{"id":1629,"date":"2023-04-12T18:50:59","date_gmt":"2023-04-12T18:50:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=1629"},"modified":"2024-10-18T20:54:16","modified_gmt":"2024-10-18T20:54:16","slug":"numerical-summaries-of-data-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/numerical-summaries-of-data-fresh-take\/","title":{"raw":"Numerical Summaries of Data: Fresh Take","rendered":"Numerical Summaries of Data: Fresh Take"},"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<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea<\/strong><\/p>\r\n<p><strong>Mean<\/strong>, <strong>median<\/strong>, and <strong>mode <\/strong>are three types of statistical measures used to analyze a set of data.<\/p>\r\n<p>The <strong>mean<\/strong>, often referred to as the \"average,\" is calculated by adding all the numbers in a data set and then dividing by the count of those numbers.<\/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\r\n\r\nThe <strong>median <\/strong>is the middle number when the data set is arranged in ascending or descending order; if the data set has an even number of observations, the median is the average of the two middle numbers.\r\n\r\n\r\n<p>The <strong>mode<\/strong>, on the other hand, is the number that occurs most frequently in a data set.<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/B1HEzNTGeZ4\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<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<\/section>\r\n<section class=\"textbox example\">Find the mean of the numbers [latex]8,12,15,9,\\text{ and }6[\/latex].<br \/>\r\n[reveal-answer q=\"682352\"]Detailed Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"682352\"]\r\n\r\n\r\n<table id=\"eip-id1168467435661\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 501.9px; height: 30px;\">Write the formula for the mean:<\/td>\r\n<td style=\"width: 333.1px; height: 30px;\">[latex]\\text{mean}={\\Large\\frac{\\text{sum of all the numbers}}{n}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 501.9px; height: 30px;\">Write the sum of the numbers in the numerator.<\/td>\r\n<td style=\"width: 333.1px; height: 30px;\">[latex]\\text{mean}={\\Large\\frac{8+12+15+9+6}{n}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 501.9px; height: 30px;\">Count how many numbers are in the set. There are [latex]5[\/latex] numbers in the set, so [latex]n=5[\/latex] .<\/td>\r\n<td style=\"width: 333.1px; height: 30px;\">[latex]\\text{mean}={\\Large\\frac{8+12+15+9+6}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 501.9px; height: 15px;\">Add the numbers in the numerator.<\/td>\r\n<td style=\"width: 333.1px; height: 15px;\">[latex]\\text{mean}={\\Large\\frac{50}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 501.9px; height: 15px;\">Then divide.<\/td>\r\n<td style=\"width: 333.1px; height: 15px;\">[latex]\\text{mean}=10[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30.8281px;\">\r\n<td style=\"width: 501.9px; height: 30.8281px;\">Check to see that the mean is 'typical': [latex]10[\/latex] is neither less than [latex]6[\/latex] nor greater than [latex]15[\/latex].<\/td>\r\n<td style=\"width: 333.1px; height: 30.8281px;\">The mean is [latex]10[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">For the past four months, Daisy\u2019s cell phone bills were [latex]\\text{\\$42.75},\\text{\\$50.12},\\text{\\$41.54},\\text{\\$48.15}[\/latex]. Find the mean cost of Daisy\u2019s cell phone bills.<br \/>\r\n[reveal-answer q=\"42782\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"42782\"]Solution\r\n\r\n\r\n<table id=\"eip-id1168469762854\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Write the formula for the mean.<\/td>\r\n<td>[latex]\\text{mean}=\\Large\\frac{\\text{sum of all the numbers}}{n}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count how many numbers are in the set. Call this [latex]n[\/latex] and write it in the denominator.<\/td>\r\n<td>[latex]\\text{mean}={\\Large\\frac{\\text{sum of all the numbers}}{4}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the sum of all the numbers in the numerator.<\/td>\r\n<td>[latex]\\text{mean}={\\Large\\frac{42.75+50.12+41.54+48.15}{4}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the fraction.<\/td>\r\n<td>[latex]\\text{mean}={\\Large\\frac{182.56}{4}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]\\text{mean}=45.64[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>Does [latex]\\text{\\$45.64}[\/latex] seem \u2018typical\u2019 of this set of numbers? Yes, it is neither less than [latex]\\text{\\$41.54}[\/latex] nor greater than [latex]\\text{\\$50.12}[\/latex].<\/p>\r\n<p>The mean cost of her cell phone bill was [latex]\\text{\\$45.64}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Find the median of [latex]12,13,19,9,11,15,\\text{and }18[\/latex].<br \/>\r\n[reveal-answer q=\"42784\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"42784\"]\r\n\r\n\r\n<table id=\"eip-id1168468416053\" class=\"unnumbered unstyled\" summary=\"The figure shows the numbers 59, 60, and 62 separated by a small space from the numbers 65, 68, and 70. Each set of three has a bracket underneath grouping the numbers together.\">\r\n<tbody>\r\n<tr>\r\n<td>List the numbers in order from smallest to largest.<\/td>\r\n<td>[latex]9, 11, 12, 13, 15, 18, 19[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count how many numbers are in the set. Call this [latex]n[\/latex] .<\/td>\r\n<td>[latex]n=7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is [latex]n[\/latex] odd or even?<\/td>\r\n<td>odd<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The median is the middle value.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221814\/CNX_BMath_Figure_05_05_009_img.png\" alt=\"The series 9, 11, 12, 13, 15, 18, 19, with 13 labeled as the median\" width=\"327\" height=\"150\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The middle is the number in the [latex]4[\/latex]th position.<\/td>\r\n<td>So the median of the data is [latex]13[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Kristen received the following scores on her weekly math quizzes:<br \/>\r\n<center>[latex]83,79,85,86,92,100,76,90,88,\\text{and }64[\/latex].<\/center>Find her median score.<br \/>\r\n[reveal-answer q=\"982119\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"982119\"]\r\n\r\n\r\n<table id=\"eip-id1168467170036\" class=\"unnumbered unstyled\" summary=\"The problem says, 'Find the median of 83, 79, 85, 86, 92, 100, 76, 90, 88, and 64.' The first step says, 'List the numbers in order from smallest to largest,' and shows 64, 76, 79, 83, 85, 86, 88, 90, 92, 100. The next step says, 'Count how many numbers are in the set. Call this n. n equals 10.' The next step asks, 'Is n odd or even? Even.' The next line says, 'The median is the two middle values, the 5th and 6th numbers.' The ordered list of numbers is shown again with the first five numbers grouped together and labeled 5 numbers and the second five numbers are grouped together and labeled 5 numbers. The next step says 'Find the mean of 85 and 86.' The mean equals the sum of 85 plus 86 divided by 2, which equals 85.5. The last line shows 'Kristen's median score is 85.5'.\">\r\n<tbody>\r\n<tr>\r\n<td>Find the median of [latex]83, 79, 85, 86, 92, 100, 76, 90, 88,\\text{ and }64[\/latex].<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>List the numbers in order from smallest to largest.<\/td>\r\n<td>[latex]64, 76, 79, 83, 85, 86, 88, 90, 92, 100[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the number of data values in the set. Call this [latex]\\mathrm{n.}[\/latex]<\/td>\r\n<td>[latex]n=10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is [latex]n[\/latex] odd or even?<\/td>\r\n<td>even<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The median is the mean of the two middle values, the [latex]5[\/latex]th and [latex]6[\/latex]th numbers.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221816\/CNX_BMath_Figure_05_05_010_img-01.png\" alt=\"The series 64, 76, 79, 83, 85, 86, 88, 90, 92, 100. The series is split into two groups of numbers, showing that 85 and 86 are the closest two to the median.\" width=\"319\" height=\"64\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the mean of [latex]85[\/latex] and [latex]86[\/latex].<\/td>\r\n<td>[latex]\\text{mean}={\\Large\\frac{85+86}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]\\text{mean}=85.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>Kristen's median score is [latex]85.5[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>The ages of the students in a statistics class are listed here:<\/p>\r\n<p style=\"text-align: center;\">[latex]19[\/latex] , [latex]20[\/latex] , [latex]23[\/latex] , [latex]23[\/latex] , [latex]38[\/latex] , [latex]21[\/latex] , [latex]19[\/latex] , [latex]21[\/latex] , [latex]19[\/latex] , [latex]21[\/latex] , [latex]20[\/latex] , [latex]43[\/latex] , [latex]20[\/latex] , [latex]23[\/latex] , [latex]17[\/latex] , [latex]21[\/latex] , [latex]21[\/latex] , [latex]20[\/latex] , [latex]29[\/latex] , [latex]18[\/latex] , [latex]28[\/latex].<\/p>\r\n<p>What is the mode?<\/p>\r\n\r\n\r\n[reveal-answer q=\"888357\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"888357\"]\r\n\r\n\r\n<p>[latex]21[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>Students listed the number of members in their household as follows:<\/p>\r\n<p style=\"text-align: center;\">[latex]6[\/latex] , [latex]2[\/latex] , [latex]5[\/latex] , [latex]6[\/latex] , [latex]3[\/latex] , [latex]7[\/latex] , [latex]5[\/latex] , [latex]6[\/latex] , [latex]5[\/latex] , [latex]3[\/latex] , [latex]4[\/latex] , [latex]4[\/latex] , [latex]5[\/latex] , [latex]7[\/latex] , [latex]6[\/latex] , [latex]4[\/latex] , [latex]5[\/latex] , [latex]2[\/latex] , [latex]1[\/latex] , [latex]5[\/latex].<\/p>\r\n<p>What is the mode?<\/p>\r\n\r\n\r\n[reveal-answer q=\"888927\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"888927\"][latex]5[\/latex][\/hidden-answer]<\/section>\r\n<h2>Range, Standard Deviation, and Variance<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea<\/strong><\/p>\r\n<p><strong>Range<\/strong>, <strong>standard deviation<\/strong>, and <strong>variance <\/strong>are three key measures of dispersion in a dataset.<\/p>\r\n<p>The <strong>range <\/strong>of a dataset is the difference between the highest and lowest values, giving a simple measure of total spread.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\text{Range } = \\text{ maximum value } \u2013 \\text{ minimum value }\u00a0 =\u00a0 \\text{ largest value } \u2013 \\text{ smallest value}[\/latex]<\/p>\r\n<p><strong>Standard deviation<\/strong>, a more complex measure, gauges the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values are close to the mean, while a high standard deviation suggests greater dispersion.<\/p>\r\n<p>A few important characteristics:<\/p>\r\n<ul>\r\n\t<li>Standard deviation is always positive. Standard deviation will be zero if all the data values are equal, and will get larger as the data spreads out.<\/li>\r\n\t<li>Standard deviation has the same units as the original data.<\/li>\r\n\t<li>Standard deviation, like the mean, can be highly influenced by outliers.<\/li>\r\n<\/ul>\r\n<p>The following formulas are used to calculate the standard deviation of a population and a sample:<\/p>\r\n<p style=\"padding-left: 60px;\"><strong>Standard deviation of a population<\/strong>: [latex]\\sigma = \\sqrt{\\dfrac{\\sum \\left(x-\\mu\\right)^2}{n}}[\/latex], where [latex]\\mu[\/latex] represents the population mean.<\/p>\r\n<p style=\"padding-left: 60px;\"><strong>Standard deviation of a sample<\/strong>: [latex]s=\\sqrt{\\dfrac{\\sum \\left(x-\\bar{x}\\right)^2}{n-1}} [\/latex], where [latex]\\bar{x}[\/latex] represents the sample mean.<\/p>\r\n<p><strong>Variance<\/strong>, often denoted by the squared units of the original data, is the average of the squared differences from the mean, effectively measuring how far each number in the set is from the mean.<\/p>\r\n<p style=\"text-align: center;\"><strong>Variance of a population<\/strong>:\u00a0 [latex]\\sigma^{2}=\\dfrac{\\sum\\left(x-\\mu\\right)^{2}}{n}[\/latex]<\/p>\r\n<p style=\"text-align: center;\"><strong>Variance of a sample<\/strong>:\u00a0 [latex]s^{2}=\\dfrac{\\sum\\left(x-\\bar{x}\\right)^{2}}{n-1}[\/latex]<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10350382&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=s7WTQ0H0Acc&amp;video_target=tpm-plugin-il9fa5lg-s7WTQ0H0Acc\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Measures+of+Variability+Range+Standard+Deviation+Variance.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMeasures of Variability (Range, Standard Deviation, Variance)\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Five-Number Summary<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea<\/strong><\/p>\r\n<p>The <strong>Five-Number Summary<\/strong> is a descriptive statistic that provides information about a dataset. It consists of five values: the minimum, the first quartile (Q1 or 25th percentile), the median (or Q2 or 50th percentile), the third quartile (Q3 or 75th percentile), and the maximum. The minimum and maximum values depict the smallest and largest numbers in the dataset respectively. The first quartile is the median of the lower half of the data (not including the overall median), the third quartile is the median of the upper half, and the median is the middle value of the entire dataset.<\/p>\r\n<p>To find the first quartile, [latex]Q1[\/latex]:<\/p>\r\n<ol>\r\n\t<li>Begin by ordering the data from smallest to largest<\/li>\r\n\t<li>Compute the locator: [latex]L = 0.25n[\/latex]<\/li>\r\n\t<li>If [latex]L[\/latex] is a decimal value:\r\n\r\n\r\n<ul>\r\n\t<li>Round up to [latex]L+[\/latex]<\/li>\r\n\t<li>Use the data value in the [latex]L+[\/latex]th position<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>If [latex]L[\/latex] is a whole number:\r\n\r\n\r\n<ul>\r\n\t<li>Find the mean of the data values in the <em>[latex]L[\/latex]<\/em>th and [latex]L+1[\/latex]th positions.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p>To find the third quartile, [latex]Q3[\/latex]:<\/p>\r\n<p>Use the same procedure as for [latex]Q1[\/latex], but with locator: [latex]L = 0.75n[\/latex]<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10350383&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=1IXJZl_JP1o&amp;video_target=tpm-plugin-dyygmajn-1IXJZl_JP1o\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/What+is+a+5+Number+Summary.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat is a 5 Number Summary?\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Boxplots<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea<\/strong><\/p>\r\n<p><strong>Box plots<\/strong>, also known as box-and-whisker plots, are graphical representations used to depict the spread and skewness of a data set.<\/p>\r\n<p>They are constructed using the five-number summary: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum.<\/p>\r\n<p>The\u00a0<strong>interquartile range<\/strong>\u00a0(sometimes denoted as IQR) is the difference between the quartiles calculated as <br \/>\r\n[latex]Q3 \u2013 Q1[\/latex].<\/p>\r\n<p>The 'box' of the plot represents the interquartile range (IQR), stretching from Q1 to Q3, and the line inside the box marks the median. 'Whiskers' extend from the box to the minimum and maximum values, showing the full spread of the data. Outliers, if any, are typically indicated as individual points beyond the whiskers.<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10350384&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=fJZv9YeQ-qQ&amp;video_target=tpm-plugin-fje6fw09-fJZv9YeQ-qQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/BOX+AND+WHISKER+PLOTS+EXPLAINED.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cBOX AND WHISKER PLOTS EXPLAINED!\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">The box plot below is based on the [latex]9[\/latex] female height data with five-number summary:<center>[latex]59[\/latex], [latex]62[\/latex], [latex]66[\/latex], [latex]69[\/latex], [latex]72[\/latex].<\/center>\u00a0<center><img class=\"alignnone wp-image-12885\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175921\/large-heights.png\" alt=\"Number line titled Heights (inches), in increments of 1 from 55-75. Above this, a vertical line indicates 59. A horizontal line connects this to the next vertical line, 62. This line forms the left side of a rectangle; a line at 66 is its right side. The line at 66 also serves as the left side of another rectangle, with a line at 69 as its right side. This line at 69 connects with a horizontal line to a final vertical line at 72.\" width=\"600\" height=\"125\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n<hr \/>\r\n<p>The box plot below is based on the household income data with five-number summary:<\/p>\r\n<center>[latex]15[\/latex], [latex]27.5[\/latex], [latex]35[\/latex], [latex]40[\/latex], [latex]50[\/latex]<\/center><center><img class=\"alignnone wp-image-12886\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175940\/large-household-incomes.png\" alt=\"Number line titled Thousands of Dollars, in increments of 5 from 0-55. Above this, a vertical line indicates 15. A horizontal line connects this to the next vertical line, 27.5. This line forms the left side of a rectangle; a line at 35 is its right side. The line at 35 also serves as the left side of another rectangle, with a line at 40 as its right side. This line at 40 connects with a horizontal line to a final vertical line at 50.\" width=\"600\" height=\"174\" \/><\/center><\/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<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p><strong>Mean<\/strong>, <strong>median<\/strong>, and <strong>mode <\/strong>are three types of statistical measures used to analyze a set of data.<\/p>\n<p>The <strong>mean<\/strong>, often referred to as the &#8220;average,&#8221; is calculated by adding all the numbers in a data set and then dividing by the count of those numbers.<\/p>\n<p style=\"text-align: center;\">[latex]\\text{mean}={\\Large\\frac{\\text{sum of values in data set}}{n}}[\/latex]<\/p>\n<p>The <strong>median <\/strong>is the middle number when the data set is arranged in ascending or descending order; if the data set has an even number of observations, the median is the average of the two middle numbers.<\/p>\n<p>The <strong>mode<\/strong>, on the other hand, is the number that occurs most frequently in a data set.<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/B1HEzNTGeZ4\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<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<\/section>\n<section class=\"textbox example\">Find the mean of the numbers [latex]8,12,15,9,\\text{ and }6[\/latex].<\/p>\n<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\">\n<table id=\"eip-id1168467435661\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\".\">\n<tbody>\n<tr style=\"height: 30px;\">\n<td style=\"width: 501.9px; height: 30px;\">Write the formula for the mean:<\/td>\n<td style=\"width: 333.1px; height: 30px;\">[latex]\\text{mean}={\\Large\\frac{\\text{sum of all the numbers}}{n}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"width: 501.9px; height: 30px;\">Write the sum of the numbers in the numerator.<\/td>\n<td style=\"width: 333.1px; height: 30px;\">[latex]\\text{mean}={\\Large\\frac{8+12+15+9+6}{n}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"width: 501.9px; height: 30px;\">Count how many numbers are in the set. There are [latex]5[\/latex] numbers in the set, so [latex]n=5[\/latex] .<\/td>\n<td style=\"width: 333.1px; height: 30px;\">[latex]\\text{mean}={\\Large\\frac{8+12+15+9+6}{5}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 501.9px; height: 15px;\">Add the numbers in the numerator.<\/td>\n<td style=\"width: 333.1px; height: 15px;\">[latex]\\text{mean}={\\Large\\frac{50}{5}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 501.9px; height: 15px;\">Then divide.<\/td>\n<td style=\"width: 333.1px; height: 15px;\">[latex]\\text{mean}=10[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30.8281px;\">\n<td style=\"width: 501.9px; height: 30.8281px;\">Check to see that the mean is &#8216;typical&#8217;: [latex]10[\/latex] is neither less than [latex]6[\/latex] nor greater than [latex]15[\/latex].<\/td>\n<td style=\"width: 333.1px; height: 30.8281px;\">The mean is [latex]10[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">For the past four months, Daisy\u2019s cell phone bills were [latex]\\text{\\$42.75},\\text{\\$50.12},\\text{\\$41.54},\\text{\\$48.15}[\/latex]. Find the mean cost of Daisy\u2019s cell phone bills.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q42782\">Show Solution<\/button><\/p>\n<div id=\"q42782\" class=\"hidden-answer\" style=\"display: none\">Solution<\/p>\n<table id=\"eip-id1168469762854\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>Write the formula for the mean.<\/td>\n<td>[latex]\\text{mean}=\\Large\\frac{\\text{sum of all the numbers}}{n}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Count how many numbers are in the set. Call this [latex]n[\/latex] and write it in the denominator.<\/td>\n<td>[latex]\\text{mean}={\\Large\\frac{\\text{sum of all the numbers}}{4}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write the sum of all the numbers in the numerator.<\/td>\n<td>[latex]\\text{mean}={\\Large\\frac{42.75+50.12+41.54+48.15}{4}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the fraction.<\/td>\n<td>[latex]\\text{mean}={\\Large\\frac{182.56}{4}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]\\text{mean}=45.64[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Does [latex]\\text{\\$45.64}[\/latex] seem \u2018typical\u2019 of this set of numbers? Yes, it is neither less than [latex]\\text{\\$41.54}[\/latex] nor greater than [latex]\\text{\\$50.12}[\/latex].<\/p>\n<p>The mean cost of her cell phone bill was [latex]\\text{\\$45.64}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Find the median of [latex]12,13,19,9,11,15,\\text{and }18[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q42784\">Show Solution<\/button><\/p>\n<div id=\"q42784\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168468416053\" class=\"unnumbered unstyled\" summary=\"The figure shows the numbers 59, 60, and 62 separated by a small space from the numbers 65, 68, and 70. Each set of three has a bracket underneath grouping the numbers together.\">\n<tbody>\n<tr>\n<td>List the numbers in order from smallest to largest.<\/td>\n<td>[latex]9, 11, 12, 13, 15, 18, 19[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Count how many numbers are in the set. Call this [latex]n[\/latex] .<\/td>\n<td>[latex]n=7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Is [latex]n[\/latex] odd or even?<\/td>\n<td>odd<\/td>\n<\/tr>\n<tr>\n<td>The median is the middle value.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221814\/CNX_BMath_Figure_05_05_009_img.png\" alt=\"The series 9, 11, 12, 13, 15, 18, 19, with 13 labeled as the median\" width=\"327\" height=\"150\" \/><\/td>\n<\/tr>\n<tr>\n<td>The middle is the number in the [latex]4[\/latex]th position.<\/td>\n<td>So the median of the data is [latex]13[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Kristen received the following scores on her weekly math quizzes:<\/p>\n<div style=\"text-align: center;\">[latex]83,79,85,86,92,100,76,90,88,\\text{and }64[\/latex].<\/div>\n<p>Find her median score.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q982119\">Show Solution<\/button><\/p>\n<div id=\"q982119\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168467170036\" class=\"unnumbered unstyled\" summary=\"The problem says, 'Find the median of 83, 79, 85, 86, 92, 100, 76, 90, 88, and 64.' The first step says, 'List the numbers in order from smallest to largest,' and shows 64, 76, 79, 83, 85, 86, 88, 90, 92, 100. The next step says, 'Count how many numbers are in the set. Call this n. n equals 10.' The next step asks, 'Is n odd or even? Even.' The next line says, 'The median is the two middle values, the 5th and 6th numbers.' The ordered list of numbers is shown again with the first five numbers grouped together and labeled 5 numbers and the second five numbers are grouped together and labeled 5 numbers. The next step says 'Find the mean of 85 and 86.' The mean equals the sum of 85 plus 86 divided by 2, which equals 85.5. The last line shows 'Kristen's median score is 85.5'.\">\n<tbody>\n<tr>\n<td>Find the median of [latex]83, 79, 85, 86, 92, 100, 76, 90, 88,\\text{ and }64[\/latex].<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>List the numbers in order from smallest to largest.<\/td>\n<td>[latex]64, 76, 79, 83, 85, 86, 88, 90, 92, 100[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Count the number of data values in the set. Call this [latex]\\mathrm{n.}[\/latex]<\/td>\n<td>[latex]n=10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Is [latex]n[\/latex] odd or even?<\/td>\n<td>even<\/td>\n<\/tr>\n<tr>\n<td>The median is the mean of the two middle values, the [latex]5[\/latex]th and [latex]6[\/latex]th numbers.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221816\/CNX_BMath_Figure_05_05_010_img-01.png\" alt=\"The series 64, 76, 79, 83, 85, 86, 88, 90, 92, 100. The series is split into two groups of numbers, showing that 85 and 86 are the closest two to the median.\" width=\"319\" height=\"64\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the mean of [latex]85[\/latex] and [latex]86[\/latex].<\/td>\n<td>[latex]\\text{mean}={\\Large\\frac{85+86}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]\\text{mean}=85.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>Kristen&#8217;s median score is [latex]85.5[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>The ages of the students in a statistics class are listed here:<\/p>\n<p style=\"text-align: center;\">[latex]19[\/latex] , [latex]20[\/latex] , [latex]23[\/latex] , [latex]23[\/latex] , [latex]38[\/latex] , [latex]21[\/latex] , [latex]19[\/latex] , [latex]21[\/latex] , [latex]19[\/latex] , [latex]21[\/latex] , [latex]20[\/latex] , [latex]43[\/latex] , [latex]20[\/latex] , [latex]23[\/latex] , [latex]17[\/latex] , [latex]21[\/latex] , [latex]21[\/latex] , [latex]20[\/latex] , [latex]29[\/latex] , [latex]18[\/latex] , [latex]28[\/latex].<\/p>\n<p>What is the mode?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q888357\">Show Solution<\/button><\/p>\n<div id=\"q888357\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]21[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>Students listed the number of members in their household as follows:<\/p>\n<p style=\"text-align: center;\">[latex]6[\/latex] , [latex]2[\/latex] , [latex]5[\/latex] , [latex]6[\/latex] , [latex]3[\/latex] , [latex]7[\/latex] , [latex]5[\/latex] , [latex]6[\/latex] , [latex]5[\/latex] , [latex]3[\/latex] , [latex]4[\/latex] , [latex]4[\/latex] , [latex]5[\/latex] , [latex]7[\/latex] , [latex]6[\/latex] , [latex]4[\/latex] , [latex]5[\/latex] , [latex]2[\/latex] , [latex]1[\/latex] , [latex]5[\/latex].<\/p>\n<p>What is the mode?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q888927\">Show Solution<\/button><\/p>\n<div id=\"q888927\" class=\"hidden-answer\" style=\"display: none\">[latex]5[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Range, Standard Deviation, and Variance<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p><strong>Range<\/strong>, <strong>standard deviation<\/strong>, and <strong>variance <\/strong>are three key measures of dispersion in a dataset.<\/p>\n<p>The <strong>range <\/strong>of a dataset is the difference between the highest and lowest values, giving a simple measure of total spread.<\/p>\n<p style=\"text-align: center;\">[latex]\\text{Range } = \\text{ maximum value } \u2013 \\text{ minimum value }\u00a0 =\u00a0 \\text{ largest value } \u2013 \\text{ smallest value}[\/latex]<\/p>\n<p><strong>Standard deviation<\/strong>, a more complex measure, gauges the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values are close to the mean, while a high standard deviation suggests greater dispersion.<\/p>\n<p>A few important characteristics:<\/p>\n<ul>\n<li>Standard deviation is always positive. Standard deviation will be zero if all the data values are equal, and will get larger as the data spreads out.<\/li>\n<li>Standard deviation has the same units as the original data.<\/li>\n<li>Standard deviation, like the mean, can be highly influenced by outliers.<\/li>\n<\/ul>\n<p>The following formulas are used to calculate the standard deviation of a population and a sample:<\/p>\n<p style=\"padding-left: 60px;\"><strong>Standard deviation of a population<\/strong>: [latex]\\sigma = \\sqrt{\\dfrac{\\sum \\left(x-\\mu\\right)^2}{n}}[\/latex], where [latex]\\mu[\/latex] represents the population mean.<\/p>\n<p style=\"padding-left: 60px;\"><strong>Standard deviation of a sample<\/strong>: [latex]s=\\sqrt{\\dfrac{\\sum \\left(x-\\bar{x}\\right)^2}{n-1}}[\/latex], where [latex]\\bar{x}[\/latex] represents the sample mean.<\/p>\n<p><strong>Variance<\/strong>, often denoted by the squared units of the original data, is the average of the squared differences from the mean, effectively measuring how far each number in the set is from the mean.<\/p>\n<p style=\"text-align: center;\"><strong>Variance of a population<\/strong>:\u00a0 [latex]\\sigma^{2}=\\dfrac{\\sum\\left(x-\\mu\\right)^{2}}{n}[\/latex]<\/p>\n<p style=\"text-align: center;\"><strong>Variance of a sample<\/strong>:\u00a0 [latex]s^{2}=\\dfrac{\\sum\\left(x-\\bar{x}\\right)^{2}}{n-1}[\/latex]<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10350382&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=s7WTQ0H0Acc&amp;video_target=tpm-plugin-il9fa5lg-s7WTQ0H0Acc\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Measures+of+Variability+Range+Standard+Deviation+Variance.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMeasures of Variability (Range, Standard Deviation, Variance)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Five-Number Summary<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p>The <strong>Five-Number Summary<\/strong> is a descriptive statistic that provides information about a dataset. It consists of five values: the minimum, the first quartile (Q1 or 25th percentile), the median (or Q2 or 50th percentile), the third quartile (Q3 or 75th percentile), and the maximum. The minimum and maximum values depict the smallest and largest numbers in the dataset respectively. The first quartile is the median of the lower half of the data (not including the overall median), the third quartile is the median of the upper half, and the median is the middle value of the entire dataset.<\/p>\n<p>To find the first quartile, [latex]Q1[\/latex]:<\/p>\n<ol>\n<li>Begin by ordering the data from smallest to largest<\/li>\n<li>Compute the locator: [latex]L = 0.25n[\/latex]<\/li>\n<li>If [latex]L[\/latex] is a decimal value:\n<ul>\n<li>Round up to [latex]L+[\/latex]<\/li>\n<li>Use the data value in the [latex]L+[\/latex]th position<\/li>\n<\/ul>\n<\/li>\n<li>If [latex]L[\/latex] is a whole number:\n<ul>\n<li>Find the mean of the data values in the <em>[latex]L[\/latex]<\/em>th and [latex]L+1[\/latex]th positions.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p>To find the third quartile, [latex]Q3[\/latex]:<\/p>\n<p>Use the same procedure as for [latex]Q1[\/latex], but with locator: [latex]L = 0.75n[\/latex]<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10350383&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=1IXJZl_JP1o&amp;video_target=tpm-plugin-dyygmajn-1IXJZl_JP1o\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/What+is+a+5+Number+Summary.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat is a 5 Number Summary?\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Boxplots<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p><strong>Box plots<\/strong>, also known as box-and-whisker plots, are graphical representations used to depict the spread and skewness of a data set.<\/p>\n<p>They are constructed using the five-number summary: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum.<\/p>\n<p>The\u00a0<strong>interquartile range<\/strong>\u00a0(sometimes denoted as IQR) is the difference between the quartiles calculated as <br \/>\n[latex]Q3 \u2013 Q1[\/latex].<\/p>\n<p>The &#8216;box&#8217; of the plot represents the interquartile range (IQR), stretching from Q1 to Q3, and the line inside the box marks the median. &#8216;Whiskers&#8217; extend from the box to the minimum and maximum values, showing the full spread of the data. Outliers, if any, are typically indicated as individual points beyond the whiskers.<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10350384&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=fJZv9YeQ-qQ&amp;video_target=tpm-plugin-fje6fw09-fJZv9YeQ-qQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/BOX+AND+WHISKER+PLOTS+EXPLAINED.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cBOX AND WHISKER PLOTS EXPLAINED!\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">The box plot below is based on the [latex]9[\/latex] female height data with five-number summary:<\/p>\n<div style=\"text-align: center;\">[latex]59[\/latex], [latex]62[\/latex], [latex]66[\/latex], [latex]69[\/latex], [latex]72[\/latex].<\/div>\n<p>\u00a0<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12885\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175921\/large-heights.png\" alt=\"Number line titled Heights (inches), in increments of 1 from 55-75. Above this, a vertical line indicates 59. A horizontal line connects this to the next vertical line, 62. This line forms the left side of a rectangle; a line at 66 is its right side. The line at 66 also serves as the left side of another rectangle, with a line at 69 as its right side. This line at 69 connects with a horizontal line to a final vertical line at 72.\" width=\"600\" height=\"125\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175921\/large-heights.png 786w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175921\/large-heights-300x63.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175921\/large-heights-768x160.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175921\/large-heights-65x14.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175921\/large-heights-225x47.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175921\/large-heights-350x73.png 350w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/div>\n<p>&nbsp;<\/p>\n<hr \/>\n<p>The box plot below is based on the household income data with five-number summary:<\/p>\n<div style=\"text-align: center;\">[latex]15[\/latex], [latex]27.5[\/latex], [latex]35[\/latex], [latex]40[\/latex], [latex]50[\/latex]<\/div>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12886\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175940\/large-household-incomes.png\" alt=\"Number line titled Thousands of Dollars, in increments of 5 from 0-55. Above this, a vertical line indicates 15. A horizontal line connects this to the next vertical line, 27.5. This line forms the left side of a rectangle; a line at 35 is its right side. The line at 35 also serves as the left side of another rectangle, with a line at 40 as its right side. This line at 40 connects with a horizontal line to a final vertical line at 50.\" width=\"600\" height=\"174\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175940\/large-household-incomes.png 778w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175940\/large-household-incomes-300x87.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175940\/large-household-incomes-768x222.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175940\/large-household-incomes-65x19.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175940\/large-household-incomes-225x65.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/04\/12175940\/large-household-incomes-350x101.png 350w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":17,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1572,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1629"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":42,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1629\/revisions"}],"predecessor-version":[{"id":15389,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1629\/revisions\/15389"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/1572"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1629\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=1629"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=1629"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=1629"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=1629"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}