails off to both sides. The second is mounded to the left and tails of to the right. The third contains two mounds and tails off to the left and right. The fourth is mounded to the right and tails off to the left. \" width=\"490\" height=\"362\" \/>\r\n\r\n[reveal-answer q=\"861785\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"861785\"]\r\n<ol>\r\n \t<li>Unimodal, symmetric<\/li>\r\n \t<li>Right-skewed (a tail of infrequent values trails out to the right of the bulk of the data)<\/li>\r\n \t<li>Bimodal<\/li>\r\n \t<li>Left-skewed (a tail of infrequent values trails out to the left of the bulk of the data)<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next activity, you'll need to calculate and interpret the mean and median in skewed distributions. Let's get some practice with these skills using data collected around the T.V. show\u00a0<em>Friends<\/em>.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Goals<\/h3>\n<p>After completing this section, you should feel comfortable performing these skills.<\/p>\n<ul>\n<li><a href=\"#IntMeanMedian\">Interpret the median of a data set.<\/a><\/li>\n<li><a href=\"#IntMeanMedian\">Interpret the mean of a data set.<\/a><\/li>\n<li><a href=\"#IdentSkew\">Identify whether a data set is left-skewed, symmetric, or right-skewed.<\/a><\/li>\n<li><a href=\"#IdentSkew\">Identify in which data set the mean is greater than, less than, or approximately equal to the median.<\/a><\/li>\n<li><a href=\"#resistant\">Identify which of the mean or median is resistant to skew.<\/a><\/li>\n<\/ul>\n<p>Click on a skill above to jump to its location in this section.<\/p>\n<\/div>\n<p>When examining the distribution of a quantitative variable using a histogram or a dotplot, we often find that the distribution follows a bell shape with a mound of observances in the middle of the distribution and even amounts of data falling to the right and left. But sometimes a distribution&#8217;s values are bunched up to one side or the other, with a few observations stretching way out to the other side. You may recall from <em>What to Know About Applications of Histograms: 3D <\/em>that there are specialized statistical terms we use for these different distribution shapes: skewness and symmetry. In this section, you&#8217;ll learn that there are certain ways the mean of the data relates to the median under these different shapes.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11184151\/Picture131.png\" alt=\"An image of three histograms: left skewed, in which the data is bunched up to the right with a long tail of data to the left; symmetric, in which the data is mounded in the center and falls away evenly to either side; and right-skewed, in which the data is bunched up to the left with a tail of data falling away to the right.\" width=\"1576\" height=\"608\" \/><\/p>\n<h2>Skewness<\/h2>\n<p>Recall that we say a quantitative variable has a <strong>right-skewed<\/strong> distribution or a <strong>positive skew<\/strong> if there is a &#8220;tail&#8221; of infrequent values on the right (upper) end of the distribution. We say a data set has an approximately <strong>symmetric<\/strong> distribution if values are similarly distributed on either side of the mean\/median. We say a data set has a <strong>left-skewed<\/strong> distribution or a <strong>negative skew<\/strong> if there is a &#8220;tail&#8221; of infrequent values on the left (lower) end of the distribution.<br \/>\n<iframe loading=\"lazy\" class=\"resizable\" src=\"https:\/\/www.geogebra.org\/m\/fvdqmz2z\" frameborder=\"0\" width=\"500\" height=\"750\"><\/iframe><br \/>\nRefresh your memory for how to describe the shape of a histogram by trying the question in the interactive example below.<\/p>\n<div class=\"textbox exercises\">\n<h3>interactive example<\/h3>\n<p>Several histograms are displayed below. Provide a description of the shape of each.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1024\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5772\/2022\/02\/11210038\/Shape_Hist.jpg\" alt=\"A group of four histograms. The first is mounded in the middle and tails off to both sides. The second is mounded to the left and tails of to the right. The third contains two mounds and tails off to the left and right. The fourth is mounded to the right and tails off to the left.\" width=\"490\" height=\"362\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q861785\">Show Answer<\/span><\/p>\n<div id=\"q861785\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Unimodal, symmetric<\/li>\n<li>Right-skewed (a tail of infrequent values trails out to the right of the bulk of the data)<\/li>\n<li>Bimodal<\/li>\n<li>Left-skewed (a tail of infrequent values trails out to the left of the bulk of the data)<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next activity, you&#8217;ll need to calculate and interpret the mean and median in skewed distributions. Let&#8217;s get some practice with these skills using data collected around the T.V. show\u00a0<em>Friends<\/em>.<\/p>\n","protected":false},"author":17533,"menu_order":31,"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-41","chapter","type-chapter","status-publish","hentry"],"part":20,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters\/41","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters\/41\/revisions"}],"predecessor-version":[{"id":584,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters\/41\/revisions\/584"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/parts\/20"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters\/41\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/media?parent=41"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapter-type?post=41"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/contributor?post=41"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/license?post=41"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}