{"id":84,"date":"2016-04-21T22:43:45","date_gmt":"2016-04-21T22:43:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstats1xmaster\/?post_type=chapter&#038;p=84"},"modified":"2020-10-21T15:16:57","modified_gmt":"2020-10-21T15:16:57","slug":"skewness-and-the-mean-median-and-mode","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstats1\/chapter\/skewness-and-the-mean-median-and-mode\/","title":{"raw":"Skewness and the Mean, Median, and Mode","rendered":"Skewness and the Mean, Median, and Mode"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul id=\"list123523\">\r\n \t<li>Recognize, describe, and calculate the measures of the center of data: mean, median, and mode.<\/li>\r\n<\/ul>\r\n<\/div>\r\nConsider the following data set.\r\n[latex]4[\/latex]; [latex]5[\/latex]; [latex]6[\/latex]; [latex]6[\/latex]; [latex]6[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]8[\/latex]; [latex]8[\/latex]; [latex]8[\/latex]; [latex]9[\/latex]; [latex]10[\/latex]\r\nThis data set can be represented by following histogram. Each interval has width one, and each value is located in the middle of an interval.\r\n\r\n<figure id=\"M06_Ch02_fig001\"><span id=\"id16811614\" data-type=\"media\" data-alt=\"This histogram matches the supplied data. It consists of 7 adjacent bars with the x-axis split into intervals of 1 from 4 to 10. The heighs of the bars peak in the middle and taper symmetrically to the right and left.\" data-display=\"block\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214237\/fig-ch02_08_01.jpg\" alt=\"This histogram matches the supplied data. It consists of 7 adjacent bars with the x-axis split into intervals of 1 from 4 to 10. The heighs of the bars peak in the middle and taper symmetrically to the right and left.\" width=\"350\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\r\n<center>Figure 1<\/center>\r\n\r\nThe histogram displays a <strong>symmetrical<\/strong> distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other. The mean, the median, and the mode are each seven for these data. <strong>In a perfectly symmetrical distribution, the mean and the median are the same.<\/strong> This example has one mode (unimodal), and the mode is the same as the mean and median. In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median.\r\n\r\nThe histogram for the data: [latex]4[\/latex]; [latex]5[\/latex]; [latex]6[\/latex]; [latex]6[\/latex]; [latex]6[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]8[\/latex] is not symmetrical. The right-hand side seems \"chopped off\" compared to the left side. A distribution of this type is called <strong>skewed to the left<\/strong> because it is pulled out to the left.\r\n\r\n<figure id=\"M06_Ch02_fig002\"><span id=\"id17014514\" data-type=\"media\" data-alt=\"This histogram matches the supplied data. It consists of 5 adjacent bars with the x-axis split into intervals of 1 from 4 to 8. The peak is to the right, and the heights of the bars taper down to the left.\" data-display=\"block\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214239\/fig-ch02_08_02.jpg\" alt=\"This histogram matches the supplied data. It consists of 5 adjacent bars with the x-axis split into intervals of 1 from 4 to 8. The peak is to the right, and the heights of the bars taper down to the left.\" width=\"350\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\r\n<center>Figure 2<\/center>\r\n\r\nThe mean is [latex]6.3[\/latex], the median is [latex]6.5[\/latex], and the mode is seven. <strong>Notice that the mean is less than the median, and they are both less than the mode.<\/strong> The mean and the median both reflect the skewing, but the mean reflects it more so.\r\nThe histogram for the data: [latex]6[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]8[\/latex]; [latex]8[\/latex]; [latex]8[\/latex]; [latex]9[\/latex]; [latex]10[\/latex], is also not symmetrical. It is <strong>skewed to the right<\/strong>.\r\n\r\n<figure id=\"M06_Ch02_fig003\"><span id=\"id17014699\" data-type=\"media\" data-alt=\"This histogram matches the supplied data. It consists of 5 adjacent bars with the x-axis split into intervals of 1 from 6 to 10. The peak is to the left, and the heights of the bars taper down to the right.\" data-display=\"block\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214242\/fig-ch02_08_03.jpg\" alt=\"This histogram matches the supplied data. It consists of 5 adjacent bars with the x-axis split into intervals of 1 from 6 to 10. The peak is to the left, and the heights of the bars taper down to the right.\" width=\"350\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\r\n<center>Figure 3<\/center>\r\n\r\nThe mean is [latex]7.7[\/latex], the median is [latex]7.5[\/latex], and the mode is seven. Of the three statistics, <strong>the mean is the largest, while the mode is the smallest<\/strong>. Again, the mean reflects the skewing the most.\r\n\r\nTo summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean.\r\n\r\nSkewness and symmetry become important when we discuss probability distributions in later chapters.\r\n\r\nHere is a video that summarizes how the mean, median and mode can help us describe the skewness of a dataset. Don't worry about the terms leptokurtic and platykurtic for this course.\r\n\r\nhttps:\/\/youtu.be\/s6N_l3Bu-Mc\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nStatistics are used to compare and sometimes identify authors. The following lists shows a simple random sample that compares the letter counts for three authors.\r\n\r\nTerry: [latex]7[\/latex]; [latex]9[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]4[\/latex]; [latex]1[\/latex]; [latex]3[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]\r\nDavis: [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]4[\/latex]; [latex]1[\/latex]; [latex]4[\/latex]; [latex]3[\/latex]; [latex]2[\/latex]; [latex]3[\/latex]; [latex]1[\/latex]\r\nMaris: [latex]2[\/latex]; [latex]3[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]6[\/latex]; [latex]6[\/latex]; [latex]6[\/latex]; [latex]8[\/latex]; [latex]3[\/latex]\r\n\r\n<ol id=\"fs-idm63711472\" data-number-style=\"lower-alpha\">\r\n \t<li>Make a dot plot for the three authors and compare the shapes.<\/li>\r\n \t<li>Calculate the mean for each.<\/li>\r\n \t<li>Calculate the median for each.<\/li>\r\n \t<li>Describe any pattern you notice between the shape and the measures of center.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"283391\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283391\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21150437\/Terrys-Letter-Count-300x88.jpeg\" alt=\"This dot plot matches the supplied data for Terry. The plot uses a number line from 1 to 10. It shows one  x over 1, two x&#039;s over 2, four x&#039;s over 3, one  x over 4, one x over 7, and one x over 9. There are no x&#039;s over the numbers 5, 6, 8, and 10.\" width=\"300\" height=\"88\" class=\"alignnone size-medium wp-image-1928\" \/>\r\nTerry\u2019s distribution has a right (positive) skew.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21150553\/Davis-Letter-Count-300x97.jpeg\" alt=\"This dot plot matches the supplied data for Davi. The plot uses a number line from 1 to 10. It shows two  x&#039;s over 1, one x over 2, five x&#039;s over 3, and two x&#039;s over 4. There are no x&#039;s over the numbers 5, 6, 7, 8, 9, and 10.\" width=\"300\" height=\"97\" class=\"alignnone size-medium wp-image-1929\" \/>\r\nDavis\u2019 distribution has a left (negative) skew\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21150557\/Maris-Letter-Count-300x77.jpeg\" alt=\"This dot plot matches the supplied data for Mari. The plot uses a number line from 1 to 10. It shows one x over 2, two x&#039;s over 3, three x&#039;s over 4, three x&#039;s over 6, and one  x over 8. There are no x&#039;s over the numbers 1, 5, 7, 9, and 10.\" width=\"300\" height=\"77\" class=\"alignnone size-medium wp-image-1930\" \/>\r\nMaris\u2019 distribution is symmetrically shaped.<\/li>\r\n\t<li>Terry\u2019s mean is [latex]3.7[\/latex], Davis\u2019 mean is [latex]2.7[\/latex], Maris\u2019 mean is [latex]4.6[\/latex].<\/li>\r\n\t<li>Terry\u2019s median is three, Davis\u2019 median is three. Maris\u2019 median is four.<\/li>\r\n\t<li>It appears that the median is always closest to the high point (the mode), while the mean tends to be farther out on the tail. In a symmetrical distribution, the mean and the median are both centrally located close to the high point of the distribution.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n<\/div>\r\n\r\n<div id=\"fs-idm10131056\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<div class=\"title\" data-label-parent=\"\" data-type=\"title\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-idm4859600\" class=\"exercise\" data-type=\"exercise\"><section>\r\n<div id=\"fs-idm31997616\" class=\"problem\" data-type=\"problem\">\r\nDiscuss the mean, median, and mode for each of the following problems. Is there a pattern between the shape and measure of the center?\r\n1.\r\n\r\n<figure id=\"fs-idp12578240\"><span id=\"fs-idp12578368\" data-type=\"media\" data-alt=\"This dot plot matches the supplied data. The plot uses a number line from 0 to 14. It shows two x's over 0, four x's over 1, three x's over 2, one x over 3, two x's over the number 4, 5, 6, and 9, and 1 x each over 10 and 14. There are no x's over the numbers 7, 8, 11, 12, and 13.\"> <img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214244\/CNX_Stats_C02_M08_033.png\" alt=\"This dot plot matches the supplied data. The plot uses a number line from 0 to 14. It shows two x's over 0, four x's over 1, three x's over 2, one x over 3, two x's over the number 4, 5, 6, and 9, and 1 x each over 10 and 14. There are no x's over the numbers 7, 8, 11, 12, and 13.\" width=\"400\" data-media-type=\"image\/png\" \/><\/span><\/figure>\r\n2.\r\n\r\n<table id=\"eip-idp39390048\" summary=\"The ages former U.S. presidents died\">\r\n<thead>\r\n<tr>\r\n<th colspan=\"2\">The Ages Former U.S Presidents Died<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]6[\/latex] [latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]3[\/latex] [latex]6[\/latex] [latex]7[\/latex] [latex]7[\/latex] [latex]7[\/latex] [latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]0[\/latex] [latex]0[\/latex] [latex]3[\/latex] [latex]3[\/latex] [latex]4[\/latex] [latex]4[\/latex] [latex]5[\/latex] [latex]6[\/latex] [latex]7[\/latex] [latex]7[\/latex] [latex]7[\/latex] [latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]0[\/latex] [latex]1[\/latex] [latex]1[\/latex] [latex]2[\/latex] [latex]3[\/latex] [latex]4[\/latex] [latex]7[\/latex] [latex]8[\/latex] [latex]8[\/latex] [latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]0[\/latex] [latex]1[\/latex] [latex]3[\/latex] [latex]5[\/latex] [latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]9[\/latex]<\/td>\r\n<td>[latex]0[\/latex] [latex]0[\/latex] [latex]3[\/latex] [latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Key: [latex]8|0 [\/latex] means [latex]80[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n3.\r\n\r\n<figure id=\"fs-idp18736080\"><span id=\"fs-idp18736208\" data-type=\"media\" data-alt=\"This is a histogram titled Hours Spent Playing Video Games on Weekends. The x-axis shows the number of hours spent playing video games with bars showing values at intervals of 5. The y-axis shows the number of students. The first bar for 0 - 4.99 hours has a height of 2. The second bar from 5 - 9.99 has a height of 3. The third bar from 10 - 14.99 has a height of 4. The fourth bar from 15 - 19.99 has a height of 7. The fifth bar from 20 - 24.99 has a height of 9.\"> <img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214246\/CNX_Stats_C02_M08_034.png\" alt=\"This is a histogram titled Hours Spent Playing Video Games on Weekends. The x-axis shows the number of hours spent playing video games with bars showing values at intervals of 5. The y-axis shows the number of students. The first bar for 0 - 4.99 hours has a height of 2. The second bar from 5 - 9.99 has a height of 3. The third bar from 10 - 14.99 has a height of 4. The fourth bar from 15 - 19.99 has a height of 7. The fifth bar from 20 - 24.99 has a height of 9.\" width=\"400\" data-media-type=\"image\/png\" \/><\/span><\/figure>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n\r\n\r\n<h1 data-type=\"title\">Concept Review<\/h1>\r\nLooking at the distribution of data can reveal a lot about the relationship between the mean, the median, and the mode. There are <u data-effect=\"underline\">three types of distributions<\/u>. A <strong data-effect=\"bold\">right (or positive) skewed<\/strong> distribution has a shape like Figure 3. A <strong data-effect=\"bold\">left (or negative) skewed<\/strong> distribution has a shape like Figure 2 . A <strong data-effect=\"bold\">symmetrical<\/strong> distribution looks like Figure 1.\r\n\r\n\r\n<div id=\"fs-idm10131056\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><\/div>\r\n\r\n\r\n\r\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul id=\"list123523\">\n<li>Recognize, describe, and calculate the measures of the center of data: mean, median, and mode.<\/li>\n<\/ul>\n<\/div>\n<p>Consider the following data set.<br \/>\n[latex]4[\/latex]; [latex]5[\/latex]; [latex]6[\/latex]; [latex]6[\/latex]; [latex]6[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]8[\/latex]; [latex]8[\/latex]; [latex]8[\/latex]; [latex]9[\/latex]; [latex]10[\/latex]<br \/>\nThis data set can be represented by following histogram. Each interval has width one, and each value is located in the middle of an interval.<\/p>\n<figure id=\"M06_Ch02_fig001\"><span id=\"id16811614\" data-type=\"media\" data-alt=\"This histogram matches the supplied data. It consists of 7 adjacent bars with the x-axis split into intervals of 1 from 4 to 10. The heighs of the bars peak in the middle and taper symmetrically to the right and left.\" data-display=\"block\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214237\/fig-ch02_08_01.jpg\" alt=\"This histogram matches the supplied data. It consists of 7 adjacent bars with the x-axis split into intervals of 1 from 4 to 10. The heighs of the bars peak in the middle and taper symmetrically to the right and left.\" width=\"350\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\n<div style=\"text-align: center;\">Figure 1<\/div>\n<p>The histogram displays a <strong>symmetrical<\/strong> distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other. The mean, the median, and the mode are each seven for these data. <strong>In a perfectly symmetrical distribution, the mean and the median are the same.<\/strong> This example has one mode (unimodal), and the mode is the same as the mean and median. In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median.<\/p>\n<p>The histogram for the data: [latex]4[\/latex]; [latex]5[\/latex]; [latex]6[\/latex]; [latex]6[\/latex]; [latex]6[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]8[\/latex] is not symmetrical. The right-hand side seems &#8220;chopped off&#8221; compared to the left side. A distribution of this type is called <strong>skewed to the left<\/strong> because it is pulled out to the left.<\/p>\n<figure id=\"M06_Ch02_fig002\"><span id=\"id17014514\" data-type=\"media\" data-alt=\"This histogram matches the supplied data. It consists of 5 adjacent bars with the x-axis split into intervals of 1 from 4 to 8. The peak is to the right, and the heights of the bars taper down to the left.\" data-display=\"block\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214239\/fig-ch02_08_02.jpg\" alt=\"This histogram matches the supplied data. It consists of 5 adjacent bars with the x-axis split into intervals of 1 from 4 to 8. The peak is to the right, and the heights of the bars taper down to the left.\" width=\"350\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\n<div style=\"text-align: center;\">Figure 2<\/div>\n<p>The mean is [latex]6.3[\/latex], the median is [latex]6.5[\/latex], and the mode is seven. <strong>Notice that the mean is less than the median, and they are both less than the mode.<\/strong> The mean and the median both reflect the skewing, but the mean reflects it more so.<br \/>\nThe histogram for the data: [latex]6[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]8[\/latex]; [latex]8[\/latex]; [latex]8[\/latex]; [latex]9[\/latex]; [latex]10[\/latex], is also not symmetrical. It is <strong>skewed to the right<\/strong>.<\/p>\n<figure id=\"M06_Ch02_fig003\"><span id=\"id17014699\" data-type=\"media\" data-alt=\"This histogram matches the supplied data. It consists of 5 adjacent bars with the x-axis split into intervals of 1 from 6 to 10. The peak is to the left, and the heights of the bars taper down to the right.\" data-display=\"block\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214242\/fig-ch02_08_03.jpg\" alt=\"This histogram matches the supplied data. It consists of 5 adjacent bars with the x-axis split into intervals of 1 from 6 to 10. The peak is to the left, and the heights of the bars taper down to the right.\" width=\"350\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\n<div style=\"text-align: center;\">Figure 3<\/div>\n<p>The mean is [latex]7.7[\/latex], the median is [latex]7.5[\/latex], and the mode is seven. Of the three statistics, <strong>the mean is the largest, while the mode is the smallest<\/strong>. Again, the mean reflects the skewing the most.<\/p>\n<p>To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean.<\/p>\n<p>Skewness and symmetry become important when we discuss probability distributions in later chapters.<\/p>\n<p>Here is a video that summarizes how the mean, median and mode can help us describe the skewness of a dataset. Don&#8217;t worry about the terms leptokurtic and platykurtic for this course.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Elementary Business Statistics | Skewness and the Mean, Median, and Mode\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/s6N_l3Bu-Mc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Statistics are used to compare and sometimes identify authors. The following lists shows a simple random sample that compares the letter counts for three authors.<\/p>\n<p>Terry: [latex]7[\/latex]; [latex]9[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]4[\/latex]; [latex]1[\/latex]; [latex]3[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]<br \/>\nDavis: [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]4[\/latex]; [latex]1[\/latex]; [latex]4[\/latex]; [latex]3[\/latex]; [latex]2[\/latex]; [latex]3[\/latex]; [latex]1[\/latex]<br \/>\nMaris: [latex]2[\/latex]; [latex]3[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]6[\/latex]; [latex]6[\/latex]; [latex]6[\/latex]; [latex]8[\/latex]; [latex]3[\/latex]<\/p>\n<ol id=\"fs-idm63711472\" data-number-style=\"lower-alpha\">\n<li>Make a dot plot for the three authors and compare the shapes.<\/li>\n<li>Calculate the mean for each.<\/li>\n<li>Calculate the median for each.<\/li>\n<li>Describe any pattern you notice between the shape and the measures of center.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283391\">Show Solution<\/span><\/p>\n<div id=\"q283391\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21150437\/Terrys-Letter-Count-300x88.jpeg\" alt=\"This dot plot matches the supplied data for Terry. The plot uses a number line from 1 to 10. It shows one  x over 1, two x&#39;s over 2, four x&#39;s over 3, one  x over 4, one x over 7, and one x over 9. There are no x&#39;s over the numbers 5, 6, 8, and 10.\" width=\"300\" height=\"88\" class=\"alignnone size-medium wp-image-1928\" \/><br \/>\nTerry\u2019s distribution has a right (positive) skew.<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21150553\/Davis-Letter-Count-300x97.jpeg\" alt=\"This dot plot matches the supplied data for Davi. The plot uses a number line from 1 to 10. It shows two  x&#39;s over 1, one x over 2, five x&#39;s over 3, and two x&#39;s over 4. There are no x&#39;s over the numbers 5, 6, 7, 8, 9, and 10.\" width=\"300\" height=\"97\" class=\"alignnone size-medium wp-image-1929\" \/><br \/>\nDavis\u2019 distribution has a left (negative) skew<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21150557\/Maris-Letter-Count-300x77.jpeg\" alt=\"This dot plot matches the supplied data for Mari. The plot uses a number line from 1 to 10. It shows one x over 2, two x&#39;s over 3, three x&#39;s over 4, three x&#39;s over 6, and one  x over 8. There are no x&#39;s over the numbers 1, 5, 7, 9, and 10.\" width=\"300\" height=\"77\" class=\"alignnone size-medium wp-image-1930\" \/><br \/>\nMaris\u2019 distribution is symmetrically shaped.<\/li>\n<li>Terry\u2019s mean is [latex]3.7[\/latex], Davis\u2019 mean is [latex]2.7[\/latex], Maris\u2019 mean is [latex]4.6[\/latex].<\/li>\n<li>Terry\u2019s median is three, Davis\u2019 median is three. Maris\u2019 median is four.<\/li>\n<li>It appears that the median is always closest to the high point (the mode), while the mean tends to be farther out on the tail. In a symmetrical distribution, the mean and the median are both centrally located close to the high point of the distribution.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-idm10131056\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<header>\n<div class=\"title\" data-label-parent=\"\" data-type=\"title\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-idm4859600\" class=\"exercise\" data-type=\"exercise\">\n<section>\n<div id=\"fs-idm31997616\" class=\"problem\" data-type=\"problem\">\nDiscuss the mean, median, and mode for each of the following problems. Is there a pattern between the shape and measure of the center?<br \/>\n1.<\/p>\n<figure id=\"fs-idp12578240\"><span id=\"fs-idp12578368\" data-type=\"media\" data-alt=\"This dot plot matches the supplied data. The plot uses a number line from 0 to 14. It shows two x's over 0, four x's over 1, three x's over 2, one x over 3, two x's over the number 4, 5, 6, and 9, and 1 x each over 10 and 14. There are no x's over the numbers 7, 8, 11, 12, and 13.\"> <img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214244\/CNX_Stats_C02_M08_033.png\" alt=\"This dot plot matches the supplied data. The plot uses a number line from 0 to 14. It shows two x's over 0, four x's over 1, three x's over 2, one x over 3, two x's over the number 4, 5, 6, and 9, and 1 x each over 10 and 14. There are no x's over the numbers 7, 8, 11, 12, and 13.\" width=\"400\" data-media-type=\"image\/png\" \/><\/span><\/figure>\n<p>2.<\/p>\n<table id=\"eip-idp39390048\" summary=\"The ages former U.S. presidents died\">\n<thead>\n<tr>\n<th colspan=\"2\">The Ages Former U.S Presidents Died<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]6[\/latex] [latex]9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]3[\/latex] [latex]6[\/latex] [latex]7[\/latex] [latex]7[\/latex] [latex]7[\/latex] [latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]0[\/latex] [latex]0[\/latex] [latex]3[\/latex] [latex]3[\/latex] [latex]4[\/latex] [latex]4[\/latex] [latex]5[\/latex] [latex]6[\/latex] [latex]7[\/latex] [latex]7[\/latex] [latex]7[\/latex] [latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]0[\/latex] [latex]1[\/latex] [latex]1[\/latex] [latex]2[\/latex] [latex]3[\/latex] [latex]4[\/latex] [latex]7[\/latex] [latex]8[\/latex] [latex]8[\/latex] [latex]9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]0[\/latex] [latex]1[\/latex] [latex]3[\/latex] [latex]5[\/latex] [latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]9[\/latex]<\/td>\n<td>[latex]0[\/latex] [latex]0[\/latex] [latex]3[\/latex] [latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Key: [latex]8|0[\/latex] means [latex]80[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>3.<\/p>\n<figure id=\"fs-idp18736080\"><span id=\"fs-idp18736208\" data-type=\"media\" data-alt=\"This is a histogram titled Hours Spent Playing Video Games on Weekends. The x-axis shows the number of hours spent playing video games with bars showing values at intervals of 5. The y-axis shows the number of students. The first bar for 0 - 4.99 hours has a height of 2. The second bar from 5 - 9.99 has a height of 3. The third bar from 10 - 14.99 has a height of 4. The fourth bar from 15 - 19.99 has a height of 7. The fifth bar from 20 - 24.99 has a height of 9.\"> <img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214246\/CNX_Stats_C02_M08_034.png\" alt=\"This is a histogram titled Hours Spent Playing Video Games on Weekends. The x-axis shows the number of hours spent playing video games with bars showing values at intervals of 5. The y-axis shows the number of students. The first bar for 0 - 4.99 hours has a height of 2. The second bar from 5 - 9.99 has a height of 3. The third bar from 10 - 14.99 has a height of 4. The fourth bar from 15 - 19.99 has a height of 7. The fifth bar from 20 - 24.99 has a height of 9.\" width=\"400\" data-media-type=\"image\/png\" \/><\/span><\/figure>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/header>\n<\/div>\n<h1 data-type=\"title\">Concept Review<\/h1>\n<p>Looking at the distribution of data can reveal a lot about the relationship between the mean, the median, and the mode. There are <u data-effect=\"underline\">three types of distributions<\/u>. A <strong data-effect=\"bold\">right (or positive) skewed<\/strong> distribution has a shape like Figure 3. A <strong data-effect=\"bold\">left (or negative) skewed<\/strong> distribution has a shape like Figure 2 . A <strong data-effect=\"bold\">symmetrical<\/strong> distribution looks like Figure 1.<\/p>\n<div id=\"fs-idm10131056\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-84\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Introductory Statistics. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44.\">http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44.<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44<\/li><li>Introductory Statistics . <strong>Authored by<\/strong>: Barbara Illowski, Susan Dean. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\">http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Elementary Business Statistics | Skewness and the Mean, Median, and Mode. <strong>Authored by<\/strong>: Janux. <strong>Provided by<\/strong>: Janux. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/s6N_l3Bu-Mc\">https:\/\/youtu.be\/s6N_l3Bu-Mc<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44.\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics \",\"author\":\"Barbara Illowski, Susan Dean\",\"organization\":\"Open 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