{"id":293,"date":"2022-02-18T23:59:02","date_gmt":"2022-02-18T23:59:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/?post_type=chapter&#038;p=293"},"modified":"2022-05-20T16:43:19","modified_gmt":"2022-05-20T16:43:19","slug":"calculating-mean-and-median-of-a-dataset-what-to-know","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/calculating-mean-and-median-of-a-dataset-what-to-know\/","title":{"raw":"Calculating the Mean and Median of a Data Set: What to Know","rendered":"Calculating the Mean and Median of a Data Set: What to Know"},"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=\"#MeanHand\">Calculate the mean of a small data set by hand.<\/a><\/li>\r\n \t<li><a href=\"#MedianHand\">Calculate the median of a small data set by hand.<\/a><\/li>\r\n \t<li><a href=\"#MeanMedianTool\">Use a data analysis tool to calculate the mean of a large data set.<\/a><\/li>\r\n \t<li><a href=\"#MeanMedianTool\">Use a data analysis tool to calculate the median of a large data set.<\/a><\/li>\r\n \t<li><a href=\"#MeanMedianHist\">Estimate the mean of a data set by examining a histogram of its distribution.<\/a><\/li>\r\n \t<li><a href=\"#MeanMedianHist\">Estimate the median of a data set by examining a histogram of its distribution.<\/a><\/li>\r\n \t<li><a href=\"#MeanMedianGroups\">Use a data analysis tool to calculate and compare the mean and median for multiple groups at once.<\/a><\/li>\r\n \t<li><a href=\"#MeanMedianGroupsHist\">Use a histogram to estimate and compare centers for multiple groups.<\/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\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\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\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\nFor example, consider this small set of data values:\r\n<p style=\"text-align: center;\">[latex]3.3\\qquad 1.2\\qquad 5.8\\qquad 10.0\\qquad 3.6\\qquad 8.7\\qquad 4.5[\/latex]<\/p>\r\nAs seen below, the sum of these values is\u00a0[latex]37.1[\/latex], and there are\u00a0[latex]7[\/latex] values. Dividing these numbers, we determine that the mean is [latex]5.3[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\bar{x}=\\dfrac{3.3+1.2+5.8+10+3.6+8.7+4.5}{7}=\\dfrac{37.1}{7}=5.3[\/latex]<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/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>\r\n<h3 id=\"MedianHand\">Median<\/h3>\r\nAnother measure of center you may recall is the <strong>median<\/strong>. This value is computed by ordering the data values and identifying the value in \u201cthe middle.\u201d\r\n\r\nIf we consider the sample data from above, ordering these values from least to greatest, we get:\r\n<p style=\"text-align: center;\">[latex]1.2\\qquad 3.3\\qquad 3.6\\qquad 4.5\\qquad 5.8\\qquad 8.7\\qquad 10.0[\/latex]<\/p>\r\nThe value\u00a0[latex]4.5[\/latex] is the \u201cmiddle number\u201d in the ordered set; we see there are three values less than\u00a0[latex]4.5[\/latex] ([latex]1.2, 3.3, 3.6[\/latex]) and three values greater than\u00a0[latex]4.5[\/latex] ([latex]5.8, 8.7, 10.0[\/latex]). The value [latex]4.5[\/latex] is the median.\r\n<p style=\"text-align: center;\">[latex]\\cancel{1.2}\\qquad \\cancel{3.3}\\qquad \\cancel{3.6}\\qquad 4.5\\qquad \\cancel{5.8}\\qquad \\cancel{8.7}\\qquad \\cancel{10.0}[\/latex]<\/p>\r\nIf there are an odd number of observations, the \"middle number\" is the number that is left alone after all of the others have been crossed out. If there are an even number of observations, the \u201cmiddle number\u201d is the mean of the middle two observations. Check out the following videos to practice finding the median. The first video is using an odd number of observations, and the second is using an even number of observations.\r\n<div class=\"textbox tryit\">\r\n<h3>finding the median using an odd numbered data set<\/h3>\r\n<span style=\"background-color: #ffff99;\">Need a video demo showing the \"counting in from the ends\" to find the middle-most number in an odd numbered set.--&gt;video starts at 2:00, it should stop around 2:48.\u00a0<\/span>\r\n\r\n[embed]https:\/\/youtu.be\/A1mQ9kD-i9I?t=120[\/embed]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>finding the median using an even numbered data set<\/h3>\r\n<span style=\"background-color: #ffff99;\">Need a video demo showing the \"counting in from the ends\" to find the middle-most number in an even numbered set.--&gt;same video but starting at 10:32 and it should stop around 11:22.<\/span>\r\n\r\n[embed]https:\/\/youtu.be\/A1mQ9kD-i9I?t=632[\/embed]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nCalculate the median of each small data set below. These are the same sets used earlier to calculate the mean.\r\n<p style=\"padding-left: 30px;\">a)\u00a0[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=\"296010\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"296010\"]\r\n\r\na)\u00a0 \u00a0The median is\u00a0[latex]5[\/latex].\r\n<p style=\"padding-left: 30px;\">Put the numbers in order:\u00a0[latex]2, 3, 4, 6, 7, 8[\/latex]. Since there are an even number of values in the set, identify and take the average of the middle two: [latex]\\cancel{2}[\/latex], [latex]\\cancel{3}[\/latex], 4, 6, [latex]\\cancel{7}[\/latex], [latex]\\cancel{8}[\/latex]. The two numbers in the center spot are\u00a0[latex]4[\/latex] and\u00a0[latex]6[\/latex].\u00a0 [latex]\\frac{4+6}{2}=5[\/latex].<\/p>\r\nb)\u00a0 The median is\u00a0[latex]4.05[\/latex].\r\n<p style=\"padding-left: 30px;\">Put the numbers in order:\u00a0[latex]1.2, 3.9, 4.2, 5.3[\/latex]. Since there are an even number of values in the set, identify and take the average of the middle two:\u00a0[latex]\\cancel{1.2}\\text{, }[\/latex]3.9, 4.2, [latex]\\cancel{5.3}\\text{, }[\/latex]. The two numbers in the center spot are\u00a0[latex]3.9[\/latex] and\u00a0[latex]4.2[\/latex]. [latex]\\frac{3.9+4.2}{2}=4.05[\/latex].<\/p>\r\nc)\u00a0 The median is\u00a0[latex]88[\/latex].\r\n<p style=\"padding-left: 30px;\">Put the numbers in order:\u00a0[latex]79, 86, 88, 92, 93[\/latex]. Since there are an odd number of values in the set, identify and take the middle number as the median: [latex]\\cancel{79}\\text{, } \\cancel{86}\\text{, } 88\\text{, } \\cancel{92}\\text{, }\\cancel{93}[\/latex]. The middle number is\u00a0[latex]88[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you try it by taking the mean and median of the small set of data below.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 1<\/h3>\r\n[ohm_question hide_question_numbers=1]240965[\/ohm_question]\r\n\r\n[reveal-answer q=\"125513\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"125513\"]Recall that the median is the mean of the middle two values.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3 id=\"MeanMedianTool\">Large data sets<\/h3>\r\n<span style=\"background-color: #ffff00;\">At this point, students will be presented with two datasets. They will be able to choose which one they would like to use to answer example questions before using the Descriptive Statistics in the data analysis tool.<\/span>\r\n\r\nWhen computing these values with a large data set, it is not efficient to do so by hand. Instead, we will rely on technology to calculate these values. Let's try that now.\r\n<div class=\"textbox\">\r\n\r\nGo to the <em>Describing and Exploring Quantitative Variables<\/em> tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\" target=\"_blank\" rel=\"noopener\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a>.\r\n<p style=\"padding-left: 30px;\">Step 1) Select the <strong>Single Group<\/strong> tab.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 2)\u00a0Locate the dropdown under <strong>Enter Data<\/strong> and select <strong>From Textbook<\/strong>.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 3) Locate the drop-down menu under <strong>Data Set<\/strong> and select <strong>Sleep Study: Average Sleep<\/strong>.<\/p>\r\n\r\n<\/div>\r\nThis data set contains the average number of hours of sleep per night for each of the\u00a0[latex]253[\/latex] students in the sleep study.\r\n\r\nNote the <strong>Descriptive Statistics<\/strong> located above the graphical display. You'll find the mean and median located in the list. Eventually, we will use all the values shown but for now we just want to record the mean and median.\u00a0To use the tool to calculate descriptive statistics for any data set, load, copy and paste, or type the data values into the \"Observations\" box of the tool, and it will automatically list the values in the \"Descriptive Statistics.\"\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 2<\/h3>\r\n[ohm_question hide_question_numbers=1]240966[\/ohm_question]\r\n\r\n[reveal-answer q=\"124361\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"124361\"]Make sure the \u201cSingle Group\u201d tab is selected at the top and \"Sleep Study: Average Sleep\" is selected under \"Data Set.\" The mean and median are given in the Descriptive Statistics.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 id=\"MeanMedianHist\">Mean and Median as the Center of Data<\/h2>\r\nThere are other ways that we can think about the mean and median as measures of center of numerical data. More specifically, the mean represents the balance point of the data, and the median represents the\u00a0[latex]50[\/latex]<sup>th<\/sup> percentile, or the value that splits the data in half (i.e., half of the data are below the median and the other half of the data are above the median).\r\n<div class=\"textbox tryit\">\r\n<h3>mean and median<\/h3>\r\n<span style=\"background-color: #99cc00;\">[Perspective Video - a 3-instructor video illustrating the mean as a balance point and the median as splitting the data in half]<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 3<\/h3>\r\n[ohm_question hide_question_numbers=1]240620[\/ohm_question]\r\n\r\n[reveal-answer q=\"509633\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"509633\"]Which choice would keep the histogram balanced if the x-axis was a scale and the heights of the columns were weights? It may be helpful to first eliminate the choices that clearly do not represent the balance point of the data.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 4<\/h3>\r\n[ohm_question hide_question_numbers=1]240623[\/ohm_question]\r\n\r\n[reveal-answer q=\"694612\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"694612\"]The median will separate the data into equal halves: half of the data will be below the median and half will be above the median. It may be helpful to first eliminate the choices that clearly do not separate the data in half.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3 id=\"MeanMedianGroups\">Using Technology to Calculate and Compare Centers Across Groups<\/h3>\r\n<span style=\"background-color: #99cc00;\">[Worked example video - a 3-instructor video providing an example like the one below for questions 5 - 7]<\/span>\r\n\r\n<\/div>\r\nAnother benefit of using technology to calculate the mean and median is that we can quickly calculate these values for multiple groups. We can do so by using the <strong>Several Groups<\/strong> tab on the\u00a0<em>Describing and Exploring Quantitative Variables<\/em> tool (the same tool you used to complete questions 2 - 4 above).\r\n<div class=\"textbox\">\r\n\r\nGo to the <em>Describing and Exploring Quantitative Variables<\/em> tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\" target=\"_blank\" rel=\"noopener\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a>.\r\n<p style=\"padding-left: 30px;\">Step 1) Select the <strong>Several Groups<\/strong> tab.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 2) Under\u00a0<strong>Enter Data<\/strong>, select\u00a0<strong>From Textbook<\/strong>.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 3) Locate the drop-down menu under <strong>Data Set<\/strong>, and select <strong>Sleep Study: Average Sleep Score<\/strong>.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 4) Change <strong>Choose Type of Plot<\/strong>\u00a0to <strong>Histogram<\/strong> if desired.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 5) Calculate the mean and median for each of the groups: \u201cOwl,\u201d \u201cLark,\u201d and \u201cNeither,\u201d and list these values in the table in question 5 below (Note: the mean and median will be automatically calculated by the technology and can be found under Descriptive Statistics).<\/p>\r\n\r\n<\/div>\r\nRecall that \u201cOwl\u201d describes the group of students who tends to stay up late, and \u201cLark\u201d describes the group who tends to wake up early. Students who did not identify as an owl nor a lark were classified in the \u201cNeither\u201d group.\r\n\r\nRecall also that we consider the mean to be the arithmetic mean (commonly called the\u00a0\"average\") of a set of numbers, while the median refers to the value that sits in the middle of the distribution with half of the values above it and half of the values below.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 5<\/h3>\r\n[ohm_question hide_question_numbers=1]240967[\/ohm_question]\r\n<div align=\"left\">\r\n\r\n[reveal-answer q=\"475952\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"475952\"]Locate these values in Descriptive Statistics in the tool. Confirm that you have Several Groups and data set Sleep Study: Average Sleep Score selected. The data in the tool may be presented in a different order than in this table.[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 6<\/h3>\r\n[ohm_question hide_question_numbers=1]240968[\/ohm_question]\r\n\r\n[reveal-answer q=\"832913\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"832913\"]The measure we commonly call an \"average\" is actually the mean of a data set. [\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 7<\/h3>\r\n[ohm_question hide_question_numbers=1]240969[\/ohm_question]\r\n\r\n[reveal-answer q=\"863794\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"863794\"]Recall that the median for each group represents the\u00a0[latex]50^{\\text{th}}[\/latex]\u00a0percentile: half of the data values are below the median and half of the data values are above the median.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3 id=\"MeanMedianGroupsHist\">Histograms<\/h3>\r\nThe following image uses histograms to compare the distribution of the variable \u201cAlcoholic Drinks Per Week\u201d for two groups of college students in this study. Based on these histograms, determine whether you believe the following statements are true or false.\r\n\r\n<img class=\"alignnone wp-image-996\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11191400\/Picture311-300x171.png\" alt=\"Two histograms. The horizontal axis is labeled &quot;Alcoholic Drinks Consumed Per Week.&quot; At the top, a legend shows that green indicates Group 1 and yellow indicates Group 2. The first graph is green. For 1-2, the count is approximately 8. For 3-4, the count is approximately 39. For 5-6, the count is approximately 20. For 7-8, the count is approximately 4. For 9-10, the count is approximately 3. For 11-12, the count is approximately 4. For 13-14, the count is approximately 2. For 24-25, the count is approximately 1. The next graph is yellow. For 3, the count is approximately 7. For 4, the count is approximately 10. For 5, the count is approximately 18. For 6-7, the count is approximately 20. For 8, the count is approximately 11. For 9, the count is approximately 10. For 10, the count is approximately 22. For 12, the count is approximately 7. For 13, the count is approximately 3. For 14, the count is approximately 1. For 15, the count is approximately 3. For 18, the count is approximately 1. For 20, the count is approximately 2.\" width=\"895\" height=\"510\" \/>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 8<\/h3>\r\n[ohm_question hide_question_numbers=1]240970[\/ohm_question]\r\n\r\n[reveal-answer q=\"807492\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"807492\"]Examine the Group 1 graph. Where does the \"balance point\" of the data appear to be? [\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 9<\/h3>\r\n[ohm_question hide_question_numbers=1]240971[\/ohm_question]\r\n\r\n[reveal-answer q=\"6273\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"6273\"]Examine both graphs. Where does the \"balance point\" of the data appear to be for each? Compare.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 10<\/h3>\r\n[ohm_question hide_question_numbers=1]240972[\/ohm_question]\r\n\r\n[reveal-answer q=\"782713\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"782713\"]Recall that the median splits the data values in half.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 11<\/h3>\r\n[ohm_question hide_question_numbers=1]240973[\/ohm_question]\r\n\r\n[reveal-answer q=\"131791\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"131791\"]Recall that the median splits the data values in half.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nIn this section, you've gained practice calculating means and medians by hand and with technology. Let's summarize where these skills showed up in the material.\r\n<ul>\r\n \t<li>In question 1, you calculated the mean and median of a data set by hand.<\/li>\r\n \t<li>In question 2, you calculated the mean and median of a data set using technology.<\/li>\r\n \t<li>In question 5, you used technology to calculate the mean and median for multiple groups.<\/li>\r\n \t<li>In questions 6 and 7, you compared the mean and median for multiple groups.<\/li>\r\n \t<li>In questions 3, 4, and 8 - 11, you estimated the mean and median by looking at the data presented in a histogram.<\/li>\r\n<\/ul>\r\nBeing able to use technology to calculate the mean and median of numerical data in order to compare groups, as well as interpreting histograms to estimate the mean and median of a data set will be necessary for completing the next activity. If you feel comfortable with these skills, please move on to the activity!","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=\"#MeanHand\">Calculate the mean of a small data set by hand.<\/a><\/li>\n<li><a href=\"#MedianHand\">Calculate the median of a small data set by hand.<\/a><\/li>\n<li><a href=\"#MeanMedianTool\">Use a data analysis tool to calculate the mean of a large data set.<\/a><\/li>\n<li><a href=\"#MeanMedianTool\">Use a data analysis tool to calculate the median of a large data set.<\/a><\/li>\n<li><a href=\"#MeanMedianHist\">Estimate the mean of a data set by examining a histogram of its distribution.<\/a><\/li>\n<li><a href=\"#MeanMedianHist\">Estimate the median of a data set by examining a histogram of its distribution.<\/a><\/li>\n<li><a href=\"#MeanMedianGroups\">Use a data analysis tool to calculate and compare the mean and median for multiple groups at once.<\/a><\/li>\n<li><a href=\"#MeanMedianGroupsHist\">Use a histogram to estimate and compare centers for multiple groups.<\/a><\/li>\n<\/ul>\n<p>Click on a skill above to jump to its location in this section.<\/p>\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>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<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<p>For example, consider this small set of data values:<\/p>\n<p style=\"text-align: center;\">[latex]3.3\\qquad 1.2\\qquad 5.8\\qquad 10.0\\qquad 3.6\\qquad 8.7\\qquad 4.5[\/latex]<\/p>\n<p>As seen below, the sum of these values is\u00a0[latex]37.1[\/latex], and there are\u00a0[latex]7[\/latex] values. Dividing these numbers, we determine that the mean is [latex]5.3[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\bar{x}=\\dfrac{3.3+1.2+5.8+10+3.6+8.7+4.5}{7}=\\dfrac{37.1}{7}=5.3[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>Examples<\/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<h3 id=\"MedianHand\">Median<\/h3>\n<p>Another measure of center you may recall is the <strong>median<\/strong>. This value is computed by ordering the data values and identifying the value in \u201cthe middle.\u201d<\/p>\n<p>If we consider the sample data from above, ordering these values from least to greatest, we get:<\/p>\n<p style=\"text-align: center;\">[latex]1.2\\qquad 3.3\\qquad 3.6\\qquad 4.5\\qquad 5.8\\qquad 8.7\\qquad 10.0[\/latex]<\/p>\n<p>The value\u00a0[latex]4.5[\/latex] is the \u201cmiddle number\u201d in the ordered set; we see there are three values less than\u00a0[latex]4.5[\/latex] ([latex]1.2, 3.3, 3.6[\/latex]) and three values greater than\u00a0[latex]4.5[\/latex] ([latex]5.8, 8.7, 10.0[\/latex]). The value [latex]4.5[\/latex] is the median.<\/p>\n<p style=\"text-align: center;\">[latex]\\cancel{1.2}\\qquad \\cancel{3.3}\\qquad \\cancel{3.6}\\qquad 4.5\\qquad \\cancel{5.8}\\qquad \\cancel{8.7}\\qquad \\cancel{10.0}[\/latex]<\/p>\n<p>If there are an odd number of observations, the &#8220;middle number&#8221; is the number that is left alone after all of the others have been crossed out. If there are an even number of observations, the \u201cmiddle number\u201d is the mean of the middle two observations. Check out the following videos to practice finding the median. The first video is using an odd number of observations, and the second is using an even number of observations.<\/p>\n<div class=\"textbox tryit\">\n<h3>finding the median using an odd numbered data set<\/h3>\n<p><span style=\"background-color: #ffff99;\">Need a video demo showing the &#8220;counting in from the ends&#8221; to find the middle-most number in an odd numbered set.&#8211;&gt;video starts at 2:00, it should stop around 2:48.\u00a0<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Mean, Median, Mode, and Range - How To Find It!\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/A1mQ9kD-i9I?start=120&#38;feature=oembed\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>finding the median using an even numbered data set<\/h3>\n<p><span style=\"background-color: #ffff99;\">Need a video demo showing the &#8220;counting in from the ends&#8221; to find the middle-most number in an even numbered set.&#8211;&gt;same video but starting at 10:32 and it should stop around 11:22.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Mean, Median, Mode, and Range - How To Find It!\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/A1mQ9kD-i9I?start=632&#38;feature=oembed\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>Calculate the median of each small data set below. These are the same sets used earlier to calculate the mean.<\/p>\n<p style=\"padding-left: 30px;\">a)\u00a0[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=\"q296010\">Show Solution<\/span><\/p>\n<div id=\"q296010\" class=\"hidden-answer\" style=\"display: none\">\n<p>a)\u00a0 \u00a0The median is\u00a0[latex]5[\/latex].<\/p>\n<p style=\"padding-left: 30px;\">Put the numbers in order:\u00a0[latex]2, 3, 4, 6, 7, 8[\/latex]. Since there are an even number of values in the set, identify and take the average of the middle two: [latex]\\cancel{2}[\/latex], [latex]\\cancel{3}[\/latex], 4, 6, [latex]\\cancel{7}[\/latex], [latex]\\cancel{8}[\/latex]. The two numbers in the center spot are\u00a0[latex]4[\/latex] and\u00a0[latex]6[\/latex].\u00a0 [latex]\\frac{4+6}{2}=5[\/latex].<\/p>\n<p>b)\u00a0 The median is\u00a0[latex]4.05[\/latex].<\/p>\n<p style=\"padding-left: 30px;\">Put the numbers in order:\u00a0[latex]1.2, 3.9, 4.2, 5.3[\/latex]. Since there are an even number of values in the set, identify and take the average of the middle two:\u00a0[latex]\\cancel{1.2}\\text{, }[\/latex]3.9, 4.2, [latex]\\cancel{5.3}\\text{, }[\/latex]. The two numbers in the center spot are\u00a0[latex]3.9[\/latex] and\u00a0[latex]4.2[\/latex]. [latex]\\frac{3.9+4.2}{2}=4.05[\/latex].<\/p>\n<p>c)\u00a0 The median is\u00a0[latex]88[\/latex].<\/p>\n<p style=\"padding-left: 30px;\">Put the numbers in order:\u00a0[latex]79, 86, 88, 92, 93[\/latex]. Since there are an odd number of values in the set, identify and take the middle number as the median: [latex]\\cancel{79}\\text{, } \\cancel{86}\\text{, } 88\\text{, } \\cancel{92}\\text{, }\\cancel{93}[\/latex]. The middle number is\u00a0[latex]88[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you try it by taking the mean and median of the small set of data below.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 1<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240965\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240965&theme=oea&iframe_resize_id=ohm240965\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q125513\">Hint<\/span><\/p>\n<div id=\"q125513\" class=\"hidden-answer\" style=\"display: none\">Recall that the median is the mean of the middle two values.<\/div>\n<\/div>\n<\/div>\n<h3 id=\"MeanMedianTool\">Large data sets<\/h3>\n<p><span style=\"background-color: #ffff00;\">At this point, students will be presented with two datasets. They will be able to choose which one they would like to use to answer example questions before using the Descriptive Statistics in the data analysis tool.<\/span><\/p>\n<p>When computing these values with a large data set, it is not efficient to do so by hand. Instead, we will rely on technology to calculate these values. Let&#8217;s try that now.<\/p>\n<div class=\"textbox\">\n<p>Go to the <em>Describing and Exploring Quantitative Variables<\/em> tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\" target=\"_blank\" rel=\"noopener\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a>.<\/p>\n<p style=\"padding-left: 30px;\">Step 1) Select the <strong>Single Group<\/strong> tab.<\/p>\n<p style=\"padding-left: 30px;\">Step 2)\u00a0Locate the dropdown under <strong>Enter Data<\/strong> and select <strong>From Textbook<\/strong>.<\/p>\n<p style=\"padding-left: 30px;\">Step 3) Locate the drop-down menu under <strong>Data Set<\/strong> and select <strong>Sleep Study: Average Sleep<\/strong>.<\/p>\n<\/div>\n<p>This data set contains the average number of hours of sleep per night for each of the\u00a0[latex]253[\/latex] students in the sleep study.<\/p>\n<p>Note the <strong>Descriptive Statistics<\/strong> located above the graphical display. You&#8217;ll find the mean and median located in the list. Eventually, we will use all the values shown but for now we just want to record the mean and median.\u00a0To use the tool to calculate descriptive statistics for any data set, load, copy and paste, or type the data values into the &#8220;Observations&#8221; box of the tool, and it will automatically list the values in the &#8220;Descriptive Statistics.&#8221;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 2<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240966\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240966&theme=oea&iframe_resize_id=ohm240966\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q124361\">Hint<\/span><\/p>\n<div id=\"q124361\" class=\"hidden-answer\" style=\"display: none\">Make sure the \u201cSingle Group\u201d tab is selected at the top and &#8220;Sleep Study: Average Sleep&#8221; is selected under &#8220;Data Set.&#8221; The mean and median are given in the Descriptive Statistics.<\/div>\n<\/div>\n<\/div>\n<h2 id=\"MeanMedianHist\">Mean and Median as the Center of Data<\/h2>\n<p>There are other ways that we can think about the mean and median as measures of center of numerical data. More specifically, the mean represents the balance point of the data, and the median represents the\u00a0[latex]50[\/latex]<sup>th<\/sup> percentile, or the value that splits the data in half (i.e., half of the data are below the median and the other half of the data are above the median).<\/p>\n<div class=\"textbox tryit\">\n<h3>mean and median<\/h3>\n<p><span style=\"background-color: #99cc00;\">[Perspective Video &#8211; a 3-instructor video illustrating the mean as a balance point and the median as splitting the data in half]<\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 3<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240620\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240620&theme=oea&iframe_resize_id=ohm240620\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q509633\">Hint<\/span><\/p>\n<div id=\"q509633\" class=\"hidden-answer\" style=\"display: none\">Which choice would keep the histogram balanced if the x-axis was a scale and the heights of the columns were weights? It may be helpful to first eliminate the choices that clearly do not represent the balance point of the data.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 4<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240623\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240623&theme=oea&iframe_resize_id=ohm240623\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q694612\">Hint<\/span><\/p>\n<div id=\"q694612\" class=\"hidden-answer\" style=\"display: none\">The median will separate the data into equal halves: half of the data will be below the median and half will be above the median. It may be helpful to first eliminate the choices that clearly do not separate the data in half.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3 id=\"MeanMedianGroups\">Using Technology to Calculate and Compare Centers Across Groups<\/h3>\n<p><span style=\"background-color: #99cc00;\">[Worked example video &#8211; a 3-instructor video providing an example like the one below for questions 5 &#8211; 7]<\/span><\/p>\n<\/div>\n<p>Another benefit of using technology to calculate the mean and median is that we can quickly calculate these values for multiple groups. We can do so by using the <strong>Several Groups<\/strong> tab on the\u00a0<em>Describing and Exploring Quantitative Variables<\/em> tool (the same tool you used to complete questions 2 &#8211; 4 above).<\/p>\n<div class=\"textbox\">\n<p>Go to the <em>Describing and Exploring Quantitative Variables<\/em> tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\" target=\"_blank\" rel=\"noopener\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a>.<\/p>\n<p style=\"padding-left: 30px;\">Step 1) Select the <strong>Several Groups<\/strong> tab.<\/p>\n<p style=\"padding-left: 30px;\">Step 2) Under\u00a0<strong>Enter Data<\/strong>, select\u00a0<strong>From Textbook<\/strong>.<\/p>\n<p style=\"padding-left: 30px;\">Step 3) Locate the drop-down menu under <strong>Data Set<\/strong>, and select <strong>Sleep Study: Average Sleep Score<\/strong>.<\/p>\n<p style=\"padding-left: 30px;\">Step 4) Change <strong>Choose Type of Plot<\/strong>\u00a0to <strong>Histogram<\/strong> if desired.<\/p>\n<p style=\"padding-left: 30px;\">Step 5) Calculate the mean and median for each of the groups: \u201cOwl,\u201d \u201cLark,\u201d and \u201cNeither,\u201d and list these values in the table in question 5 below (Note: the mean and median will be automatically calculated by the technology and can be found under Descriptive Statistics).<\/p>\n<\/div>\n<p>Recall that \u201cOwl\u201d describes the group of students who tends to stay up late, and \u201cLark\u201d describes the group who tends to wake up early. Students who did not identify as an owl nor a lark were classified in the \u201cNeither\u201d group.<\/p>\n<p>Recall also that we consider the mean to be the arithmetic mean (commonly called the\u00a0&#8220;average&#8221;) of a set of numbers, while the median refers to the value that sits in the middle of the distribution with half of the values above it and half of the values below.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 5<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240967\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240967&theme=oea&iframe_resize_id=ohm240967\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q475952\">Hint<\/span><\/p>\n<div id=\"q475952\" class=\"hidden-answer\" style=\"display: none\">Locate these values in Descriptive Statistics in the tool. Confirm that you have Several Groups and data set Sleep Study: Average Sleep Score selected. The data in the tool may be presented in a different order than in this table.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 6<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240968\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240968&theme=oea&iframe_resize_id=ohm240968\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q832913\">Hint<\/span><\/p>\n<div id=\"q832913\" class=\"hidden-answer\" style=\"display: none\">The measure we commonly call an &#8220;average&#8221; is actually the mean of a data set. <\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 7<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240969\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240969&theme=oea&iframe_resize_id=ohm240969\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q863794\">Hint<\/span><\/p>\n<div id=\"q863794\" class=\"hidden-answer\" style=\"display: none\">Recall that the median for each group represents the\u00a0[latex]50^{\\text{th}}[\/latex]\u00a0percentile: half of the data values are below the median and half of the data values are above the median.<\/div>\n<\/div>\n<\/div>\n<h3 id=\"MeanMedianGroupsHist\">Histograms<\/h3>\n<p>The following image uses histograms to compare the distribution of the variable \u201cAlcoholic Drinks Per Week\u201d for two groups of college students in this study. Based on these histograms, determine whether you believe the following statements are true or false.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-996\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11191400\/Picture311-300x171.png\" alt=\"Two histograms. The horizontal axis is labeled &quot;Alcoholic Drinks Consumed Per Week.&quot; At the top, a legend shows that green indicates Group 1 and yellow indicates Group 2. The first graph is green. For 1-2, the count is approximately 8. For 3-4, the count is approximately 39. For 5-6, the count is approximately 20. For 7-8, the count is approximately 4. For 9-10, the count is approximately 3. For 11-12, the count is approximately 4. For 13-14, the count is approximately 2. For 24-25, the count is approximately 1. The next graph is yellow. For 3, the count is approximately 7. For 4, the count is approximately 10. For 5, the count is approximately 18. For 6-7, the count is approximately 20. For 8, the count is approximately 11. For 9, the count is approximately 10. For 10, the count is approximately 22. For 12, the count is approximately 7. For 13, the count is approximately 3. For 14, the count is approximately 1. For 15, the count is approximately 3. For 18, the count is approximately 1. For 20, the count is approximately 2.\" width=\"895\" height=\"510\" \/><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 8<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240970\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240970&theme=oea&iframe_resize_id=ohm240970\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q807492\">Hint<\/span><\/p>\n<div id=\"q807492\" class=\"hidden-answer\" style=\"display: none\">Examine the Group 1 graph. Where does the &#8220;balance point&#8221; of the data appear to be? <\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 9<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240971\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240971&theme=oea&iframe_resize_id=ohm240971\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q6273\">Hint<\/span><\/p>\n<div id=\"q6273\" class=\"hidden-answer\" style=\"display: none\">Examine both graphs. Where does the &#8220;balance point&#8221; of the data appear to be for each? Compare.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 10<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240972\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240972&theme=oea&iframe_resize_id=ohm240972\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q782713\">Hint<\/span><\/p>\n<div id=\"q782713\" class=\"hidden-answer\" style=\"display: none\">Recall that the median splits the data values in half.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 11<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240973\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240973&theme=oea&iframe_resize_id=ohm240973\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q131791\">Hint<\/span><\/p>\n<div id=\"q131791\" class=\"hidden-answer\" style=\"display: none\">Recall that the median splits the data values in half.<\/div>\n<\/div>\n<\/div>\n<h2>Summary<\/h2>\n<p>In this section, you&#8217;ve gained practice calculating means and medians by hand and with technology. Let&#8217;s summarize where these skills showed up in the material.<\/p>\n<ul>\n<li>In question 1, you calculated the mean and median of a data set by hand.<\/li>\n<li>In question 2, you calculated the mean and median of a data set using technology.<\/li>\n<li>In question 5, you used technology to calculate the mean and median for multiple groups.<\/li>\n<li>In questions 6 and 7, you compared the mean and median for multiple groups.<\/li>\n<li>In questions 3, 4, and 8 &#8211; 11, you estimated the mean and median by looking at the data presented in a histogram.<\/li>\n<\/ul>\n<p>Being able to use technology to calculate the mean and median of numerical data in order to compare groups, as well as interpreting histograms to estimate the mean and median of a data set will be necessary for completing the next activity. If you feel comfortable with these skills, please move on to the activity!<\/p>\n","protected":false},"author":175116,"menu_order":29,"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-293","chapter","type-chapter","status-publish","hentry"],"part":1252,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/293","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/users\/175116"}],"version-history":[{"count":17,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/293\/revisions"}],"predecessor-version":[{"id":1257,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/293\/revisions\/1257"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/parts\/1252"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/293\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/media?parent=293"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapter-type?post=293"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/contributor?post=293"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/license?post=293"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}