{"id":339,"date":"2022-06-17T15:36:52","date_gmt":"2022-06-17T15:36:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/applications-of-histograms-what-to-know\/"},"modified":"2022-06-17T15:45:30","modified_gmt":"2022-06-17T15:45:30","slug":"applications-of-histograms-what-to-know","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/applications-of-histograms-what-to-know\/","title":{"raw":"Applications of Histograms: Learn It 1","rendered":"Applications of Histograms: Learn It 1"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Goals<\/h3>\r\n<ul>\r\n \t<li>Describe the shape of the distribution of a quantitative variable.<\/li>\r\n \t<li>Describe the center of the distribution of a quantitative variable.<\/li>\r\n \t<li>Describe the spread of the distribution of a quantitative variable.<\/li>\r\n \t<li>Identify any outliers in the distribution of a quantitative variable.<\/li>\r\n \t<li>Identify a graphical display given its description.<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the previous section and activity, you learned how to use graphs (histograms and dotplots) to visualize the distribution of a quantitative variable. By displaying the data in a graph, you were able to answer questions about the distribution, and you began to develop strategies for choosing the most helpful graph to use for a given situation.\r\n\r\nIn the upcoming activity, you'll need to use a histogram to describe the distribution of a quantitative variable and answer questions about it. Let's prepare for that now\u00a0by learning about the four features used to describe a quantitative distribution: shape, center, spread, and the presence of outliers.\r\n\r\n<img class=\"aligncenter size-full wp-image-3012\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2021\/10\/10123642\/512px-Adelie_Penguin2.jpg\" alt=\"4 penguins, their wings outspread, running over a rock in front of water with an iceberg floating in the background\" width=\"512\" height=\"341\" \/>\r\n<h2>Let\u2019s study penguins!<\/h2>\r\nThe data for this example includes the species and size measurements of\u00a0[latex]342[\/latex] penguins found foraging near Palmer Station, Antarctica.[footnote]Horst, A., Hill, A., &amp; Gorman, K. (n.d.). <em>palmerpenguins<\/em>. Github. https:\/\/allisonhorst.github.io\/palmerpenguins\/[\/footnote] The following table contains\u00a0[latex]10[\/latex] observations.\r\n<table style=\"height: 624px;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 58.7188px; text-align: center;\" colspan=\"8\"><strong>Species and Size of Penguins<\/strong><strong>\r\n<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"height: 11px; width: 58.7188px; text-align: center;\"><strong>species<\/strong><\/td>\r\n<td style=\"height: 11px; width: 71.1562px; text-align: center;\"><strong>island<\/strong><\/td>\r\n<td style=\"height: 11px; width: 117.344px; text-align: center;\"><strong>bill_length_mm<\/strong><\/td>\r\n<td style=\"height: 11px; width: 112.891px; text-align: center;\"><strong>bill_depth_mm<\/strong><\/td>\r\n<td style=\"height: 11px; width: 143.125px; text-align: center;\"><strong>flipper_length_mm<\/strong><\/td>\r\n<td style=\"height: 11px; width: 106.719px; text-align: center;\"><strong>body_mass_g<\/strong><\/td>\r\n<td style=\"height: 11px; width: 48px; text-align: center;\"><strong>sex<\/strong><\/td>\r\n<td style=\"height: 11px; width: 35.6094px; text-align: center;\"><strong>year<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 58.7188px; text-align: center;\">Adelie<\/td>\r\n<td style=\"height: 12px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\r\n<td style=\"height: 12px; width: 117.344px; text-align: center;\">[latex]39.1[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 112.891px; text-align: center;\">[latex]18.7[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 143.125px; text-align: center;\">[latex]181[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 106.719px; text-align: center;\">[latex]3,750[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 48px; text-align: center;\">male<\/td>\r\n<td style=\"height: 12px; width: 35.6094px; text-align: center;\">2007<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"height: 17px; width: 58.7188px; text-align: center;\">Adelie<\/td>\r\n<td style=\"height: 17px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\r\n<td style=\"height: 17px; width: 117.344px; text-align: center;\">[latex]39.5[\/latex]<\/td>\r\n<td style=\"height: 17px; width: 112.891px; text-align: center;\">[latex]17.4[\/latex]<\/td>\r\n<td style=\"height: 17px; width: 143.125px; text-align: center;\">[latex]186[\/latex]<\/td>\r\n<td style=\"height: 17px; width: 106.719px; text-align: center;\">[latex]3,800[\/latex]<\/td>\r\n<td style=\"height: 17px; width: 48px; text-align: center;\">female<\/td>\r\n<td style=\"height: 17px; width: 35.6094px; text-align: center;\">2007<\/td>\r\n<\/tr>\r\n<tr style=\"height: 73px;\">\r\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\r\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\r\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">[latex]40.3[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">[latex]18[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">[latex]195[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">[latex]3,250[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 48px; text-align: center;\">female<\/td>\r\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\r\n<\/tr>\r\n<tr style=\"height: 73px;\">\r\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\r\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\r\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">N\/A<\/td>\r\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">N\/A<\/td>\r\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">N\/A<\/td>\r\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">N\/A<\/td>\r\n<td style=\"height: 73px; width: 48px; text-align: center;\">N\/A<\/td>\r\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\r\n<\/tr>\r\n<tr style=\"height: 73px;\">\r\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\r\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\r\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">[latex]36.7[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">[latex]19.3[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">[latex]193[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">[latex]3,450[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 48px; text-align: center;\">female<\/td>\r\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\r\n<\/tr>\r\n<tr style=\"height: 73px;\">\r\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\r\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\r\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">[latex]39.3[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">[latex]20.6[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">[latex]190[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">[latex]3,650[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 48px; text-align: center;\">male<\/td>\r\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\r\n<\/tr>\r\n<tr style=\"height: 73px;\">\r\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\r\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\r\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">[latex]38.9[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">[latex]17.8[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">[latex]181[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">[latex]3,625[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 48px; text-align: center;\">female<\/td>\r\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\r\n<\/tr>\r\n<tr style=\"height: 73px;\">\r\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\r\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\r\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">[latex]39.2[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">[latex]19.6[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">[latex]195[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">[latex]4,675[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 48px; text-align: center;\">male<\/td>\r\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\r\n<\/tr>\r\n<tr style=\"height: 73px;\">\r\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\r\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\r\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">[latex]34.1[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">[latex]18.1[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">[latex]193[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">[latex]3,475[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 48px; text-align: center;\">N\/A<\/td>\r\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\r\n<\/tr>\r\n<tr style=\"height: 73px;\">\r\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\r\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\r\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">[latex]42[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">[latex]20.2[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">[latex]190[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">[latex]4,250[\/latex]<\/td>\r\n<td style=\"height: 73px; width: 48px; text-align: center;\">N\/A<\/td>\r\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe following is the <strong>data dictionary<\/strong> for the variables in the table:\r\n<ul>\r\n \t<li><em><strong>species<\/strong>: <\/em>A factor denoting penguin species (Ad\u00e9lie, Chinstrap, or Gentoo)<\/li>\r\n \t<li><em><strong>island<\/strong>: <\/em>A factor denoting the island in Palmer Archipelago, Antarctica (Biscoe, Dream, or Torgersen)<\/li>\r\n \t<li><em><strong>bill_length_mm<\/strong>:<\/em> A number denoting bill length (millimeters)<\/li>\r\n \t<li><em><strong>bill_depth_mm<\/strong>:<\/em> A number denoting bill depth (millimeters)<\/li>\r\n \t<li><em><strong>flipper_length_mm<\/strong>:<\/em> An integer denoting flipper length (millimeters)<\/li>\r\n \t<li><em><strong>body_mass_g<\/strong>:<\/em> An integer denoting body mass (grams)<\/li>\r\n \t<li><em><strong>sex<\/strong>:<\/em> A factor denoting penguin sex (female or male)<\/li>\r\n \t<li><em><strong>year<\/strong>:<\/em> An integer denoting the study year (2007, 2008, or 2009)<\/li>\r\n<\/ul>\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\r\nSelect the data set <strong>Penguins \u2013 Body Mass<\/strong> and make a <strong>Histogram<\/strong> of the variable <em>body_mass_g<\/em>, the total body weight in grams. <strong>Select Binwidth for Histogram<\/strong>: 500.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 1<\/h3>\r\n[ohm_question hide_question_numbers=1]240614[\/ohm_question]\r\n\r\n[reveal-answer q=\"377964\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"377964\"]Make sure you have selected Single Group and the correct data set: Penguins - Body Mass.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 2<\/h3>\r\n[ohm_question hide_question_numbers=1]240868[\/ohm_question]\r\n\r\n[reveal-answer q=\"268778\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"268778\"]The height of the bars indicates frequency. The width indicates an interval of variable values.[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following activity, you will use the histogram to describe features of the distribution of a quantitative variable. The features used to describe the distribution of a quantitative variable are the <strong>shape,<\/strong> <strong>center,<\/strong> <strong>spread,<\/strong> and <strong>presence of outliers<\/strong>. You will learn to summarize all of these features in a single description, but for now let's discuss them one by one.\r\n<div class=\"textbox tryit\">\r\n<h3>determining shape, center, spread, and the presence outliers<\/h3>\r\n<span style=\"background-color: #99cc00;\">[Perspective video - a 3-instructor video describing how to determine shape, center (just a visual estimation -- we don't talk about mean\/median yet!), spread (just the range for now!), and the presence of outliers (just the appearance for now - we don't use statistical methods yet!) -- emphasize skew, modality, and center. Emphasize what the term \"<strong><em>distribution<\/em><\/strong>\" of a quantitative variable means. It would be very cool to have an animation in which the numbers from a data table lift from the table en masse and sprinkle down onto a histogram, all falling into place in the distribution.]<\/span>\r\n\r\n<\/div>\r\n<h2 id=\"Shape\">Shape<\/h2>\r\nLet\u2019s begin with the <strong>shape. <\/strong>The description of shape includes two parts: (1) overall pattern (left skewed, right skewed, symmetric) and (2) the number of peaks (unimodal, bimodal, multimodal).\r\n<h3>Skew<\/h3>\r\nLet\u2019s take a look at the first component. The overall pattern can be described as one of the following:\r\n\r\n<img class=\"alignnone wp-image-973\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11184151\/Picture131-300x116.png\" alt=\"Three bar graphs are shown, depicting different ways data can be distributed. The graph on the left shows left-skewed data, where there is little to no data on the left, and a steep increase in the amount of data as we move to the right, creating a long \u201ctail\u201d on the left. The middle graph shows symmetric data, where most of the data lies in the center of the graph, with a steep decrease of data as you move to the right or left. The graph on the right shows right-skewed data, where there is little to no data on the right, and a steep increase in the amount of data as we move to the left, creating a long \u201ctail\u201d to the right.\" width=\"892\" height=\"345\" \/>\r\n<ul>\r\n \t<li><strong>Symmetric:<\/strong> The left and right sides of the distribution (closely) mirror each other. If you drew a vertical line down the center of the distribution and folded the distribution in half, the left and right sides would closely match one another.<\/li>\r\n \t<li><strong>Left skewed:<\/strong> The distribution has a longer tail to the left.<\/li>\r\n \t<li><strong>Right skewed:<\/strong> The distribution has a longer tail to the right.<\/li>\r\n<\/ul>\r\n<h3>Modality<\/h3>\r\nIn addition to the overall pattern, the description of shape also includes the number of peaks. This is also known as the <strong>modality<\/strong>. The modality can be described as one of the following:\r\n\r\n<img class=\"alignnone wp-image-974\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11184156\/Picture141-300x178.png\" alt=\"Four bar graphs are shown, depicting different modes of data. The first graph is unimodal, where the data has one peak it is centered around. The second graph is bimodal, where there are two peaks of data with a trough in the middle separating the two peaks. The third graph is multimodal, where there are multiple peaks of data with troughs between them (in this example, there are three peaks of data). The last graph is uniform, showing a consistent spread of data with no distinct peaks.\" width=\"900\" height=\"534\" \/>\r\n<ul>\r\n \t<li><strong>Unimodal:<\/strong> There is one prominent peak.<\/li>\r\n \t<li><strong>Bimodal:<\/strong> There are two prominent peaks.<\/li>\r\n \t<li><strong>Multimodal:<\/strong> There are three or more prominent peaks.<\/li>\r\n \t<li><strong>Uniform:<\/strong> There are no prominent peaks.<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWhat two elements make up a description of the\u00a0<strong>shape<\/strong> of a distribution?\r\n\r\n[reveal-answer q=\"950195\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"950195\"]Skew and modality are both used to describe the shape of a distribution[\/hidden-answer]\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]240870[\/ohm_question]\r\n\r\n[reveal-answer q=\"137505\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"137505\"]Select all answers that apply. There may be more than one. See the description of <em>shape<\/em> above for guidance.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 id=\"Center\">Center<\/h2>\r\nThe next feature is the <strong>center<\/strong>. The <strong>center<\/strong> describes the location of the middle of the distribution. The center is a number that describes a typical value. For example, one way to think about the center is that it could be the point in the distribution where about half of the observations are below it and half are above it. For now, we will use the histogram to get an approximate value of the center. (In a later lesson, you will learn statistics used to describe the center more precisely.)\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n&nbsp;\r\n\r\nWhat is one way to think about the center as the location of the middle of a distribution?\r\n\r\na) The center is always the value that splits the data in half.\r\n\r\nb) The center is always the value indicated by the tallest bar in the graph.\r\n\r\nc) The center is always the middle number between the highest and lowest values on the horizontal axis.\r\n\r\n[reveal-answer q=\"371864\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"371864\"](a) The center, as the middle of a distribution, is always the value that splits the data in half, with\u00a0[latex]50[\/latex]% of the data above this value and\u00a0[latex]50[\/latex]% of it below. This is not necessarily the most frequently appearing value nor the middle-most, but it can be.[\/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]240871[\/ohm_question]\r\n\r\n[reveal-answer q=\"702095\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"702095\"]Which value appears to split the data in half?[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 id=\"Spread\">Spread<\/h2>\r\nNext, let\u2019s approximate the spread. The <strong>spread<\/strong> is a measure of how much the values in a data set tend to differ from one another. One way we can find the spread is by finding the minimum and maximum values in the data and calculating the difference between them. This difference is called the <strong>range.<\/strong>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nHow can we find the range of a distribution?\r\n\r\na) By finding the minimum and maximum values on the graph's horizontal axis and\u00a0calculating the difference between them.\r\n\r\nb)\u00a0By finding the minimum and maximum values on the graph's vertical axis and\u00a0calculating the difference between them.\r\n\r\nc)\u00a0By finding the minimum and maximum values in the data and calculating the difference between them.\r\n\r\n[reveal-answer q=\"162261\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"162261\"](c) We can calculate the range by taking the difference between the minimum and maximum values in the data. Note that choice (a) would generally permit you to make an estimation of the range but using minimum and maximum values in the data will always return an accurate result.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 5<\/h3>\r\n[ohm_question hide_question_numbers=1]240872[\/ohm_question]\r\n\r\n[reveal-answer q=\"561443\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"561443\"]To calculate range, find the difference between the minimum and maximum data values.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 id=\"Outliers\">Outliers<\/h2>\r\nThe last feature in the description is the presence of outliers. <strong>Outliers <\/strong>are observations in the data that are unusual and outside the general pattern of the rest of the observations in the distribution. When working with a univariate (\"one variable\") distribution for a quantitative variable, an outlier is an observation that has an unusually high or unusually low value. It is good practice to make note of outliers, as these observations can sometimes influence the statistical results (e.g., the value of the range).\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWhy is it a good practice to make note of outliers?\r\n\r\na) These observations can sometimes influence the location of the middle of the data.\r\n\r\nb) These observations can sometimes influence the value of the range.\r\n\r\nc) These observations can sometimes influence the modality of the graph.\r\n\r\n[reveal-answer q=\"644229\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"644229\"](b) These observations can sometimes influence the value of the range. The middle number of the data will always be the middle number despite the presence of an unusually high or low value. [\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 6<\/h3>\r\n[ohm_question hide_question_numbers=1]240873[\/ohm_question]\r\n\r\n[reveal-answer q=\"378911\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"378911\"]Do there appear to be any unusually low or high values in the data?[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 id=\"Identifying Graphical Displays\">Identifying Graphical Displays<\/h2>\r\nWhen visually assessing graphical displays of the distribution of a quantitative variable, we want to feel comfortable summarizing the graph with a description that includes all four characteristics: shape, center, spread, and the presence of outliers. It can be challenging to identify these features when the graph doesn't indicate a clear tendency, but with time and practice, you'll find that the accuracy of your predictions will improve. Question 7 below will help you get started.\r\n<div class=\"textbox tryit\">\r\n<h3>describing the characteristics of a graph<\/h3>\r\n<span style=\"background-color: #99cc00;\"><strong>[Worked Example \u2014 a 3-instructors worked example of describing the four characteristics of a graph with less than super-clear features. ]<\/strong><\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 7<\/h3>\r\n<img class=\"aligncenter wp-image-3337\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2021\/10\/18012001\/7.png\" alt=\"\" width=\"688\" height=\"232\" \/>\r\n\r\n[ohm_question hide_question_numbers=1]240615[\/ohm_question]\r\n\r\n[reveal-answer q=\"716060\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"716060\"]What do <em>you<\/em> think?[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nIn this lesson, you dove deeply into using a histogram to describe the distribution of quantitative variables and to answer questions about a distribution. You examined whether the shape of a distribution was symmetrical or skewed left or right, and you identified a distribution's modality. You practiced estimating the center and spread using a histogram, and you noted the presence of possible outliers that could affect the range or shape of the distribution. Let's summarize the skills you saw in each question.\r\n<ul>\r\n \t<li>In question 1, you used technology to make a histogram of a quantitative variable.<\/li>\r\n \t<li>In question 2, you used a histogram to answer questions about the distribution of a quantitative variable.<\/li>\r\n \t<li>In question 3, you described the shape of a distribution.<\/li>\r\n \t<li>In question 4 , you described the center of a distribution.<\/li>\r\n \t<li>In question 5, you described the spread of a distribution.<\/li>\r\n \t<li>In question 6, you identified outliers in a distribution.<\/li>\r\n \t<li>In question 7, you matched the description of a distribution to the graphical display.<\/li>\r\n<\/ul>\r\nHopefully, you are beginning to feel more confident at describing the characteristics of a distribution of a quantitative variable such as a histogram. If so, it's time to move on to the next activity in Forming Connections, where you'll put your new skills to work analyzing and describing distributions of quantitative variables.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Goals<\/h3>\n<ul>\n<li>Describe the shape of the distribution of a quantitative variable.<\/li>\n<li>Describe the center of the distribution of a quantitative variable.<\/li>\n<li>Describe the spread of the distribution of a quantitative variable.<\/li>\n<li>Identify any outliers in the distribution of a quantitative variable.<\/li>\n<li>Identify a graphical display given its description.<\/li>\n<\/ul>\n<\/div>\n<p>In the previous section and activity, you learned how to use graphs (histograms and dotplots) to visualize the distribution of a quantitative variable. By displaying the data in a graph, you were able to answer questions about the distribution, and you began to develop strategies for choosing the most helpful graph to use for a given situation.<\/p>\n<p>In the upcoming activity, you&#8217;ll need to use a histogram to describe the distribution of a quantitative variable and answer questions about it. Let&#8217;s prepare for that now\u00a0by learning about the four features used to describe a quantitative distribution: shape, center, spread, and the presence of outliers.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-3012\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2021\/10\/10123642\/512px-Adelie_Penguin2.jpg\" alt=\"4 penguins, their wings outspread, running over a rock in front of water with an iceberg floating in the background\" width=\"512\" height=\"341\" \/><\/p>\n<h2>Let\u2019s study penguins!<\/h2>\n<p>The data for this example includes the species and size measurements of\u00a0[latex]342[\/latex] penguins found foraging near Palmer Station, Antarctica.<a class=\"footnote\" title=\"Horst, A., Hill, A., &amp; Gorman, K. (n.d.). palmerpenguins. Github. https:\/\/allisonhorst.github.io\/palmerpenguins\/\" id=\"return-footnote-339-1\" href=\"#footnote-339-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> The following table contains\u00a0[latex]10[\/latex] observations.<\/p>\n<table style=\"height: 624px;\">\n<tbody>\n<tr>\n<td style=\"width: 58.7188px; text-align: center;\" colspan=\"8\"><strong>Species and Size of Penguins<\/strong><strong><br \/>\n<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"height: 11px; width: 58.7188px; text-align: center;\"><strong>species<\/strong><\/td>\n<td style=\"height: 11px; width: 71.1562px; text-align: center;\"><strong>island<\/strong><\/td>\n<td style=\"height: 11px; width: 117.344px; text-align: center;\"><strong>bill_length_mm<\/strong><\/td>\n<td style=\"height: 11px; width: 112.891px; text-align: center;\"><strong>bill_depth_mm<\/strong><\/td>\n<td style=\"height: 11px; width: 143.125px; text-align: center;\"><strong>flipper_length_mm<\/strong><\/td>\n<td style=\"height: 11px; width: 106.719px; text-align: center;\"><strong>body_mass_g<\/strong><\/td>\n<td style=\"height: 11px; width: 48px; text-align: center;\"><strong>sex<\/strong><\/td>\n<td style=\"height: 11px; width: 35.6094px; text-align: center;\"><strong>year<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 58.7188px; text-align: center;\">Adelie<\/td>\n<td style=\"height: 12px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\n<td style=\"height: 12px; width: 117.344px; text-align: center;\">[latex]39.1[\/latex]<\/td>\n<td style=\"height: 12px; width: 112.891px; text-align: center;\">[latex]18.7[\/latex]<\/td>\n<td style=\"height: 12px; width: 143.125px; text-align: center;\">[latex]181[\/latex]<\/td>\n<td style=\"height: 12px; width: 106.719px; text-align: center;\">[latex]3,750[\/latex]<\/td>\n<td style=\"height: 12px; width: 48px; text-align: center;\">male<\/td>\n<td style=\"height: 12px; width: 35.6094px; text-align: center;\">2007<\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"height: 17px; width: 58.7188px; text-align: center;\">Adelie<\/td>\n<td style=\"height: 17px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\n<td style=\"height: 17px; width: 117.344px; text-align: center;\">[latex]39.5[\/latex]<\/td>\n<td style=\"height: 17px; width: 112.891px; text-align: center;\">[latex]17.4[\/latex]<\/td>\n<td style=\"height: 17px; width: 143.125px; text-align: center;\">[latex]186[\/latex]<\/td>\n<td style=\"height: 17px; width: 106.719px; text-align: center;\">[latex]3,800[\/latex]<\/td>\n<td style=\"height: 17px; width: 48px; text-align: center;\">female<\/td>\n<td style=\"height: 17px; width: 35.6094px; text-align: center;\">2007<\/td>\n<\/tr>\n<tr style=\"height: 73px;\">\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">[latex]40.3[\/latex]<\/td>\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">[latex]18[\/latex]<\/td>\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">[latex]195[\/latex]<\/td>\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">[latex]3,250[\/latex]<\/td>\n<td style=\"height: 73px; width: 48px; text-align: center;\">female<\/td>\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\n<\/tr>\n<tr style=\"height: 73px;\">\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">N\/A<\/td>\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">N\/A<\/td>\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">N\/A<\/td>\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">N\/A<\/td>\n<td style=\"height: 73px; width: 48px; text-align: center;\">N\/A<\/td>\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\n<\/tr>\n<tr style=\"height: 73px;\">\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">[latex]36.7[\/latex]<\/td>\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">[latex]19.3[\/latex]<\/td>\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">[latex]193[\/latex]<\/td>\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">[latex]3,450[\/latex]<\/td>\n<td style=\"height: 73px; width: 48px; text-align: center;\">female<\/td>\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\n<\/tr>\n<tr style=\"height: 73px;\">\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">[latex]39.3[\/latex]<\/td>\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">[latex]20.6[\/latex]<\/td>\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">[latex]190[\/latex]<\/td>\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">[latex]3,650[\/latex]<\/td>\n<td style=\"height: 73px; width: 48px; text-align: center;\">male<\/td>\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\n<\/tr>\n<tr style=\"height: 73px;\">\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">[latex]38.9[\/latex]<\/td>\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">[latex]17.8[\/latex]<\/td>\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">[latex]181[\/latex]<\/td>\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">[latex]3,625[\/latex]<\/td>\n<td style=\"height: 73px; width: 48px; text-align: center;\">female<\/td>\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\n<\/tr>\n<tr style=\"height: 73px;\">\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">[latex]39.2[\/latex]<\/td>\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">[latex]19.6[\/latex]<\/td>\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">[latex]195[\/latex]<\/td>\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">[latex]4,675[\/latex]<\/td>\n<td style=\"height: 73px; width: 48px; text-align: center;\">male<\/td>\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\n<\/tr>\n<tr style=\"height: 73px;\">\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">[latex]34.1[\/latex]<\/td>\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">[latex]18.1[\/latex]<\/td>\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">[latex]193[\/latex]<\/td>\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">[latex]3,475[\/latex]<\/td>\n<td style=\"height: 73px; width: 48px; text-align: center;\">N\/A<\/td>\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\n<\/tr>\n<tr style=\"height: 73px;\">\n<td style=\"height: 73px; width: 58.7188px; text-align: center;\">Adelie<\/td>\n<td style=\"height: 73px; width: 71.1562px; text-align: center;\">Torgersen<\/td>\n<td style=\"height: 73px; width: 117.344px; text-align: center;\">[latex]42[\/latex]<\/td>\n<td style=\"height: 73px; width: 112.891px; text-align: center;\">[latex]20.2[\/latex]<\/td>\n<td style=\"height: 73px; width: 143.125px; text-align: center;\">[latex]190[\/latex]<\/td>\n<td style=\"height: 73px; width: 106.719px; text-align: center;\">[latex]4,250[\/latex]<\/td>\n<td style=\"height: 73px; width: 48px; text-align: center;\">N\/A<\/td>\n<td style=\"height: 73px; width: 35.6094px; text-align: center;\">2007<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The following is the <strong>data dictionary<\/strong> for the variables in the table:<\/p>\n<ul>\n<li><em><strong>species<\/strong>: <\/em>A factor denoting penguin species (Ad\u00e9lie, Chinstrap, or Gentoo)<\/li>\n<li><em><strong>island<\/strong>: <\/em>A factor denoting the island in Palmer Archipelago, Antarctica (Biscoe, Dream, or Torgersen)<\/li>\n<li><em><strong>bill_length_mm<\/strong>:<\/em> A number denoting bill length (millimeters)<\/li>\n<li><em><strong>bill_depth_mm<\/strong>:<\/em> A number denoting bill depth (millimeters)<\/li>\n<li><em><strong>flipper_length_mm<\/strong>:<\/em> An integer denoting flipper length (millimeters)<\/li>\n<li><em><strong>body_mass_g<\/strong>:<\/em> An integer denoting body mass (grams)<\/li>\n<li><em><strong>sex<\/strong>:<\/em> A factor denoting penguin sex (female or male)<\/li>\n<li><em><strong>year<\/strong>:<\/em> An integer denoting the study year (2007, 2008, or 2009)<\/li>\n<\/ul>\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>Select the data set <strong>Penguins \u2013 Body Mass<\/strong> and make a <strong>Histogram<\/strong> of the variable <em>body_mass_g<\/em>, the total body weight in grams. <strong>Select Binwidth for Histogram<\/strong>: 500.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 1<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240614\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240614&theme=oea&iframe_resize_id=ohm240614\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q377964\">Hint<\/span><\/p>\n<div id=\"q377964\" class=\"hidden-answer\" style=\"display: none\">Make sure you have selected Single Group and the correct data set: Penguins &#8211; Body Mass.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 2<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240868\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240868&theme=oea&iframe_resize_id=ohm240868\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q268778\">Hint<\/span><\/p>\n<div id=\"q268778\" class=\"hidden-answer\" style=\"display: none\">The height of the bars indicates frequency. The width indicates an interval of variable values.<\/div>\n<\/div>\n<\/div>\n<p>In the following activity, you will use the histogram to describe features of the distribution of a quantitative variable. The features used to describe the distribution of a quantitative variable are the <strong>shape,<\/strong> <strong>center,<\/strong> <strong>spread,<\/strong> and <strong>presence of outliers<\/strong>. You will learn to summarize all of these features in a single description, but for now let&#8217;s discuss them one by one.<\/p>\n<div class=\"textbox tryit\">\n<h3>determining shape, center, spread, and the presence outliers<\/h3>\n<p><span style=\"background-color: #99cc00;\">[Perspective video &#8211; a 3-instructor video describing how to determine shape, center (just a visual estimation &#8212; we don&#8217;t talk about mean\/median yet!), spread (just the range for now!), and the presence of outliers (just the appearance for now &#8211; we don&#8217;t use statistical methods yet!) &#8212; emphasize skew, modality, and center. Emphasize what the term &#8220;<strong><em>distribution<\/em><\/strong>&#8221; of a quantitative variable means. It would be very cool to have an animation in which the numbers from a data table lift from the table en masse and sprinkle down onto a histogram, all falling into place in the distribution.]<\/span><\/p>\n<\/div>\n<h2 id=\"Shape\">Shape<\/h2>\n<p>Let\u2019s begin with the <strong>shape. <\/strong>The description of shape includes two parts: (1) overall pattern (left skewed, right skewed, symmetric) and (2) the number of peaks (unimodal, bimodal, multimodal).<\/p>\n<h3>Skew<\/h3>\n<p>Let\u2019s take a look at the first component. The overall pattern can be described as one of the following:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-973\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11184151\/Picture131-300x116.png\" alt=\"Three bar graphs are shown, depicting different ways data can be distributed. The graph on the left shows left-skewed data, where there is little to no data on the left, and a steep increase in the amount of data as we move to the right, creating a long \u201ctail\u201d on the left. The middle graph shows symmetric data, where most of the data lies in the center of the graph, with a steep decrease of data as you move to the right or left. The graph on the right shows right-skewed data, where there is little to no data on the right, and a steep increase in the amount of data as we move to the left, creating a long \u201ctail\u201d to the right.\" width=\"892\" height=\"345\" \/><\/p>\n<ul>\n<li><strong>Symmetric:<\/strong> The left and right sides of the distribution (closely) mirror each other. If you drew a vertical line down the center of the distribution and folded the distribution in half, the left and right sides would closely match one another.<\/li>\n<li><strong>Left skewed:<\/strong> The distribution has a longer tail to the left.<\/li>\n<li><strong>Right skewed:<\/strong> The distribution has a longer tail to the right.<\/li>\n<\/ul>\n<h3>Modality<\/h3>\n<p>In addition to the overall pattern, the description of shape also includes the number of peaks. This is also known as the <strong>modality<\/strong>. The modality can be described as one of the following:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-974\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11184156\/Picture141-300x178.png\" alt=\"Four bar graphs are shown, depicting different modes of data. The first graph is unimodal, where the data has one peak it is centered around. The second graph is bimodal, where there are two peaks of data with a trough in the middle separating the two peaks. The third graph is multimodal, where there are multiple peaks of data with troughs between them (in this example, there are three peaks of data). The last graph is uniform, showing a consistent spread of data with no distinct peaks.\" width=\"900\" height=\"534\" \/><\/p>\n<ul>\n<li><strong>Unimodal:<\/strong> There is one prominent peak.<\/li>\n<li><strong>Bimodal:<\/strong> There are two prominent peaks.<\/li>\n<li><strong>Multimodal:<\/strong> There are three or more prominent peaks.<\/li>\n<li><strong>Uniform:<\/strong> There are no prominent peaks.<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>What two elements make up a description of the\u00a0<strong>shape<\/strong> of a distribution?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q950195\">Show Solution<\/span><\/p>\n<div id=\"q950195\" class=\"hidden-answer\" style=\"display: none\">Skew and modality are both used to describe the shape of a distribution<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 3<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240870\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240870&theme=oea&iframe_resize_id=ohm240870\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q137505\">Hint<\/span><\/p>\n<div id=\"q137505\" class=\"hidden-answer\" style=\"display: none\">Select all answers that apply. There may be more than one. See the description of <em>shape<\/em> above for guidance.<\/div>\n<\/div>\n<\/div>\n<h2 id=\"Center\">Center<\/h2>\n<p>The next feature is the <strong>center<\/strong>. The <strong>center<\/strong> describes the location of the middle of the distribution. The center is a number that describes a typical value. For example, one way to think about the center is that it could be the point in the distribution where about half of the observations are below it and half are above it. For now, we will use the histogram to get an approximate value of the center. (In a later lesson, you will learn statistics used to describe the center more precisely.)<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>&nbsp;<\/p>\n<p>What is one way to think about the center as the location of the middle of a distribution?<\/p>\n<p>a) The center is always the value that splits the data in half.<\/p>\n<p>b) The center is always the value indicated by the tallest bar in the graph.<\/p>\n<p>c) The center is always the middle number between the highest and lowest values on the horizontal axis.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q371864\">Show Solution<\/span><\/p>\n<div id=\"q371864\" class=\"hidden-answer\" style=\"display: none\">(a) The center, as the middle of a distribution, is always the value that splits the data in half, with\u00a0[latex]50[\/latex]% of the data above this value and\u00a0[latex]50[\/latex]% of it below. This is not necessarily the most frequently appearing value nor the middle-most, but it can be.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 4<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240871\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240871&theme=oea&iframe_resize_id=ohm240871\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q702095\">Hint<\/span><\/p>\n<div id=\"q702095\" class=\"hidden-answer\" style=\"display: none\">Which value appears to split the data in half?<\/div>\n<\/div>\n<\/div>\n<h2 id=\"Spread\">Spread<\/h2>\n<p>Next, let\u2019s approximate the spread. The <strong>spread<\/strong> is a measure of how much the values in a data set tend to differ from one another. One way we can find the spread is by finding the minimum and maximum values in the data and calculating the difference between them. This difference is called the <strong>range.<\/strong><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>How can we find the range of a distribution?<\/p>\n<p>a) By finding the minimum and maximum values on the graph&#8217;s horizontal axis and\u00a0calculating the difference between them.<\/p>\n<p>b)\u00a0By finding the minimum and maximum values on the graph&#8217;s vertical axis and\u00a0calculating the difference between them.<\/p>\n<p>c)\u00a0By finding the minimum and maximum values in the data and calculating the difference between them.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q162261\">Show Solution<\/span><\/p>\n<div id=\"q162261\" class=\"hidden-answer\" style=\"display: none\">(c) We can calculate the range by taking the difference between the minimum and maximum values in the data. Note that choice (a) would generally permit you to make an estimation of the range but using minimum and maximum values in the data will always return an accurate result.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 5<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240872\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240872&theme=oea&iframe_resize_id=ohm240872\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q561443\">Hint<\/span><\/p>\n<div id=\"q561443\" class=\"hidden-answer\" style=\"display: none\">To calculate range, find the difference between the minimum and maximum data values.<\/div>\n<\/div>\n<\/div>\n<h2 id=\"Outliers\">Outliers<\/h2>\n<p>The last feature in the description is the presence of outliers. <strong>Outliers <\/strong>are observations in the data that are unusual and outside the general pattern of the rest of the observations in the distribution. When working with a univariate (&#8220;one variable&#8221;) distribution for a quantitative variable, an outlier is an observation that has an unusually high or unusually low value. It is good practice to make note of outliers, as these observations can sometimes influence the statistical results (e.g., the value of the range).<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Why is it a good practice to make note of outliers?<\/p>\n<p>a) These observations can sometimes influence the location of the middle of the data.<\/p>\n<p>b) These observations can sometimes influence the value of the range.<\/p>\n<p>c) These observations can sometimes influence the modality of the graph.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q644229\">Show Solution<\/span><\/p>\n<div id=\"q644229\" class=\"hidden-answer\" style=\"display: none\">(b) These observations can sometimes influence the value of the range. The middle number of the data will always be the middle number despite the presence of an unusually high or low value. <\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 6<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240873\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240873&theme=oea&iframe_resize_id=ohm240873\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q378911\">Hint<\/span><\/p>\n<div id=\"q378911\" class=\"hidden-answer\" style=\"display: none\">Do there appear to be any unusually low or high values in the data?<\/div>\n<\/div>\n<\/div>\n<h2 id=\"Identifying Graphical Displays\">Identifying Graphical Displays<\/h2>\n<p>When visually assessing graphical displays of the distribution of a quantitative variable, we want to feel comfortable summarizing the graph with a description that includes all four characteristics: shape, center, spread, and the presence of outliers. It can be challenging to identify these features when the graph doesn&#8217;t indicate a clear tendency, but with time and practice, you&#8217;ll find that the accuracy of your predictions will improve. Question 7 below will help you get started.<\/p>\n<div class=\"textbox tryit\">\n<h3>describing the characteristics of a graph<\/h3>\n<p><span style=\"background-color: #99cc00;\"><strong>[Worked Example \u2014 a 3-instructors worked example of describing the four characteristics of a graph with less than super-clear features. ]<\/strong><\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 7<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3337\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2021\/10\/18012001\/7.png\" alt=\"\" width=\"688\" height=\"232\" \/><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm240615\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240615&theme=oea&iframe_resize_id=ohm240615\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q716060\">Hint<\/span><\/p>\n<div id=\"q716060\" class=\"hidden-answer\" style=\"display: none\">What do <em>you<\/em> think?<\/div>\n<\/div>\n<\/div>\n<h2>Summary<\/h2>\n<p>In this lesson, you dove deeply into using a histogram to describe the distribution of quantitative variables and to answer questions about a distribution. You examined whether the shape of a distribution was symmetrical or skewed left or right, and you identified a distribution&#8217;s modality. You practiced estimating the center and spread using a histogram, and you noted the presence of possible outliers that could affect the range or shape of the distribution. Let&#8217;s summarize the skills you saw in each question.<\/p>\n<ul>\n<li>In question 1, you used technology to make a histogram of a quantitative variable.<\/li>\n<li>In question 2, you used a histogram to answer questions about the distribution of a quantitative variable.<\/li>\n<li>In question 3, you described the shape of a distribution.<\/li>\n<li>In question 4 , you described the center of a distribution.<\/li>\n<li>In question 5, you described the spread of a distribution.<\/li>\n<li>In question 6, you identified outliers in a distribution.<\/li>\n<li>In question 7, you matched the description of a distribution to the graphical display.<\/li>\n<\/ul>\n<p>Hopefully, you are beginning to feel more confident at describing the characteristics of a distribution of a quantitative variable such as a histogram. If so, it&#8217;s time to move on to the next activity in Forming Connections, where you&#8217;ll put your new skills to work analyzing and describing distributions of quantitative variables.<\/p>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-339-1\">Horst, A., Hill, A., &amp; Gorman, K. (n.d.). <em>palmerpenguins<\/em>. Github. https:\/\/allisonhorst.github.io\/palmerpenguins\/ <a href=\"#return-footnote-339-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":17533,"menu_order":19,"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-339","chapter","type-chapter","status-publish","hentry"],"part":160,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters\/339","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters\/339\/revisions"}],"predecessor-version":[{"id":379,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters\/339\/revisions\/379"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/parts\/160"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters\/339\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/media?parent=339"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapter-type?post=339"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/contributor?post=339"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/license?post=339"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}