1 00:00:00,090 --> 00:00:04,890 Finally, we're going to be looking at the ANOVA process, which provides us with a statistical test 2 00:00:04,890 --> 00:00:08,890 to determine whether two or more population means are equal to one another. 3 00:00:08,910 --> 00:00:13,770 Now this ANOVA word comes from analysis of variance. 4 00:00:13,770 --> 00:00:19,590 We just take the and the O and the VA to create ANOVA, but if you hear ANOVA, just think analysis 5 00:00:19,590 --> 00:00:20,430 of variance. 6 00:00:20,430 --> 00:00:27,150 What we're trying to do is analyze the variance within and between samples from different populations. 7 00:00:27,150 --> 00:00:31,500 So the simplest way to get our arms around ANOVA is to work through an example. 8 00:00:31,530 --> 00:00:38,430 Let's say that we manage all of the customer service call centers for a large company and we have one 9 00:00:38,430 --> 00:00:43,200 customer service center in the northeast, one in the Northwest and one in the Southeast. 10 00:00:43,200 --> 00:00:49,860 And what we want to know is whether mean call time differs across these three different call centers. 11 00:00:49,890 --> 00:00:53,970 Now, the issue here comes back to this idea of variance. 12 00:00:53,970 --> 00:00:57,750 Let's just focus in for a second on the call center in the northeast. 13 00:00:57,780 --> 00:01:04,290 We know that if we take samples of calls from that Northeast call center, we're going to get a bunch 14 00:01:04,290 --> 00:01:10,500 of different samples, each of which has its own mean and standard deviation, which means there's variance 15 00:01:10,500 --> 00:01:13,350 among samples within the Northeast call center. 16 00:01:13,350 --> 00:01:18,240 There's also variance within the Northwest call center and variance within the southeast call center. 17 00:01:18,240 --> 00:01:23,640 And then, of course, we could have variance between the different call centers depending on which 18 00:01:23,640 --> 00:01:26,130 sample we pick from each center. 19 00:01:26,130 --> 00:01:32,910 So what we're trying to do here is analyze how much of the variance is coming from within the sample 20 00:01:32,910 --> 00:01:37,410 that we take from a call center versus between the samples that we take from each call center. 21 00:01:37,410 --> 00:01:44,790 Now, the set of hypotheses that we're going to be testing is always here that the null hypothesis is 22 00:01:44,790 --> 00:01:47,340 that the three means are equivalent. 23 00:01:47,340 --> 00:01:53,940 So the mean from the Northeast call center is equal to the mean from the Northwest call center, which 24 00:01:53,940 --> 00:01:58,350 is equal to the mean from the southeast call center. 25 00:01:58,350 --> 00:02:03,540 In other words, mean call time for the entire population of calls coming out of the Northeast call 26 00:02:03,540 --> 00:02:09,300 center is equivalent to mean call time for the entire population of calls coming out of the northwest 27 00:02:09,300 --> 00:02:10,020 call center. 28 00:02:10,020 --> 00:02:13,500 And those are both equivalent to population mean for the southeast call center. 29 00:02:13,500 --> 00:02:20,040 So by default here, this is our null hypothesis, which means then that our alternative hypothesis, 30 00:02:20,040 --> 00:02:25,800 what we're trying to analyze is whether there's actually any difference in call time across call centers. 31 00:02:25,800 --> 00:02:32,010 So if we say in our null hypothesis that call times are all equivalent, our alternative hypothesis 32 00:02:32,010 --> 00:02:34,860 is that these call times are not equivalent. 33 00:02:35,280 --> 00:02:40,770 What we've done to try to investigate this set of hypotheses is we've taken a sample from each of the 34 00:02:40,770 --> 00:02:43,410 call centers of ten call times each. 35 00:02:43,410 --> 00:02:49,050 Let's say that these call times are in minutes, and now we want to use a statistical test to see if 36 00:02:49,050 --> 00:02:51,570 the call times are different across call centers. 37 00:02:51,570 --> 00:02:56,820 So to use ANOVA to do this, the first thing that we need to figure out is how many different groups 38 00:02:56,820 --> 00:03:00,150 we have and how many different data points we have within each group. 39 00:03:00,150 --> 00:03:04,890 So here we have three groups, the northeast, northwest and southeast. 40 00:03:04,890 --> 00:03:07,650 So we'll say that M is equal to three. 41 00:03:07,890 --> 00:03:11,880 We've taken a sample of ten calls from each call center. 42 00:03:11,880 --> 00:03:16,080 So within each group we have an equal ten. 43 00:03:16,110 --> 00:03:18,120 The sample size is N equals ten. 44 00:03:18,120 --> 00:03:22,720 So we have M equals three groups, each containing N equals ten data points. 45 00:03:22,740 --> 00:03:27,660 Our next step then, is to calculate the mean within each group. 46 00:03:27,660 --> 00:03:33,120 So for the Northeast group here, we add up these ten data points and then divide by ten to find the 47 00:03:33,120 --> 00:03:34,590 mean of the Northeast. 48 00:03:34,590 --> 00:03:41,490 So we could think about this here as our sample mean, we could call it x bar for the Northeast. 49 00:03:41,490 --> 00:03:44,040 We do the same thing for the Northwest. 50 00:03:44,040 --> 00:03:49,530 And so we have a sample mean for the northwest and we do the same thing for the southeast. 51 00:03:49,530 --> 00:03:52,710 So we have a sample mean for the southeast. 52 00:03:52,710 --> 00:03:58,080 So mean call time in the northeast is 3 minutes mean call time in the northwest is 6 minutes mean call 53 00:03:58,080 --> 00:04:00,120 time in the southeast is 4 minutes. 54 00:04:00,150 --> 00:04:05,040 Now, of course, just like with all of these statistical tests, the fact that the sample means are 55 00:04:05,040 --> 00:04:10,290 different doesn't tell us that we can reject the null and lend support to the alternative hypothesis 56 00:04:10,290 --> 00:04:12,240 that the population means are different. 57 00:04:12,240 --> 00:04:16,079 Just because the sample means are different doesn't automatically mean the populations are different. 58 00:04:16,079 --> 00:04:22,050 We need to show with statistical significance that these sample means are different enough such that 59 00:04:22,050 --> 00:04:28,620 we can justify making a conclusion that the associated population means are likely to be different as 60 00:04:28,620 --> 00:04:29,130 well. 61 00:04:29,160 --> 00:04:35,130 Now, once we have the mean within each group and remember, we can use this analysis of variance process 62 00:04:35,130 --> 00:04:39,330 on any number of groups as long as we're using two or more groups. 63 00:04:39,330 --> 00:04:43,110 So this is a one way ANOVA that we're performing on three groups. 64 00:04:43,110 --> 00:04:48,120 There are also two way ANOVA and lots of other ANOVA variations, but right now we're just going to 65 00:04:48,120 --> 00:04:54,510 focus on one way ANOVA as we happen to be using three groups here and we'll look at how to work through 66 00:04:54,510 --> 00:04:57,570 an example with this particular kind of test. 67 00:04:57,600 --> 00:04:59,730 Now back to the means that we. 68 00:04:59,860 --> 00:05:03,040 We've calculated we have the sample mean for each group. 69 00:05:03,040 --> 00:05:06,010 The next thing we need to find is this grand mean. 70 00:05:06,010 --> 00:05:11,500 And if you remember before, when we talked about combinations of random variables, we said that the 71 00:05:11,500 --> 00:05:14,860 mean of the sum was equal to the sum of the means. 72 00:05:14,860 --> 00:05:21,370 So to find this grand mean, all we have to do is take the mean of these three sample means so we can 73 00:05:21,370 --> 00:05:24,640 take three plus six is nine plus four is 13. 74 00:05:24,640 --> 00:05:32,440 13 divided by three Sample means is 4.33 repeating so the grand mean is 4.3, three or four and a third. 75 00:05:32,440 --> 00:05:37,600 We could also have found this value by adding up all 30 data points here. 76 00:05:37,600 --> 00:05:40,540 Three groups times ten data points each is 30 data points. 77 00:05:40,540 --> 00:05:45,490 We could have added up all 30 of these individual values and divided by 30, and we would have found 78 00:05:45,490 --> 00:05:49,930 this exact same value for the grand mean so we can find it either of those two ways. 79 00:05:49,930 --> 00:05:55,330 But since we have to calculate the mean of each group anyway, it's a little faster to find the means 80 00:05:55,330 --> 00:05:57,790 first and then just add these means together. 81 00:05:57,790 --> 00:06:00,850 Divide by the number of means to get this grand mean. 82 00:06:00,850 --> 00:06:03,460 So this is part one of our ANOVA process. 83 00:06:03,460 --> 00:06:04,750 We have three more parts. 84 00:06:04,750 --> 00:06:07,120 First, we're going to find total sum of squares. 85 00:06:07,120 --> 00:06:11,830 Then we're going to find what's called sum of squares within and then sum of squares between. 86 00:06:11,830 --> 00:06:14,860 And then we'll talk about how to bring all of those parts together. 87 00:06:14,860 --> 00:06:21,250 So looking at total sum of squares here, these are the calculations already done for us in software 88 00:06:21,250 --> 00:06:22,060 in a spreadsheet. 89 00:06:22,060 --> 00:06:26,350 But the formula that we use to find all of these values is this one here. 90 00:06:26,350 --> 00:06:32,320 So we take the sum of each individual data point minus the grand mean. 91 00:06:32,320 --> 00:06:40,240 So we'll say sum x, sub data point minus the grand mean, and then we square that quantity and then 92 00:06:40,240 --> 00:06:42,520 we add up all those squared values here. 93 00:06:42,520 --> 00:06:44,590 So you can see that we're getting a sum of squares. 94 00:06:44,590 --> 00:06:49,330 The fact that we use the grand mean reminds us that we're calculating here total sum of squares. 95 00:06:49,330 --> 00:06:55,000 In other words, we take this first value from the Northeast here too, and we subtract the grand mean, 96 00:06:55,000 --> 00:06:55,930 the value we get. 97 00:06:55,960 --> 00:06:59,530 We've recorded right here in our table, -2.33. 98 00:06:59,530 --> 00:07:07,510 If we look at the last value for the Northeast three down here, we take three -4.33 and we get -1.33. 99 00:07:07,510 --> 00:07:09,580 And we've recorded that here in our table. 100 00:07:09,730 --> 00:07:17,770 If we take the first value for southeast, we take five minus the grand mean 4.33 and we get 0.67. 101 00:07:17,770 --> 00:07:23,230 And then if we look at the last value for the southeast here we take four minus the grand mean 4.33 102 00:07:23,230 --> 00:07:25,690 and we get -0.33. 103 00:07:25,690 --> 00:07:33,640 So this whole left side of the total sum of squares table gets us all of these values right here, just 104 00:07:33,640 --> 00:07:36,460 the x abi minus the grand mean values. 105 00:07:36,460 --> 00:07:40,210 The right side of this table is what we get when we square those values. 106 00:07:40,210 --> 00:07:47,860 So we're just taking -2.33 and squaring it to get this 5.44 or down in the lower right here we're taking 107 00:07:47,860 --> 00:07:51,460 -0.33 and squaring it to get positive 0.11. 108 00:07:51,460 --> 00:07:54,580 So we square all of those values. 109 00:07:54,580 --> 00:08:03,640 That gives us now this value each squared value and then to find the sum, the whole thing. 110 00:08:04,300 --> 00:08:12,610 Here, we add up everything from this right hand side, all 30 of these squared values, and we get 111 00:08:12,610 --> 00:08:18,310 this value right here, 196.667 or 196 and two thirds. 112 00:08:18,310 --> 00:08:21,100 That is our total sum of squares value. 113 00:08:21,100 --> 00:08:29,080 And we should know also that the degrees of freedom for total sum of squares is always m times n minus 114 00:08:29,080 --> 00:08:32,559 one, where m in our case is three and is equal to ten. 115 00:08:32,559 --> 00:08:35,289 So M is the number of groups and is the sample size. 116 00:08:35,289 --> 00:08:37,990 That's our degrees of freedom for total sum of squares. 117 00:08:37,990 --> 00:08:44,049 Now we need sum of squares within, and the calculation here is really similar here. 118 00:08:44,049 --> 00:08:50,500 Basically the idea is that we're taking each data point, so we'll call each data point X and we subtract 119 00:08:50,500 --> 00:08:53,080 from that the mean for its own data set. 120 00:08:53,080 --> 00:08:55,480 So we'll just call this the mean x sub pi. 121 00:08:55,510 --> 00:09:00,630 And once we find each of those differences, we square those values and then we add them all up. 122 00:09:00,640 --> 00:09:02,890 In other words, we take each data point. 123 00:09:02,890 --> 00:09:06,490 So looking here at the Northeast, we start at the top of that column. 124 00:09:06,490 --> 00:09:09,040 So the first data point for the Northeast is two. 125 00:09:09,070 --> 00:09:10,510 We subtract from that. 126 00:09:10,510 --> 00:09:12,700 This data sets own mean. 127 00:09:12,700 --> 00:09:14,290 It's mean is three. 128 00:09:14,290 --> 00:09:16,390 The mean for the Northeast is three. 129 00:09:16,390 --> 00:09:18,430 So we take two minus three. 130 00:09:18,430 --> 00:09:19,810 That's a negative one. 131 00:09:19,810 --> 00:09:21,580 We square that, we get a positive one. 132 00:09:21,580 --> 00:09:26,500 And so you see those two values recorded here, negative one for the first value for the Northeast and 133 00:09:26,500 --> 00:09:27,430 then the square of it. 134 00:09:27,430 --> 00:09:33,040 Positive one here on the right hand side, if we take the last value in the Southeast sample. 135 00:09:33,040 --> 00:09:38,800 So this four at the bottom of the Southeast column, we take four, we subtract its own mean. 136 00:09:38,800 --> 00:09:40,660 The mean for the Southeast is four. 137 00:09:40,660 --> 00:09:44,320 So we take four minus four and we get a difference of zero. 138 00:09:44,350 --> 00:09:46,420 We square that value and we still get zero. 139 00:09:46,420 --> 00:09:50,860 And we see those two zeros recorded here at the bottom of the Southeast column and then the bottom of 140 00:09:50,860 --> 00:09:52,240 this southeast column. 141 00:09:52,240 --> 00:09:53,590 These are the differences. 142 00:09:53,590 --> 00:09:55,240 These are the squared values. 143 00:09:55,240 --> 00:10:02,470 So to get this whole left hand side of this sum of squares within chart, we find these differences 144 00:10:02,500 --> 00:10:04,480 to get this whole right hand side. 145 00:10:04,480 --> 00:10:09,520 We square those differences and we record those values here on the right hand side. 146 00:10:09,520 --> 00:10:13,090 Then we add up all 30 of these squared values. 147 00:10:13,090 --> 00:10:22,240 We take the sum of those squares and the sum of squares is therefore this entire value here or in this 148 00:10:22,240 --> 00:10:31,000 specific example, 150 the degrees of freedom for sum of squares within is m times quantity and minus 149 00:10:31,000 --> 00:10:31,570 one. 150 00:10:31,570 --> 00:10:35,290 And then the last thing we have to do is calculate sum of squares between. 151 00:10:35,290 --> 00:10:44,260 So for sum of squares between, for every single data point, we take the mean for the set and we subtract. 152 00:10:44,880 --> 00:10:45,930 The grand mien. 153 00:10:45,930 --> 00:10:50,610 And then we square those values and then we take the full sum. 154 00:10:50,610 --> 00:10:54,930 In other words, for the first data point here for the Northeast, the two at the top of the Northeast 155 00:10:54,930 --> 00:11:00,990 column, we don't actually use this to at all, but since we're looking at a data point for the Northeast, 156 00:11:00,990 --> 00:11:03,240 we need to use the mean for the Northeast. 157 00:11:03,240 --> 00:11:05,220 So the mean is three. 158 00:11:05,220 --> 00:11:09,690 So for this data point right here, we take the mean associated with that group. 159 00:11:09,690 --> 00:11:13,770 The mean is three we subtract the grand mean of 4.33. 160 00:11:13,770 --> 00:11:17,520 So three -4.33 is -1.33. 161 00:11:17,520 --> 00:11:23,580 And we see that value right here recorded in the left hand side of the sum of squares between table. 162 00:11:23,580 --> 00:11:31,530 And then we square that -1.33 and we get a positive 1.77 and we see that positive squared value here. 163 00:11:31,530 --> 00:11:37,920 Now realize that we're always going to get the same value for every data point in the Northeast Group 164 00:11:37,920 --> 00:11:42,870 because we're always using the same mean for the Northeast and the same grand mean. 165 00:11:42,870 --> 00:11:48,630 We're just making this calculation ten times for all of the ten data points in the Northeast sample. 166 00:11:48,630 --> 00:11:55,020 So this northeast column over here where we find this difference right here, we see that same value 167 00:11:55,020 --> 00:11:58,110 in the entire Northeast column, that -1.33. 168 00:11:58,290 --> 00:12:01,170 The same is true for the entire Northwest column. 169 00:12:01,170 --> 00:12:06,330 So even though we have these ten data points here, we're always just using the mean for the Northwest, 170 00:12:06,330 --> 00:12:07,260 which is six. 171 00:12:07,260 --> 00:12:15,270 We take six minus the grand mean of 4.33 and we get 1.67 and we see that value being recorded for every 172 00:12:15,270 --> 00:12:17,970 data point in the sample for the Northwest. 173 00:12:17,970 --> 00:12:22,770 So we see the same value repeated ten times for northeast, the same value repeated ten times for the 174 00:12:22,770 --> 00:12:25,950 Northwest and the same value repeated ten times for the Southeast. 175 00:12:25,950 --> 00:12:31,770 And then we square all those values to get the right hand side of this table, to get this part of our 176 00:12:31,770 --> 00:12:32,760 calculation. 177 00:12:32,760 --> 00:12:37,890 And so, of course, we see identical values throughout the entire Northeast column, through the entire 178 00:12:37,890 --> 00:12:41,040 Northwest column and through the entire Southeast column. 179 00:12:41,040 --> 00:12:47,820 And then we sum up all 30 of these values on the right hand side of the sum of squares between table 180 00:12:48,060 --> 00:12:51,330 to get this entire value here. 181 00:12:51,330 --> 00:12:58,890 Or in our specific case, the sum there is 46.67, and the degrees of freedom for sum of squares between 182 00:12:58,890 --> 00:13:01,140 is a minus one. 183 00:13:01,140 --> 00:13:07,470 Now, that was a lot of numbers that we just threw out, but really the concept that's being conveyed 184 00:13:07,470 --> 00:13:09,660 here is actually pretty simple. 185 00:13:09,660 --> 00:13:16,050 If we just go back for a second to this sum of squares within chart, basically this whole middle area 186 00:13:16,050 --> 00:13:22,050 in red here, all we're trying to look at is the amount of variance within each of these individual 187 00:13:22,050 --> 00:13:22,680 groups. 188 00:13:22,680 --> 00:13:28,440 If we focus in here on just the Northeast, we know the mean of the Northeast is three and then we have 189 00:13:28,440 --> 00:13:30,600 ten data points for the Northeast. 190 00:13:30,600 --> 00:13:37,410 So if we want to think about finding the total variance in this Northeast sample, we would just take 191 00:13:37,410 --> 00:13:39,390 each of the data points for Northeast. 192 00:13:39,390 --> 00:13:41,970 We would subtract the mean for the Northeast. 193 00:13:41,970 --> 00:13:46,170 We square that value and we essentially have a square error. 194 00:13:46,170 --> 00:13:52,200 We have a certain amount of variance for each of these data points when we do that same thing. 195 00:13:52,200 --> 00:14:01,080 So we can think about the sum of everything in this column right here as all of the variance within 196 00:14:01,080 --> 00:14:02,340 the Northeast Group. 197 00:14:02,490 --> 00:14:06,420 Everything in this column is all of the variance within the Northwest Group. 198 00:14:06,450 --> 00:14:10,650 And everything in this column is all of the variance within the Southeast Group. 199 00:14:10,680 --> 00:14:16,380 So if we then add up all three of those together, we're sort of getting a figure or an idea of the 200 00:14:16,380 --> 00:14:22,230 total variance that we see within inside of each of these individual groups. 201 00:14:22,230 --> 00:14:29,100 So the point then is that this calculation is giving us a picture into the variance inside of each group. 202 00:14:29,250 --> 00:14:34,110 This calculation is giving us a picture into the variance in between these three groups. 203 00:14:34,110 --> 00:14:36,060 And then this is total variance. 204 00:14:36,060 --> 00:14:40,560 And what we notice is that in fact the sum of. 205 00:14:41,240 --> 00:14:49,760 Some have squares within and some have squares between is equivalent to total sum of squares. 206 00:14:49,760 --> 00:14:55,160 When we add 150 to 46 and two thirds, we get 196 and two thirds. 207 00:14:55,160 --> 00:14:58,790 So these two variances add up to total variance. 208 00:14:58,790 --> 00:15:02,570 Furthermore, the degrees of freedom will add up the same way. 209 00:15:02,570 --> 00:15:07,130 So if we add up the degrees of freedom for sum of squares within and sum of squares between, we will 210 00:15:07,130 --> 00:15:08,900 get the degrees of freedom. 211 00:15:08,900 --> 00:15:10,460 For total sum of squares. 212 00:15:10,760 --> 00:15:11,780 We'll see that here. 213 00:15:11,780 --> 00:15:19,460 If we take m times n minus one plus degrees of freedom from sum of squares between M minus one. 214 00:15:19,460 --> 00:15:28,070 If we then expand this, we get m n minus m plus m minus one. 215 00:15:28,070 --> 00:15:35,750 We get these two values to cancel minus M and plus m, and we get m n minus one or the degrees of freedom 216 00:15:35,750 --> 00:15:37,160 for total sum of squares. 217 00:15:37,370 --> 00:15:42,320 Now, once we have these three values, for total sum of squares, sum of squares within and sum of 218 00:15:42,320 --> 00:15:46,100 squares between, we can calculate our test statistic value. 219 00:15:46,100 --> 00:15:53,450 And when we use this process of ANOVA on a one way analysis of variance, we will use an F statistic 220 00:15:53,450 --> 00:15:59,480 which we calculate this way, where we take sum of squares between divided by its own degrees of freedom, 221 00:15:59,480 --> 00:16:05,570 and then we look at the ratio between that and sum of squares within divided by its own degrees of freedom. 222 00:16:05,570 --> 00:16:12,350 So in our case, if we plug in what we already know here, we said that sum of squares between was 46 223 00:16:12,350 --> 00:16:17,690 and two thirds, so 46.667 approximately. 224 00:16:17,690 --> 00:16:19,700 Remember, in our case we have three groups. 225 00:16:19,700 --> 00:16:20,990 So M is equal to three. 226 00:16:20,990 --> 00:16:22,940 M minus one is two. 227 00:16:23,360 --> 00:16:31,280 That's our degrees of freedom for sum of squares between and then sum of squares within we have as 150 228 00:16:31,280 --> 00:16:34,280 and then we divide that by its own degrees of freedom. 229 00:16:34,280 --> 00:16:38,540 MX times n minus one while M is three, KN is ten. 230 00:16:38,540 --> 00:16:43,490 So here we get ten minus one or nine, nine minus three is 27. 231 00:16:43,940 --> 00:16:50,390 And when we use a calculator to find this value, we get approximately 4.200. 232 00:16:50,390 --> 00:16:57,320 So what we have here with this F statistic is actually a ratio of chi squared distributions. 233 00:16:57,320 --> 00:17:03,710 We talked about chi squared tests before the numerator of our F statistic formula here is actually a 234 00:17:03,710 --> 00:17:09,020 chi squared distribution with M minus one degrees of freedom, and the denominator of our F statistic 235 00:17:09,020 --> 00:17:14,329 is actually a chi squared distribution with m times n minus one degrees of freedom, which means that 236 00:17:14,329 --> 00:17:18,740 the F distribution is just a ratio of two chi squared distributions. 237 00:17:18,740 --> 00:17:25,970 Now, when it comes to calculating this F statistic, what we can say is that if the numerator is significantly 238 00:17:25,970 --> 00:17:33,170 larger than the denominator, so in our case our numerator is approximately 23, our denominator is 239 00:17:33,170 --> 00:17:34,700 approximately five. 240 00:17:34,700 --> 00:17:40,250 If we're just doing rough math here, if the numerator is significantly larger, what that means is 241 00:17:40,250 --> 00:17:45,920 that more variation is explained by the variation between groups than by the variation within groups. 242 00:17:45,920 --> 00:17:51,320 Here we're dividing essentially this idea of the variation between groups by essentially this idea of 243 00:17:51,320 --> 00:17:52,820 the variation within groups. 244 00:17:52,820 --> 00:17:58,730 So we can see because 23 is significantly larger than five, that may be what this f statistic is telling 245 00:17:58,730 --> 00:18:05,300 us is that we can explain more of the variation as a difference between groups then within the groups 246 00:18:05,300 --> 00:18:07,550 themselves or just noise in the data. 247 00:18:07,550 --> 00:18:12,110 Whereas if we have the opposite scenario, if the denominator here is significantly larger than the 248 00:18:12,110 --> 00:18:18,410 numerator, then that means that we find more variation within the groups themselves as opposed to between 249 00:18:18,410 --> 00:18:23,030 the groups, which might mean that there isn't necessarily a huge difference between the groups and 250 00:18:23,030 --> 00:18:26,690 therefore that we might have trouble rejecting the null hypothesis. 251 00:18:26,720 --> 00:18:33,080 It'll be easier for us to reject the null hypothesis when the numerator here is larger than the denominator, 252 00:18:33,080 --> 00:18:38,720 because when the numerator is larger than the denominator, we're going to have a larger F statistic 253 00:18:38,720 --> 00:18:43,850 and the larger the F statistic, the more likely it is that we can reject the null and therefore lend 254 00:18:43,850 --> 00:18:46,520 support to our alternative hypothesis. 255 00:18:46,670 --> 00:18:51,470 But no matter what we have in the numerator and denominator of our F statistic, what it really comes 256 00:18:51,470 --> 00:18:56,840 down to is the actual F statistic that we calculated in this case about 4.2. 257 00:18:56,870 --> 00:19:01,760 Our next step is to look up the critical value in our F table. 258 00:19:01,760 --> 00:19:08,060 Let's assume here that at the beginning, when we set up our hypothesis statements that we chose a significance 259 00:19:08,060 --> 00:19:10,790 level, an alpha value of five. 260 00:19:11,210 --> 00:19:13,100 So alpha is 0.05. 261 00:19:13,100 --> 00:19:18,260 There's a different F table for every level of significance alpha. 262 00:19:18,260 --> 00:19:25,910 So if we choose alpha equal to 0.05, then we need to go find an F table for alpha equals 0.05. 263 00:19:25,940 --> 00:19:27,830 These are very easy to find online. 264 00:19:27,830 --> 00:19:34,850 You can quickly find one for alpha equals 0.05, for alpha equals 0.1, all of the common alpha values. 265 00:19:34,850 --> 00:19:40,430 So this table here is an F table for. 266 00:19:40,580 --> 00:19:41,930 This alpha value here. 267 00:19:41,930 --> 00:19:44,420 Alpha is 0.05. 268 00:19:44,420 --> 00:19:50,390 And what we have to do then is cross reference the degrees of freedom in our numerator with the degrees 269 00:19:50,390 --> 00:19:52,160 of freedom in our denominator. 270 00:19:52,190 --> 00:19:56,670 In other words, we're looking at two compared to 27. 271 00:19:56,690 --> 00:20:01,730 Now I've modified this table so that you can see the beginning of at the top of it, where both the 272 00:20:01,730 --> 00:20:04,190 degrees of freedom are equal to one. 273 00:20:04,190 --> 00:20:06,800 So this is always the top left hand corner of the table. 274 00:20:06,800 --> 00:20:12,440 And then I've skipped a bunch of rows to jump down to the rows where we can see that the denominator 275 00:20:12,440 --> 00:20:14,240 degree of freedom is 27. 276 00:20:14,240 --> 00:20:17,900 So in our case, denominator degrees of freedom is 27. 277 00:20:17,900 --> 00:20:23,360 We see that here and numerator degrees of freedom is two. 278 00:20:23,660 --> 00:20:31,460 So we find that here and we just find their intersection in our F table for this particular alpha level. 279 00:20:31,460 --> 00:20:39,590 And we see that we get a critical value of 3.35 for all we have to do now is compare our F statistic 280 00:20:39,590 --> 00:20:40,790 to the critical value. 281 00:20:40,790 --> 00:20:47,960 In our case, 4.2 is greater than the critical value of 3.354. 282 00:20:48,200 --> 00:20:53,090 And whenever the F statistic that we calculate is greater than the critical value, as always, that 283 00:20:53,090 --> 00:20:59,330 means that we can reject the null hypothesis, which means that we do in fact lend support to the alternative 284 00:20:59,330 --> 00:21:06,170 hypothesis that there is actually a statistically significant difference between these three groups 285 00:21:06,170 --> 00:21:11,750 and that the differences that we're seeing is not just caused by variation within the groups or noise 286 00:21:11,750 --> 00:21:12,830 within the data. 287 00:21:12,830 --> 00:21:19,130 If we get to this point and we realize that we don't have a statistically significant F statistic, 288 00:21:19,130 --> 00:21:23,210 we can try to increase the value of the F statistic. 289 00:21:23,210 --> 00:21:30,020 We can try to do that either by increasing sample size or we can try to do that by reducing the variation 290 00:21:30,020 --> 00:21:32,570 within each of these groups. 291 00:21:32,600 --> 00:21:37,640 Oftentimes that can mean breaking groups down into more homogeneous categories. 292 00:21:37,640 --> 00:21:44,570 Maybe, for instance, in our Northeast office there are two divisions one division that handles only 293 00:21:44,570 --> 00:21:49,430 English speaking calls and another division that handles calls from international language speakers. 294 00:21:49,430 --> 00:21:54,980 We could hypothesize maybe that the international calls take longer because they have to be routed to 295 00:21:54,980 --> 00:21:58,040 the appropriate call center representative who can speak that language. 296 00:21:58,040 --> 00:22:04,100 So maybe the fact that there's those two departments within this Northeast office is causing more variation 297 00:22:04,100 --> 00:22:10,520 in this Northeast data, and we can reduce some of that variation if we segment those two divisions 298 00:22:10,520 --> 00:22:12,680 into separate groups, something like that. 299 00:22:12,680 --> 00:22:18,380 So those are a couple of things we can look at if we don't find significance when we calculate this 300 00:22:18,380 --> 00:22:23,780 F statistic and we want to try running the whole test again, Keep in mind, of course, that we walked 301 00:22:23,780 --> 00:22:25,970 through this whole process by hand. 302 00:22:25,970 --> 00:22:31,490 We used a spreadsheet to calculate these numbers, but ultimately what it came down to was calculating 303 00:22:31,490 --> 00:22:38,030 this F statistic by hand and then looking up our F statistic in comparison to a critical value that 304 00:22:38,030 --> 00:22:39,290 we found in a table. 305 00:22:39,290 --> 00:22:43,040 Typically, though, that's not the way that we run this ANOVA process. 306 00:22:43,040 --> 00:22:48,110 Of course, typically these days we're going to do this whole thing with software, which means we'll 307 00:22:48,110 --> 00:22:54,890 be able to feed in just our raw data and the software will immediately be able to give us all of this 308 00:22:54,890 --> 00:22:56,630 information we spent time calculating. 309 00:22:56,630 --> 00:23:01,730 We'll be able to get the mean for each group, the grand mean, and instantly see total sum of squares, 310 00:23:01,730 --> 00:23:04,100 sum of squares within and sum of squares between. 311 00:23:04,100 --> 00:23:08,570 We'll know immediately the value of our F statistic and the software. 312 00:23:08,570 --> 00:23:14,480 Instead of looking this statistic up in an F table, it'll calculate automatically the probability, 313 00:23:14,480 --> 00:23:20,840 the p value of finding an F value that's greater than or equal to the F statistic that we found. 314 00:23:20,840 --> 00:23:26,030 If that probability is less than or equal to the alpha value that we set, then the conclusion will 315 00:23:26,030 --> 00:23:27,680 be that we can reject the null. 316 00:23:27,680 --> 00:23:32,270 But even if we're using software, this is what's actually going on behind the scenes. 317 00:23:32,270 --> 00:23:36,020 So this ANOVA process is extremely powerful. 318 00:23:36,020 --> 00:23:38,210 There are many different ways that we can use it. 319 00:23:38,210 --> 00:23:45,620 This is just one example of a one way ANOVA with three groups, but while it is extremely powerful, 320 00:23:45,620 --> 00:23:48,740 it does of course also have its own limitations. 321 00:23:48,740 --> 00:23:54,920 For example, this F test that we ran told us that we could reject this null hypothesis, which means 322 00:23:54,920 --> 00:23:59,570 that we can lend support to the idea that the population means are in fact different. 323 00:23:59,570 --> 00:24:04,130 That mean call time is different across these three call centers. 324 00:24:04,130 --> 00:24:08,120 But notice that what it didn't tell us is which means we're different. 325 00:24:08,120 --> 00:24:14,000 So we don't know if we should conclude that mean call time is different for each of these call centers. 326 00:24:14,000 --> 00:24:15,350 Or maybe just that. 327 00:24:15,350 --> 00:24:22,340 For example, the Northeast and the Southeast have similar mean call times and maybe the Northwest is 328 00:24:22,340 --> 00:24:24,920 the only call center that happens to be an outlier. 329 00:24:24,920 --> 00:24:30,710 Maybe that's the only call center that's causing us to make this conclusion that the population means 330 00:24:30,710 --> 00:24:32,480 are different, that they're not the same. 331 00:24:32,480 --> 00:24:38,930 So to actually figure out which of these call centers is causing the difference, we might need to run 332 00:24:38,930 --> 00:24:40,400 another set of tests. 333 00:24:40,700 --> 00:24:46,460 But at the very least, this analysis of variance process and understanding of this process gives us 334 00:24:46,460 --> 00:24:48,560 a great place to start.