{"id":1628,"date":"2023-04-12T18:51:01","date_gmt":"2023-04-12T18:51:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=1628"},"modified":"2024-10-18T20:54:13","modified_gmt":"2024-10-18T20:54:13","slug":"numerical-summaries-of-data-learn-it-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/numerical-summaries-of-data-learn-it-1\/","title":{"raw":"Numerical Summaries of Data: Learn It 1","rendered":"Numerical Summaries of Data: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Find the average, middle value, and most common value in a set of data<\/li>\r\n\t<li>Calculate how spread out the data is using the range and standard deviation<\/li>\r\n\t<li>Identify the parts of a five-number summary for a set of data and create a box plot<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Mean, Median, and Mode<\/h2>\r\n<p>When we talk about the \"center\" of a data set in statistics, we are often referring to measures of central tendency, which summarize a key aspect of the distribution of the data. The <strong>mean<\/strong> and <strong>median<\/strong> are two such measures, each providing a different perspective on the data.\u00a0 Understanding which measure to use gives us insight into the true nature of the data's central tendency.<\/p>\r\n<h3>Mean<\/h3>\r\n<p>In analyzing quantitative data, the measure of center will be one key component. The mean, calculated as the average of all values, can be influenced by outliers and skewed distributions.\u00a0<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>mean<\/h3>\r\n<p>The <strong>mean<\/strong> of a set of [latex]n[\/latex] numbers is the arithmetic average of the numbers.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]\\text{mean}={\\Large\\frac{\\text{sum of values in data set}}{n}}[\/latex]<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Calculate the Mean of a Set of Numbers<\/strong><\/p>\r\n<ol id=\"eip-id1168468272241\" class=\"stepwise\">\r\n\t<li>Write the formula for the mean:<br \/>\r\n<p style=\"text-align: center;\">[latex]\\text{mean}={\\Large\\frac{\\text{sum of values in data set}}{n}}[\/latex]<\/p>\r\n<\/li>\r\n\t<li>Find the sum of all the values in the set. Write the sum in the numerator.<\/li>\r\n\t<li>Count the number, [latex]n[\/latex], of values in the set. Write this number in the denominator.<\/li>\r\n\t<li>Simplify the fraction.<\/li>\r\n\t<li>Check to see that the mean is reasonable. It should be greater than the least number and less than the greatest number in the set.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<h3>Median<\/h3>\r\n<p>The median, the middle value when all observations are ordered, is more robust to outliers and provides a better representation of the \"typical\" value in a skewed dataset.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>median<\/h3>\r\n<p>The <strong>median<\/strong> of a set of data values is the middle value.<\/p>\r\n<p>&nbsp;<\/p>\r\n<ul id=\"fs-id2455647\">\r\n\t<li>Half the data values are less than or equal to the median.<\/li>\r\n\t<li>Half the data values are greater than or equal to the median.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Find the Median of a Set of Numbers<\/strong><\/p>\r\n<ol id=\"eip-id1168466010714\" class=\"stepwise\">\r\n\t<li>List the numbers from smallest to largest.<\/li>\r\n\t<li>Count how many numbers are in the set. Call this [latex]n[\/latex].<\/li>\r\n\t<li>Is [latex]n[\/latex] odd or even?<br \/>\r\n<ul id=\"fs-id1733078\">\r\n\t<li>If [latex]n[\/latex] is an odd number, the median is the middle value.<\/li>\r\n\t<li>If [latex]n[\/latex] is an even number, the median is the mean of the two middle values.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">Let's consider this small set of data values:\r\n\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\n<p>a) Calculate the mean of the data set.<\/p>\r\n<p>Mean [latex]= 4.6[\/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]4.6[\/latex].[\/hidden-answer]<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>b) Calculate the median of the data set.<\/p>\r\n<p>Median [latex]= 3.6[\/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]= 3.6[\/latex].[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2047[\/ohm2_question]<\/section>\r\n<h3>Mode<\/h3>\r\n<p>In addition to mean and median, the <strong>mode<\/strong> is another measure of central tendency that identifies the most common or frequent value in a data set. It can be particularly useful in understanding the distribution of categorical data, where numerical averages are not applicable.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>mode<\/h3>\r\n<p>The mode of a set of numbers is the number with the highest frequency.<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Find the Mode of a Set of Numbers<\/strong><\/p>\r\n<ol id=\"eip-id1168466097152\" class=\"stepwise\">\r\n\t<li>List the data values in numerical order.<\/li>\r\n\t<li>Count the number of times each value appears.<\/li>\r\n\t<li>The mode is the value with the highest frequency.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<p>Some data sets do not have a mode because no value appears more than any other. And some data sets have more than one mode. In a given set, if two or more data values have the same highest frequency, we say they are all modes.<\/p>\r\n<section class=\"textbox example\">Statistics exam scores for 2020 students are as follows:\r\n\r\n<p style=\"text-align: center;\">[latex]50, 53, 59, 59, 63, 63, 72, 72, 72, 72, 72, 76, 78, 81, 83, 84, 84, 84, 90, 93[\/latex]<\/p>\r\n<p>Find the mode.<br \/>\r\n[reveal-answer q=\"262645\"]Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"262645\"]<\/p>\r\n<p>The mode score is [latex]72[\/latex].<\/p>\r\n<p>[\/hidden-answer]<br \/>\r\n[reveal-answer q=\"4\"]Show Detailed Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"4\"]<\/p>\r\n<p>Let's create a frequency table for the data set.<\/p>\r\n<table style=\"border-collapse: collapse; width: 50.0016%; height: 156px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 2.38095%; height: 12px;\">Score<\/td>\r\n<td style=\"width: 2.38095%; height: 12px;\">Frequency<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]50[\/latex]<\/td>\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]53[\/latex]<\/td>\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]59[\/latex]<\/td>\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]63[\/latex]<\/td>\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 2.38095%; height: 12px;\"><strong>[latex]72[\/latex]<\/strong><\/td>\r\n<td style=\"width: 2.38095%; height: 12px;\"><strong>[latex]5[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]76[\/latex]<\/td>\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]78[\/latex]<\/td>\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]81[\/latex]<\/td>\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]83[\/latex]<\/td>\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]84[\/latex]<\/td>\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]90[\/latex]<\/td>\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]93[\/latex]<\/td>\r\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>Based on the frequency table, the most frequent score is [latex]72[\/latex], which occurs five times. Therefore, <strong>mode = [latex] 72[\/latex].<\/strong><\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">The data lists the heights (in inches) of students in a statistics class. Identify the mode.\r\n\r\n<table id=\"fs-id3456019\" class=\"unnumbered\" summary=\"A table is shown with eight columns and four rows. The numbers 56, 61, 63, 64, 65, 66, 67, 67, 60, 62, 63, 64, 65, 66, 67, 70, 60, 63, 63, 64, 66, 66, 67, 74, 61, 63, 64, 65. 66, 67, 67 are listed in individual cells.\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]56[\/latex]<\/td>\r\n<td>[latex]61[\/latex]<\/td>\r\n<td>[latex]63[\/latex]<\/td>\r\n<td>[latex]64[\/latex]<\/td>\r\n<td>[latex]65[\/latex]<\/td>\r\n<td>[latex]66[\/latex]<\/td>\r\n<td>[latex]67[\/latex]<\/td>\r\n<td>[latex]67[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]60[\/latex]<\/td>\r\n<td>[latex]62[\/latex]<\/td>\r\n<td>[latex]63[\/latex]<\/td>\r\n<td>[latex]64[\/latex]<\/td>\r\n<td>[latex]65[\/latex]<\/td>\r\n<td>[latex]66[\/latex]<\/td>\r\n<td>[latex]67[\/latex]<\/td>\r\n<td>[latex]70[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]60[\/latex]<\/td>\r\n<td>[latex]63[\/latex]<\/td>\r\n<td>[latex]63[\/latex]<\/td>\r\n<td>[latex]64[\/latex]<\/td>\r\n<td>[latex]66[\/latex]<\/td>\r\n<td>[latex]66[\/latex]<\/td>\r\n<td>[latex]67[\/latex]<\/td>\r\n<td>[latex]74[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]61[\/latex]<\/td>\r\n<td>[latex]63[\/latex]<\/td>\r\n<td>[latex]64[\/latex]<\/td>\r\n<td>[latex]65[\/latex]<\/td>\r\n<td>[latex]66[\/latex]<\/td>\r\n<td>[latex]67[\/latex]<\/td>\r\n<td>[latex]67[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[reveal-answer q=\"165458\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"165458\"]<br \/>\r\nList each number with its frequency.<\/p>\r\n<center><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221822\/CNX_BMath_Figure_05_05_012_img.png\" alt=\"A table is shown with 2 rows. The first row is labeled Number and the second is labeled Frequency. Going across, the Numbers are 56, 60, 61, 62, 63, 64, 65, 66, 67, 70, 74. Going across, the Frequencies are 1, 2, 2, 1, 5, 4, 3, 5, 6, 1, 1.\" width=\"466\" height=\"59\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n<p>Now look for the highest frequency. The highest frequency is [latex]6[\/latex], which corresponds to the height [latex]67[\/latex] inches. So the mode of this set of heights is [latex]67[\/latex] inches.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox youChoose\">[videopicker divId=\"tnh-video-picker\" title=\"Mean, Median, and Mode\" label=\"Select Video\"]<br \/>\r\n[videooption displayName=\"Math Antics - Mean, Median and Mode\" value=\"https:\/\/youtu.be\/B1HEzNTGeZ4\"][videooption displayName=\"Finding mean, median, and mode; Descriptive statistics; Probability and Statistics - Khan Academy\" value=\"https:\/\/youtu.be\/k3aKKasOmIw\"] [videooption displayName=\"Mean, Median, Mode, and Range - Math with Mr. J\" value=\"\/\/plugin.3playmedia.com\/show?mf=12431189&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=e3uY2LraXts&amp;video_target=tpm-plugin-0o0fo34j-e3uY2LraXts\"]<br \/>\r\n[\/videopicker]\r\n\r\n<p>&nbsp;<\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Math+Antics+-+Mean%2C+Median+and+Mode.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMath Antics - Mean, Median and Mode\u201d here (opens in new window).<\/a><\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Finding+mean%2C+median%2C+and+mode+_+Descriptive+statistics+_+Probability+and+Statistics+_+Khan+Academy.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding mean, median, and mode | Descriptive statistics | Probability and Statistics | Khan Academy\u201d here (opens in new window).<\/a><\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Mean%2C+Median%2C+Mode%2C+and+Range+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMean, Median, Mode, and Range | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Find the average, middle value, and most common value in a set of data<\/li>\n<li>Calculate how spread out the data is using the range and standard deviation<\/li>\n<li>Identify the parts of a five-number summary for a set of data and create a box plot<\/li>\n<\/ul>\n<\/section>\n<h2>Mean, Median, and Mode<\/h2>\n<p>When we talk about the &#8220;center&#8221; of a data set in statistics, we are often referring to measures of central tendency, which summarize a key aspect of the distribution of the data. The <strong>mean<\/strong> and <strong>median<\/strong> are two such measures, each providing a different perspective on the data.\u00a0 Understanding which measure to use gives us insight into the true nature of the data&#8217;s central tendency.<\/p>\n<h3>Mean<\/h3>\n<p>In analyzing quantitative data, the measure of center will be one key component. The mean, calculated as the average of all values, can be influenced by outliers and skewed distributions.\u00a0<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>mean<\/h3>\n<p>The <strong>mean<\/strong> of a set of [latex]n[\/latex] numbers is the arithmetic average of the numbers.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\text{mean}={\\Large\\frac{\\text{sum of values in data set}}{n}}[\/latex]<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Calculate the Mean of a Set of Numbers<\/strong><\/p>\n<ol id=\"eip-id1168468272241\" class=\"stepwise\">\n<li>Write the formula for the mean:\n<p style=\"text-align: center;\">[latex]\\text{mean}={\\Large\\frac{\\text{sum of values in data set}}{n}}[\/latex]<\/p>\n<\/li>\n<li>Find the sum of all the values in the set. Write the sum in the numerator.<\/li>\n<li>Count the number, [latex]n[\/latex], of values in the set. Write this number in the denominator.<\/li>\n<li>Simplify the fraction.<\/li>\n<li>Check to see that the mean is reasonable. It should be greater than the least number and less than the greatest number in the set.<\/li>\n<\/ol>\n<\/section>\n<h3>Median<\/h3>\n<p>The median, the middle value when all observations are ordered, is more robust to outliers and provides a better representation of the &#8220;typical&#8221; value in a skewed dataset.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>median<\/h3>\n<p>The <strong>median<\/strong> of a set of data values is the middle value.<\/p>\n<p>&nbsp;<\/p>\n<ul id=\"fs-id2455647\">\n<li>Half the data values are less than or equal to the median.<\/li>\n<li>Half the data values are greater than or equal to the median.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Find the Median of a Set of Numbers<\/strong><\/p>\n<ol id=\"eip-id1168466010714\" class=\"stepwise\">\n<li>List the numbers from smallest to largest.<\/li>\n<li>Count how many numbers are in the set. Call this [latex]n[\/latex].<\/li>\n<li>Is [latex]n[\/latex] odd or even?\n<ul id=\"fs-id1733078\">\n<li>If [latex]n[\/latex] is an odd number, the median is the middle value.<\/li>\n<li>If [latex]n[\/latex] is an even number, the median is the mean of the two middle values.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">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>a) Calculate the mean of the data set.<\/p>\n<p>Mean [latex]= 4.6[\/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]4.6[\/latex].<\/p><\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>b) Calculate the median of the data set.<\/p>\n<p>Median [latex]= 3.6[\/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]= 3.6[\/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<h3>Mode<\/h3>\n<p>In addition to mean and median, the <strong>mode<\/strong> is another measure of central tendency that identifies the most common or frequent value in a data set. It can be particularly useful in understanding the distribution of categorical data, where numerical averages are not applicable.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>mode<\/h3>\n<p>The mode of a set of numbers is the number with the highest frequency.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Find the Mode of a Set of Numbers<\/strong><\/p>\n<ol id=\"eip-id1168466097152\" class=\"stepwise\">\n<li>List the data values in numerical order.<\/li>\n<li>Count the number of times each value appears.<\/li>\n<li>The mode is the value with the highest frequency.<\/li>\n<\/ol>\n<\/section>\n<p>Some data sets do not have a mode because no value appears more than any other. And some data sets have more than one mode. In a given set, if two or more data values have the same highest frequency, we say they are all modes.<\/p>\n<section class=\"textbox example\">Statistics exam scores for 2020 students are as follows:<\/p>\n<p style=\"text-align: center;\">[latex]50, 53, 59, 59, 63, 63, 72, 72, 72, 72, 72, 76, 78, 81, 83, 84, 84, 84, 90, 93[\/latex]<\/p>\n<p>Find the mode.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q262645\">Solution<\/button><\/p>\n<div id=\"q262645\" class=\"hidden-answer\" style=\"display: none\">\n<p>The mode score is [latex]72[\/latex].<\/p>\n<\/div>\n<\/div>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4\">Show Detailed Solution<\/button><\/p>\n<div id=\"q4\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let&#8217;s create a frequency table for the data set.<\/p>\n<table style=\"border-collapse: collapse; width: 50.0016%; height: 156px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"width: 2.38095%; height: 12px;\">Score<\/td>\n<td style=\"width: 2.38095%; height: 12px;\">Frequency<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 2.38095%; height: 12px;\">[latex]50[\/latex]<\/td>\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 2.38095%; height: 12px;\">[latex]53[\/latex]<\/td>\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 2.38095%; height: 12px;\">[latex]59[\/latex]<\/td>\n<td style=\"width: 2.38095%; height: 12px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 2.38095%; height: 12px;\">[latex]63[\/latex]<\/td>\n<td style=\"width: 2.38095%; height: 12px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 2.38095%; height: 12px;\"><strong>[latex]72[\/latex]<\/strong><\/td>\n<td style=\"width: 2.38095%; height: 12px;\"><strong>[latex]5[\/latex]<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 2.38095%; height: 12px;\">[latex]76[\/latex]<\/td>\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 2.38095%; height: 12px;\">[latex]78[\/latex]<\/td>\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 2.38095%; height: 12px;\">[latex]81[\/latex]<\/td>\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 2.38095%; height: 12px;\">[latex]83[\/latex]<\/td>\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 2.38095%; height: 12px;\">[latex]84[\/latex]<\/td>\n<td style=\"width: 2.38095%; height: 12px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 2.38095%; height: 12px;\">[latex]90[\/latex]<\/td>\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 2.38095%; height: 12px;\">[latex]93[\/latex]<\/td>\n<td style=\"width: 2.38095%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Based on the frequency table, the most frequent score is [latex]72[\/latex], which occurs five times. Therefore, <strong>mode = [latex]72[\/latex].<\/strong><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">The data lists the heights (in inches) of students in a statistics class. Identify the mode.<\/p>\n<table id=\"fs-id3456019\" class=\"unnumbered\" summary=\"A table is shown with eight columns and four rows. The numbers 56, 61, 63, 64, 65, 66, 67, 67, 60, 62, 63, 64, 65, 66, 67, 70, 60, 63, 63, 64, 66, 66, 67, 74, 61, 63, 64, 65. 66, 67, 67 are listed in individual cells.\">\n<tbody>\n<tr valign=\"top\">\n<td>[latex]56[\/latex]<\/td>\n<td>[latex]61[\/latex]<\/td>\n<td>[latex]63[\/latex]<\/td>\n<td>[latex]64[\/latex]<\/td>\n<td>[latex]65[\/latex]<\/td>\n<td>[latex]66[\/latex]<\/td>\n<td>[latex]67[\/latex]<\/td>\n<td>[latex]67[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]60[\/latex]<\/td>\n<td>[latex]62[\/latex]<\/td>\n<td>[latex]63[\/latex]<\/td>\n<td>[latex]64[\/latex]<\/td>\n<td>[latex]65[\/latex]<\/td>\n<td>[latex]66[\/latex]<\/td>\n<td>[latex]67[\/latex]<\/td>\n<td>[latex]70[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]60[\/latex]<\/td>\n<td>[latex]63[\/latex]<\/td>\n<td>[latex]63[\/latex]<\/td>\n<td>[latex]64[\/latex]<\/td>\n<td>[latex]66[\/latex]<\/td>\n<td>[latex]66[\/latex]<\/td>\n<td>[latex]67[\/latex]<\/td>\n<td>[latex]74[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]61[\/latex]<\/td>\n<td>[latex]63[\/latex]<\/td>\n<td>[latex]64[\/latex]<\/td>\n<td>[latex]65[\/latex]<\/td>\n<td>[latex]66[\/latex]<\/td>\n<td>[latex]67[\/latex]<\/td>\n<td>[latex]67[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q165458\">Show Solution<\/button><\/p>\n<div id=\"q165458\" class=\"hidden-answer\" style=\"display: none\">\nList each number with its frequency.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221822\/CNX_BMath_Figure_05_05_012_img.png\" alt=\"A table is shown with 2 rows. The first row is labeled Number and the second is labeled Frequency. Going across, the Numbers are 56, 60, 61, 62, 63, 64, 65, 66, 67, 70, 74. Going across, the Frequencies are 1, 2, 2, 1, 5, 4, 3, 5, 6, 1, 1.\" width=\"466\" height=\"59\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>Now look for the highest frequency. The highest frequency is [latex]6[\/latex], which corresponds to the height [latex]67[\/latex] inches. So the mode of this set of heights is [latex]67[\/latex] inches.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox youChoose\">\n<div id=\"tnh-video-picker\" class=\"videoPicker\">\n<h3>Mean, Median, and Mode<\/h3>\n<form><label>Select Video:<\/label><select name=\"video\"><option value=\"https:\/\/www.youtube.com\/embed\/B1HEzNTGeZ4\">Math Antics &#8211; Mean, Median and Mode<\/option><option value=\"https:\/\/www.youtube.com\/embed\/k3aKKasOmIw\">Finding mean, median, and mode; Descriptive statistics; Probability and Statistics &#8211; Khan Academy<\/option><option value=\"\/\/plugin.3playmedia.com\/show?mf=12431189&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=e3uY2LraXts&amp;video_target=tpm-plugin-0o0fo34j-e3uY2LraXts\">Mean, Median, Mode, and Range &#8211; Math with Mr. J<\/option><\/select><\/form>\n<div class=\"videoContainer\"><iframe src=\"https:\/\/www.youtube.com\/embed\/B1HEzNTGeZ4\" allowfullscreen><\/iframe><\/div>\n<p>&nbsp;<\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Math+Antics+-+Mean%2C+Median+and+Mode.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMath Antics &#8211; Mean, Median and Mode\u201d here (opens in new window).<\/a><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Finding+mean%2C+median%2C+and+mode+_+Descriptive+statistics+_+Probability+and+Statistics+_+Khan+Academy.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding mean, median, and mode | Descriptive statistics | Probability and Statistics | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Mean%2C+Median%2C+Mode%2C+and+Range+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMean, Median, Mode, and Range | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Math Antics - Mean, Median and Mode\",\"author\":\"mathantics\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/B1HEzNTGeZ4\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Finding mean, median, and mode | Descriptive statistics | Probability and Statistics | Khan Academy\",\"author\":\"Khan Academy\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/k3aKKasOmIw\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Mean, Median, Mode, and Range | Math with Mr. J\",\"author\":\"Math with Mr. J\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/e3uY2LraXts\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1572,"module-header":"learn_it","content_attributions":[{"type":"copyrighted_video","description":"Math Antics - Mean, Median and Mode","author":"mathantics","organization":"","url":"https:\/\/youtu.be\/B1HEzNTGeZ4","project":"","license":"arr","license_terms":""},{"type":"copyrighted_video","description":"Finding mean, median, and mode | Descriptive statistics | Probability and Statistics | Khan Academy","author":"Khan Academy","organization":"","url":"https:\/\/youtu.be\/k3aKKasOmIw","project":"","license":"arr","license_terms":""},{"type":"copyrighted_video","description":"Mean, Median, Mode, and Range | Math with Mr. J","author":"Math with Mr. J","organization":"","url":"https:\/\/youtu.be\/e3uY2LraXts","project":"","license":"arr","license_terms":""}],"internal_book_links":[],"video_content":[{"divId":"tnh-video-picker","title":"Mean, Median, and Mode","label":"Select Video","video_collection":[{"displayName":"Math Antics - Mean, Median and Mode","value":"https:\/\/www.youtube.com\/embed\/B1HEzNTGeZ4"},{"displayName":"Finding mean, median, and mode; 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