{"id":113,"date":"2017-04-15T03:16:59","date_gmt":"2017-04-15T03:16:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/conceptstest1\/chapter\/standard-deviation-1-of-4\/"},"modified":"2017-07-21T13:22:25","modified_gmt":"2017-07-21T13:22:25","slug":"standard-deviation-1-of-4","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/chapter\/standard-deviation-1-of-4\/","title":{"raw":"Standard Deviation (1 of 4)","rendered":"Standard Deviation (1 of 4)"},"content":{"raw":"&nbsp;\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Use mean and standard deviation to describe a distribution.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Introduction<\/h3>\r\nIn the section \"Distributions for Quantitative Data,\" we discussed the spread of a distribution in terms of a <em>typical range <\/em>of values. In \"Quantifying Variability Relative to the Median,\" we made this idea more precise with the interquartile range, IQR. The IQR gives us a measure of spread about the median. We defined a typical range of values about the median as the values between the first and third quartiles.\r\n\r\nNow we want to develop a numerical measure of spread that we can use with the mean. In constructing a measure of spread about the mean, we want to compute how far a \"typical\" number is away from the mean.\r\n<h3>Measuring Spread about the Mean<\/h3>\r\nLet\u2019s consider the sample data set 2, 2, 4, 5, 6, 7, 9. The mean of this data set is\r\n<p style=\"text-align: center\">[latex]\\stackrel{\u00af}{x}=\\frac{2&amp;plus;2&amp;plus;4&amp;plus;5&amp;plus;6&amp;plus;7&amp;plus;9\\text{}}{7}\\text{}=\\text{}\\frac{35}{7}=5[\/latex]<\/p>\r\nHere is a dotplot of this data set with the mean marked by the vertical blue line.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031639\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image2.png\" alt=\"Dotplot of data set with the mean marked by vertical blue line\" width=\"722\" height=\"146\" \/>\r\n\r\nWe can see that some data is close to the mean and some data is further from the mean.\r\n\r\nSince we want to see how the data points deviate from the mean, we determine how far each point is from the mean. We compute the difference between each of these values and the mean. These differences are called the <em>deviations from the mean <\/em>for each point.\r\n<table>\r\n<tbody>\r\n<tr class=\"oli_table\" style=\"height: 15px\">\r\n<td style=\"height: 15px\" align=\"center\">2 \u2212 5 = \u22123<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15px\">\r\n<td style=\"height: 15px\" align=\"center\">2 \u2212 5 = \u22123<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15px\">\r\n<td style=\"height: 15px\" align=\"center\">4 \u2212 5 = \u22121<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15px\">\r\n<td style=\"height: 15px\" align=\"center\">5 \u2212 5 = 0<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15.5156px\">\r\n<td style=\"height: 15.5156px\" align=\"center\">6 \u2212 5 = 1<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15px\">\r\n<td style=\"height: 15px\" align=\"center\">7 \u2212 5 = 2<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15px\">\r\n<td style=\"height: 15px\" align=\"center\">9 \u2212 5 = 4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen visualized on a dotplot, these differences are viewed as distances between each point and the mean. A negative difference indicates that the data point is to the <em>left<\/em> of the mean (shown in blue on the graph below). A positive difference indicates that the data point is to the <em>right<\/em> of the mean (shown in green on the graph below).\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031641\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image3.png\" alt=\"Dotplot where negative differences are shown as data points to the left of the mean; positive differences are shown as data points to the right\" width=\"537\" height=\"161\" \/>\r\n\r\nOur goal is to develop a single measurement that summarizes a typical distance from the mean. Before we continue, let\u2019s practice determining the distance of a single data point from the mean.\r\n<div class=\"textbox exercises\">\r\n<h3>Learn By Doing<\/h3>\r\nThe two questions below refer to the following dotplot. The mean is 9 and it is marked by the vertical blue line.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/images\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image4.png\" alt=\"Dotplot where the average is 9, used as an exercise to show typical distance between data points and the mean.\" width=\"561\" height=\"202\" \/>\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3449\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3450\r\n\r\n<\/div>\r\nSince we want to determine how far a typical number is away from the mean, we might try to average these numbers. However, if we add them all up, we will get 0 (try it). Getting 0 with this procedure (finding differences from mean and adding them all together) is no accident - it <em>always <\/em>produces 0. We have to overcome this problem.\r\n\r\nRecall that we are trying to find the typical <em>distance <\/em>between data points and the mean. It therefore makes sense to take the absolute value of each of these differences.\r\n<table>\r\n<tbody>\r\n<tr class=\"oli_table\" style=\"height: 15px\">\r\n<td style=\"height: 15px\" align=\"center\">| 2 \u2212 5 | = | \u22123 | = 3<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15px\">\r\n<td style=\"height: 15px\" align=\"center\">| 2 \u2212 5 | = | \u22123 | = 3<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15px\">\r\n<td style=\"height: 15px\" align=\"center\">| 4 \u2212 5 | = | \u22121 | = 1<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15.875px\">\r\n<td style=\"height: 15.875px\" align=\"center\">| 5 \u2212 5 | = | 0 | = 0<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15px\">\r\n<td style=\"height: 15px\" align=\"center\">| 6 \u2212 5 | = | 1 | = 1<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15px\">\r\n<td style=\"height: 15px\" align=\"center\">| 7 \u2212 5 | = | 2 | = 2<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15px\">\r\n<td style=\"height: 15px\" align=\"center\">| 9 \u2212 5 | = | 4 | = 4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow we can compute the average of these deviations. There are seven data points, so we add these seven distances and divide by 7. The result is a measure of spread about the mean called the <strong>average deviation from the mean (ADM)<\/strong>.\r\n<p style=\"text-align: center\">[latex]\\frac{3&amp;plus;3&amp;plus;1&amp;plus;0&amp;plus;1&amp;plus;2&amp;plus;4\\text{}}{7}\\text{}=\\text{}\\frac{14}{7}=2[\/latex]<\/p>\r\nWe can indicate this average deviation on a dotplot with a graphic similar to a boxplot as follows.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031643\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image7.png\" alt=\"Dotplot with graphic showing average deviation\" width=\"708\" height=\"178\" \/>\r\n\r\nThe shaded box in the middle is centered at the mean. It extends left and right a distance of 1 average deviation from the mean. Because the average deviation about the mean for this data set is 2, the box starts at 3 (because 5 \u2212 2 = 3) and ends at 7 (because 5 + 2 = 7). In this way, we can use the ADM to define a typical range of values about the mean. Notice that this typical range of values (within 1 ADM of the mean) contains more than half of the values in the data set.\r\n\r\nThe goal of the next Learn By Doing exercise is to improve our intuition of what the ADM measures. We use the following simulation to investigate how the ADM responds to changes in a data set.\r\n\r\nInstructions for adding or removing data points:\r\n<ul>\r\n \t<li>To add a point, move the slider to the value you want, then click the + sign.<\/li>\r\n \t<li>To remove a point, move the slider to the value you want, then click the - sign.<\/li>\r\n \t<li>To reset the simulation to a blank screen, click the button in the upper left corner that says <strong>Reset<\/strong>.<\/li>\r\n<\/ul>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/interactives\/standard_deviation_1_of_4\/standarddeviation1of4.html\" target=\"new\">Click here to open this simulation in its own window.<\/a>\r\n\r\n<iframe id=\"_i_4a\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/interactives\/standard_deviation_1_of_4\/standarddeviation1of4.html\" width=\"550\" height=\"650\"><\/iframe>\r\n<div class=\"textbox exercises\">\r\n<h3>Learn By Doing<\/h3>\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3451\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Learn By Doing<\/h3>\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3452\r\n\r\n<\/div>\r\nBefore we continue, let\u2019s summarize our main points:\r\n<ul>\r\n \t<li>The ADM (average distance from the mean) is a measurement of spread about the mean. More precisely, ADM measures the average distance of the data from the mean.<\/li>\r\n \t<li>We can use the ADM to define a typical range of values about the mean. We mark the mean, then we mark 1 ADM below the mean and 1 ADM above the mean. This interval is centered at the mean and captures typical values about the mean.<\/li>\r\n<\/ul>\r\nUsing these two ideas, we can estimate the ADM by looking at a graph of the distribution of data. We practice this important skill in the next Learn By Doing.\r\n<div class=\"textbox exercises\">\r\n<h3>Learn By Doing<\/h3>\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3453\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3454\r\n\r\n<\/div>\r\nIn the next example, we compare the ADM as a measure of spread to the other ways we have measured spread.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\n<h2>Measuring Variability in Different Ways<\/h2>\r\nThe following dotplots show the potassium content in 76 cereals. Compare children\u2019s cereals to adult cereals. Which type of cereal has more variability in potassium content?\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031645\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image10.png\" alt=\"Dotplots showing potassium content of 76 children\u2019s and adult cereals. \" width=\"622\" height=\"216\" \/>\r\n\r\nWe can visually see that there is more variability in the potassium content of the adult cereals than in the children\u2019s cereals. We can measure this spread in three ways:\r\n<ul>\r\n \t<li>Using overall range: The range of potassium content is larger for the adult cereals than for the children\u2019s cereals. The children\u2019s cereal set has a range of 90 (because 110 \u2212 20 = 90), whereas the adult cereal set has a range of 315 (because 330 \u2212 15 = 315).<\/li>\r\n<\/ul>\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031647\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image11.png\" alt=\"Overall range of potassium content for both adult and children's of cereals\" width=\"671\" height=\"357\" \/>\r\n<ul>\r\n \t<li>Using IQR: The IQR of the adult cereal set is larger than the IQR of the children\u2019s cereal set. The adult cereal set has an IQR of 80, since for that set Q1 = 80 and Q3 = 160. The children\u2019s cereal set has an IQR of 30, since for that set Q1 = 35 and Q3 = 65. Notice here we use the median as a measure of center. The median is marked with a red line. IQR measures spread about the median.<\/li>\r\n<\/ul>\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031649\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image12.png\" alt=\"Boxplots of potassium content using IQR\" width=\"670\" height=\"433\" \/>\r\n<ul>\r\n \t<li>Using ADM: The children\u2019s cereal data set has an ADM of 22. The adult cereal data set has an ADM of 55. Notice here we use the mean as a measure of center. The mean is marked with a blue line. ADM measures spread about the mean.<\/li>\r\n<\/ul>\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031652\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image13.png\" alt=\"ADMs of child and adult cereal data sets\" \/>\r\n\r\n<\/div>\r\nBased on the preceding example, we might expect the data set with the larger range to also have the larger ADM. This is not true, as we illustrate in the next example.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\n<h2>Comparing Range and ADM<\/h2>\r\nWhich data set has more variability? Our answer to this question depends on how we measure variability.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031654\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image14.png\" alt=\"Dotplot using overall range to measure spread\" width=\"495\" height=\"242\" \/>\r\n\r\nWe can see that the overall range is larger for data set b.\r\n<ul>\r\n \t<li>Data set a: range = 100 \u2212 0 = 100<\/li>\r\n \t<li>Data set b: range = 200 \u2212 0 = 200<\/li>\r\n<\/ul>\r\nIf we use overall range to measure spread, we will say that data set b has more variability.\r\n\r\nDoes our answer change if we use ADM to measure spread? Yes!\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031656\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image15.png\" alt=\"Hat plots using ADM to measure spread\" width=\"489\" height=\"260\" \/>\r\n<ul>\r\n \t<li>Data set a: ADM = 41<\/li>\r\n \t<li>Data set b: ADM = 23<\/li>\r\n<\/ul>\r\nMost of the data in data set a is located away from the mean, so the ADM is large: 41. Compare this to data set b. Most of the data in data set b is located close to the mean, so the ADM is small: 23.\r\n\r\nIf we use ADM as a measure of spread, we will say that data set a has more variability.\r\n\r\n<\/div>\r\nThe ADM is a reasonable measure of spread about the mean, but there is another measure that is used much more often: the standard deviation (SD). The standard deviation behaves very much like the average deviation. So all of the work we have done on this page is useful in understanding standard deviation. We discuss standard deviation next.\r\n\r\n&nbsp;","rendered":"<p>&nbsp;<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use mean and standard deviation to describe a distribution.<\/li>\n<\/ul>\n<\/div>\n<h3>Introduction<\/h3>\n<p>In the section &#8220;Distributions for Quantitative Data,&#8221; we discussed the spread of a distribution in terms of a <em>typical range <\/em>of values. In &#8220;Quantifying Variability Relative to the Median,&#8221; we made this idea more precise with the interquartile range, IQR. The IQR gives us a measure of spread about the median. We defined a typical range of values about the median as the values between the first and third quartiles.<\/p>\n<p>Now we want to develop a numerical measure of spread that we can use with the mean. In constructing a measure of spread about the mean, we want to compute how far a &#8220;typical&#8221; number is away from the mean.<\/p>\n<h3>Measuring Spread about the Mean<\/h3>\n<p>Let\u2019s consider the sample data set 2, 2, 4, 5, 6, 7, 9. The mean of this data set is<\/p>\n<p style=\"text-align: center\">[latex]\\stackrel{\u00af}{x}=\\frac{2&plus;2&plus;4&plus;5&plus;6&plus;7&plus;9\\text{}}{7}\\text{}=\\text{}\\frac{35}{7}=5[\/latex]<\/p>\n<p>Here is a dotplot of this data set with the mean marked by the vertical blue line.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031639\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image2.png\" alt=\"Dotplot of data set with the mean marked by vertical blue line\" width=\"722\" height=\"146\" \/><\/p>\n<p>We can see that some data is close to the mean and some data is further from the mean.<\/p>\n<p>Since we want to see how the data points deviate from the mean, we determine how far each point is from the mean. We compute the difference between each of these values and the mean. These differences are called the <em>deviations from the mean <\/em>for each point.<\/p>\n<table>\n<tbody>\n<tr class=\"oli_table\" style=\"height: 15px\">\n<td style=\"height: 15px\" align=\"center\">2 \u2212 5 = \u22123<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15px\">\n<td style=\"height: 15px\" align=\"center\">2 \u2212 5 = \u22123<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15px\">\n<td style=\"height: 15px\" align=\"center\">4 \u2212 5 = \u22121<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15px\">\n<td style=\"height: 15px\" align=\"center\">5 \u2212 5 = 0<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15.5156px\">\n<td style=\"height: 15.5156px\" align=\"center\">6 \u2212 5 = 1<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15px\">\n<td style=\"height: 15px\" align=\"center\">7 \u2212 5 = 2<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15px\">\n<td style=\"height: 15px\" align=\"center\">9 \u2212 5 = 4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When visualized on a dotplot, these differences are viewed as distances between each point and the mean. A negative difference indicates that the data point is to the <em>left<\/em> of the mean (shown in blue on the graph below). A positive difference indicates that the data point is to the <em>right<\/em> of the mean (shown in green on the graph below).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031641\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image3.png\" alt=\"Dotplot where negative differences are shown as data points to the left of the mean; positive differences are shown as data points to the right\" width=\"537\" height=\"161\" \/><\/p>\n<p>Our goal is to develop a single measurement that summarizes a typical distance from the mean. Before we continue, let\u2019s practice determining the distance of a single data point from the mean.<\/p>\n<div class=\"textbox exercises\">\n<h3>Learn By Doing<\/h3>\n<p>The two questions below refer to the following dotplot. The mean is 9 and it is marked by the vertical blue line.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/images\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image4.png\" alt=\"Dotplot where the average is 9, used as an exercise to show typical distance between data points and the mean.\" width=\"561\" height=\"202\" \/><br \/>\n\t<iframe id=\"lumen_assessment_3449\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3449&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3449\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3450\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3450&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3450\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<p>Since we want to determine how far a typical number is away from the mean, we might try to average these numbers. However, if we add them all up, we will get 0 (try it). Getting 0 with this procedure (finding differences from mean and adding them all together) is no accident &#8211; it <em>always <\/em>produces 0. We have to overcome this problem.<\/p>\n<p>Recall that we are trying to find the typical <em>distance <\/em>between data points and the mean. It therefore makes sense to take the absolute value of each of these differences.<\/p>\n<table>\n<tbody>\n<tr class=\"oli_table\" style=\"height: 15px\">\n<td style=\"height: 15px\" align=\"center\">| 2 \u2212 5 | = | \u22123 | = 3<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15px\">\n<td style=\"height: 15px\" align=\"center\">| 2 \u2212 5 | = | \u22123 | = 3<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15px\">\n<td style=\"height: 15px\" align=\"center\">| 4 \u2212 5 | = | \u22121 | = 1<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15.875px\">\n<td style=\"height: 15.875px\" align=\"center\">| 5 \u2212 5 | = | 0 | = 0<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15px\">\n<td style=\"height: 15px\" align=\"center\">| 6 \u2212 5 | = | 1 | = 1<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15px\">\n<td style=\"height: 15px\" align=\"center\">| 7 \u2212 5 | = | 2 | = 2<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15px\">\n<td style=\"height: 15px\" align=\"center\">| 9 \u2212 5 | = | 4 | = 4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now we can compute the average of these deviations. There are seven data points, so we add these seven distances and divide by 7. The result is a measure of spread about the mean called the <strong>average deviation from the mean (ADM)<\/strong>.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{3&plus;3&plus;1&plus;0&plus;1&plus;2&plus;4\\text{}}{7}\\text{}=\\text{}\\frac{14}{7}=2[\/latex]<\/p>\n<p>We can indicate this average deviation on a dotplot with a graphic similar to a boxplot as follows.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031643\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image7.png\" alt=\"Dotplot with graphic showing average deviation\" width=\"708\" height=\"178\" \/><\/p>\n<p>The shaded box in the middle is centered at the mean. It extends left and right a distance of 1 average deviation from the mean. Because the average deviation about the mean for this data set is 2, the box starts at 3 (because 5 \u2212 2 = 3) and ends at 7 (because 5 + 2 = 7). In this way, we can use the ADM to define a typical range of values about the mean. Notice that this typical range of values (within 1 ADM of the mean) contains more than half of the values in the data set.<\/p>\n<p>The goal of the next Learn By Doing exercise is to improve our intuition of what the ADM measures. We use the following simulation to investigate how the ADM responds to changes in a data set.<\/p>\n<p>Instructions for adding or removing data points:<\/p>\n<ul>\n<li>To add a point, move the slider to the value you want, then click the + sign.<\/li>\n<li>To remove a point, move the slider to the value you want, then click the &#8211; sign.<\/li>\n<li>To reset the simulation to a blank screen, click the button in the upper left corner that says <strong>Reset<\/strong>.<\/li>\n<\/ul>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/interactives\/standard_deviation_1_of_4\/standarddeviation1of4.html\" target=\"new\">Click here to open this simulation in its own window.<\/a><\/p>\n<p><iframe loading=\"lazy\" id=\"_i_4a\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/interactives\/standard_deviation_1_of_4\/standarddeviation1of4.html\" width=\"550\" height=\"650\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Learn By Doing<\/h3>\n<p>\t<iframe id=\"lumen_assessment_3451\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3451&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3451\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Learn By Doing<\/h3>\n<p>\t<iframe id=\"lumen_assessment_3452\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3452&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3452\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<p>Before we continue, let\u2019s summarize our main points:<\/p>\n<ul>\n<li>The ADM (average distance from the mean) is a measurement of spread about the mean. More precisely, ADM measures the average distance of the data from the mean.<\/li>\n<li>We can use the ADM to define a typical range of values about the mean. We mark the mean, then we mark 1 ADM below the mean and 1 ADM above the mean. This interval is centered at the mean and captures typical values about the mean.<\/li>\n<\/ul>\n<p>Using these two ideas, we can estimate the ADM by looking at a graph of the distribution of data. We practice this important skill in the next Learn By Doing.<\/p>\n<div class=\"textbox exercises\">\n<h3>Learn By Doing<\/h3>\n<p>\t<iframe id=\"lumen_assessment_3453\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3453&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3453\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3454\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3454&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3454\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<p>In the next example, we compare the ADM as a measure of spread to the other ways we have measured spread.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<h2>Measuring Variability in Different Ways<\/h2>\n<p>The following dotplots show the potassium content in 76 cereals. Compare children\u2019s cereals to adult cereals. Which type of cereal has more variability in potassium content?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031645\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image10.png\" alt=\"Dotplots showing potassium content of 76 children\u2019s and adult cereals.\" width=\"622\" height=\"216\" \/><\/p>\n<p>We can visually see that there is more variability in the potassium content of the adult cereals than in the children\u2019s cereals. We can measure this spread in three ways:<\/p>\n<ul>\n<li>Using overall range: The range of potassium content is larger for the adult cereals than for the children\u2019s cereals. The children\u2019s cereal set has a range of 90 (because 110 \u2212 20 = 90), whereas the adult cereal set has a range of 315 (because 330 \u2212 15 = 315).<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031647\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image11.png\" alt=\"Overall range of potassium content for both adult and children's of cereals\" width=\"671\" height=\"357\" \/><\/p>\n<ul>\n<li>Using IQR: The IQR of the adult cereal set is larger than the IQR of the children\u2019s cereal set. The adult cereal set has an IQR of 80, since for that set Q1 = 80 and Q3 = 160. The children\u2019s cereal set has an IQR of 30, since for that set Q1 = 35 and Q3 = 65. Notice here we use the median as a measure of center. The median is marked with a red line. IQR measures spread about the median.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031649\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image12.png\" alt=\"Boxplots of potassium content using IQR\" width=\"670\" height=\"433\" \/><\/p>\n<ul>\n<li>Using ADM: The children\u2019s cereal data set has an ADM of 22. The adult cereal data set has an ADM of 55. Notice here we use the mean as a measure of center. The mean is marked with a blue line. ADM measures spread about the mean.<\/li>\n<\/ul>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031652\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image13.png\" alt=\"ADMs of child and adult cereal data sets\" \/><\/p>\n<\/div>\n<p>Based on the preceding example, we might expect the data set with the larger range to also have the larger ADM. This is not true, as we illustrate in the next example.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<h2>Comparing Range and ADM<\/h2>\n<p>Which data set has more variability? Our answer to this question depends on how we measure variability.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031654\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image14.png\" alt=\"Dotplot using overall range to measure spread\" width=\"495\" height=\"242\" \/><\/p>\n<p>We can see that the overall range is larger for data set b.<\/p>\n<ul>\n<li>Data set a: range = 100 \u2212 0 = 100<\/li>\n<li>Data set b: range = 200 \u2212 0 = 200<\/li>\n<\/ul>\n<p>If we use overall range to measure spread, we will say that data set b has more variability.<\/p>\n<p>Does our answer change if we use ADM to measure spread? Yes!<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031656\/m2_summarizing_data_topic_2_4_Topic2_4StandardDeviation1of4_image15.png\" alt=\"Hat plots using ADM to measure spread\" width=\"489\" height=\"260\" \/><\/p>\n<ul>\n<li>Data set a: ADM = 41<\/li>\n<li>Data set b: ADM = 23<\/li>\n<\/ul>\n<p>Most of the data in data set a is located away from the mean, so the ADM is large: 41. Compare this to data set b. Most of the data in data set b is located close to the mean, so the ADM is small: 23.<\/p>\n<p>If we use ADM as a measure of spread, we will say that data set a has more variability.<\/p>\n<\/div>\n<p>The ADM is a reasonable measure of spread about the mean, but there is another measure that is used much more often: the standard deviation (SD). The standard deviation behaves very much like the average deviation. So all of the work we have done on this page is useful in understanding standard deviation. We discuss standard deviation next.<\/p>\n<p>&nbsp;<\/p>\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-113\">\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>Concepts in Statistics. <strong>Provided by<\/strong>: Open Learning Initiative. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/oli.cmu.edu\">http:\/\/oli.cmu.edu<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":163,"menu_order":20,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Concepts in Statistics\",\"author\":\"\",\"organization\":\"Open Learning Initiative\",\"url\":\"http:\/\/oli.cmu.edu\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"79d74466-4e4f-4565-9797-6b7ada3035f7","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-113","chapter","type-chapter","status-web-only","hentry"],"part":43,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/113","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/users\/163"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/113\/revisions"}],"predecessor-version":[{"id":1577,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/113\/revisions\/1577"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/parts\/43"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/113\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/media?parent=113"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapter-type?post=113"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/contributor?post=113"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/license?post=113"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}