{"id":1198,"date":"2021-06-27T15:21:44","date_gmt":"2021-06-27T19:21:44","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&p=1198"},"modified":"2025-06-24T17:45:56","modified_gmt":"2025-06-24T21:45:56","slug":"12-2-main-idea-behind-one-way-anova","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/12-2-main-idea-behind-one-way-anova\/","title":{"raw":"12.2 Main Idea Behind One-Way ANOVA","rendered":"12.2 Main Idea Behind One-Way ANOVA"},"content":{"raw":"Let [latex]\\mu_1, \\mu_2, \\dots , \\mu_k[\/latex] be [latex]k[\/latex] population means. The hypotheses of one-way ANOVA are formulated as\r\n\r\n[latex]H_0[\/latex]: all means are equal, i.e., [latex]\\mu_1 = \\mu_2 = \\dots = \\mu_k[\/latex]\r\n\r\n[latex]H_a[\/latex]: not all the means are equal.\r\n\r\nThe alternative hypothesis [latex]H_{a}[\/latex]\u00a0means there exists at least one pair of means that are not equal. Do not<\/strong> write as [latex]H_{a}: \\mu_1 \\neq \\mu_2 \\neq \\dots \\neq \\mu_k[\/latex]. Do not<\/strong> write \u201cat least one mean is different from the others\u201d since it sounds like at least one mean is different from the others while all the others are the same. Both are just two special cases of what \"not all the means are equal\" means.<\/a>\r\n\r\n[caption id=\"attachment_2911\" align=\"aligncenter\" width=\"817\"]\"Three Figure 12.1<\/strong>: One-Way ANOVA Based on k Independent Samples. [Image Description (See Appendix D Figure 12.1)<\/a>][\/caption]ANOVA F tests are based on [latex]k[\/latex] independent, simple random samples from [latex]k[\/latex] populations. If [latex]H_0: \\mu_1 = \\mu_2 = \\dots = \\mu_k[\/latex] is true, the sample means [latex] \\bar{x}_1, \\bar{x}_2, \\dots, \\bar{x}_k[\/latex] should be close to one another and hence, the variation among sample means should be small. Therefore, we should reject\u00a0 [latex]H_0[\/latex] if the sample means are very different from one another (meaning the variation among the sample means would be large).\r\n

Quantifying Variation<\/strong><\/h2>\r\nThe total variation of the data (SST: the total sum of squares) is quantified as the sum of squared distances from each observation to the overall mean, i.e., [latex]SST = \\sum (x_{ij} - \\bar{x})^2[\/latex], where [latex]x_{ij}[\/latex] is the [latex]j[\/latex]th observation of sample [latex]i[\/latex], [latex]\\bar{x} = \\frac{\\sum x_{ij}}{n}[\/latex] \u00a0is the overall mean, and [latex]n = n_1 + n_2 + \\dots + n_k[\/latex] is the overall sample size.\r\n\r\nThe treatment sum of squares quantifies the variation of the sample means:\r\n

[latex]SSTR = \\sum\\limits_{i=1}^{k} n_i (\\bar{x}_i - \\bar{x})^2.[\/latex]<\/p>\r\n\u00a0SSTR quantifies the so-called between-group variation. For this reason,\u00a0\u00a0we should reject [latex]H_0: \\mu_1 = \\mu_2 = \\dots = \\mu_k[\/latex] if SSTR is too large. However, SSTR is only considered \"large\" if it is large relative to the next measure of variation.\r\n\r\nThe within-group variation can be quantified as the sum of squared distances from each observation to the mean of its sample group, i.e.,\r\n

[latex]SSE = \\sum (x_{ij} - \\bar{x}_i)^2 = \\sum_\\limits{i=1}^k (n_i - 1) s_i^2.[\/latex]<\/p>\r\nIn practice, software is used to calculate all these sums of squares and other ANOVA calculations.\r\n\r\nThe total variation SST can be shown as:\r\n

[latex]SST = SSTR + SSE = \\text{between group variation + within group variation}.[\/latex]<\/p>\r\nThe relationship between SSTR (between group variation) and SSE (within-group variation) will be illustrated in the following example.\r\n\r\nA person recorded waiting times each time he called either Uber or Taxi service from his house and again each time he called either service from work. His results are summarized in the following table (red values correspond to waiting times for Uber and blue values correspond to Taxi):\r\n

Table 12.2<\/strong>: Waiting Time for Uber (red) and Taxi (blue) Called from Home and Work<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
Home<\/th>\r\n1<\/span><\/td>\r\n2<\/span><\/td>\r\n3<\/span><\/td>\r\n3<\/span><\/td>\r\n4<\/span><\/td>\r\n5<\/span><\/td>\r\n6<\/span><\/td>\r\n7<\/span><\/td>\r\n8<\/span><\/td>\r\n8<\/span><\/td>\r\n9<\/span><\/td>\r\n10<\/span><\/td>\r\n<\/tr>\r\n
Work<\/th>\r\n1<\/span><\/td>\r\n2<\/span><\/td>\r\n3<\/span><\/td>\r\n3<\/span><\/td>\r\n4<\/span><\/td>\r\n5<\/span><\/td>\r\n6<\/span><\/td>\r\n7<\/span><\/td>\r\n8<\/span><\/td>\r\n8<\/span><\/td>\r\n9<\/span><\/td>\r\n10<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe define the waiting times called from Home as Data Set 1 and those called from Work as Data Set 2. The figure below shows two data sets. Each set consists of two populations: waiting time for Uber (red circle with mean [latex]\\mu_1[\/latex]) and waiting time for Taxi (blue cross with mean [latex]\\mu_2[\/latex]).<\/a>\r\n\r\n[caption id=\"attachment_1202\" align=\"aligncenter\" width=\"416\"]\"Two Figure 12.2<\/strong>: Waiting Time for Uber (red) and Taxi (blue) Called from Home (Data Set 1) and Work (Data Set 2). [Image Description (See Appendix D Figure 12.2)<\/a>][\/caption]\r\n
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Exercise: Quantify Variation<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nBased on the two data sets shown above, answer these questions.\r\n