{"id":1369,"date":"2023-06-22T02:20:35","date_gmt":"2023-06-22T02:20:35","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/anova-learn-it-1\/"},"modified":"2025-05-16T22:56:10","modified_gmt":"2025-05-16T22:56:10","slug":"anova-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/anova-learn-it-1\/","title":{"raw":"ANOVA - Learn It 1","rendered":"ANOVA &#8211; Learn It 1"},"content":{"raw":"<section>\r\n<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Perform a one-way ANOVA hypothesis test&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12801,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:10}\">Complete a one-way ANOVA hypothesis test<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Write the conclusion of a one-way ANOVA hypothesis test in context of the problem&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12801,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:10}\">Write the conclusion of a one-way ANOVA hypothesis test in context of the problem<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<\/section>\r\n<p><img class=\"wp-image-1341 alignright\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2023\/04\/24193417\/Screen-Shot-2022-11-28-at-10.33.00-AM.png\" alt=\"A large potted plant on a stool with additional potted plants on the ground behind it.\" width=\"162\" height=\"350\" \/><\/p>\r\n<p class=\"para\" style=\"margin: 6.0pt 0in 6.0pt 0in;\">Suppose a researcher wants to investigate the effect of the amount of fertilizer on the height of a common houseplant. More specifically, the researcher is interested in determining if there is a difference in the mean height of plants between those receiving one of the following three different fertilizer levels: high, medium, and low.<\/p>\r\n<p class=\"para\" style=\"margin: 6.0pt 0in 6.0pt 0in;\">The following data are the simulated results of this controlled experiment.<\/p>\r\n<p class=\"para\" style=\"margin: 6.0pt 0in 6.0pt 0in;\">(Note that this small data set is used to introduce the concept and make calculations easier. When conducting an ANOVA, larger sample sizes are usually needed to meet assumptions.)<\/p>\r\n<div align=\"center\">\r\n<table style=\"width: 338px;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 92.4479px;\"><strong>Fertilizer Level<\/strong><\/td>\r\n<td style=\"width: 205.33px;\"><strong>Height of Plant (inches)<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 92.4479px;\">Low<\/td>\r\n<td style=\"width: 205.33px;\">23.2, 20.9, 21.5, 25.3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 92.4479px;\">Medium<\/td>\r\n<td style=\"width: 205.33px;\">24.6, 27.7, 22.5, 30.1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 92.4479px;\">High<\/td>\r\n<td style=\"width: 205.33px;\">29.2, 30.2, 31.1, 33.6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: left;\">As we saw previously, conducting a one-way ANOVA involves comparing the variation <em>within<\/em> each of the groups to the variation <em>between<\/em> each of the groups. When the variation <em>between<\/em> each of the groups is significantly larger than the variation <em>within<\/em> each of the groups, we might conclude that there is a statistically significant difference among the means.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>ANOVA table<\/h3>\r\n<p style=\"text-align: left;\">In an ANOVA table, the calculation illustrating the total variation <em>within<\/em> the groups of interest is known as the <strong>error sum of squares (SSError)<\/strong>. The calculation illustrating the total variation <em>between<\/em> the groups is known as the <strong>group sum of squares (SSGroup)<\/strong>.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: left;\">Two other essential columns found in an ANOVA table are the <strong>degrees of freedom<\/strong> <strong>(df)<\/strong> and the <strong>mean square<\/strong>. <strong>\u00a0<\/strong><\/p>\r\n<\/section>\r\n<p style=\"text-align: left;\">The following table illustrates how these values are calculated for each of the given sources: Group or Error (i.e., <em>between<\/em> and <em>within<\/em>).<\/p>\r\n<p style=\"text-align: left;\">When calculating these values, it is important to know that represents the number of groups being considered and represents the total number of data values among all groups.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>ANOVA table<\/h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 56.9271px;\"><strong>Source<\/strong><\/td>\r\n<td style=\"width: 148.993px;\"><strong>Degrees of Freedom (df)<\/strong><\/td>\r\n<td style=\"width: 69.0625px;\"><strong>Sum of Squares<\/strong><\/td>\r\n<td style=\"width: 210px;\"><strong>Mean Square<\/strong><\/td>\r\n<td style=\"width: 212.795px;\"><strong>F-Statistic<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 56.9271px;\">Group<\/td>\r\n<td style=\"width: 148.993px;\">\r\n<p>[latex]k-1[\/latex]<\/p>\r\n<p>(The number of groups minus 1)<\/p>\r\n<\/td>\r\n<td style=\"width: 69.0625px;\">SSGroup<\/td>\r\n<td style=\"width: 210px;\">[latex]\\dfrac{\\text{SSGroup}}{k-1}[\/latex]<\/td>\r\n<td style=\"width: 212.795px;\">[latex]\\dfrac{\\text{MSGroup}}{\\text{MSError}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 56.9271px;\">Error<\/td>\r\n<td style=\"width: 148.993px;\">\r\n<p>[latex]N-k[\/latex]<\/p>\r\n<p>(The total number of data points minus the number of groups)<\/p>\r\n<\/td>\r\n<td style=\"width: 69.0625px;\">SSError<\/td>\r\n<td style=\"width: 210px;\">[latex]\\dfrac{\\text{SSError}}{N-k}[\/latex]<\/td>\r\n<td style=\"width: 212.795px;\">\u00a0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 56.9271px;\">Total<\/td>\r\n<td style=\"width: 148.993px;\">\r\n<p>\u00a0[latex]N-1[\/latex]<\/p>\r\n<p>(The total number of data points minus 1)<\/p>\r\n<\/td>\r\n<td style=\"width: 69.0625px;\">SSGroup + SSError<\/td>\r\n<td style=\"width: 210px;\">\u00a0<\/td>\r\n<td style=\"width: 212.795px;\">\u00a0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>\r\n<div style=\"text-align: left;\" align=\"center\">\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2188[\/ohm2_question]<\/section>\r\n<\/div>\r\n<\/div>","rendered":"<section>\n<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Perform a one-way ANOVA hypothesis test&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12801,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:10}\">Complete a one-way ANOVA hypothesis test<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Write the conclusion of a one-way ANOVA hypothesis test in context of the problem&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12801,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:10}\">Write the conclusion of a one-way ANOVA hypothesis test in context of the problem<\/span><\/li>\n<\/ul>\n<\/section>\n<\/section>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1341 alignright\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2023\/04\/24193417\/Screen-Shot-2022-11-28-at-10.33.00-AM.png\" alt=\"A large potted plant on a stool with additional potted plants on the ground behind it.\" width=\"162\" height=\"350\" \/><\/p>\n<p class=\"para\" style=\"margin: 6.0pt 0in 6.0pt 0in;\">Suppose a researcher wants to investigate the effect of the amount of fertilizer on the height of a common houseplant. More specifically, the researcher is interested in determining if there is a difference in the mean height of plants between those receiving one of the following three different fertilizer levels: high, medium, and low.<\/p>\n<p class=\"para\" style=\"margin: 6.0pt 0in 6.0pt 0in;\">The following data are the simulated results of this controlled experiment.<\/p>\n<p class=\"para\" style=\"margin: 6.0pt 0in 6.0pt 0in;\">(Note that this small data set is used to introduce the concept and make calculations easier. When conducting an ANOVA, larger sample sizes are usually needed to meet assumptions.)<\/p>\n<div style=\"margin: auto;\">\n<table style=\"width: 338px;\">\n<tbody>\n<tr>\n<td style=\"width: 92.4479px;\"><strong>Fertilizer Level<\/strong><\/td>\n<td style=\"width: 205.33px;\"><strong>Height of Plant (inches)<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 92.4479px;\">Low<\/td>\n<td style=\"width: 205.33px;\">23.2, 20.9, 21.5, 25.3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 92.4479px;\">Medium<\/td>\n<td style=\"width: 205.33px;\">24.6, 27.7, 22.5, 30.1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 92.4479px;\">High<\/td>\n<td style=\"width: 205.33px;\">29.2, 30.2, 31.1, 33.6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: left;\">As we saw previously, conducting a one-way ANOVA involves comparing the variation <em>within<\/em> each of the groups to the variation <em>between<\/em> each of the groups. When the variation <em>between<\/em> each of the groups is significantly larger than the variation <em>within<\/em> each of the groups, we might conclude that there is a statistically significant difference among the means.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>ANOVA table<\/h3>\n<p style=\"text-align: left;\">In an ANOVA table, the calculation illustrating the total variation <em>within<\/em> the groups of interest is known as the <strong>error sum of squares (SSError)<\/strong>. The calculation illustrating the total variation <em>between<\/em> the groups is known as the <strong>group sum of squares (SSGroup)<\/strong>.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">Two other essential columns found in an ANOVA table are the <strong>degrees of freedom<\/strong> <strong>(df)<\/strong> and the <strong>mean square<\/strong>. <strong>\u00a0<\/strong><\/p>\n<\/section>\n<p style=\"text-align: left;\">The following table illustrates how these values are calculated for each of the given sources: Group or Error (i.e., <em>between<\/em> and <em>within<\/em>).<\/p>\n<p style=\"text-align: left;\">When calculating these values, it is important to know that represents the number of groups being considered and represents the total number of data values among all groups.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>ANOVA table<\/h3>\n<table>\n<tbody>\n<tr>\n<td style=\"width: 56.9271px;\"><strong>Source<\/strong><\/td>\n<td style=\"width: 148.993px;\"><strong>Degrees of Freedom (df)<\/strong><\/td>\n<td style=\"width: 69.0625px;\"><strong>Sum of Squares<\/strong><\/td>\n<td style=\"width: 210px;\"><strong>Mean Square<\/strong><\/td>\n<td style=\"width: 212.795px;\"><strong>F-Statistic<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 56.9271px;\">Group<\/td>\n<td style=\"width: 148.993px;\">\n[latex]k-1[\/latex]<\/p>\n<p>(The number of groups minus 1)<\/p>\n<\/td>\n<td style=\"width: 69.0625px;\">SSGroup<\/td>\n<td style=\"width: 210px;\">[latex]\\dfrac{\\text{SSGroup}}{k-1}[\/latex]<\/td>\n<td style=\"width: 212.795px;\">[latex]\\dfrac{\\text{MSGroup}}{\\text{MSError}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 56.9271px;\">Error<\/td>\n<td style=\"width: 148.993px;\">\n[latex]N-k[\/latex]<\/p>\n<p>(The total number of data points minus the number of groups)<\/p>\n<\/td>\n<td style=\"width: 69.0625px;\">SSError<\/td>\n<td style=\"width: 210px;\">[latex]\\dfrac{\\text{SSError}}{N-k}[\/latex]<\/td>\n<td style=\"width: 212.795px;\">\u00a0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 56.9271px;\">Total<\/td>\n<td style=\"width: 148.993px;\">\n[latex]N-1[\/latex]<\/p>\n<p>(The total number of data points minus 1)<\/p>\n<\/td>\n<td style=\"width: 69.0625px;\">SSGroup + SSError<\/td>\n<td style=\"width: 210px;\">\u00a0<\/td>\n<td style=\"width: 212.795px;\">\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<div style=\"text-align: left; margin: auto;\">\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2188\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2188&theme=lumen&iframe_resize_id=ohm2188&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<\/div>\n","protected":false},"author":8,"menu_order":21,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1348,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1369"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":11,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1369\/revisions"}],"predecessor-version":[{"id":6845,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1369\/revisions\/6845"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1348"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1369\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1369"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1369"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1369"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1369"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}