{"id":341,"date":"2022-02-21T17:53:26","date_gmt":"2022-02-21T17:53:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/?post_type=chapter&#038;p=341"},"modified":"2022-05-20T16:46:42","modified_gmt":"2022-05-20T16:46:42","slug":"interpreting-the-mean-and-median-of-a-dataset-what-to-know","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/interpreting-the-mean-and-median-of-a-dataset-what-to-know\/","title":{"raw":"Interpreting the Mean and Median of a Data Set: What to Know","rendered":"Interpreting the Mean and Median of a Data Set: What to Know"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Goals<\/h3>\r\nAfter completing this section, you should feel comfortable performing these skills.\r\n<ul>\r\n \t<li><a href=\"#IntMeanMedian\">Interpret the median of a data set.<\/a><\/li>\r\n \t<li><a href=\"#IntMeanMedian\">Interpret the mean of a data set.<\/a><\/li>\r\n \t<li><a href=\"#IdentSkew\">Identify whether a data set is left-skewed, symmetric, or right-skewed.<\/a><\/li>\r\n \t<li><a href=\"#IdentSkew\">Identify in which data set the mean is greater than, less than, or approximately equal to the median.<\/a><\/li>\r\n \t<li><a href=\"#resistant\">Identify which of the mean or median is resistant to skew.<\/a><\/li>\r\n<\/ul>\r\nClick on a skill above to jump to its location in this section.\r\n\r\n<\/div>\r\nWhen examining the distribution of a quantitative variable using a histogram or a dotplot, we often find that the distribution follows a bell shape with a mound of observances in the middle of the distribution and even amounts of data falling to the right and left. But sometimes a distribution's values are bunched up to one side or the other, with a few observations stretching way out to the other side. You may recall from <em>What to Know About Applications of Histograms: 3D <\/em>that there are specialized statistical terms we use for these different distribution shapes: skewness and symmetry. In this section, you'll learn that there are certain ways the mean of the data relates to the median under these different shapes.\r\n\r\n<img class=\"aligncenter size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11184151\/Picture131.png\" alt=\"An image of three histograms: left skewed, in which the data is bunched up to the right with a long tail of data to the left; symmetric, in which the data is mounded in the center and falls away evenly to either side; and right-skewed, in which the data is bunched up to the left with a tail of data falling away to the right.\" width=\"1576\" height=\"608\" \/>\r\n<h2>Skewness<\/h2>\r\nRecall that we say a quantitative variable has a <strong>right-skewed<\/strong> distribution or a <strong>positive skew<\/strong> if there is a \"tail\" of infrequent values on the right (upper) end of the distribution. We say a data set has an approximately <strong>symmetric<\/strong> distribution if values are similarly distributed on either side of the mean\/median. We say a data set has a <strong>left-skewed<\/strong> distribution or a <strong>negative skew<\/strong> if there is a \"tail\" of infrequent values on the left (lower) end of the distribution.\r\n<div class=\"textbox tryit\">\r\n<h3>skewed distributions<\/h3>\r\n<span style=\"background-color: #ffff99;\">I'd like an animation here (super simple) of a data set that moves from right skew to symmetry to left skew with a slider students can manipulate. The labels would change over the slider: right skew \/ roughly symmetric \/ roughly symmetric \/ left skew.<\/span>\r\n\r\n<\/div>\r\nRefresh your memory for how to describe the shape of a histogram by trying the question in the interactive example below.\r\n<div class=\"textbox exercises\">\r\n<h3>interactive example<\/h3>\r\nSeveral histograms are displayed below. Provide a description of the shape of each.\r\n\r\n<img class=\"aligncenter size-full wp-image-1024\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5772\/2022\/02\/11210038\/Shape_Hist.jpg\" alt=\"A group of four histograms. The first is mounded in the middle and tails off to both sides. The second is mounded to the left and tails of to the right. The third contains two mounds and tails off to the left and right. The fourth is mounded to the right and tails off to the left. \" width=\"490\" height=\"362\" \/>\r\n\r\n[reveal-answer q=\"861785\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"861785\"]\r\n<ol>\r\n \t<li>Unimodal, symmetric<\/li>\r\n \t<li>Right-skewed (a tail of infrequent values trails out to the right of the bulk of the data)<\/li>\r\n \t<li>Bimodal<\/li>\r\n \t<li>Left-skewed (a tail of infrequent values trails out to the left of the bulk of the data)<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next activity, you'll need to calculate and interpret the mean and median in skewed distributions. Let's get some practice with these skills using data collected around the T.V. show\u00a0<em>Friends<\/em>.\r\n<h3 id=\"IntMeanMedian\">Mean and Median<\/h3>\r\n<em>Friends<\/em> was a popular American television show that aired from 1994 to 2004. The show followed a group of six friends living in New York City and chronicled their relationships and day-to-day adventures. The show became known in popular culture for its comedy and for the closeness of its cast.[footnote]Encyclopedia Britannica. (n.d.). Friends. In <em>Encyclopedia Britannica.com<\/em>. https:\/\/www.britannica.com\/topic\/Friends[\/footnote]\r\n\r\nThe following table lists the number of U.S. viewers of each episode of the\u00a0[latex]10[\/latex]<sup>th<\/sup> and final season of Friends.[footnote]Mock, T. (2020). <em>A weekly data project aimed at the R ecosystem<\/em>. TidyTuesday. https:\/\/github.com\/rfordatascience\/tidytuesday\/blob\/master\/data\/2020\/2020-09-08\/readme.md#friends_infocsv[\/footnote]\r\n<div align=\"left\">\r\n<table><caption class=\"center\"><span style=\"text-transform: uppercase;\">Friends Final Season Viewers by episode<\/span><strong>\r\n<\/strong><\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>Episode Number<\/strong><\/td>\r\n<td style=\"text-align: center;\"><strong>Episode Title<\/strong><\/td>\r\n<td style=\"text-align: center;\"><strong>Air Date<\/strong><\/td>\r\n<td style=\"text-align: center;\"><strong>U.S. Viewers (Millions)<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>1<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One After Joey and Rachel Kiss<\/td>\r\n<td style=\"text-align: center;\">9\/25\/03<\/td>\r\n<td style=\"text-align: center;\">[latex]24.54[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>2<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One Where Ross Is Fine<\/td>\r\n<td style=\"text-align: center;\">10\/2\/03<\/td>\r\n<td style=\"text-align: center;\">[latex]22.38[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>3<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One with Ross's Tan<\/td>\r\n<td style=\"text-align: center;\">10\/9\/03<\/td>\r\n<td style=\"text-align: center;\">[latex]21.87[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>4<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One with the Cake<\/td>\r\n<td style=\"text-align: center;\">10\/23\/03<\/td>\r\n<td style=\"text-align: center;\">[latex]18.77[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>5<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One Where Rachel's Sister Babysits<\/td>\r\n<td style=\"text-align: center;\">10\/30\/03<\/td>\r\n<td style=\"text-align: center;\">[latex]19.37[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>6<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One with Ross's Grant<\/td>\r\n<td style=\"text-align: center;\">11\/6\/03<\/td>\r\n<td style=\"text-align: center;\">[latex]20.38[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>7<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One with the Home Study<\/td>\r\n<td style=\"text-align: center;\">11\/13\/03<\/td>\r\n<td style=\"text-align: center;\">[latex]20.21[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>8<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One with the Late Thanksgiving<\/td>\r\n<td style=\"text-align: center;\">11\/20\/03<\/td>\r\n<td style=\"text-align: center;\">[latex]20.66[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>9<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One with the Birth Mother<\/td>\r\n<td style=\"text-align: center;\">1\/8\/04<\/td>\r\n<td style=\"text-align: center;\">[latex]25.49[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>10<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One Where Chandler Gets Caught<\/td>\r\n<td style=\"text-align: center;\">1\/15\/04<\/td>\r\n<td style=\"text-align: center;\">[latex]26.68[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>11<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One Where the Stripper Cries<\/td>\r\n<td style=\"text-align: center;\">2\/5\/04<\/td>\r\n<td style=\"text-align: center;\">[latex]24.91[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>12<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One with Phoebe's Wedding<\/td>\r\n<td style=\"text-align: center;\">2\/12\/04<\/td>\r\n<td style=\"text-align: center;\">[latex]25.9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>13<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One Where Joey Speaks French<\/td>\r\n<td style=\"text-align: center;\">2\/19\/04<\/td>\r\n<td style=\"text-align: center;\">[latex]24.27[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>14<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One with Princess Consuela<\/td>\r\n<td style=\"text-align: center;\">2\/26\/04<\/td>\r\n<td style=\"text-align: center;\">[latex]22.83[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>15<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One Where Estelle Dies<\/td>\r\n<td style=\"text-align: center;\">4\/22\/04<\/td>\r\n<td style=\"text-align: center;\">[latex]22.64[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>16<\/strong><\/td>\r\n<td style=\"text-align: center;\">The One with Rachel's Going Away Party<\/td>\r\n<td style=\"text-align: center;\">4\/29\/04<\/td>\r\n<td style=\"text-align: center;\">[latex]24.51[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>17<\/strong><\/td>\r\n<td style=\"text-align: center;\">The Last One*<\/td>\r\n<td style=\"text-align: center;\">5\/6\/04<\/td>\r\n<td style=\"text-align: center;\">[latex]52.46[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>18<\/strong><\/td>\r\n<td style=\"text-align: center;\">The Last One*<\/td>\r\n<td style=\"text-align: center;\">5\/6\/04<\/td>\r\n<td style=\"text-align: center;\">[latex]52.46[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table class=\"fin-table gridded\"><caption class=\"center\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0*Note: the final two episodes aired back-to-back on the same night\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/caption>\r\n<thead><\/thead>\r\n<\/table>\r\n<span style=\"font-size: 1rem; text-align: initial;\">We'll use technology to analyze this data set.<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n\r\nGo to the <em>Describing and Exploring Quantitative Variables<\/em> tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\" target=\"_blank\" rel=\"noopener\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a>.\r\n<p style=\"padding-left: 30px;\">Step 1) Select the <strong>Single Group<\/strong> tab.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 2) Locate the drop-down menu under <strong>Enter Data<\/strong> and select <strong>Your Own<\/strong>.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 3) Under\u00a0<strong>Do you have<\/strong>, select\u00a0<strong>Individual Observations<\/strong>.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 4)\u00a0Under <strong>Name of Variable<\/strong>, type \u201cU.S. Viewers (Millions).\u201d<\/p>\r\n<p style=\"padding-left: 30px;\">Step 5) Cut and paste or enter the data presented in the above table for U.S. Viewers (Millions).<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>interactive example<\/h3>\r\nIn the following questions, you'll need to interpret the mean and median of the data set. Recall that we think of the mean as the \"average\" data value and the median as the 50th percentile, the value that splits the data in half.\u00a0If needed, you may return to\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/calculating-mean-and-median-of-a-data set-what-to-know\/\"><em>Calculating the Mean and Median of a Data Set: What to Know<\/em><\/a> for a refresher of these interpretations of mean and median.\r\n\r\nExample: Let's say the mean of a data set is given as 10.5 and the median as 11. Which of the following statements are true? Explain.\r\n<ol>\r\n \t<li>The median tells us a typical value for this data set. That is, if we took all the values and spread them evenly about, each value would be about 11.<\/li>\r\n \t<li>About half the data values fall below 11 and half fall above.<\/li>\r\n \t<li>The most common data value appearing is 10.5.<\/li>\r\n \t<li>A typical data value for this set is 10.5. That is, if we distributed the sum of all the values evenly, each value would be about 10.5.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"679737\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"679737\"]\r\n<ol>\r\n \t<li>False. The median represents the 50th percentile, with about half the values falling above 11 and half below.<\/li>\r\n \t<li>True. The median is 11.<\/li>\r\n \t<li>The mode tells us the most common data value. Neither the mean nor the median gives us that information.<\/li>\r\n \t<li>True. The mean is 10.5, which we can consider to be the \"average\" data value.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow it's your turn to use the data set for the TV show\u00a0<em>Friends<\/em> to answer the questions below.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 1<\/h3>\r\n[ohm_question hide_question_numbers=1]241087[\/ohm_question]\r\n\r\n[reveal-answer q=\"253029\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"253029\"]The median will be located in Descriptive Statistics.[\/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]241088[\/ohm_question]\r\n\r\n[reveal-answer q=\"497957\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"497957\"]The median is the\u00a0[latex]50[\/latex]<sup>th<\/sup> percentile and splits the data in half.[\/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]241089[\/ohm_question]\r\n\r\n[reveal-answer q=\"704874\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"704874\"]The mean will be located in Descriptive Statistics[\/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]241090[\/ohm_question]\r\n\r\n[reveal-answer q=\"203004\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"203004\"]The mean is what we think of as the \"average\" value in the set.[\/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]241092[\/ohm_question]\r\n\r\n[reveal-answer q=\"412918\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"412918\"]Use Descriptive Statistics in the tool to compare them.[\/hidden-answer]\r\n\r\n<\/div>\r\nFor this question, use the following histogram of the Season 10 Friends viewership data.\r\n\r\n<strong><img class=\"alignnone wp-image-1010\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11194844\/Picture43-300x112.png\" alt=\"A histogram labeled &quot;US Viewers (Millions)&quot; on the x-axis and &quot;Count&quot; on the y-axis. The x-axis is numbered in increments of five from 15 to 55 and the y-axis is numbered in increments of 1 from 0 to 4. For 18-19, the count is 1. For 19-20, the count is 1. For 20-21, the count is 3. For 21-22, the count is 1. For 22-23, the count is 3. For 24-25, the count is 4. For 25-26, the count is 2. For 26-27, the count is 1. For 52-53, the count is 2. For all other ranges, the count is 0.\" width=\"892\" height=\"333\" \/><\/strong>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 6<\/h3>\r\n[ohm_question hide_question_numbers=1]241093[\/ohm_question]\r\n\r\n[reveal-answer q=\"755933\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"755933\"]Refer to the definitions at the beginning of this assignment.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 7<\/h3>\r\n[ohm_question hide_question_numbers=1]241094[\/ohm_question]\r\n\r\n[reveal-answer q=\"875490\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"875490\"]Consider the implications that the shape of the graph has for the size of the mean relative to the median.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 8<\/h3>\r\n[ohm_question hide_question_numbers=1]241096[\/ohm_question]\r\n\r\n[reveal-answer q=\"55553\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"55553\"]Refer to the table to locate specific episodes.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 9<\/h3>\r\n[ohm_question hide_question_numbers=1]241097[\/ohm_question]\r\n\r\n[reveal-answer q=\"972576\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"972576\"]What do <em>you<\/em> think?[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Mean and Median Under Skew<\/h3>\r\n<div class=\"textbox tryit\">\r\n<h3>effects of skew on mean and median<\/h3>\r\n<span style=\"background-color: #99cc00;\">[Perspective video -- a 3-instructor video that shows how to think about the tail and the two outliers in the data above together with the fact that the mean is larger than the median to begin to understand that the mean tends to be pulled to the right of the median under a right skew.]\u00a0<\/span>\r\n\r\n<\/div>\r\nFor each of the plots of data below, choose the description that matches the shape of the data\u2019s distribution, and then select the choice that gives the relationship between the mean and median for those data. Base your answers on the understanding you established in Questions 1 - 9 about the direction the mean was pulled in under the skewness in the data set.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 10<\/h3>\r\n[ohm_question hide_question_numbers=1]241101[\/ohm_question]\r\n\r\n[reveal-answer q=\"16821\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"16821\"]Refer to the definitions at the top of the page and your answer to Question 7 for guidance.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 11<\/h3>\r\n[ohm_question hide_question_numbers=1]241102[\/ohm_question]\r\n\r\n[reveal-answer q=\"738598\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"738598\"]Refer to the definitions at the top of the page and your answer to Question 7 for guidance.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 12<\/h3>\r\n[ohm_question hide_question_numbers=1]241103[\/ohm_question]\r\n\r\n[reveal-answer q=\"627907\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"627907\"]Refer to the definitions at the top of the page and your answer to Question 7 for guidance.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Resistance<\/h3>\r\n<div class=\"textbox tryit\">\r\n<h3>Resistant and Nonresistant Measures of Center<\/h3>\r\n<span style=\"background-color: #99cc00;\">[Worked example - a 3-instructor video showing a symmetric data set with the mean and median identical, then, skewing the distribution to show what happens to the mean while the median remains in place.]<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 13<\/h3>\r\n[ohm_question hide_question_numbers=1]241104[\/ohm_question]\r\n\r\n[reveal-answer q=\"56189\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"56189\"]\r\n\r\nConsider which measure (mean or median) seemed to be \"pulled\" in the direction of the tail in the skewed distributions and which did not.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nHopefully, you have noticed that when a distribution is symmetric, the mean and median occupy the same value. But under a skew, the mean is \"pulled\" in the direction of the outliers: greater than the median in the case of positive (right) skew, and less than the median in the case of negative (left) skew. It appears that the mean is affected by the presence of outliers while the median is not.\r\n<h3>Looking ahead<\/h3>\r\nBroadly speaking, we consider a value in a data set to be an outlier if that value is unusual or extreme, given the other values in the data set.\r\n\r\nSuppose you have two groups of people:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Group 1 is made up of five professional basketball players, and Group 2 is made up of four professional basketball players and one kindergartener.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Dataset 1 contains the number of three-pointers each person in Group 1 can make in one minute. Dataset 2 contains the number of three-pointers each person in Group 2 can make in an hour.<\/li>\r\n<\/ul>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 14<\/h3>\r\n[ohm_question hide_question_numbers=1]241105[\/ohm_question]\r\n\r\n[reveal-answer q=\"785920\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"785920\"]Imagine a dotplot of the observations in each data set.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nIn this section, you've learned about skewed distributions vs. symmetric distributions and how skew affects the mean of a data distribution. You also got some practice calculating and interpreting the mean and median of a data set. Let's summarize where these skills showed up in the material.\r\n<ul>\r\n \t<li>In Question 1, you calculated the median of a data set, and interpreted the median in Question 2.<\/li>\r\n \t<li>In Question 3, you calculated the mean of a data set, and interpreted the mean in Question 4.<\/li>\r\n \t<li>In Question 5, you began to see how the mean and median relate in a distribution.<\/li>\r\n \t<li>In Questions 6, and 10 - 13, you used statistical terms for skew and extreme values to describe the features of a data set, and began to make connections between the mean and median under differently shaped distributions.<\/li>\r\n \t<li>In Questions 7 -9, you interpreted the mean and median to make connections between them and the data distribution.<\/li>\r\n \t<li>In Question 13, you identified which of the mean or median is resistant to skew.<\/li>\r\n<\/ul>\r\nBeing able to interpret the mean and median with regard to the shape of a distribution and the presence of outliers will be essential skills to use when assessing claims made about data that rely on measures of center. If you feel comfortable with these skills, please move on to the activity!","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Goals<\/h3>\n<p>After completing this section, you should feel comfortable performing these skills.<\/p>\n<ul>\n<li><a href=\"#IntMeanMedian\">Interpret the median of a data set.<\/a><\/li>\n<li><a href=\"#IntMeanMedian\">Interpret the mean of a data set.<\/a><\/li>\n<li><a href=\"#IdentSkew\">Identify whether a data set is left-skewed, symmetric, or right-skewed.<\/a><\/li>\n<li><a href=\"#IdentSkew\">Identify in which data set the mean is greater than, less than, or approximately equal to the median.<\/a><\/li>\n<li><a href=\"#resistant\">Identify which of the mean or median is resistant to skew.<\/a><\/li>\n<\/ul>\n<p>Click on a skill above to jump to its location in this section.<\/p>\n<\/div>\n<p>When examining the distribution of a quantitative variable using a histogram or a dotplot, we often find that the distribution follows a bell shape with a mound of observances in the middle of the distribution and even amounts of data falling to the right and left. But sometimes a distribution&#8217;s values are bunched up to one side or the other, with a few observations stretching way out to the other side. You may recall from <em>What to Know About Applications of Histograms: 3D <\/em>that there are specialized statistical terms we use for these different distribution shapes: skewness and symmetry. In this section, you&#8217;ll learn that there are certain ways the mean of the data relates to the median under these different shapes.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11184151\/Picture131.png\" alt=\"An image of three histograms: left skewed, in which the data is bunched up to the right with a long tail of data to the left; symmetric, in which the data is mounded in the center and falls away evenly to either side; and right-skewed, in which the data is bunched up to the left with a tail of data falling away to the right.\" width=\"1576\" height=\"608\" \/><\/p>\n<h2>Skewness<\/h2>\n<p>Recall that we say a quantitative variable has a <strong>right-skewed<\/strong> distribution or a <strong>positive skew<\/strong> if there is a &#8220;tail&#8221; of infrequent values on the right (upper) end of the distribution. We say a data set has an approximately <strong>symmetric<\/strong> distribution if values are similarly distributed on either side of the mean\/median. We say a data set has a <strong>left-skewed<\/strong> distribution or a <strong>negative skew<\/strong> if there is a &#8220;tail&#8221; of infrequent values on the left (lower) end of the distribution.<\/p>\n<div class=\"textbox tryit\">\n<h3>skewed distributions<\/h3>\n<p><span style=\"background-color: #ffff99;\">I&#8217;d like an animation here (super simple) of a data set that moves from right skew to symmetry to left skew with a slider students can manipulate. The labels would change over the slider: right skew \/ roughly symmetric \/ roughly symmetric \/ left skew.<\/span><\/p>\n<\/div>\n<p>Refresh your memory for how to describe the shape of a histogram by trying the question in the interactive example below.<\/p>\n<div class=\"textbox exercises\">\n<h3>interactive example<\/h3>\n<p>Several histograms are displayed below. Provide a description of the shape of each.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1024\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5772\/2022\/02\/11210038\/Shape_Hist.jpg\" alt=\"A group of four histograms. The first is mounded in the middle and tails off to both sides. The second is mounded to the left and tails of to the right. The third contains two mounds and tails off to the left and right. The fourth is mounded to the right and tails off to the left.\" width=\"490\" height=\"362\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q861785\">Show Answer<\/span><\/p>\n<div id=\"q861785\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Unimodal, symmetric<\/li>\n<li>Right-skewed (a tail of infrequent values trails out to the right of the bulk of the data)<\/li>\n<li>Bimodal<\/li>\n<li>Left-skewed (a tail of infrequent values trails out to the left of the bulk of the data)<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next activity, you&#8217;ll need to calculate and interpret the mean and median in skewed distributions. Let&#8217;s get some practice with these skills using data collected around the T.V. show\u00a0<em>Friends<\/em>.<\/p>\n<h3 id=\"IntMeanMedian\">Mean and Median<\/h3>\n<p><em>Friends<\/em> was a popular American television show that aired from 1994 to 2004. The show followed a group of six friends living in New York City and chronicled their relationships and day-to-day adventures. The show became known in popular culture for its comedy and for the closeness of its cast.<a class=\"footnote\" title=\"Encyclopedia Britannica. (n.d.). Friends. In Encyclopedia Britannica.com. https:\/\/www.britannica.com\/topic\/Friends\" id=\"return-footnote-341-1\" href=\"#footnote-341-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<p>The following table lists the number of U.S. viewers of each episode of the\u00a0[latex]10[\/latex]<sup>th<\/sup> and final season of Friends.<a class=\"footnote\" title=\"Mock, T. (2020). A weekly data project aimed at the R ecosystem. TidyTuesday. https:\/\/github.com\/rfordatascience\/tidytuesday\/blob\/master\/data\/2020\/2020-09-08\/readme.md#friends_infocsv\" id=\"return-footnote-341-2\" href=\"#footnote-341-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/p>\n<div style=\"text-align: left;\">\n<table>\n<caption class=\"center\"><span style=\"text-transform: uppercase;\">Friends Final Season Viewers by episode<\/span><strong><br \/>\n<\/strong><\/caption>\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><strong>Episode Number<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>Episode Title<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>Air Date<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>U.S. Viewers (Millions)<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>1<\/strong><\/td>\n<td style=\"text-align: center;\">The One After Joey and Rachel Kiss<\/td>\n<td style=\"text-align: center;\">9\/25\/03<\/td>\n<td style=\"text-align: center;\">[latex]24.54[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>2<\/strong><\/td>\n<td style=\"text-align: center;\">The One Where Ross Is Fine<\/td>\n<td style=\"text-align: center;\">10\/2\/03<\/td>\n<td style=\"text-align: center;\">[latex]22.38[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>3<\/strong><\/td>\n<td style=\"text-align: center;\">The One with Ross&#8217;s Tan<\/td>\n<td style=\"text-align: center;\">10\/9\/03<\/td>\n<td style=\"text-align: center;\">[latex]21.87[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>4<\/strong><\/td>\n<td style=\"text-align: center;\">The One with the Cake<\/td>\n<td style=\"text-align: center;\">10\/23\/03<\/td>\n<td style=\"text-align: center;\">[latex]18.77[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>5<\/strong><\/td>\n<td style=\"text-align: center;\">The One Where Rachel&#8217;s Sister Babysits<\/td>\n<td style=\"text-align: center;\">10\/30\/03<\/td>\n<td style=\"text-align: center;\">[latex]19.37[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>6<\/strong><\/td>\n<td style=\"text-align: center;\">The One with Ross&#8217;s Grant<\/td>\n<td style=\"text-align: center;\">11\/6\/03<\/td>\n<td style=\"text-align: center;\">[latex]20.38[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>7<\/strong><\/td>\n<td style=\"text-align: center;\">The One with the Home Study<\/td>\n<td style=\"text-align: center;\">11\/13\/03<\/td>\n<td style=\"text-align: center;\">[latex]20.21[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>8<\/strong><\/td>\n<td style=\"text-align: center;\">The One with the Late Thanksgiving<\/td>\n<td style=\"text-align: center;\">11\/20\/03<\/td>\n<td style=\"text-align: center;\">[latex]20.66[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>9<\/strong><\/td>\n<td style=\"text-align: center;\">The One with the Birth Mother<\/td>\n<td style=\"text-align: center;\">1\/8\/04<\/td>\n<td style=\"text-align: center;\">[latex]25.49[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>10<\/strong><\/td>\n<td style=\"text-align: center;\">The One Where Chandler Gets Caught<\/td>\n<td style=\"text-align: center;\">1\/15\/04<\/td>\n<td style=\"text-align: center;\">[latex]26.68[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>11<\/strong><\/td>\n<td style=\"text-align: center;\">The One Where the Stripper Cries<\/td>\n<td style=\"text-align: center;\">2\/5\/04<\/td>\n<td style=\"text-align: center;\">[latex]24.91[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>12<\/strong><\/td>\n<td style=\"text-align: center;\">The One with Phoebe&#8217;s Wedding<\/td>\n<td style=\"text-align: center;\">2\/12\/04<\/td>\n<td style=\"text-align: center;\">[latex]25.9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>13<\/strong><\/td>\n<td style=\"text-align: center;\">The One Where Joey Speaks French<\/td>\n<td style=\"text-align: center;\">2\/19\/04<\/td>\n<td style=\"text-align: center;\">[latex]24.27[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>14<\/strong><\/td>\n<td style=\"text-align: center;\">The One with Princess Consuela<\/td>\n<td style=\"text-align: center;\">2\/26\/04<\/td>\n<td style=\"text-align: center;\">[latex]22.83[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>15<\/strong><\/td>\n<td style=\"text-align: center;\">The One Where Estelle Dies<\/td>\n<td style=\"text-align: center;\">4\/22\/04<\/td>\n<td style=\"text-align: center;\">[latex]22.64[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>16<\/strong><\/td>\n<td style=\"text-align: center;\">The One with Rachel&#8217;s Going Away Party<\/td>\n<td style=\"text-align: center;\">4\/29\/04<\/td>\n<td style=\"text-align: center;\">[latex]24.51[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>17<\/strong><\/td>\n<td style=\"text-align: center;\">The Last One*<\/td>\n<td style=\"text-align: center;\">5\/6\/04<\/td>\n<td style=\"text-align: center;\">[latex]52.46[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>18<\/strong><\/td>\n<td style=\"text-align: center;\">The Last One*<\/td>\n<td style=\"text-align: center;\">5\/6\/04<\/td>\n<td style=\"text-align: center;\">[latex]52.46[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"fin-table gridded\">\n<caption class=\"center\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0*Note: the final two episodes aired back-to-back on the same night\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/caption>\n<thead><\/thead>\n<\/table>\n<p><span style=\"font-size: 1rem; text-align: initial;\">We&#8217;ll use technology to analyze this data set.<\/span><\/p>\n<\/div>\n<div class=\"textbox\">\n<p>Go to the <em>Describing and Exploring Quantitative Variables<\/em> tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\" target=\"_blank\" rel=\"noopener\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a>.<\/p>\n<p style=\"padding-left: 30px;\">Step 1) Select the <strong>Single Group<\/strong> tab.<\/p>\n<p style=\"padding-left: 30px;\">Step 2) Locate the drop-down menu under <strong>Enter Data<\/strong> and select <strong>Your Own<\/strong>.<\/p>\n<p style=\"padding-left: 30px;\">Step 3) Under\u00a0<strong>Do you have<\/strong>, select\u00a0<strong>Individual Observations<\/strong>.<\/p>\n<p style=\"padding-left: 30px;\">Step 4)\u00a0Under <strong>Name of Variable<\/strong>, type \u201cU.S. Viewers (Millions).\u201d<\/p>\n<p style=\"padding-left: 30px;\">Step 5) Cut and paste or enter the data presented in the above table for U.S. Viewers (Millions).<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>interactive example<\/h3>\n<p>In the following questions, you&#8217;ll need to interpret the mean and median of the data set. Recall that we think of the mean as the &#8220;average&#8221; data value and the median as the 50th percentile, the value that splits the data in half.\u00a0If needed, you may return to\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/calculating-mean-and-median-of-a-data set-what-to-know\/\"><em>Calculating the Mean and Median of a Data Set: What to Know<\/em><\/a> for a refresher of these interpretations of mean and median.<\/p>\n<p>Example: Let&#8217;s say the mean of a data set is given as 10.5 and the median as 11. Which of the following statements are true? Explain.<\/p>\n<ol>\n<li>The median tells us a typical value for this data set. That is, if we took all the values and spread them evenly about, each value would be about 11.<\/li>\n<li>About half the data values fall below 11 and half fall above.<\/li>\n<li>The most common data value appearing is 10.5.<\/li>\n<li>A typical data value for this set is 10.5. That is, if we distributed the sum of all the values evenly, each value would be about 10.5.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q679737\">Show Answer<\/span><\/p>\n<div id=\"q679737\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>False. The median represents the 50th percentile, with about half the values falling above 11 and half below.<\/li>\n<li>True. The median is 11.<\/li>\n<li>The mode tells us the most common data value. Neither the mean nor the median gives us that information.<\/li>\n<li>True. The mean is 10.5, which we can consider to be the &#8220;average&#8221; data value.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Now it&#8217;s your turn to use the data set for the TV show\u00a0<em>Friends<\/em> to answer the questions below.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 1<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241087\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241087&theme=oea&iframe_resize_id=ohm241087\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q253029\">Hint<\/span><\/p>\n<div id=\"q253029\" class=\"hidden-answer\" style=\"display: none\">The median will be located in Descriptive Statistics.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 2<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241088\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241088&theme=oea&iframe_resize_id=ohm241088\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q497957\">Hint<\/span><\/p>\n<div id=\"q497957\" class=\"hidden-answer\" style=\"display: none\">The median is the\u00a0[latex]50[\/latex]<sup>th<\/sup> percentile and splits the data in half.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 3<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241089\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241089&theme=oea&iframe_resize_id=ohm241089\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q704874\">Hint<\/span><\/p>\n<div id=\"q704874\" class=\"hidden-answer\" style=\"display: none\">The mean will be located in Descriptive Statistics<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 4<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241090\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241090&theme=oea&iframe_resize_id=ohm241090\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q203004\">Hint<\/span><\/p>\n<div id=\"q203004\" class=\"hidden-answer\" style=\"display: none\">The mean is what we think of as the &#8220;average&#8221; value in the set.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 5<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241092\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241092&theme=oea&iframe_resize_id=ohm241092\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q412918\">Hint<\/span><\/p>\n<div id=\"q412918\" class=\"hidden-answer\" style=\"display: none\">Use Descriptive Statistics in the tool to compare them.<\/div>\n<\/div>\n<\/div>\n<p>For this question, use the following histogram of the Season 10 Friends viewership data.<\/p>\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1010\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11194844\/Picture43-300x112.png\" alt=\"A histogram labeled &quot;US Viewers (Millions)&quot; on the x-axis and &quot;Count&quot; on the y-axis. The x-axis is numbered in increments of five from 15 to 55 and the y-axis is numbered in increments of 1 from 0 to 4. For 18-19, the count is 1. For 19-20, the count is 1. For 20-21, the count is 3. For 21-22, the count is 1. For 22-23, the count is 3. For 24-25, the count is 4. For 25-26, the count is 2. For 26-27, the count is 1. For 52-53, the count is 2. For all other ranges, the count is 0.\" width=\"892\" height=\"333\" \/><\/strong><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 6<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241093\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241093&theme=oea&iframe_resize_id=ohm241093\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q755933\">Hint<\/span><\/p>\n<div id=\"q755933\" class=\"hidden-answer\" style=\"display: none\">Refer to the definitions at the beginning of this assignment.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 7<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241094\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241094&theme=oea&iframe_resize_id=ohm241094\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q875490\">Hint<\/span><\/p>\n<div id=\"q875490\" class=\"hidden-answer\" style=\"display: none\">Consider the implications that the shape of the graph has for the size of the mean relative to the median.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 8<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241096\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241096&theme=oea&iframe_resize_id=ohm241096\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q55553\">Hint<\/span><\/p>\n<div id=\"q55553\" class=\"hidden-answer\" style=\"display: none\">Refer to the table to locate specific episodes.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 9<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241097\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241097&theme=oea&iframe_resize_id=ohm241097\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q972576\">Hint<\/span><\/p>\n<div id=\"q972576\" class=\"hidden-answer\" style=\"display: none\">What do <em>you<\/em> think?<\/div>\n<\/div>\n<\/div>\n<h3>Mean and Median Under Skew<\/h3>\n<div class=\"textbox tryit\">\n<h3>effects of skew on mean and median<\/h3>\n<p><span style=\"background-color: #99cc00;\">[Perspective video &#8212; a 3-instructor video that shows how to think about the tail and the two outliers in the data above together with the fact that the mean is larger than the median to begin to understand that the mean tends to be pulled to the right of the median under a right skew.]\u00a0<\/span><\/p>\n<\/div>\n<p>For each of the plots of data below, choose the description that matches the shape of the data\u2019s distribution, and then select the choice that gives the relationship between the mean and median for those data. Base your answers on the understanding you established in Questions 1 &#8211; 9 about the direction the mean was pulled in under the skewness in the data set.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 10<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241101\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241101&theme=oea&iframe_resize_id=ohm241101\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q16821\">Hint<\/span><\/p>\n<div id=\"q16821\" class=\"hidden-answer\" style=\"display: none\">Refer to the definitions at the top of the page and your answer to Question 7 for guidance.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 11<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241102\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241102&theme=oea&iframe_resize_id=ohm241102\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q738598\">Hint<\/span><\/p>\n<div id=\"q738598\" class=\"hidden-answer\" style=\"display: none\">Refer to the definitions at the top of the page and your answer to Question 7 for guidance.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 12<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241103\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241103&theme=oea&iframe_resize_id=ohm241103\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q627907\">Hint<\/span><\/p>\n<div id=\"q627907\" class=\"hidden-answer\" style=\"display: none\">Refer to the definitions at the top of the page and your answer to Question 7 for guidance.<\/div>\n<\/div>\n<\/div>\n<h3>Resistance<\/h3>\n<div class=\"textbox tryit\">\n<h3>Resistant and Nonresistant Measures of Center<\/h3>\n<p><span style=\"background-color: #99cc00;\">[Worked example &#8211; a 3-instructor video showing a symmetric data set with the mean and median identical, then, skewing the distribution to show what happens to the mean while the median remains in place.]<\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 13<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241104\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241104&theme=oea&iframe_resize_id=ohm241104\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q56189\">Hint<\/span><\/p>\n<div id=\"q56189\" class=\"hidden-answer\" style=\"display: none\">\n<p>Consider which measure (mean or median) seemed to be &#8220;pulled&#8221; in the direction of the tail in the skewed distributions and which did not.\n<\/p><\/div>\n<\/div>\n<\/div>\n<p>Hopefully, you have noticed that when a distribution is symmetric, the mean and median occupy the same value. But under a skew, the mean is &#8220;pulled&#8221; in the direction of the outliers: greater than the median in the case of positive (right) skew, and less than the median in the case of negative (left) skew. It appears that the mean is affected by the presence of outliers while the median is not.<\/p>\n<h3>Looking ahead<\/h3>\n<p>Broadly speaking, we consider a value in a data set to be an outlier if that value is unusual or extreme, given the other values in the data set.<\/p>\n<p>Suppose you have two groups of people:<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Group 1 is made up of five professional basketball players, and Group 2 is made up of four professional basketball players and one kindergartener.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Dataset 1 contains the number of three-pointers each person in Group 1 can make in one minute. Dataset 2 contains the number of three-pointers each person in Group 2 can make in an hour.<\/li>\n<\/ul>\n<div class=\"textbox key-takeaways\">\n<h3>question 14<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241105\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241105&theme=oea&iframe_resize_id=ohm241105\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q785920\">Hint<\/span><\/p>\n<div id=\"q785920\" class=\"hidden-answer\" style=\"display: none\">Imagine a dotplot of the observations in each data set.<\/div>\n<\/div>\n<\/div>\n<h2>Summary<\/h2>\n<p>In this section, you&#8217;ve learned about skewed distributions vs. symmetric distributions and how skew affects the mean of a data distribution. You also got some practice calculating and interpreting the mean and median of a data set. Let&#8217;s summarize where these skills showed up in the material.<\/p>\n<ul>\n<li>In Question 1, you calculated the median of a data set, and interpreted the median in Question 2.<\/li>\n<li>In Question 3, you calculated the mean of a data set, and interpreted the mean in Question 4.<\/li>\n<li>In Question 5, you began to see how the mean and median relate in a distribution.<\/li>\n<li>In Questions 6, and 10 &#8211; 13, you used statistical terms for skew and extreme values to describe the features of a data set, and began to make connections between the mean and median under differently shaped distributions.<\/li>\n<li>In Questions 7 -9, you interpreted the mean and median to make connections between them and the data distribution.<\/li>\n<li>In Question 13, you identified which of the mean or median is resistant to skew.<\/li>\n<\/ul>\n<p>Being able to interpret the mean and median with regard to the shape of a distribution and the presence of outliers will be essential skills to use when assessing claims made about data that rely on measures of center. If you feel comfortable with these skills, please move on to the activity!<\/p>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-341-1\">Encyclopedia Britannica. (n.d.). Friends. In <em>Encyclopedia Britannica.com<\/em>. https:\/\/www.britannica.com\/topic\/Friends <a href=\"#return-footnote-341-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-341-2\">Mock, T. (2020). <em>A weekly data project aimed at the R ecosystem<\/em>. TidyTuesday. https:\/\/github.com\/rfordatascience\/tidytuesday\/blob\/master\/data\/2020\/2020-09-08\/readme.md#friends_infocsv <a href=\"#return-footnote-341-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":175116,"menu_order":47,"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-341","chapter","type-chapter","status-publish","hentry"],"part":1252,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/341","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/users\/175116"}],"version-history":[{"count":17,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/341\/revisions"}],"predecessor-version":[{"id":1260,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/341\/revisions\/1260"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/parts\/1252"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/341\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/media?parent=341"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapter-type?post=341"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/contributor?post=341"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/license?post=341"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}