{"id":840,"date":"2023-03-20T19:17:19","date_gmt":"2023-03-20T19:17:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/understand-means-and-medians-as-measures-of-centers-learn-it-1\/"},"modified":"2025-05-11T22:31:16","modified_gmt":"2025-05-11T22:31:16","slug":"understand-means-and-medians-as-measures-of-centers-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/understand-means-and-medians-as-measures-of-centers-learn-it-1\/","title":{"raw":"Measures of Center: Learn It 1","rendered":"Measures of Center: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Name and compare the measures of center shown in a graph&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4865,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:0,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Name and compare the measures of center shown in a graph<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Calculating Mean and Median<\/h2>\r\n<p>Mean\u00a0and median are two different ways to define the center of a data set. Depending on the data and its distribution, one measure of center might be most informative or most representative of the \u201ctypical\u201d value. In analyzing quantitative data, the measure of center will be one key component.<\/p>\r\n<h3>Mean &amp; Median<\/h3>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>mean<\/h3>\r\n<p>The\u00a0<strong>mean<\/strong> of a data set is also commonly known as the average of a data set.<\/p>\r\n<p>The symbol we use to denote mean differs depending on whether we are discussing a sample or a population.<\/p>\r\n<ul>\r\n\t<li>Notation for mean of a population:\u00a0[latex]{\\mu}[\/latex]\u00a0(pronounced \u201cmu\u201d)<\/li>\r\n\t<li>Notation for mean of a sample of observations: [latex]\\stackrel{\u00af}{x}[\/latex] (pronounced \u201cx-bar\u201d)<\/li>\r\n<\/ul>\r\n<p>To calculate the mean, we add all the data values and divide by the number of data points.<\/p>\r\n<p>Formula for mean:\u00a0[latex]{\\mu}\\text{ or }\\bar{x}=\\dfrac{\\sum{x}}{n}[\/latex]<\/p>\r\n<p>where [latex]{\\mu}\\text{ or }\\bar{x}[\/latex] is the mean, [latex]\\sum[\/latex] is the symbol for sum (add up the data values), [latex]x[\/latex] represents the data values, and [latex]n[\/latex] represents the number of data values.<\/p>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>median<\/h3>\r\n<p>The\u00a0<strong>median<\/strong> of a data set is the value \u201cin the middle\u201d after all of the values have been arranged in ascending order.<\/p>\r\n<ul>\r\n\t<li>If there are an odd number of terms, take the one in the middle as the median.<\/li>\r\n\t<li>If there are an even number of terms, take the mean of the two in the middle.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Let's consider this small set of data that represents the cost per day of snacks for one individual:<\/p>\r\n<p style=\"text-align: center;\">[latex]$3.30\\qquad $0.80\\qquad $5.80\\qquad $10.00\\qquad $3.60\\qquad $8.70\\qquad $0[\/latex]<\/p>\r\n<p><strong>a) Calculate the mean of the data set.<\/strong><\/p>\r\n<p>Mean = [latex]$4.60[\/latex]<\/p>\r\n<p>[reveal-answer q=\"682352\"]Detailed Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"682352\"]First, you need to find the sum by adding all the values.<\/p>\r\n<p style=\"text-align: center;\">[latex]3.3+.8+5.8+10+3.6+8.7+0=32.2[\/latex]<\/p>\r\n<p>Next, count how many values were in the data set. Here, there are [latex]7[\/latex] values (zero is still a value).<\/p>\r\n<p>Then, divide the sum of these numbers by how many values there are.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\bar{x}=\\dfrac{3.3+.8+5.8+10+3.6+8.7+0}{7}=\\dfrac{32.2}{7}=4.6[\/latex]<\/p>\r\n<p>From this calculation, we determine that the mean is [latex]$4.60[\/latex].[\/hidden-answer]<\/p>\r\n<p><strong>b) Calculate the median of the data set.<\/strong><\/p>\r\n<p>Median = [latex]$3.60[\/latex]<\/p>\r\n<p>[reveal-answer q=\"662645\"]Detailed Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"662645\"]First, you need to arrange the data set in ascending order:<\/p>\r\n<p style=\"text-align: center;\">[latex]0\\qquad 0.8\\qquad 3.3\\qquad 3.6\\qquad 5.8\\qquad8.7\\qquad 10.0[\/latex]<\/p>\r\n<p>[latex]3.6[\/latex] is the center data value. There are three data values below [latex]3.6[\/latex] and three data values above [latex]3.6[\/latex].<\/p>\r\n<p>So, Median = [latex]$3.60[\/latex].[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2047[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Name and compare the measures of center shown in a graph&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4865,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:0,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Name and compare the measures of center shown in a graph<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Calculating Mean and Median<\/h2>\n<p>Mean\u00a0and median are two different ways to define the center of a data set. Depending on the data and its distribution, one measure of center might be most informative or most representative of the \u201ctypical\u201d value. In analyzing quantitative data, the measure of center will be one key component.<\/p>\n<h3>Mean &amp; Median<\/h3>\n<section class=\"textbox keyTakeaway\">\n<h3>mean<\/h3>\n<p>The\u00a0<strong>mean<\/strong> of a data set is also commonly known as the average of a data set.<\/p>\n<p>The symbol we use to denote mean differs depending on whether we are discussing a sample or a population.<\/p>\n<ul>\n<li>Notation for mean of a population:\u00a0[latex]{\\mu}[\/latex]\u00a0(pronounced \u201cmu\u201d)<\/li>\n<li>Notation for mean of a sample of observations: [latex]\\stackrel{\u00af}{x}[\/latex] (pronounced \u201cx-bar\u201d)<\/li>\n<\/ul>\n<p>To calculate the mean, we add all the data values and divide by the number of data points.<\/p>\n<p>Formula for mean:\u00a0[latex]{\\mu}\\text{ or }\\bar{x}=\\dfrac{\\sum{x}}{n}[\/latex]<\/p>\n<p>where [latex]{\\mu}\\text{ or }\\bar{x}[\/latex] is the mean, [latex]\\sum[\/latex] is the symbol for sum (add up the data values), [latex]x[\/latex] represents the data values, and [latex]n[\/latex] represents the number of data values.<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>median<\/h3>\n<p>The\u00a0<strong>median<\/strong> of a data set is the value \u201cin the middle\u201d after all of the values have been arranged in ascending order.<\/p>\n<ul>\n<li>If there are an odd number of terms, take the one in the middle as the median.<\/li>\n<li>If there are an even number of terms, take the mean of the two in the middle.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">\n<p>Let&#8217;s consider this small set of data that represents the cost per day of snacks for one individual:<\/p>\n<p style=\"text-align: center;\">[latex]$3.30\\qquad $0.80\\qquad $5.80\\qquad $10.00\\qquad $3.60\\qquad $8.70\\qquad $0[\/latex]<\/p>\n<p><strong>a) Calculate the mean of the data set.<\/strong><\/p>\n<p>Mean = [latex]$4.60[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q682352\">Detailed Solution<\/button><\/p>\n<div id=\"q682352\" class=\"hidden-answer\" style=\"display: none\">First, you need to find the sum by adding all the values.<\/p>\n<p style=\"text-align: center;\">[latex]3.3+.8+5.8+10+3.6+8.7+0=32.2[\/latex]<\/p>\n<p>Next, count how many values were in the data set. Here, there are [latex]7[\/latex] values (zero is still a value).<\/p>\n<p>Then, divide the sum of these numbers by how many values there are.<\/p>\n<p style=\"text-align: center;\">[latex]\\bar{x}=\\dfrac{3.3+.8+5.8+10+3.6+8.7+0}{7}=\\dfrac{32.2}{7}=4.6[\/latex]<\/p>\n<p>From this calculation, we determine that the mean is [latex]$4.60[\/latex].<\/p><\/div>\n<\/div>\n<p><strong>b) Calculate the median of the data set.<\/strong><\/p>\n<p>Median = [latex]$3.60[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q662645\">Detailed Solution<\/button><\/p>\n<div id=\"q662645\" class=\"hidden-answer\" style=\"display: none\">First, you need to arrange the data set in ascending order:<\/p>\n<p style=\"text-align: center;\">[latex]0\\qquad 0.8\\qquad 3.3\\qquad 3.6\\qquad 5.8\\qquad8.7\\qquad 10.0[\/latex]<\/p>\n<p>[latex]3.6[\/latex] is the center data value. There are three data values below [latex]3.6[\/latex] and three data values above [latex]3.6[\/latex].<\/p>\n<p>So, Median = [latex]$3.60[\/latex].<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2047\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2047&theme=lumen&iframe_resize_id=ohm2047&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":13,"menu_order":6,"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\/840"}],"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":8,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/840\/revisions"}],"predecessor-version":[{"id":6623,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/840\/revisions\/6623"}],"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\/840\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=840"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=840"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=840"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=840"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}