{"id":23,"date":"2022-05-20T16:59:02","date_gmt":"2022-05-20T16:59:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/calculating-mean-and-median-of-a-dataset-what-to-know\/"},"modified":"2022-07-18T00:31:33","modified_gmt":"2022-07-18T00:31:33","slug":"calculating-mean-and-median-of-a-dataset-what-to-know","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/calculating-mean-and-median-of-a-dataset-what-to-know\/","title":{"raw":"Calculating the Mean and Median of a Data Set: Learn It 1","rendered":"Calculating 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>Calculate the mean of a small data set by hand.<\/li>\r\n \t<li>Calculate the median of a small data set by hand.<\/li>\r\n \t<li>Use a data analysis tool to calculate the mean of a large data set.<\/li>\r\n \t<li>Use a data analysis tool to calculate the median of a large data set.<\/li>\r\n \t<li>Estimate the mean of a data set by examining a histogram of its distribution.<\/li>\r\n \t<li>Estimate the median of a data set by examining a histogram of its distribution.<\/li>\r\n \t<li>Use a data analysis tool to calculate and compare the mean and median for multiple groups at once.<\/li>\r\n \t<li>Use a histogram to estimate and compare centers for multiple groups.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<img class=\"alignnone wp-image-3033\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2021\/12\/10180418\/Screen-Shot-2022-02-10-at-12.16.27-PM.png\" alt=\"On a green background, mean is defined as the sum of all the values, then divided by the total number of values. The values given are 2, 8, 5, 3, 6, 9. All values are added up and divided by 6 resulting in a mean of 5.5.\" width=\"398\" height=\"224\" \/><img class=\"alignnone wp-image-3036\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2021\/12\/10181916\/Screen-Shot-2022-02-10-at-1.18.51-PM.png\" alt=\"On a teal background, median is defined as the value in the middle after all of the values have been arranged in ascending order. The same values 2, 8, 5, 3, 6, 9 are placed in order 2, 3, 5, 6, 8, 9. There are two values in the middle, 5 and 6, resulting in a median of 5.5.\" width=\"398\" height=\"224\" \/>\r\n\r\nMean\r\n\r\n<hr \/>\r\n\r\n&nbsp;\r\n\r\nIn the next activity, you'll need to use technology to calculate the mean and median of numerical data in order to compare groups. You will also need to interpret histograms to estimate the mean and median of a data set. Get some practice with these skills in this section.\r\n<h2>Calculating Mean and Median<\/h2>\r\n<div class=\"textbox examples\">\r\n<h3>Recall<\/h3>\r\nYou've probably seen the terms\u00a0<em>mean<\/em> and\u00a0<em>median<\/em> before. Before we discuss in detail how to calculate them, take a minute to see if you can recall doing it before. Do you remember how these terms are defined?\r\n\r\nCore skill: [reveal-answer q=\"223483\"]Define the terms <em>mean\u00a0<\/em>and\u00a0<em>median<\/em>[\/reveal-answer]\r\n[hidden-answer a=\"223483\"]Here are two quick tips to help remember what these are.\r\n\r\n<strong>Mean:\u00a0<\/strong>the arithmetic mean of a list of numbers, commonly called the \"average.\" Divide the sum of the values by the number of values in the list.\r\n\r\n<strong>Median:<\/strong> the \"middlemost\" number. Put the numbers in order from least to greatest then count in from the ends. If there are an odd number of terms, take the one in the middle as the median. If there are an even number of terms, take the mean of the two in the middle.\r\n\r\nDid you recall them from a previous math experience? If not, that's perfectly fine! Read on to learn everything you'll need to know.[\/hidden-answer]\r\n\r\n<\/div>\r\nIn most of the activities that follow, we'll be using technology to calculate the mean and median of large data sets. A close look at the notation of the mean and a refresher of how to calculate the median, though, will be helpful before you move forward. Much of the statistics presented in this course involves mathematical notation. The symbols used can appear similar, which causes some confusion until they become familiar to you. It will be helpful to spend some time getting to know the mathematical notation for the mean and practice calculating means and medians of small data sets by hand.\r\n<h3>Mean<\/h3>\r\n<div class=\"textbox tryit\">\r\n<h3>Calculating the Mean by Hand<\/h3>\r\n<span style=\"background-color: #ffff99;\">Need a video demo of the notation and translation of it to the example given below (students know how to get the average -- it's the notation for which they have no existing cognitive anchor).<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Mean<\/h3>\r\n\r\n<hr \/>\r\n\r\nYou may recall from a previous mathematics or statistics class that the <strong>mean<\/strong> of a data set can be computed by summing the data values and dividing by the number of values,\r\n<p style=\"text-align: center;\">[latex]\\text{mean } = \\dfrac{\\text{sum of data values}}{\\text{total number of data values}}[\/latex]<\/p>\r\nor more formally,\r\n<p style=\"text-align: center;\">[latex]\\bar{x}=\\dfrac{\\sum{x}}{n}[\/latex]<\/p>\r\nwhere [latex]\\bar{x}[\/latex] is the mean, [latex]\\sum[\/latex] is the symbol for \u201csum of,\u201d [latex]x[\/latex] represents the data values, and [latex]n[\/latex] is the total number of data values.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nLet's consider this small set of data values:\r\n<p style=\"text-align: center;\">[latex]3.3\\qquad 0.8\\qquad 5.8\\qquad 10.0\\qquad 3.6\\qquad 8.7\\qquad 0[\/latex]<\/p>\r\nFirst, you need to find the sum by adding all the values.\r\n<p style=\"text-align: center;\">[latex]3.3+.8+5.8+10+3.6+8.7+0=32.2[\/latex]<\/p>\r\nNext, count how many values were in the data set. Here there are\u00a0[latex]7[\/latex] values (zero is still a value1).\r\n\r\nThen, divide the sum these numbers by how many values there are.\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\nFrom this calculation, we determine that the mean is [latex]4.6[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It 1<\/h3>\r\nCalculate the mean of each small data set below.\r\n<p style=\"padding-left: 30px;\">a) [latex]7, 4, 8, 2, 3, 6[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">b)\u00a0[latex]1.2, 3.9, 5.3, 4.2[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">c)\u00a0[latex]79, 86, 92, 93, 88[\/latex]<\/p>\r\n[reveal-answer q=\"568579\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"568579\"]\r\n\r\na)\u00a0 \u00a0[latex]\\bar{x}=\\dfrac{\\sum{x}}{n}=\\dfrac{\\text{sum of the values}}{\\text{number of the values}}=\\dfrac{7+4+8+2+3+6}{6}=5[\/latex]\r\n\r\nb)\u00a0 \u00a0[latex]\\bar{x}=\\dfrac{\\sum{x}}{n}=\\dfrac{\\text{sum of the values}}{\\text{number of the values}}=\\dfrac{1.2+3.9+5.3+4.2}{4}=3.65[\/latex]\r\n\r\nc)\u00a0 \u00a0[latex]\\bar{x}=\\dfrac{\\sum{x}}{n}=\\dfrac{\\text{sum of the values}}{\\text{number of the values}}=\\dfrac{79+86+92+93+88}{5}=87.6[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","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>Calculate the mean of a small data set by hand.<\/li>\n<li>Calculate the median of a small data set by hand.<\/li>\n<li>Use a data analysis tool to calculate the mean of a large data set.<\/li>\n<li>Use a data analysis tool to calculate the median of a large data set.<\/li>\n<li>Estimate the mean of a data set by examining a histogram of its distribution.<\/li>\n<li>Estimate the median of a data set by examining a histogram of its distribution.<\/li>\n<li>Use a data analysis tool to calculate and compare the mean and median for multiple groups at once.<\/li>\n<li>Use a histogram to estimate and compare centers for multiple groups.<\/li>\n<\/ul>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-3033\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2021\/12\/10180418\/Screen-Shot-2022-02-10-at-12.16.27-PM.png\" alt=\"On a green background, mean is defined as the sum of all the values, then divided by the total number of values. The values given are 2, 8, 5, 3, 6, 9. All values are added up and divided by 6 resulting in a mean of 5.5.\" width=\"398\" height=\"224\" \/><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-3036\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2021\/12\/10181916\/Screen-Shot-2022-02-10-at-1.18.51-PM.png\" alt=\"On a teal background, median is defined as the value in the middle after all of the values have been arranged in ascending order. The same values 2, 8, 5, 3, 6, 9 are placed in order 2, 3, 5, 6, 8, 9. There are two values in the middle, 5 and 6, resulting in a median of 5.5.\" width=\"398\" height=\"224\" \/><\/p>\n<p>Mean<\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p>In the next activity, you&#8217;ll need to use technology to calculate the mean and median of numerical data in order to compare groups. You will also need to interpret histograms to estimate the mean and median of a data set. Get some practice with these skills in this section.<\/p>\n<h2>Calculating Mean and Median<\/h2>\n<div class=\"textbox examples\">\n<h3>Recall<\/h3>\n<p>You&#8217;ve probably seen the terms\u00a0<em>mean<\/em> and\u00a0<em>median<\/em> before. Before we discuss in detail how to calculate them, take a minute to see if you can recall doing it before. Do you remember how these terms are defined?<\/p>\n<p>Core skill: <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q223483\">Define the terms <em>mean\u00a0<\/em>and\u00a0<em>median<\/em><\/span><\/p>\n<div id=\"q223483\" class=\"hidden-answer\" style=\"display: none\">Here are two quick tips to help remember what these are.<\/p>\n<p><strong>Mean:\u00a0<\/strong>the arithmetic mean of a list of numbers, commonly called the &#8220;average.&#8221; Divide the sum of the values by the number of values in the list.<\/p>\n<p><strong>Median:<\/strong> the &#8220;middlemost&#8221; number. Put the numbers in order from least to greatest then count in from the ends. If there are an odd number of terms, take the one in the middle as the median. If there are an even number of terms, take the mean of the two in the middle.<\/p>\n<p>Did you recall them from a previous math experience? If not, that&#8217;s perfectly fine! Read on to learn everything you&#8217;ll need to know.<\/p><\/div>\n<\/div>\n<\/div>\n<p>In most of the activities that follow, we&#8217;ll be using technology to calculate the mean and median of large data sets. A close look at the notation of the mean and a refresher of how to calculate the median, though, will be helpful before you move forward. Much of the statistics presented in this course involves mathematical notation. The symbols used can appear similar, which causes some confusion until they become familiar to you. It will be helpful to spend some time getting to know the mathematical notation for the mean and practice calculating means and medians of small data sets by hand.<\/p>\n<h3>Mean<\/h3>\n<div class=\"textbox tryit\">\n<h3>Calculating the Mean by Hand<\/h3>\n<p><span style=\"background-color: #ffff99;\">Need a video demo of the notation and translation of it to the example given below (students know how to get the average &#8212; it&#8217;s the notation for which they have no existing cognitive anchor).<\/span><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Mean<\/h3>\n<hr \/>\n<p>You may recall from a previous mathematics or statistics class that the <strong>mean<\/strong> of a data set can be computed by summing the data values and dividing by the number of values,<\/p>\n<p style=\"text-align: center;\">[latex]\\text{mean } = \\dfrac{\\text{sum of data values}}{\\text{total number of data values}}[\/latex]<\/p>\n<p>or more formally,<\/p>\n<p style=\"text-align: center;\">[latex]\\bar{x}=\\dfrac{\\sum{x}}{n}[\/latex]<\/p>\n<p>where [latex]\\bar{x}[\/latex] is the mean, [latex]\\sum[\/latex] is the symbol for \u201csum of,\u201d [latex]x[\/latex] represents the data values, and [latex]n[\/latex] is the total number of data values.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Let&#8217;s consider this small set of data values:<\/p>\n<p style=\"text-align: center;\">[latex]3.3\\qquad 0.8\\qquad 5.8\\qquad 10.0\\qquad 3.6\\qquad 8.7\\qquad 0[\/latex]<\/p>\n<p>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\u00a0[latex]7[\/latex] values (zero is still a value1).<\/p>\n<p>Then, divide the sum 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]4.6[\/latex].<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It 1<\/h3>\n<p>Calculate the mean of each small data set below.<\/p>\n<p style=\"padding-left: 30px;\">a) [latex]7, 4, 8, 2, 3, 6[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">b)\u00a0[latex]1.2, 3.9, 5.3, 4.2[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">c)\u00a0[latex]79, 86, 92, 93, 88[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q568579\">Show Solution<\/span><\/p>\n<div id=\"q568579\" class=\"hidden-answer\" style=\"display: none\">\n<p>a)\u00a0 \u00a0[latex]\\bar{x}=\\dfrac{\\sum{x}}{n}=\\dfrac{\\text{sum of the values}}{\\text{number of the values}}=\\dfrac{7+4+8+2+3+6}{6}=5[\/latex]<\/p>\n<p>b)\u00a0 \u00a0[latex]\\bar{x}=\\dfrac{\\sum{x}}{n}=\\dfrac{\\text{sum of the values}}{\\text{number of the values}}=\\dfrac{1.2+3.9+5.3+4.2}{4}=3.65[\/latex]<\/p>\n<p>c)\u00a0 \u00a0[latex]\\bar{x}=\\dfrac{\\sum{x}}{n}=\\dfrac{\\text{sum of the values}}{\\text{number of the values}}=\\dfrac{79+86+92+93+88}{5}=87.6[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":17533,"menu_order":7,"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-23","chapter","type-chapter","status-publish","hentry"],"part":20,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters\/23","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":5,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters\/23\/revisions"}],"predecessor-version":[{"id":557,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters\/23\/revisions\/557"}],"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\/23\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/media?parent=23"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapter-type?post=23"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/contributor?post=23"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/license?post=23"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}