1 00:00:00,180 --> 00:00:06,090 Earlier in the course in the joint distribution section, we talked about covariance and Pearson's correlation 2 00:00:06,090 --> 00:00:06,870 coefficient. 3 00:00:06,900 --> 00:00:12,780 Now we want to revisit that idea of the correlation coefficient to talk about that and the residual 4 00:00:12,780 --> 00:00:15,280 which are both key parts of regression. 5 00:00:15,300 --> 00:00:17,000 The focus of this section. 6 00:00:17,010 --> 00:00:22,680 So if you remember Pearson's correlation coefficient, which we often indicate with this variable R 7 00:00:22,710 --> 00:00:25,260 is equal to one over n minus one. 8 00:00:25,260 --> 00:00:31,920 Where n is that sample size multiplied by the sum of these two products here x sub minus the mean of 9 00:00:31,920 --> 00:00:38,880 x divided by the sample standard deviation of x and then y sub by minus the mean of y divided by sample 10 00:00:38,880 --> 00:00:39,660 standard deviation. 11 00:00:39,660 --> 00:00:46,920 With respect to y, realize here that we could also write this formula this way because both of these 12 00:00:46,920 --> 00:00:50,480 values here are basically the Z scores for X and Y. 13 00:00:50,490 --> 00:00:56,700 If we look at these here, we recognize them as the formula for a Z score X minus the mean of X divided 14 00:00:56,700 --> 00:01:00,180 by its standard deviation is the Z score for X. 15 00:01:00,180 --> 00:01:05,730 So we can replace each of these fractions with the z score of x, the z score of Y. 16 00:01:05,760 --> 00:01:09,360 So this is another way to write the correlation coefficient. 17 00:01:09,360 --> 00:01:15,450 And then this formula here is the one that we used earlier when we talked about the correlation coefficient 18 00:01:15,450 --> 00:01:16,680 and covariance. 19 00:01:16,710 --> 00:01:22,230 All three of these get us to the same place so we can use whichever formula is most convenient. 20 00:01:22,260 --> 00:01:30,720 Remember that the value of the correlation coefficient always falls on the interval -1 to 1, so we'll 21 00:01:30,720 --> 00:01:36,330 never find a value for the correlation coefficient that is less than negative one or greater than positive 22 00:01:36,330 --> 00:01:36,780 one. 23 00:01:36,780 --> 00:01:42,210 The correlation coefficient will always take on a value somewhere in the interval, negative one to 24 00:01:42,240 --> 00:01:43,170 positive one. 25 00:01:43,170 --> 00:01:49,620 And remember that the value of this correlation coefficient just tells us how strong the relationship 26 00:01:49,620 --> 00:01:52,470 is between these two values x and y. 27 00:01:52,470 --> 00:01:59,790 So if we think about a scale for the correlation coefficient, so we could almost sketch a number line 28 00:01:59,790 --> 00:02:05,970 like this where the correlation coefficient is negative one over here, positive one over here and zero. 29 00:02:05,970 --> 00:02:13,620 At this point the closer we are to negative one for the correlation coefficient, the more we say that 30 00:02:13,620 --> 00:02:17,940 a regression line with a negative slope perfectly describes the data. 31 00:02:17,940 --> 00:02:25,230 So if the correlation coefficient is exactly negative one, it means that all of the points in the scatterplot, 32 00:02:25,230 --> 00:02:28,500 all the points in the data set lie along the regression line. 33 00:02:28,500 --> 00:02:29,100 Exactly. 34 00:02:29,100 --> 00:02:31,380 They all lie directly on the line. 35 00:02:31,380 --> 00:02:33,480 None of the points are away from the line. 36 00:02:33,480 --> 00:02:40,200 So for instance, if we say that this is the regression line, maybe it looks something like this, 37 00:02:40,200 --> 00:02:47,220 then all of the points in the scatterplot lie directly on this line. 38 00:02:47,220 --> 00:02:50,100 There are no points away from the line at all. 39 00:02:50,100 --> 00:02:58,050 Whereas if the correlation coefficient is equal to positive one, that means that a regression line 40 00:02:58,050 --> 00:03:06,900 with a positive slope perfectly describes the relationship in the data, perfectly describes the position 41 00:03:06,900 --> 00:03:08,820 of all the points in the scatterplot. 42 00:03:08,820 --> 00:03:16,320 So that means that all of the points in our scatterplot lie exactly on this regression line. 43 00:03:16,320 --> 00:03:25,320 If the correlation coefficient is exactly zero, that means that a line doesn't do a good job at all 44 00:03:25,320 --> 00:03:32,730 of describing the relationship in the data, which means our scatterplot might look something like this, 45 00:03:32,730 --> 00:03:34,410 kind of like a blob. 46 00:03:34,410 --> 00:03:37,830 There's no obvious linear relationship here at all. 47 00:03:37,830 --> 00:03:45,600 We can't really identify any line that starts to describe some kind of trend in this scatterplot and 48 00:03:45,600 --> 00:03:48,210 if we lie somewhere in between. 49 00:03:48,210 --> 00:03:57,180 So if R is between zero and negative one, that means that our regression line has a negative slope. 50 00:03:57,180 --> 00:04:01,800 So it might look something like this, but that it doesn't perfectly describe the data. 51 00:04:01,800 --> 00:04:08,880 So we might have some points along the line, but we'll also have some points away from the line like 52 00:04:08,880 --> 00:04:09,420 this. 53 00:04:09,420 --> 00:04:12,210 But we see this general negative trend. 54 00:04:12,210 --> 00:04:18,089 And then of course that means that a correlation coefficient between zero and positive one. 55 00:04:18,089 --> 00:04:23,910 So somewhere over here means that the regression line is going to have a positive slope. 56 00:04:23,910 --> 00:04:30,390 So maybe something like that, but that the data will not lie all along the line. 57 00:04:30,390 --> 00:04:37,020 So we might have some points on the line, but we'll also have some points away from the line like this. 58 00:04:37,110 --> 00:04:43,470 Don't confuse the value of the correlation coefficient r with the slope of the regression line. 59 00:04:43,470 --> 00:04:50,100 So imagine here all of these regression lines, this regression line, this one we already sketched 60 00:04:50,100 --> 00:04:52,860 and this one right here. 61 00:04:53,010 --> 00:04:59,550 If we had scatter plots for each of these regression lines where all the points in the scatterplot. 62 00:04:59,940 --> 00:05:05,910 Perfectly along the regression line, then each of these scatter plots would have a correlation coefficient 63 00:05:05,910 --> 00:05:06,900 of one. 64 00:05:06,900 --> 00:05:11,060 But you can see that the slope of the regression line is different for each plot. 65 00:05:11,070 --> 00:05:17,460 In other words, a correlation coefficient between zero and one just means that the regression line 66 00:05:17,490 --> 00:05:23,430 has a positive slope and a correlation coefficient between zero and negative. 67 00:05:23,430 --> 00:05:30,930 One means that the regression line has a negative slope, but the value of the correlation coefficient 68 00:05:30,930 --> 00:05:33,910 doesn't actually tell us the slope of the regression line. 69 00:05:33,930 --> 00:05:40,020 All it tells us is that the closer we are to positive one, the more tightly clustered the points are 70 00:05:40,020 --> 00:05:45,240 around a regression line with a positive slope, and the closer the correlation coefficient is to negative 71 00:05:45,240 --> 00:05:51,240 one, the more tightly clustered the points in the scatterplot are along a regression line with a negative 72 00:05:51,240 --> 00:05:51,810 slope. 73 00:05:51,840 --> 00:05:57,660 Now remember that to calculate the correlation coefficient, we can use any of these formulas here. 74 00:05:57,660 --> 00:06:04,260 If we use this third one, we can create a small chart to quickly make this calculation for us. 75 00:06:04,260 --> 00:06:06,690 And here's what that chart should look like. 76 00:06:06,690 --> 00:06:14,490 So if we have raw data given by these columns X and Y, so these are the columns with our raw data, 77 00:06:14,490 --> 00:06:19,890 the values of X are zero two, four, six, eight, ten and 12, and the values of Y that correspond 78 00:06:19,890 --> 00:06:22,880 to each of those X values are given in this column here. 79 00:06:22,890 --> 00:06:26,790 In other words, we have coordinate points here x, y. 80 00:06:26,790 --> 00:06:30,540 So one coordinate point in the scatterplot is 00.8. 81 00:06:30,540 --> 00:06:36,390 Another coordinate point is to one, another coordinate point is four, 0.2, etc. So we have all these 82 00:06:36,390 --> 00:06:37,350 coordinate points. 83 00:06:37,350 --> 00:06:39,840 You can see that there are seven data points. 84 00:06:39,840 --> 00:06:45,540 And so if we're using this formula to find the correlation coefficient, you can see here that we're 85 00:06:45,540 --> 00:06:51,180 going to need the mean of X and the mean of Y, which means that we sum all the values of X, we get 86 00:06:51,180 --> 00:06:56,010 42 and then we divide by the number of data points we have, which is seven. 87 00:06:56,010 --> 00:06:58,230 We have seven data points here. 88 00:06:58,230 --> 00:07:01,680 So we divide 42 by seven and we get a mean of six. 89 00:07:01,770 --> 00:07:06,420 We sum all the values for y divide by seven and we get a mean of 0.8. 90 00:07:06,630 --> 00:07:12,840 So that means that the mean of x here is six, the mean of Y is 0.8. 91 00:07:12,840 --> 00:07:19,680 So we have those two values and then for each value of x, we need to find x sub B minus the mean and 92 00:07:19,680 --> 00:07:23,340 for each value of why we need to find y sub B minus the mean. 93 00:07:23,340 --> 00:07:29,010 So these next two columns here to get this negative six, we take this particular value of x zero and 94 00:07:29,010 --> 00:07:34,260 we subtract the mean so zero minus six to get negative four, we take two minus six. 95 00:07:34,260 --> 00:07:37,230 To get negative two, we take four minus six, etc.. 96 00:07:37,230 --> 00:07:45,870 So to find each of these values here, we subtract the corresponding mean from each data point and we 97 00:07:45,870 --> 00:07:47,430 get these two columns. 98 00:07:47,430 --> 00:07:52,800 Now our formula tells us that we have to multiply each of those values together, so each x value multiplied 99 00:07:52,800 --> 00:07:53,940 by each y value. 100 00:07:53,940 --> 00:07:59,310 So negative six times zero gives us six, negative four times 0.2 gives us -0.8. 101 00:07:59,310 --> 00:08:00,960 So we find that product. 102 00:08:00,960 --> 00:08:08,100 In other words, when we multiply these two together, we get this product here and then summing all 103 00:08:08,100 --> 00:08:09,300 of those products. 104 00:08:09,300 --> 00:08:17,310 So to find this sum or the entire numerator here, we take the sum of all of these products and we see 105 00:08:17,310 --> 00:08:18,990 that sum right there. 106 00:08:18,990 --> 00:08:22,050 So 1.6 is the value of our numerator. 107 00:08:22,050 --> 00:08:28,710 And then in the denominator we again have these x sub minus the mean of x values, but we need to square 108 00:08:28,710 --> 00:08:29,070 them. 109 00:08:29,070 --> 00:08:34,200 So we use this column here to square all the values in this column. 110 00:08:34,200 --> 00:08:41,460 In other words, we are getting this squared value here in this column and we're getting this squared 111 00:08:41,460 --> 00:08:45,780 value here in this column, and then we need those sums. 112 00:08:45,780 --> 00:08:53,880 So to get this sum here and then separately this sum here, we need to add all the values in each column. 113 00:08:53,880 --> 00:08:57,480 So we get that sum and that sum. 114 00:08:57,480 --> 00:09:03,000 And then taking the square root of the product of those two is the same as taking the square root of 115 00:09:03,000 --> 00:09:07,770 each of them individually and then multiplying them together so we can take the square root of both 116 00:09:07,770 --> 00:09:09,030 of these values. 117 00:09:09,030 --> 00:09:12,600 So we get those square roots here and here. 118 00:09:12,600 --> 00:09:16,890 And if we multiply these together, we get the denominator. 119 00:09:16,890 --> 00:09:22,860 So this entire denominator here is the product of these two values here. 120 00:09:22,860 --> 00:09:29,250 So if we multiply these two together and then we take 1.6 divided by the product of these two, we get 121 00:09:29,250 --> 00:09:30,510 the correlation coefficient. 122 00:09:30,510 --> 00:09:39,570 R And so the correlation coefficient for this data set here of x and Y values this raw data, our correlation 123 00:09:39,570 --> 00:09:42,630 coefficient is 0.101. 124 00:09:42,630 --> 00:09:49,530 So the fact that this value is positive tells us that the regression line through this scatterplot is 125 00:09:49,530 --> 00:09:51,540 going to have a positive slope. 126 00:09:51,690 --> 00:09:56,760 But the fact that this value is very close to zero instead of positive one. 127 00:09:56,760 --> 00:09:59,580 So the value is maybe right about here means. 128 00:09:59,810 --> 00:10:02,210 The correlation is very weak. 129 00:10:02,210 --> 00:10:06,980 We don't have a strong correlation like we do over here where all the points in the scatterplot lie 130 00:10:06,980 --> 00:10:08,630 exactly on the regression line. 131 00:10:08,630 --> 00:10:14,120 Instead, the points are very spread out away from the regression line, but the regression line does 132 00:10:14,120 --> 00:10:15,680 have a positive slope. 133 00:10:15,680 --> 00:10:22,940 So if we replace this table with a scatterplot of the raw data, then we can see the relationship we 134 00:10:22,940 --> 00:10:23,870 were talking about. 135 00:10:23,870 --> 00:10:28,610 Here's the regression line through the scatter plot of that raw data that we were just looking at. 136 00:10:28,610 --> 00:10:33,200 We can see that the regression line does have a positive slope because as we move to the right, the 137 00:10:33,200 --> 00:10:34,220 line moves up. 138 00:10:34,220 --> 00:10:41,120 So we've got a positive slope, but the data is very spread out away from the regression line. 139 00:10:41,120 --> 00:10:45,470 It's not tightly packed or tightly clustered close to the regression line. 140 00:10:45,470 --> 00:10:51,230 So it does make sense that we would find a correlation coefficient that is very close to zero, but 141 00:10:51,230 --> 00:10:55,370 still positive to indicate the positive slope of the regression line. 142 00:10:55,730 --> 00:11:03,140 That being said, as a general rule of thumb, we do classify the strength of the correlation based 143 00:11:03,140 --> 00:11:05,630 on the value of this correlation coefficient. 144 00:11:05,630 --> 00:11:11,960 So in particular, if the correlation coefficient is specifically between negative one and -0.07, we 145 00:11:11,960 --> 00:11:14,480 tend to call that a strong negative correlation. 146 00:11:14,480 --> 00:11:19,760 Whereas if we're kind of in that mid negative range between negative point seven and -0.3, we would 147 00:11:19,760 --> 00:11:22,130 call that a moderate negative correlation. 148 00:11:22,130 --> 00:11:28,130 And then if the correlation coefficient is between -0.3 and zero, we would call that a weak negative 149 00:11:28,130 --> 00:11:28,880 correlation. 150 00:11:28,880 --> 00:11:33,050 And then we have the mirror image of those relationships on the positive side. 151 00:11:33,050 --> 00:11:38,780 So between zero and 0.3, we have a weak positive correlation between 0.3 and point seven, a moderate 152 00:11:38,780 --> 00:11:42,800 positive correlation and between 0.7 and one a strong positive correlation. 153 00:11:42,800 --> 00:11:50,210 So our correlation coefficient of 0.101 would fall within this range right here. 154 00:11:50,210 --> 00:11:54,620 And so we would say that this data has a weak positive correlation. 155 00:11:54,620 --> 00:11:57,470 Now related to the correlation coefficient. 156 00:11:57,470 --> 00:12:00,590 At this point we need to talk about the residual as well. 157 00:12:00,590 --> 00:12:05,330 So the residual for any data point, In other words, every data point has a residual. 158 00:12:05,330 --> 00:12:11,240 The residual for any data point in a scatterplot is the difference between the actual value. 159 00:12:11,240 --> 00:12:17,990 So the data point itself and what the regression line predicts at that same value of the independent 160 00:12:17,990 --> 00:12:18,560 variable. 161 00:12:18,560 --> 00:12:26,600 So if we say here in this scatterplot that this is the horizontal x axis, this is the vertical y axis 162 00:12:26,720 --> 00:12:34,310 here we could look at as an example this data point here, its actual value is two, but it's predicted 163 00:12:34,310 --> 00:12:37,310 value is what we find along the regression line. 164 00:12:37,310 --> 00:12:43,580 If we follow this data point down to its corresponding point along the regression line. 165 00:12:43,580 --> 00:12:46,760 So it looks like the predicted value. 166 00:12:46,760 --> 00:12:48,620 So this is the. 167 00:12:49,440 --> 00:12:50,610 Actual value. 168 00:12:50,610 --> 00:12:53,280 And then this value right here is the. 169 00:12:55,200 --> 00:12:56,390 Predicted value. 170 00:12:56,400 --> 00:13:01,520 It looks like this predicted value is maybe about 0.85, something like that, roughly. 171 00:13:01,530 --> 00:13:06,000 So the residual is the difference between that actual value and the predicted value. 172 00:13:06,030 --> 00:13:06,930 We can write it this way. 173 00:13:06,930 --> 00:13:13,500 The residual is actual minus predicted or mathematically we would write that this way because we often 174 00:13:13,500 --> 00:13:16,140 also think about the residual as error. 175 00:13:16,140 --> 00:13:21,810 And so we represent the residual with the variable E And then the actual value here is the actual y 176 00:13:21,810 --> 00:13:22,890 value of the data point. 177 00:13:22,890 --> 00:13:30,390 Remember, this data point here is a coordinate point x, y, and so it has its actual y value, but 178 00:13:30,390 --> 00:13:33,930 then the corresponding y value along the regression line. 179 00:13:33,930 --> 00:13:38,310 Remember that we write the regression line or the equation of the regression line. 180 00:13:38,310 --> 00:13:46,020 As for y hat, the equation of the regression line is y hat equals m x plus B, where m is the slope 181 00:13:46,020 --> 00:13:48,840 of the regression line and B is the y intercept. 182 00:13:48,840 --> 00:13:53,160 So the predicted value along the regression line is always given by y hat. 183 00:13:53,160 --> 00:14:00,900 So if we look at the residuals for all the points in our scatterplot, we would show them like this. 184 00:14:00,900 --> 00:14:04,230 These are all the residuals. 185 00:14:04,230 --> 00:14:05,040 It's this difference. 186 00:14:05,040 --> 00:14:10,290 We could also think about it as distance, the distance between each point and the regression line. 187 00:14:10,290 --> 00:14:15,000 So the points that are closer to the regression line will have a smaller residual. 188 00:14:15,000 --> 00:14:19,110 The points that are further from the regression line will have a larger residual. 189 00:14:19,110 --> 00:14:24,540 If the point in the scatterplot is below the regression line, its residual will be negative. 190 00:14:24,540 --> 00:14:29,220 So everything down here, all four of these points here will have a. 191 00:14:30,000 --> 00:14:38,520 Negative residual, whereas these three points up here will have a positive residual because of their 192 00:14:38,520 --> 00:14:41,040 position in relation to the regression line. 193 00:14:41,040 --> 00:14:48,210 And now that we understand this idea of the residual, we can define the regression line in a new way. 194 00:14:48,240 --> 00:14:54,870 What we can say is that for any regression line, any y hat equation representing a regression line, 195 00:14:54,870 --> 00:15:00,420 we know that the sum of all the residuals is equal to zero. 196 00:15:00,450 --> 00:15:04,550 That is another way to define or identify the regression line. 197 00:15:04,560 --> 00:15:11,400 The regression line will always be the line that makes the sum of all of these distances equal to zero. 198 00:15:11,430 --> 00:15:14,310 That also means that the mean. 199 00:15:15,260 --> 00:15:17,020 Of the residuals is equal to zero. 200 00:15:17,030 --> 00:15:22,790 And in general, that should make sense because the idea of the regression line very, very broadly 201 00:15:22,790 --> 00:15:27,710 is that it's trying to balance out all of the data points in the scatterplot. 202 00:15:27,710 --> 00:15:33,440 And if we're creating balance in the scatterplot, we can think about that as trying to balance all 203 00:15:33,440 --> 00:15:37,670 of these residual values, the distance of each point to the regression line. 204 00:15:37,670 --> 00:15:40,040 And so the mean of the residuals is going to be zero. 205 00:15:40,070 --> 00:15:41,970 The sum of the residuals is going to be zero. 206 00:15:41,990 --> 00:15:47,120 Now, we've already looked at in the past how to calculate the equation of the regression line. 207 00:15:47,120 --> 00:16:00,530 The equation of this regression line in particular is y hat equal to 0.0 143x plus 0.7143. 208 00:16:00,530 --> 00:16:06,680 Both of these values for M and B are rounded to four decimal places, but this is the regression line 209 00:16:06,680 --> 00:16:07,430 equation. 210 00:16:07,430 --> 00:16:14,150 And so what we can do once we have the regression line equation is create a table of actual and predicted 211 00:16:14,150 --> 00:16:21,440 values and then use those to calculate the residual for each data point so we can create a chart. 212 00:16:21,440 --> 00:16:25,310 These first two columns are all of the raw data that we looked at earlier. 213 00:16:25,310 --> 00:16:31,940 These are all of the X values and then these are all of the Y values or the actual values for our equation 214 00:16:31,940 --> 00:16:32,570 here. 215 00:16:32,570 --> 00:16:39,590 The predicted values are the values we get from our y hat equation, our regression line equation. 216 00:16:39,590 --> 00:16:44,300 What we do is we take each value of x and we plug it into the y hat equation. 217 00:16:44,300 --> 00:16:49,520 So you can see here, if we plug in x equals zero, then this term goes away and we're left with y hat 218 00:16:49,520 --> 00:16:51,650 is equal to 0.7143. 219 00:16:51,650 --> 00:16:53,870 And we see that here 0.7143. 220 00:16:54,080 --> 00:16:59,990 So we're getting these predicted values out of the y hat equation and then we know that the residual 221 00:16:59,990 --> 00:17:04,400 we can calculate as Y minus. 222 00:17:05,180 --> 00:17:06,020 Y hat. 223 00:17:06,020 --> 00:17:08,270 And so we find the value of the residual. 224 00:17:08,270 --> 00:17:10,190 We find the value of the area. 225 00:17:10,190 --> 00:17:16,220 And we see here, in fact, these three positive values that represent the three points above the regression 226 00:17:16,220 --> 00:17:21,000 line and the four negative values that represent the four points below the regression line. 227 00:17:21,020 --> 00:17:26,839 Now, the reason that we're looking at this chart is because we can use it to illustrate these two facts 228 00:17:26,839 --> 00:17:32,150 here that we just talked about, that the sum of the residuals is zero and that the mean of the residuals 229 00:17:32,150 --> 00:17:32,750 is zero. 230 00:17:32,780 --> 00:17:40,610 Because if we create a new scatterplot where we have our X values along the horizontal axis, just like 231 00:17:40,610 --> 00:17:47,410 up here, but now along the vertical axis, we plot the value of the residual. 232 00:17:47,420 --> 00:17:52,610 So notice here that in our vertical axis, the value of zero is here in the middle. 233 00:17:52,610 --> 00:17:57,400 And so we have some negative values below zero and some positive values above zero. 234 00:17:57,410 --> 00:18:05,420 If we plot each of these values of the residual in this new set of coordinate axes, we get this scatterplot 235 00:18:05,420 --> 00:18:05,900 here. 236 00:18:05,900 --> 00:18:08,090 Ignore the yellow line for a second. 237 00:18:08,090 --> 00:18:10,880 We get this scatterplot with all these new points. 238 00:18:10,880 --> 00:18:16,940 What we see is that they all correspond in terms of their position to the scatterplot up here. 239 00:18:16,940 --> 00:18:19,370 We've just changed the vertical axis. 240 00:18:19,370 --> 00:18:25,760 So now this point, instead of being plotted with a vertical value of two, is being plotted with a 241 00:18:25,760 --> 00:18:28,820 vertical value of about 1.17. 242 00:18:28,820 --> 00:18:30,500 We see that point here. 243 00:18:30,500 --> 00:18:36,800 These two points down here are the -0.57 and the -0.60. 244 00:18:36,830 --> 00:18:44,600 In other words, up here in this scatterplot, all of these points are plotted as x, y coordinate pairs 245 00:18:44,600 --> 00:18:52,250 down here in this scatterplot, all of these points are plotted as x, y hat coordinate pairs. 246 00:18:52,990 --> 00:18:57,940 And when we use this new set of coordinate points, when our coordinate points are given by X and Y 247 00:18:57,940 --> 00:19:05,350 hat and we use that set of coordinate points to find a new regression line equation, that new regression 248 00:19:05,350 --> 00:19:09,910 line equation is this line here, we'll call it y hat of the residuals. 249 00:19:09,910 --> 00:19:14,270 That new regression line equation is the equation y hat equals zero. 250 00:19:14,290 --> 00:19:20,440 In other words, another way to put this is that if in this plot, the points in the scatterplot are 251 00:19:20,440 --> 00:19:25,870 pretty well spaced out with a bunch above, this y equals zero line and a bunch below this y equals 252 00:19:25,870 --> 00:19:26,620 zero line. 253 00:19:26,620 --> 00:19:34,240 That means that a linear model is going to do an okay job describing the trend in the original scatterplot 254 00:19:34,240 --> 00:19:36,070 with the original set of data. 255 00:19:36,070 --> 00:19:41,590 So if you remember, we said before that this regression line, this line of best fit, this line of 256 00:19:41,590 --> 00:19:46,000 least squares, we have a bunch of different names for this regression line, this trend line. 257 00:19:46,000 --> 00:19:53,470 We said before that we were always trying to minimize the sum of squares so we would look at each data 258 00:19:53,470 --> 00:20:01,090 point in the scatterplot and we would think about creating a square between that point and the regression 259 00:20:01,090 --> 00:20:01,450 line. 260 00:20:01,450 --> 00:20:10,330 So we would have a square like this and then we would have a square like this, and then this one up 261 00:20:10,330 --> 00:20:11,800 here would be really, really big. 262 00:20:11,800 --> 00:20:14,890 This square here would look something like that. 263 00:20:14,890 --> 00:20:18,130 Let's just go ahead and make this big one. 264 00:20:18,130 --> 00:20:19,450 So we have this big square. 265 00:20:19,450 --> 00:20:25,330 We said before that we were trying to minimize the sum of all of these square areas. 266 00:20:25,330 --> 00:20:32,290 Well, now that we know that the side of each square can be described as the residual E, if we just 267 00:20:32,290 --> 00:20:41,830 say that e sub n represents all of the residuals and we square that value, this is all of the squared 268 00:20:41,830 --> 00:20:45,970 residuals and this is the sum of all of the squared residuals. 269 00:20:45,970 --> 00:20:52,720 This is the value we're trying to minimize to find the very best fitting line through the data, through 270 00:20:52,720 --> 00:20:57,880 the scatterplot, to find the equation of the regression line, to find the equation of the trend line, 271 00:20:57,880 --> 00:21:00,100 the line of best fit, the line of least squares. 272 00:21:00,100 --> 00:21:06,520 This is the value we're trying to minimize when we describe that value in terms of this new idea we've 273 00:21:06,520 --> 00:21:12,850 introduced, which is the idea of the residual Thinking about this in the context of the residual also 274 00:21:12,850 --> 00:21:19,060 gives us a better idea of why we look at the squares instead of just the pure distances. 275 00:21:19,060 --> 00:21:24,190 In other words, instead of just pure actual minus predicted the distance from the data point to the 276 00:21:24,190 --> 00:21:25,030 regression line. 277 00:21:25,030 --> 00:21:31,300 We instead look at the squared area, and the reason is because in the context of the residual here 278 00:21:31,300 --> 00:21:36,310 we get a positive residual for any points that are above the least squares line. 279 00:21:36,310 --> 00:21:41,110 We get a negative residual for any points that are below the least squares line. 280 00:21:41,110 --> 00:21:46,630 And if we have a bunch of points above and below the line, those positive and negative distances would 281 00:21:46,630 --> 00:21:51,100 cancel each other out if we didn't square the values to turn them into areas. 282 00:21:51,100 --> 00:21:58,180 So if for this point we got a residual of positive one, and if for this point we got a residual of 283 00:21:58,180 --> 00:22:03,130 negative one, those two things would cancel each other out completely and we wouldn't necessarily be 284 00:22:03,130 --> 00:22:05,290 optimizing for either of those data points. 285 00:22:05,290 --> 00:22:09,550 When we square the residuals, we get two pieces of squared area. 286 00:22:09,550 --> 00:22:12,280 They both count as positive pieces of area. 287 00:22:12,280 --> 00:22:18,040 And so we can make sure to account for minimizing both of those areas when we're building the equation 288 00:22:18,040 --> 00:22:19,390 of our regression line. 289 00:22:19,390 --> 00:22:25,900 So we're always trying to minimize the sum of the squared residuals or the sum of these square areas. 290 00:22:25,900 --> 00:22:32,350 So that process of trying to minimize the residuals by minimizing the squares of the residuals, that's 291 00:22:32,350 --> 00:22:38,500 where we get that name least squares line or line of least squares, or we often call this process least 292 00:22:38,500 --> 00:22:39,520 squares regression. 293 00:22:39,520 --> 00:22:44,650 We're just trying to minimize the total area of all of these squares.