{"id":310,"date":"2021-07-14T15:59:11","date_gmt":"2021-07-14T15:59:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/outliers\/"},"modified":"2023-12-05T09:48:39","modified_gmt":"2023-12-05T09:48:39","slug":"outliers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/outliers\/","title":{"raw":"What is an Outlier?","rendered":"What is an Outlier?"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<section>\r\n<ul>\r\n \t<li>Identify outliers graphically from a given scatterplot<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<div>\r\n<p id=\"element-3819\" class=\" \">In some data sets, there are values\u00a0<strong>(observed data points)<\/strong>\u00a0called\u00a0<span id=\"term226\" data-type=\"term\">outliers<\/span>.\u00a0<strong>Outliers are observed data points that are far from the least-squares line.<\/strong>\u00a0They have large \"errors\", where the \"error\" or residual is the vertical distance from the line to the point.<\/p>\r\n<p id=\"eip-502\" class=\" \">Outliers need to be examined closely. Sometimes, for some reason or another, they should not be included in the analysis of the data. It is possible that an outlier is a result of erroneous data. Other times, an outlier may hold valuable information about the population under study and should remain included in the data. The key is to examine carefully what causes a data point to be an outlier.<\/p>\r\n\r\n<\/div>\r\nThe following video gives an introduction to the idea of an outlier in a set of data.\r\n\r\n<iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7115060&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=o8Q0i9VzQZA&amp;video_target=tpm-plugin-olov0mtq-o8Q0i9VzQZA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<div>\r\n\r\nThe\u00a0<em>IQR<\/em> can help to determine potential outliers. A value is suspected to be a potential outlier if it is less than (1.5)(<em>IQR<\/em>) below the first quartile or more than (1.5)(<em>IQR<\/em>) above the third quartile. Potential outliers always require further investigation.\r\n\r\n<hr \/>\r\n<p id=\"eip-224\" class=\" \">Besides outliers, a sample may contain one or a few points that are called\u00a0<span id=\"term227\" data-type=\"term\">influential points<\/span>. Influential points are observed data points that are far from the other observed data points in the horizontal direction. These points may have a big effect on the slope of the regression line. To begin to identify an influential point, you can remove it from the data set and see if the slope of the regression line is changed significantly.<\/p>\r\n<p id=\"eip-755\" class=\" \">Computers and many calculators can be used to identify outliers from the data. Computer output for regression analysis will often identify both outliers and influential points so that you can examine them.<\/p>\r\n\r\n<h2 data-type=\"title\">Identifying Outliers<\/h2>\r\n<p id=\"eip-511\" class=\" \">We could guess at outliers by looking at a graph of the scatterplot and best fit-line. However, we would like some guidelines as to how far away a point needs to be in order to be considered an outlier.\u00a0<strong>As a rough rule of thumb, we can flag any point that is located further than two standard deviations above or below the best-fit line as an outlier<\/strong>. The standard deviation used is the standard deviation of the residuals or errors.<\/p>\r\n<p class=\" \">We can do this visually in the scatter plot by drawing an extra pair of lines that are two standard deviations above and below the best-fit line. Any data points that are outside this extra pair of lines are flagged as potential outliers. Or we can do this numerically by calculating each residual and comparing it to twice the standard deviation. On the TI-83, 83+, or 84+, the graphical approach is easier. The graphical procedure is shown first, followed by the numerical calculations. You would generally need to use only one of these methods.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 1<\/h3>\r\n<div id=\"id1170949520848\" data-type=\"problem\">\r\n<div class=\"os-problem-container \">\r\n<p id=\"element-631\" class=\" \">In the\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/the-regression-equation\/\" target=\"_blank\" rel=\"noopener\" data-page-slug=\"12-3-the-regression-equation\" data-page-uuid=\"fd60680f-dbb7-4a97-84b8-f352c3a6c141\" data-page-fragment=\"element-22\">third exam\/final exam example<\/a>\u00a0(example 2), you can determine if there is an outlier or not. If there is an outlier, as an exercise, delete it and fit the remaining data to a new line. For this example, the new line ought to fit the remaining data better. This means the\u00a0<strong>SSE<\/strong>\u00a0should be smaller and the correlation coefficient ought to be closer to 1 or -1.\r\n[reveal-answer q=\"827525\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"827525\"]<\/p>\r\n<strong>Graphical Identification of Outliers<\/strong>\r\nWith the TI-83, 83+, 84+ graphing calculators, it is easy to identify the outliers graphically and visually. If we were to measure the vertical distance from any data point to the corresponding point on the line of best fit and that distance was equal to 2s or more, then we would consider the data point to be \"too far\" from the line of best fit. We need to find and graph the lines that are two standard deviations below and above the regression line. Any points that are outside these two lines are outliers. We will call these lines Y2 and Y3:\r\n\r\nAs we did with the equation of the regression line and the correlation coefficient, we will use technology to calculate this standard deviation for us. Using the <strong>LinRegTTest<\/strong> with this data, scroll down through the output screens to find <strong><em>s<\/em> = 16.412.<\/strong>\r\n\r\nLine Y2 = \u2013173.5 + 4.83<em>x<\/em> \u20132(16.4) and line Y3 = \u2013173.5 + 4.83<em>x<\/em> + 2(16.4)\r\n\r\nwhere [latex]\\hat{y}[\/latex] =\u00a0\u2013173.5 + 4.83<em>x<\/em> is the line of best fit. Y2 and Y3 have the same slope as the line of best fit.\r\n\r\nGraph the scatterplot with the best fit line in equation Y1, then enter the two extra lines as Y2 and Y3 in the \"Y=\" equation editor and press ZOOM 9. You will find that the only data point that is not between lines Y2 and Y3 is the point <em>x<\/em> = 65, <em>y<\/em> = 175. On the calculator screen, it is just barely outside these lines. The outlier is the student who had a grade of 65 on the third exam and 175 on the final exam; this point is further than two standard deviations away from the best-fit line.\r\n\r\nSometimes a point is so close to the lines used to flag outliers on the graph that it is difficult to tell if the point is between or outside the lines. On a computer, enlarging the graph may help; on a small calculator screen, zooming in may make the graph clearer. Note that when the graph does not give a clear enough picture, you can use numerical comparisons to identify outliers.\r\n\r\n<img class=\"aligncenter wp-image-2318 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11173522\/ae0a876fb8c16c7862344a01a539f3431eef9269.jpeg\" alt=\"The scatter plot of exam scores with a line of best fit.Two yellow dashed lines run parallel to the line of best fit. The dashed lines run above and below the best fit line at equal distances. One data point falls outside the boundary created by the dashed lines\u2014it is an outlier.\" width=\"487\" height=\"312\" \/>\r\n<p class=\" \">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it 1<\/h3>\r\nIdentify the potential outlier in the scatter plot. The standard deviation of the residuals or errors is approximately 8.6.\r\n\r\n<img class=\"aligncenter wp-image-2319 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11173623\/7be87771990d07958c334c3b413a1f8131a7c98d.png\" alt=\"A scatter plot with a line of best fit. Most of the dots are near the line. One is below the line at the point (6,58).\" width=\"488\" height=\"323\" \/>\r\n\r\n<\/div>\r\nIn the table below,\u00a0the first two columns are the third exam and final exam data. The third column shows the predicted\u00a0[latex]\\hat{y}[\/latex]\u00a0values calculated from the line of best fit: [latex]\\hat{y}[\/latex]\u00a0= \u2013173.5 + 4.83<em data-effect=\"italics\">x<\/em>. The residuals, or errors, have been calculated in the fourth column of the table: observed\u00a0<em data-effect=\"italics\">y<\/em>\u00a0value\u2212predicted y value =\u00a0<em data-effect=\"italics\">y<\/em>\u00a0\u2212 [latex]\\hat{y}[\/latex].\r\n<p id=\"element-973\" class=\" \"><em data-effect=\"italics\">s<\/em>\u00a0is the standard deviation of all the\u00a0<em data-effect=\"italics\">y<\/em>\u00a0\u2212 [latex]\\hat{y}[\/latex] =\u00a0<em data-effect=\"italics\">\u03b5<\/em>\u00a0values where\u00a0<em data-effect=\"italics\">n<\/em>\u00a0= the total number of data points. If each residual is calculated and squared, and the results are added, we get the SSE. The standard deviation of the residuals is calculated from the SSE as:<\/p>\r\n[latex]s = {\\sqrt{\\dfrac{SSE}{n - 2}}}[\/latex]\r\n<h4><strong>Note<\/strong><\/h4>\r\nWe divide by (<em data-effect=\"italics\">n<\/em>\u00a0\u2013 2) because the regression model involves two estimates.\r\n<p id=\"element-42\" class=\"finger \">Rather than calculate the value of\u00a0<em data-effect=\"italics\">s<\/em>\u00a0ourselves, we can find\u00a0<em data-effect=\"italics\">s<\/em>\u00a0using the computer or calculator. For this example, the calculator function LinRegTTest found\u00a0<em data-effect=\"italics\">s<\/em>\u00a0= 16.4 as the standard deviation of the residuals\u00a0<span id=\"set-list2\" data-type=\"list\" data-list-type=\"labeled-item\" data-display=\"inline\"><span data-type=\"item\">35<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">\u201317<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">16<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">\u20136<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">\u201319<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">9<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">3<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">\u20131<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">\u201310<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">\u20139<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">\u20131.<\/span><\/span><\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%; height: 164px;\" border=\"1\">\r\n<thead>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 10px;\"><em>x<\/em><\/td>\r\n<td style=\"width: 25%; height: 10px;\"><em>y<\/em><\/td>\r\n<td style=\"width: 25%; height: 10px;\"><em>[latex]\\hat{y}[\/latex]<\/em><\/td>\r\n<td style=\"width: 25%; height: 10px;\"><em>y -\u00a0[latex]\\hat{y}[\/latex]<\/em><\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px;\">65<\/td>\r\n<td style=\"width: 25%; height: 14px;\">175<\/td>\r\n<td style=\"width: 25%; height: 14px;\">140<\/td>\r\n<td style=\"width: 25%; height: 14px;\">175 \u2013 140 = 35<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px;\">67<\/td>\r\n<td style=\"width: 25%; height: 14px;\">133<\/td>\r\n<td style=\"width: 25%; height: 14px;\">150<\/td>\r\n<td style=\"width: 25%; height: 14px;\">133 \u2013 150= \u201317<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px;\">71<\/td>\r\n<td style=\"width: 25%; height: 14px;\">185<\/td>\r\n<td style=\"width: 25%; height: 14px;\">169<\/td>\r\n<td style=\"width: 25%; height: 14px;\">185 \u2013 169 = 16<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px;\">71<\/td>\r\n<td style=\"width: 25%; height: 14px;\">163<\/td>\r\n<td style=\"width: 25%; height: 14px;\">169<\/td>\r\n<td style=\"width: 25%; height: 14px;\">163 \u2013 169 = \u20136<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px;\">66<\/td>\r\n<td style=\"width: 25%; height: 14px;\">126<\/td>\r\n<td style=\"width: 25%; height: 14px;\">145<\/td>\r\n<td style=\"width: 25%; height: 14px;\">126 \u2013 145 = \u201319<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px;\">75<\/td>\r\n<td style=\"width: 25%; height: 14px;\">198<\/td>\r\n<td style=\"width: 25%; height: 14px;\">189<\/td>\r\n<td style=\"width: 25%; height: 14px;\">198 \u2013 189 = 9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px;\">67<\/td>\r\n<td style=\"width: 25%; height: 14px;\">153<\/td>\r\n<td style=\"width: 25%; height: 14px;\">150<\/td>\r\n<td style=\"width: 25%; height: 14px;\">153 \u2013 150 = 3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px;\">70<\/td>\r\n<td style=\"width: 25%; height: 14px;\">163<\/td>\r\n<td style=\"width: 25%; height: 14px;\">164<\/td>\r\n<td style=\"width: 25%; height: 14px;\">163 \u2013 164 = \u20131<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px;\">71<\/td>\r\n<td style=\"width: 25%; height: 14px;\">159<\/td>\r\n<td style=\"width: 25%; height: 14px;\">169<\/td>\r\n<td style=\"width: 25%; height: 14px;\">159 \u2013 169 = \u201310<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px;\">69<\/td>\r\n<td style=\"width: 25%; height: 14px;\">151<\/td>\r\n<td style=\"width: 25%; height: 14px;\">160<\/td>\r\n<td style=\"width: 25%; height: 14px;\">151 \u2013 160 = \u20139<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px;\">69<\/td>\r\n<td style=\"width: 25%; height: 14px;\">159<\/td>\r\n<td style=\"width: 25%; height: 14px;\">160<\/td>\r\n<td style=\"width: 25%; height: 14px;\">159 \u2013 160 = \u20131<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section id=\"eip-212\" data-depth=\"1\">\r\n<p id=\"element-508\" class=\" \">We are looking for all data points for which the residual is greater than 2<em data-effect=\"italics\">s<\/em>\u00a0= 2(16.4) = 32.8 or less than \u201332.8. Compare these values to the residuals in column four of the table. The only such data point is the student who had a grade of 65 on the third exam and 175 on the final exam; the residual for this student is 35.<\/p>\r\n\r\n<\/section>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<section>\n<ul>\n<li>Identify outliers graphically from a given scatterplot<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<div>\n<p id=\"element-3819\" class=\"\">In some data sets, there are values\u00a0<strong>(observed data points)<\/strong>\u00a0called\u00a0<span id=\"term226\" data-type=\"term\">outliers<\/span>.\u00a0<strong>Outliers are observed data points that are far from the least-squares line.<\/strong>\u00a0They have large &#8220;errors&#8221;, where the &#8220;error&#8221; or residual is the vertical distance from the line to the point.<\/p>\n<p id=\"eip-502\" class=\"\">Outliers need to be examined closely. Sometimes, for some reason or another, they should not be included in the analysis of the data. It is possible that an outlier is a result of erroneous data. Other times, an outlier may hold valuable information about the population under study and should remain included in the data. The key is to examine carefully what causes a data point to be an outlier.<\/p>\n<\/div>\n<p>The following video gives an introduction to the idea of an outlier in a set of data.<\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7115060&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=o8Q0i9VzQZA&amp;video_target=tpm-plugin-olov0mtq-o8Q0i9VzQZA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<div>\n<p>The\u00a0<em>IQR<\/em> can help to determine potential outliers. A value is suspected to be a potential outlier if it is less than (1.5)(<em>IQR<\/em>) below the first quartile or more than (1.5)(<em>IQR<\/em>) above the third quartile. Potential outliers always require further investigation.<\/p>\n<hr \/>\n<p id=\"eip-224\" class=\"\">Besides outliers, a sample may contain one or a few points that are called\u00a0<span id=\"term227\" data-type=\"term\">influential points<\/span>. Influential points are observed data points that are far from the other observed data points in the horizontal direction. These points may have a big effect on the slope of the regression line. To begin to identify an influential point, you can remove it from the data set and see if the slope of the regression line is changed significantly.<\/p>\n<p id=\"eip-755\" class=\"\">Computers and many calculators can be used to identify outliers from the data. Computer output for regression analysis will often identify both outliers and influential points so that you can examine them.<\/p>\n<h2 data-type=\"title\">Identifying Outliers<\/h2>\n<p id=\"eip-511\" class=\"\">We could guess at outliers by looking at a graph of the scatterplot and best fit-line. However, we would like some guidelines as to how far away a point needs to be in order to be considered an outlier.\u00a0<strong>As a rough rule of thumb, we can flag any point that is located further than two standard deviations above or below the best-fit line as an outlier<\/strong>. The standard deviation used is the standard deviation of the residuals or errors.<\/p>\n<p class=\"\">We can do this visually in the scatter plot by drawing an extra pair of lines that are two standard deviations above and below the best-fit line. Any data points that are outside this extra pair of lines are flagged as potential outliers. Or we can do this numerically by calculating each residual and comparing it to twice the standard deviation. On the TI-83, 83+, or 84+, the graphical approach is easier. The graphical procedure is shown first, followed by the numerical calculations. You would generally need to use only one of these methods.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example 1<\/h3>\n<div id=\"id1170949520848\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"element-631\" class=\"\">In the\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/the-regression-equation\/\" target=\"_blank\" rel=\"noopener\" data-page-slug=\"12-3-the-regression-equation\" data-page-uuid=\"fd60680f-dbb7-4a97-84b8-f352c3a6c141\" data-page-fragment=\"element-22\">third exam\/final exam example<\/a>\u00a0(example 2), you can determine if there is an outlier or not. If there is an outlier, as an exercise, delete it and fit the remaining data to a new line. For this example, the new line ought to fit the remaining data better. This means the\u00a0<strong>SSE<\/strong>\u00a0should be smaller and the correlation coefficient ought to be closer to 1 or -1.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q827525\">Show Answer<\/span><\/p>\n<div id=\"q827525\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Graphical Identification of Outliers<\/strong><br \/>\nWith the TI-83, 83+, 84+ graphing calculators, it is easy to identify the outliers graphically and visually. If we were to measure the vertical distance from any data point to the corresponding point on the line of best fit and that distance was equal to 2s or more, then we would consider the data point to be &#8220;too far&#8221; from the line of best fit. We need to find and graph the lines that are two standard deviations below and above the regression line. Any points that are outside these two lines are outliers. We will call these lines Y2 and Y3:<\/p>\n<p>As we did with the equation of the regression line and the correlation coefficient, we will use technology to calculate this standard deviation for us. Using the <strong>LinRegTTest<\/strong> with this data, scroll down through the output screens to find <strong><em>s<\/em> = 16.412.<\/strong><\/p>\n<p>Line Y2 = \u2013173.5 + 4.83<em>x<\/em> \u20132(16.4) and line Y3 = \u2013173.5 + 4.83<em>x<\/em> + 2(16.4)<\/p>\n<p>where [latex]\\hat{y}[\/latex] =\u00a0\u2013173.5 + 4.83<em>x<\/em> is the line of best fit. Y2 and Y3 have the same slope as the line of best fit.<\/p>\n<p>Graph the scatterplot with the best fit line in equation Y1, then enter the two extra lines as Y2 and Y3 in the &#8220;Y=&#8221; equation editor and press ZOOM 9. You will find that the only data point that is not between lines Y2 and Y3 is the point <em>x<\/em> = 65, <em>y<\/em> = 175. On the calculator screen, it is just barely outside these lines. The outlier is the student who had a grade of 65 on the third exam and 175 on the final exam; this point is further than two standard deviations away from the best-fit line.<\/p>\n<p>Sometimes a point is so close to the lines used to flag outliers on the graph that it is difficult to tell if the point is between or outside the lines. On a computer, enlarging the graph may help; on a small calculator screen, zooming in may make the graph clearer. Note that when the graph does not give a clear enough picture, you can use numerical comparisons to identify outliers.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2318 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11173522\/ae0a876fb8c16c7862344a01a539f3431eef9269.jpeg\" alt=\"The scatter plot of exam scores with a line of best fit.Two yellow dashed lines run parallel to the line of best fit. The dashed lines run above and below the best fit line at equal distances. One data point falls outside the boundary created by the dashed lines\u2014it is an outlier.\" width=\"487\" height=\"312\" \/><\/p>\n<p class=\"\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it 1<\/h3>\n<p>Identify the potential outlier in the scatter plot. The standard deviation of the residuals or errors is approximately 8.6.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2319 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11173623\/7be87771990d07958c334c3b413a1f8131a7c98d.png\" alt=\"A scatter plot with a line of best fit. Most of the dots are near the line. One is below the line at the point (6,58).\" width=\"488\" height=\"323\" \/><\/p>\n<\/div>\n<p>In the table below,\u00a0the first two columns are the third exam and final exam data. The third column shows the predicted\u00a0[latex]\\hat{y}[\/latex]\u00a0values calculated from the line of best fit: [latex]\\hat{y}[\/latex]\u00a0= \u2013173.5 + 4.83<em data-effect=\"italics\">x<\/em>. The residuals, or errors, have been calculated in the fourth column of the table: observed\u00a0<em data-effect=\"italics\">y<\/em>\u00a0value\u2212predicted y value =\u00a0<em data-effect=\"italics\">y<\/em>\u00a0\u2212 [latex]\\hat{y}[\/latex].<\/p>\n<p id=\"element-973\" class=\"\"><em data-effect=\"italics\">s<\/em>\u00a0is the standard deviation of all the\u00a0<em data-effect=\"italics\">y<\/em>\u00a0\u2212 [latex]\\hat{y}[\/latex] =\u00a0<em data-effect=\"italics\">\u03b5<\/em>\u00a0values where\u00a0<em data-effect=\"italics\">n<\/em>\u00a0= the total number of data points. If each residual is calculated and squared, and the results are added, we get the SSE. The standard deviation of the residuals is calculated from the SSE as:<\/p>\n<p>[latex]s = {\\sqrt{\\dfrac{SSE}{n - 2}}}[\/latex]<\/p>\n<h4><strong>Note<\/strong><\/h4>\n<p>We divide by (<em data-effect=\"italics\">n<\/em>\u00a0\u2013 2) because the regression model involves two estimates.<\/p>\n<p id=\"element-42\" class=\"finger\">Rather than calculate the value of\u00a0<em data-effect=\"italics\">s<\/em>\u00a0ourselves, we can find\u00a0<em data-effect=\"italics\">s<\/em>\u00a0using the computer or calculator. For this example, the calculator function LinRegTTest found\u00a0<em data-effect=\"italics\">s<\/em>\u00a0= 16.4 as the standard deviation of the residuals\u00a0<span id=\"set-list2\" data-type=\"list\" data-list-type=\"labeled-item\" data-display=\"inline\"><span data-type=\"item\">35<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">\u201317<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">16<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">\u20136<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">\u201319<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">9<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">3<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">\u20131<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">\u201310<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">\u20139<span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">\u20131.<\/span><\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 164px;\">\n<thead>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 10px;\"><em>x<\/em><\/td>\n<td style=\"width: 25%; height: 10px;\"><em>y<\/em><\/td>\n<td style=\"width: 25%; height: 10px;\"><em>[latex]\\hat{y}[\/latex]<\/em><\/td>\n<td style=\"width: 25%; height: 10px;\"><em>y &#8211;\u00a0[latex]\\hat{y}[\/latex]<\/em><\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px;\">65<\/td>\n<td style=\"width: 25%; height: 14px;\">175<\/td>\n<td style=\"width: 25%; height: 14px;\">140<\/td>\n<td style=\"width: 25%; height: 14px;\">175 \u2013 140 = 35<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px;\">67<\/td>\n<td style=\"width: 25%; height: 14px;\">133<\/td>\n<td style=\"width: 25%; height: 14px;\">150<\/td>\n<td style=\"width: 25%; height: 14px;\">133 \u2013 150= \u201317<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px;\">71<\/td>\n<td style=\"width: 25%; height: 14px;\">185<\/td>\n<td style=\"width: 25%; height: 14px;\">169<\/td>\n<td style=\"width: 25%; height: 14px;\">185 \u2013 169 = 16<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px;\">71<\/td>\n<td style=\"width: 25%; height: 14px;\">163<\/td>\n<td style=\"width: 25%; height: 14px;\">169<\/td>\n<td style=\"width: 25%; height: 14px;\">163 \u2013 169 = \u20136<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px;\">66<\/td>\n<td style=\"width: 25%; height: 14px;\">126<\/td>\n<td style=\"width: 25%; height: 14px;\">145<\/td>\n<td style=\"width: 25%; height: 14px;\">126 \u2013 145 = \u201319<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px;\">75<\/td>\n<td style=\"width: 25%; height: 14px;\">198<\/td>\n<td style=\"width: 25%; height: 14px;\">189<\/td>\n<td style=\"width: 25%; height: 14px;\">198 \u2013 189 = 9<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px;\">67<\/td>\n<td style=\"width: 25%; height: 14px;\">153<\/td>\n<td style=\"width: 25%; height: 14px;\">150<\/td>\n<td style=\"width: 25%; height: 14px;\">153 \u2013 150 = 3<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px;\">70<\/td>\n<td style=\"width: 25%; height: 14px;\">163<\/td>\n<td style=\"width: 25%; height: 14px;\">164<\/td>\n<td style=\"width: 25%; height: 14px;\">163 \u2013 164 = \u20131<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px;\">71<\/td>\n<td style=\"width: 25%; height: 14px;\">159<\/td>\n<td style=\"width: 25%; height: 14px;\">169<\/td>\n<td style=\"width: 25%; height: 14px;\">159 \u2013 169 = \u201310<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px;\">69<\/td>\n<td style=\"width: 25%; height: 14px;\">151<\/td>\n<td style=\"width: 25%; height: 14px;\">160<\/td>\n<td style=\"width: 25%; height: 14px;\">151 \u2013 160 = \u20139<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px;\">69<\/td>\n<td style=\"width: 25%; height: 14px;\">159<\/td>\n<td style=\"width: 25%; height: 14px;\">160<\/td>\n<td style=\"width: 25%; height: 14px;\">159 \u2013 160 = \u20131<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section id=\"eip-212\" data-depth=\"1\">\n<p id=\"element-508\" class=\"\">We are looking for all data points for which the residual is greater than 2<em data-effect=\"italics\">s<\/em>\u00a0= 2(16.4) = 32.8 or less than \u201332.8. Compare these values to the residuals in column four of the table. The only such data point is the student who had a grade of 65 on the third exam and 175 on the final exam; the residual for this student is 35.<\/p>\n<\/section>\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-310\">\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>Measures of the Location of Data. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/2-3-measures-of-the-location-of-the-data\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/2-3-measures-of-the-location-of-the-data<\/a>. <strong>Project<\/strong>: Introductory Statistics. <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>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/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>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Outliers -- 1.5 x IQR (Improved!) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/o8Q0i9VzQZA\">https:\/\/youtu.be\/o8Q0i9VzQZA<\/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":169134,"menu_order":25,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Outliers -- 1.5 x IQR (Improved!) \",\"author\":\"\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/o8Q0i9VzQZA\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"cc\",\"description\":\"Measures of the Location of Data\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/2-3-measures-of-the-location-of-the-data\",\"project\":\"Introductory Statistics\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-310","chapter","type-chapter","status-publish","hentry"],"part":303,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/310","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":27,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/310\/revisions"}],"predecessor-version":[{"id":3990,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/310\/revisions\/3990"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/303"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/310\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=310"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=310"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=310"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=310"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}