1 00:00:06,480 --> 00:00:12,100 Last time, I mean Tuesday, we discussed box plot 2 00:00:12,100 --> 00:00:19,540 and we introduced how can we use box plot to 3 00:00:19,540 --> 00:00:24,160 determine if any point is suspected to be an 4 00:00:24,160 --> 00:00:28,280 outlier by using the lower limit and upper limit. 5 00:00:29,460 --> 00:00:32,980 And we mentioned last time that if any point is 6 00:00:32,980 --> 00:00:38,580 below the lower limit or is above the upper limit, 7 00:00:39,200 --> 00:00:43,000 that point is considered to be an outlier. So 8 00:00:43,000 --> 00:00:46,940 that's one of the usage of the backsplat. I mean, 9 00:00:47,000 --> 00:00:51,360 for this specific example, we mentioned last time 10 00:00:51,360 --> 00:00:56,910 27 is an outlier. And also here you can tell also 11 00:00:56,910 --> 00:01:02,770 the data are right skewed because the right tail 12 00:01:02,770 --> 00:01:06,090 exactly is much longer than the left tail. I mean 13 00:01:06,090 --> 00:01:10,450 the distance between or from the median and the 14 00:01:10,450 --> 00:01:14,150 maximum value is bigger or larger than the 15 00:01:14,150 --> 00:01:16,950 distance from the median to the smallest value. 16 00:01:17,450 --> 00:01:20,870 That means the data is not symmetric, it's quite 17 00:01:20,870 --> 00:01:24,050 skewed to the right. In this case, you cannot use 18 00:01:24,050 --> 00:01:29,370 the mean or the range as a measure of spread and 19 00:01:29,370 --> 00:01:31,730 median and, I'm sorry, mean as a measure of 20 00:01:31,730 --> 00:01:36,130 tendency. Because these measures are affected by 21 00:01:36,130 --> 00:01:39,450 outcomes. In this case, you have to use the median 22 00:01:39,450 --> 00:01:43,690 instead of the mean and IQR instead of the range 23 00:01:43,690 --> 00:01:48,090 because IQR is the mid-spread of the data because 24 00:01:48,090 --> 00:01:52,790 we just take the range from Q3 to Q1. That means 25 00:01:52,790 --> 00:01:57,450 we ignore The data below Q1 and data after Q3. 26 00:01:57,970 --> 00:02:01,370 That means IQR is not affected by outlier and it's 27 00:02:01,370 --> 00:02:04,610 better to use it instead of R, of the range. 28 00:02:07,470 --> 00:02:10,950 If the data has an outlier, it's better just to 29 00:02:10,950 --> 00:02:13,990 make a star or circle for the box plot because 30 00:02:13,990 --> 00:02:17,250 this one mentioned that that point is an outlier. 31 00:02:18,390 --> 00:02:21,390 Sometimes outlier is maximum value or the largest 32 00:02:21,390 --> 00:02:25,000 value you have. sometimes maybe the minimum value. 33 00:02:25,520 --> 00:02:28,480 So it depends on the data. For this example, 27, 34 00:02:28,720 --> 00:02:33,360 which was the maximum, is an outlier. But zero is 35 00:02:33,360 --> 00:02:36,520 not outlier in this case, because zero is above 36 00:02:36,520 --> 00:02:41,500 the lower limit. Let's move to the next topic, 37 00:02:42,140 --> 00:02:48,060 which talks about covariance and correlation. 38 00:02:48,960 --> 00:02:51,740 Later, we'll talk in more details about 39 00:02:53,020 --> 00:02:56,060 Correlation and regression, that's when maybe 40 00:02:56,060 --> 00:03:02,840 chapter 11 or 12. But here we just show how can we 41 00:03:02,840 --> 00:03:05,420 compute the covariance of the correlation 42 00:03:05,420 --> 00:03:10,220 coefficient and what's the meaning of that value 43 00:03:10,220 --> 00:03:15,840 we have. The covariance means it measures the 44 00:03:15,840 --> 00:03:21,090 strength of the linear relationship between two 45 00:03:21,090 --> 00:03:25,410 numerical variables. That means if the data set is 46 00:03:25,410 --> 00:03:29,770 numeric, I mean if both variables are numeric, in 47 00:03:29,770 --> 00:03:33,050 this case we can use the covariance to measure the 48 00:03:33,050 --> 00:03:38,390 strength of the linear association or relationship 49 00:03:38,390 --> 00:03:42,310 between two numerical variables. Now the formula 50 00:03:42,310 --> 00:03:45,330 is used to compute the covariance given by this 51 00:03:45,330 --> 00:03:52,540 one. It's summation of the product of xi minus x 52 00:03:52,540 --> 00:03:56,380 bar, yi minus y bar, divided by n minus 1. 53 00:03:59,660 --> 00:04:03,120 So we need first to compute the means of x and y, 54 00:04:03,620 --> 00:04:07,680 then find x minus x bar times y minus y bar, then 55 00:04:07,680 --> 00:04:11,160 sum all of these values, then divide by n minus 1. 56 00:04:12,870 --> 00:04:17,770 The covariance only concerned with the strength of 57 00:04:17,770 --> 00:04:23,370 the relationship. By using the sign of the 58 00:04:23,370 --> 00:04:27,010 covariance, you can tell if there exists positive 59 00:04:27,010 --> 00:04:31,070 or negative relationship between the two 60 00:04:31,070 --> 00:04:33,710 variables. For example, if the covariance between 61 00:04:33,710 --> 00:04:42,760 x and y is positive, that means x and y move In 62 00:04:42,760 --> 00:04:48,080 the same direction. It means that if X goes up, Y 63 00:04:48,080 --> 00:04:52,260 will go in the same position. If X goes down, also 64 00:04:52,260 --> 00:04:55,660 Y goes down. For example, suppose we are 65 00:04:55,660 --> 00:04:57,920 interested in the relationship between consumption 66 00:04:57,920 --> 00:05:02,440 and income. We know that if income increases, if 67 00:05:02,440 --> 00:05:07,160 income goes up, if your salary goes up, that means 68 00:05:07,160 --> 00:05:13,510 consumption also will go up. So that means they go 69 00:05:13,510 --> 00:05:18,650 in the same or move in the same position. So for 70 00:05:18,650 --> 00:05:20,690 sure, the covariance between X and Y should be 71 00:05:20,690 --> 00:05:25,550 positive. On the other hand, if the covariance 72 00:05:25,550 --> 00:05:31,110 between X and Y is negative, that means X goes up. 73 00:05:32,930 --> 00:05:36,370 Y will go to the same, to the opposite direction. 74 00:05:36,590 --> 00:05:40,090 I mean they move to opposite direction. That means 75 00:05:40,090 --> 00:05:42,230 there exists negative relationship between X and 76 00:05:42,230 --> 00:05:47,630 Y. For example, you score in statistics a number 77 00:05:47,630 --> 00:05:55,220 of missing classes. If you miss more classes, it 78 00:05:55,220 --> 00:05:59,860 means your score will go down so as x increases y 79 00:05:59,860 --> 00:06:04,820 will go down so there is positive relationship or 80 00:06:04,820 --> 00:06:08,720 negative relationship between x and y i mean x 81 00:06:08,720 --> 00:06:12,020 goes up the other go in the same direction 82 00:06:12,020 --> 00:06:16,500 sometimes 83 00:06:16,500 --> 00:06:21,800 there is exist no relationship between x and y In 84 00:06:21,800 --> 00:06:24,780 that case, covariance between x and y equals zero. 85 00:06:24,880 --> 00:06:31,320 For example, your score in statistics and your 86 00:06:31,320 --> 00:06:31,700 weight. 87 00:06:34,540 --> 00:06:36,760 It makes sense that there is no relationship 88 00:06:36,760 --> 00:06:42,680 between your weight and your score. In this case, 89 00:06:43,580 --> 00:06:46,760 we are saying x and y are independent. I mean, 90 00:06:46,840 --> 00:06:50,790 they are uncorrelated. Because as one variable 91 00:06:50,790 --> 00:06:56,010 increases, the other maybe go up or go down. Or 92 00:06:56,010 --> 00:06:59,690 maybe constant. So that means there exists no 93 00:06:59,690 --> 00:07:02,390 relationship between the two variables. In that 94 00:07:02,390 --> 00:07:05,950 case, the covariance between x and y equals zero. 95 00:07:06,450 --> 00:07:09,210 Now, by using the covariance, you can determine 96 00:07:09,210 --> 00:07:12,710 the direction of the relationship. I mean, you can 97 00:07:12,710 --> 00:07:14,850 just figure out if the relation is positive or 98 00:07:14,850 --> 00:07:18,980 negative. But you cannot tell exactly the strength 99 00:07:18,980 --> 00:07:22,100 of the relationship. I mean you cannot tell if 100 00:07:22,100 --> 00:07:27,640 they exist. strong moderate or weak relationship 101 00:07:27,640 --> 00:07:31,040 just you can tell there exists positive or 102 00:07:31,040 --> 00:07:33,520 negative or maybe the relationship does not exist 103 00:07:33,520 --> 00:07:36,580 but you cannot tell the exact strength of the 104 00:07:36,580 --> 00:07:40,120 relationship by using the value of the covariance 105 00:07:40,120 --> 00:07:43,060 I mean the size of the covariance does not tell 106 00:07:43,060 --> 00:07:48,520 anything about the strength so generally speaking 107 00:07:48,520 --> 00:07:53,150 covariance between x and y measures the strength 108 00:07:53,150 --> 00:07:58,590 of two numerical variables, and you only tell if 109 00:07:58,590 --> 00:08:01,270 there exists positive or negative relationship, 110 00:08:01,770 --> 00:08:04,510 but you cannot tell anything about the strength of 111 00:08:04,510 --> 00:08:06,910 the relationship. Any questions? 112 00:08:09,610 --> 00:08:15,210 So let me ask you just to summarize what I said so 113 00:08:15,210 --> 00:08:21,100 far. Just give me the summary or conclusion. of 114 00:08:21,100 --> 00:08:24,670 the covariance. The value of the covariance 115 00:08:24,670 --> 00:08:26,810 determine if the relationship between the 116 00:08:26,810 --> 00:08:29,410 variables are positive or negative or there is no 117 00:08:29,410 --> 00:08:31,970 relationship that when the covariance is more than 118 00:08:31,970 --> 00:08:34,170 zero, the meaning that it's positive, the 119 00:08:34,170 --> 00:08:36,930 relationship is positive and one variable go up, 120 00:08:37,290 --> 00:08:39,590 another go up and vice versa. And when the 121 00:08:39,590 --> 00:08:41,810 covariance is less than zero, there is negative 122 00:08:41,810 --> 00:08:44,250 relationship and the meaning that when one 123 00:08:44,250 --> 00:08:47,490 variable go up, the other goes down and vice versa 124 00:08:47,490 --> 00:08:50,550 and when the covariance equals zero, there is no 125 00:08:50,550 --> 00:08:53,350 relationship between the variables. And what's 126 00:08:53,350 --> 00:08:54,930 about the strength? 127 00:08:57,950 --> 00:09:03,450 So just tell the direction, not the strength of 128 00:09:03,450 --> 00:09:08,610 the relationship. Now, in order to determine both 129 00:09:08,610 --> 00:09:12,110 the direction and the strength, we can use the 130 00:09:12,110 --> 00:09:17,580 coefficient of correlation. The coefficient of 131 00:09:17,580 --> 00:09:20,320 correlation measures the relative strength of the 132 00:09:20,320 --> 00:09:22,780 linear relationship between two numerical 133 00:09:22,780 --> 00:09:27,940 variables. The simplest formula that can be used 134 00:09:27,940 --> 00:09:31,220 to compute the correlation coefficient is given by 135 00:09:31,220 --> 00:09:34,440 this one. Maybe this is the easiest formula you 136 00:09:34,440 --> 00:09:38,060 can use. I mean, it's shortcut formula for 137 00:09:38,060 --> 00:09:40,860 computation. There are many other formulas to 138 00:09:40,860 --> 00:09:44,490 compute the correlation. This one is the easiest 139 00:09:44,490 --> 00:09:52,570 one. R is just sum of xy minus n, n is the sample 140 00:09:52,570 --> 00:09:57,570 size, times x bar is the sample mean, y bar is the 141 00:09:57,570 --> 00:10:01,090 sample mean for y, because here we have two 142 00:10:01,090 --> 00:10:06,250 variables, divided by square root, don't forget 143 00:10:06,250 --> 00:10:11,490 the square root, of two quantities. One conserved 144 00:10:11,490 --> 00:10:15,710 for x and the other for y. The first one, sum of x 145 00:10:15,710 --> 00:10:18,850 squared minus nx bar squared. The other one is 146 00:10:18,850 --> 00:10:21,830 similar just for the other variables, sum y 147 00:10:21,830 --> 00:10:26,090 squared minus ny bar squared. So in order to 148 00:10:26,090 --> 00:10:28,650 determine the value of R, we need, 149 00:10:32,170 --> 00:10:35,890 suppose for example, we have x and y, theta x and 150 00:10:35,890 --> 00:10:36,110 y. 151 00:10:40,350 --> 00:10:44,730 x is called explanatory 152 00:10:44,730 --> 00:10:54,390 variable and 153 00:10:54,390 --> 00:11:04,590 y is called response variable 154 00:11:04,590 --> 00:11:07,970 sometimes x is called independent 155 00:11:21,760 --> 00:11:25,320 For example, suppose we are talking about 156 00:11:25,320 --> 00:11:32,280 consumption and 157 00:11:32,280 --> 00:11:36,700 input. And we are interested in the relationship 158 00:11:36,700 --> 00:11:41,360 between these two variables. Now, except for the 159 00:11:41,360 --> 00:11:44,900 variable or the independent, this one affects the 160 00:11:44,900 --> 00:11:49,840 other variable. As we mentioned, as your income 161 00:11:49,840 --> 00:11:53,800 increases, your consumption will go in the same 162 00:11:53,800 --> 00:11:59,580 direction, increases also. Income causes Y, or 163 00:11:59,580 --> 00:12:04,340 income affects Y. In this case, income is your X. 164 00:12:06,180 --> 00:12:07,780 Most of the time we use 165 00:12:10,790 --> 00:12:15,590 And Y for independent. So in this case, the 166 00:12:15,590 --> 00:12:19,370 response variable or your outcome or the dependent 167 00:12:19,370 --> 00:12:23,110 variable is your consumption. So Y is consumption, 168 00:12:23,530 --> 00:12:29,150 X is income. So now in order to determine the 169 00:12:29,150 --> 00:12:32,950 correlation coefficient, we have the data of X and 170 00:12:32,950 --> 00:12:33,210 Y. 171 00:12:36,350 --> 00:12:39,190 The values of X, I mean the number of pairs of X 172 00:12:39,190 --> 00:12:41,990 should be equal to the number of pairs of Y. So if 173 00:12:41,990 --> 00:12:44,930 we have ten observations for X, we should have the 174 00:12:44,930 --> 00:12:50,010 same number of observations for Y. It's pairs. X1, 175 00:12:50,090 --> 00:12:54,750 Y1, X2, Y2, and so on. Now, the formula to compute 176 00:12:54,750 --> 00:13:04,170 R, the shortcut formula is sum of XY minus N times 177 00:13:04,970 --> 00:13:09,630 x bar, y bar, divided by the square root of two 178 00:13:09,630 --> 00:13:12,770 quantities. The first one, sum of x squared minus 179 00:13:12,770 --> 00:13:17,270 n x bar. The other one, sum of y squared minus ny 180 00:13:17,270 --> 00:13:21,710 y squared. So the first thing we have to do is to 181 00:13:21,710 --> 00:13:24,210 find the mean for each x and y. 182 00:13:28,230 --> 00:13:37,210 So first step, compute x bar and y bar. Next, if 183 00:13:37,210 --> 00:13:41,690 you look here, we have x and y, x times y. So we 184 00:13:41,690 --> 00:13:48,870 need to compute the product of x times y. So just 185 00:13:48,870 --> 00:13:53,870 for example, suppose x is 10, y is 5. So x times y 186 00:13:53,870 --> 00:13:54,970 is 50. 187 00:13:57,810 --> 00:13:59,950 In addition to that, you have to compute 188 00:14:06,790 --> 00:14:12,470 100 x squared and y squared. It's like 125. 189 00:14:14,810 --> 00:14:18,870 Do the same calculations for the rest of the data 190 00:14:18,870 --> 00:14:22,290 you have. We have other data here, so we have to 191 00:14:22,290 --> 00:14:25,410 compute the same for the others. 192 00:14:28,470 --> 00:14:33,250 Then finally, just add xy, x squared, y squared. 193 00:14:35,910 --> 00:14:40,830 The values you have here in this formula, in order 194 00:14:40,830 --> 00:14:44,830 to compute the coefficient. 195 00:14:54,250 --> 00:15:00,070 Now, this value ranges between minus one and plus 196 00:15:00,070 --> 00:15:00,370 one. 197 00:15:06,520 --> 00:15:10,800 So it's between minus one and plus one. That means 198 00:15:10,800 --> 00:15:15,840 it's never smaller 199 00:15:15,840 --> 00:15:20,200 than minus one or greater than one. It's between 200 00:15:20,200 --> 00:15:21,480 minus one and plus one. 201 00:15:24,360 --> 00:15:28,300 Make sense? I mean if your value is suppose you 202 00:15:28,300 --> 00:15:34,520 did mistake for any of these computations and R 203 00:15:34,520 --> 00:15:41,710 might be 1.15, 115. That means there is an error. 204 00:15:42,270 --> 00:15:45,870 Or for example, if R is negative 1.5, that means 205 00:15:45,870 --> 00:15:49,610 there is a mistake. So you have to find or figure 206 00:15:49,610 --> 00:15:55,670 out what is that mistake. So that's simple 207 00:15:55,670 --> 00:15:59,090 calculations. Usually in the exam, we will give 208 00:15:59,090 --> 00:16:01,350 the formula for the correlation coefficient, as we 209 00:16:01,350 --> 00:16:03,590 mentioned before. In addition to that, we will 210 00:16:03,590 --> 00:16:04,330 give the summation. 211 00:16:07,780 --> 00:16:12,720 The sum of xy is given, sum x squared and sum y 212 00:16:12,720 --> 00:16:18,320 squared. Also sum of x and sum of y, in order to 213 00:16:18,320 --> 00:16:22,320 determine the means of x and y. For example, 214 00:16:22,520 --> 00:16:26,860 suppose I give sum of xi and i goes from 1 to 10 215 00:16:26,860 --> 00:16:31,760 is 700, for example. You have to know that the 216 00:16:31,760 --> 00:16:38,720 sample size is 10, so x bar. is 700 divided by 10, 217 00:16:39,320 --> 00:16:46,180 so it's 7. Then use the curve to compute the 218 00:16:46,180 --> 00:16:52,000 coefficient of correlation. Questions? I think 219 00:16:52,000 --> 00:16:55,900 straightforward, maybe the easiest topic in this 220 00:16:55,900 --> 00:17:02,980 book is to compute the coefficient of correlation 221 00:17:02,980 --> 00:17:09,070 or correlation coefficient. Now my question is, do 222 00:17:09,070 --> 00:17:13,090 you think outliers affect the correlation 223 00:17:13,090 --> 00:17:13,690 coefficient? 224 00:17:17,010 --> 00:17:23,210 We said last time outliers affect the mean, the 225 00:17:23,210 --> 00:17:28,310 range, the variance. Now the question is, do 226 00:17:28,310 --> 00:17:33,510 outliers affect the correlation? 227 00:17:37,410 --> 00:17:38,170 Y. 228 00:17:43,830 --> 00:17:51,330 Exactly. The formula for R has X bar in it or Y 229 00:17:51,330 --> 00:17:56,670 bar. So it means outliers affect 230 00:17:56,670 --> 00:18:01,210 the correlation coefficient. So the answer is yes. 231 00:18:03,470 --> 00:18:06,410 Here we have x bar and y bar. Also, there is 232 00:18:06,410 --> 00:18:10,690 another formula to compute R. That formula is 233 00:18:10,690 --> 00:18:13,370 given by covariance between x and y. 234 00:18:17,510 --> 00:18:21,930 These two formulas are quite similar. I mean, by 235 00:18:21,930 --> 00:18:26,070 using this one, we can end with this formula. So 236 00:18:26,070 --> 00:18:33,090 this formula depends on this x is y. standard 237 00:18:33,090 --> 00:18:36,170 deviations of X and Y. That means outlier will 238 00:18:36,170 --> 00:18:42,530 affect the correlation coefficient. So in case of 239 00:18:42,530 --> 00:18:45,670 outliers, R could be changed. 240 00:18:51,170 --> 00:18:55,530 That formula is called simple correlation 241 00:18:55,530 --> 00:18:58,790 coefficient. On the other hand, we have population 242 00:18:58,790 --> 00:19:02,200 correlation coefficient. If you remember last 243 00:19:02,200 --> 00:19:08,940 time, we used X bar as the sample mean and mu as 244 00:19:08,940 --> 00:19:14,460 population mean. Also, S square as sample variance 245 00:19:14,460 --> 00:19:18,740 and sigma square as population variance. Here, R 246 00:19:18,740 --> 00:19:24,360 is used as sample coefficient of correlation and 247 00:19:24,360 --> 00:19:29,420 rho, this Greek letter pronounced as rho. Rho is 248 00:19:29,420 --> 00:19:35,160 used for population coefficient of correlation. 249 00:19:37,640 --> 00:19:42,040 There are some features of R or Rho. The first one 250 00:19:42,040 --> 00:19:47,960 is unity-free. R or Rho is unity-free. That means 251 00:19:47,960 --> 00:19:54,900 if X represents... 252 00:19:54,900 --> 00:19:58,960 And let's assume that the correlation between X 253 00:19:58,960 --> 00:20:02,040 and Y equals 0.75. 254 00:20:04,680 --> 00:20:07,260 Now, in this case, there is no unity. You cannot 255 00:20:07,260 --> 00:20:13,480 say 0.75 years or 0.75 kilograms. It's unity-free. 256 00:20:13,940 --> 00:20:17,840 There is no unit for the correlation coefficient, 257 00:20:18,020 --> 00:20:21,120 the same as Cv. If you remember Cv, the 258 00:20:21,120 --> 00:20:24,320 coefficient of correlation, also this one is unity 259 00:20:24,320 --> 00:20:30,500 -free. The second feature of R ranges between 260 00:20:30,500 --> 00:20:36,740 minus one and plus one. As I mentioned, R lies 261 00:20:36,740 --> 00:20:42,340 between minus one and plus one. Now, by using the 262 00:20:42,340 --> 00:20:48,100 value of R, you can determine two things. Number 263 00:20:48,100 --> 00:20:53,360 one, we can determine the direction. and strength 264 00:20:53,360 --> 00:20:56,940 by using the sign you can determine if there 265 00:20:56,940 --> 00:21:03,980 exists positive or negative so sign of R determine 266 00:21:03,980 --> 00:21:08,040 negative or positive relationship the direction 267 00:21:08,040 --> 00:21:17,840 the absolute value of R I mean absolute of R I 268 00:21:17,840 --> 00:21:21,980 mean ignore the sign So the absolute value of R 269 00:21:21,980 --> 00:21:24,100 determines the strength. 270 00:21:27,700 --> 00:21:30,760 So by using the sine of R, you can determine the 271 00:21:30,760 --> 00:21:35,680 direction, either positive or negative. By using 272 00:21:35,680 --> 00:21:37,740 the absolute value, you can determine the 273 00:21:37,740 --> 00:21:43,500 strength. We can split the strength into two 274 00:21:43,500 --> 00:21:52,810 parts, either strong, moderate, or weak. So weak, 275 00:21:53,770 --> 00:21:59,130 moderate, and strong by using the absolute value 276 00:21:59,130 --> 00:22:03,870 of R. The closer to minus one, if R is close to 277 00:22:03,870 --> 00:22:07,010 minus one, the stronger the negative relationship 278 00:22:07,010 --> 00:22:09,430 between X and Y. For example, imagine 279 00:22:22,670 --> 00:22:26,130 And as we mentioned, R ranges between minus 1 and 280 00:22:26,130 --> 00:22:26,630 plus 1. 281 00:22:30,070 --> 00:22:35,710 So if R is close to minus 1, it's a strong 282 00:22:35,710 --> 00:22:41,250 relationship. Strong linked relationship. The 283 00:22:41,250 --> 00:22:45,190 closer to 1, the stronger the positive 284 00:22:45,190 --> 00:22:49,230 relationship. I mean, if R is close. Strong 285 00:22:49,230 --> 00:22:54,480 positive. So strong in either direction, either to 286 00:22:54,480 --> 00:22:57,640 the left side or to the right side. Strong 287 00:22:57,640 --> 00:23:00,280 negative. On the other hand, there exists strong 288 00:23:00,280 --> 00:23:05,940 negative relationship. Positive. Positive. If R is 289 00:23:05,940 --> 00:23:10,640 close to zero, weak. Here we can say there exists 290 00:23:10,640 --> 00:23:15,940 weak relationship between X and Y. 291 00:23:19,260 --> 00:23:25,480 If R is close to 0.5 or 292 00:23:25,480 --> 00:23:32,320 minus 0.5, you can say there exists positive 293 00:23:32,320 --> 00:23:38,840 -moderate or negative-moderate relationship. So 294 00:23:38,840 --> 00:23:42,200 you can split or you can divide the strength of 295 00:23:42,200 --> 00:23:44,540 the relationship between X and Y into three parts. 296 00:23:45,860 --> 00:23:50,700 Strong, close to minus one of Plus one, weak, 297 00:23:51,060 --> 00:23:59,580 close to zero, moderate, close to 0.5. 0.5 is 298 00:23:59,580 --> 00:24:04,580 halfway between 0 and 1, and minus 0.5 is also 299 00:24:04,580 --> 00:24:09,040 halfway between minus 1 and 0. Now for example, 300 00:24:09,920 --> 00:24:15,580 what's about if R equals minus 0.5? Suppose R1 is 301 00:24:15,580 --> 00:24:16,500 minus 0.5. 302 00:24:20,180 --> 00:24:27,400 strong negative or equal minus point eight strong 303 00:24:27,400 --> 00:24:33,540 negative which is more strong nine nine because 304 00:24:33,540 --> 00:24:39,670 this value is close closer to minus one than Minus 305 00:24:39,670 --> 00:24:44,070 0.8. Even this value is greater than minus 0.9, 306 00:24:44,530 --> 00:24:50,870 but minus 0.9 is close to minus 1, more closer to 307 00:24:50,870 --> 00:24:56,910 minus 1 than minus 0.8. On the other hand, if R 308 00:24:56,910 --> 00:25:01,190 equals 0.75, that means there exists positive 309 00:25:01,190 --> 00:25:06,970 relationship. If R equals 0.85, also there exists 310 00:25:06,970 --> 00:25:13,540 positive. But R2 is stronger than R1, because 0.85 311 00:25:13,540 --> 00:25:20,980 is closer to plus 1 than 0.7. So we can say that 312 00:25:20,980 --> 00:25:23,960 there exists strong relationship between X and Y, 313 00:25:24,020 --> 00:25:27,260 and this relationship is positive. So again, by 314 00:25:27,260 --> 00:25:32,530 using the sign, you can tell the direction. The 315 00:25:32,530 --> 00:25:35,910 absolute value can tell the strength of the 316 00:25:35,910 --> 00:25:39,870 relationship between X and Y. So there are five 317 00:25:39,870 --> 00:25:44,150 features of R, unity-free. Ranges between minus 318 00:25:44,150 --> 00:25:47,750 one and plus one. Closer to minus one, it means 319 00:25:47,750 --> 00:25:51,950 stronger negative. Closer to plus one, stronger 320 00:25:51,950 --> 00:25:56,410 positive. Close to zero, it means there is no 321 00:25:56,410 --> 00:26:00,790 relationship. Or the weaker, the relationship 322 00:26:00,790 --> 00:26:13,240 between X and Y. By using scatter plots, we 323 00:26:13,240 --> 00:26:18,160 can construct a scatter plot by plotting the Y 324 00:26:18,160 --> 00:26:24,060 values versus the X values. Y in the vertical axis 325 00:26:24,060 --> 00:26:28,400 and X in the horizontal axis. If you look 326 00:26:28,400 --> 00:26:34,500 carefully at graph number one and three, We see 327 00:26:34,500 --> 00:26:42,540 that all the points lie on the straight line, 328 00:26:44,060 --> 00:26:48,880 either this way or the other way. If all the 329 00:26:48,880 --> 00:26:52,320 points lie on the straight line, it means they 330 00:26:52,320 --> 00:26:56,970 exist perfectly. not even strong it's perfect 331 00:26:56,970 --> 00:27:02,710 relationship either negative or positive so this 332 00:27:02,710 --> 00:27:07,530 one perfect negative negative 333 00:27:07,530 --> 00:27:14,090 because x increases y goes down decline so if x is 334 00:27:14,090 --> 00:27:19,590 for example five maybe y is supposed to twenty if 335 00:27:19,590 --> 00:27:25,510 x increased to seven maybe y is fifteen So if X 336 00:27:25,510 --> 00:27:29,290 increases, in this case, Y declines or decreases, 337 00:27:29,850 --> 00:27:34,290 it means there exists negative relationship. On 338 00:27:34,290 --> 00:27:40,970 the other hand, the left corner here, positive 339 00:27:40,970 --> 00:27:44,710 relationship, because X increases, Y also goes up. 340 00:27:45,970 --> 00:27:48,990 And perfect, because all the points lie on the 341 00:27:48,990 --> 00:27:52,110 straight line. So it's perfect, positive, perfect, 342 00:27:52,250 --> 00:27:57,350 negative relationship. So it's straightforward to 343 00:27:57,350 --> 00:27:59,550 determine if it's perfect by using scatterplot. 344 00:28:02,230 --> 00:28:04,950 Also, by scatterplot, you can tell the direction 345 00:28:04,950 --> 00:28:09,270 of the relationship. For the second scatterplot, 346 00:28:09,630 --> 00:28:12,270 it seems to be that there exists negative 347 00:28:12,270 --> 00:28:13,730 relationship between X and Y. 348 00:28:16,850 --> 00:28:21,030 In this one, also there exists a relationship 349 00:28:24,730 --> 00:28:32,170 positive which one is strong more strong this 350 00:28:32,170 --> 00:28:37,110 one is stronger because the points are close to 351 00:28:37,110 --> 00:28:40,710 the straight line much more than the other scatter 352 00:28:40,710 --> 00:28:43,410 plot so you can say there exists negative 353 00:28:43,410 --> 00:28:45,810 relationship but that one is stronger than the 354 00:28:45,810 --> 00:28:49,550 other one this one is positive but the points are 355 00:28:49,550 --> 00:28:55,400 scattered around the straight line so you can tell 356 00:28:55,400 --> 00:29:00,000 the direction and sometimes sometimes not all the 357 00:29:00,000 --> 00:29:04,640 time you can tell the strength sometimes it's very 358 00:29:04,640 --> 00:29:07,960 clear that the relation is strong if the points 359 00:29:07,960 --> 00:29:11,480 are very close straight line that means the 360 00:29:11,480 --> 00:29:15,940 relation is strong now the other one the last one 361 00:29:15,940 --> 00:29:23,850 here As X increases, Y stays at the same value. 362 00:29:23,970 --> 00:29:29,450 For example, if Y is 20 and X is 1. X is 1, Y is 363 00:29:29,450 --> 00:29:33,870 20. X increases to 2, for example. Y is still 20. 364 00:29:34,650 --> 00:29:37,230 So that means there is no relationship between X 365 00:29:37,230 --> 00:29:41,830 and Y. It's a constant. Y equals a constant value. 366 00:29:42,690 --> 00:29:50,490 Whatever X is, Y will have constant value. So that 367 00:29:50,490 --> 00:29:54,790 means there is no relationship between X and Y. 368 00:29:56,490 --> 00:30:01,850 Let's see how can we compute the correlation 369 00:30:01,850 --> 00:30:07,530 between two variables. For example, suppose we 370 00:30:07,530 --> 00:30:12,150 have data for father's height and son's height. 371 00:30:13,370 --> 00:30:16,510 And we are interested to see if father's height 372 00:30:16,510 --> 00:30:21,730 affects his son's height. So we have data for 10 373 00:30:21,730 --> 00:30:28,610 observations here. Father number one, his height 374 00:30:28,610 --> 00:30:38,570 is 64 inches. And you know that inch equals 2 375 00:30:38,570 --> 00:30:39,230 .5. 376 00:30:43,520 --> 00:30:52,920 So X is 64, Sun's height is 65. X is 68, Sun's 377 00:30:52,920 --> 00:30:58,820 height is 67 and so on. Sometimes, if the dataset 378 00:30:58,820 --> 00:31:02,600 is small enough, as in this example, we have just 379 00:31:02,600 --> 00:31:08,640 10 observations, you can tell the direction. I 380 00:31:08,640 --> 00:31:12,060 mean, you can say, yes, for this specific example, 381 00:31:12,580 --> 00:31:15,280 there exists positive relationship between x and 382 00:31:15,280 --> 00:31:20,820 y. But if the data set is large, it's very hard to 383 00:31:20,820 --> 00:31:22,620 figure out if the relation is positive or 384 00:31:22,620 --> 00:31:26,400 negative. So we have to find or to compute the 385 00:31:26,400 --> 00:31:29,700 coefficient of correlation in order to see there 386 00:31:29,700 --> 00:31:32,940 exists positive, negative, strong, weak, or 387 00:31:32,940 --> 00:31:37,820 moderate. but again you can tell from this simple 388 00:31:37,820 --> 00:31:40,280 example yes there is a positive relationship 389 00:31:40,280 --> 00:31:44,660 because just if you pick random numbers here for 390 00:31:44,660 --> 00:31:49,240 example 64 father's height his son's height 65 if 391 00:31:49,240 --> 00:31:54,600 we move up here to 70 for father's height his 392 00:31:54,600 --> 00:32:00,160 son's height is going to be 72 so as father 393 00:32:00,160 --> 00:32:05,020 heights increases Also, son's height increases. 394 00:32:06,320 --> 00:32:11,700 For example, 77, father's height. His son's height 395 00:32:11,700 --> 00:32:15,160 is 76. So that means there exists positive 396 00:32:15,160 --> 00:32:19,740 relationship. Make sense? But again, for large 397 00:32:19,740 --> 00:32:20,780 data, you cannot tell that. 398 00:32:31,710 --> 00:32:36,090 If, again, by using this data, small data, you can 399 00:32:36,090 --> 00:32:40,730 determine just the length, the strength, I'm 400 00:32:40,730 --> 00:32:43,490 sorry, the strength of a relationship or the 401 00:32:43,490 --> 00:32:47,590 direction of the relationship. Just pick any 402 00:32:47,590 --> 00:32:51,030 number at random. For example, if we pick this 403 00:32:51,030 --> 00:32:51,290 number. 404 00:32:55,050 --> 00:33:00,180 Father's height is 68, his son's height is 70. Now 405 00:33:00,180 --> 00:33:02,180 suppose we pick another number that is greater 406 00:33:02,180 --> 00:33:05,840 than 68, then let's see what will happen. For 407 00:33:05,840 --> 00:33:11,060 father's height 70, his son's height increases up 408 00:33:11,060 --> 00:33:17,160 to 72. Similarly, 72 father's height, his son's 409 00:33:17,160 --> 00:33:22,060 height 75. So that means X increases, Y also 410 00:33:22,060 --> 00:33:25,740 increases. So that means there exists both of 411 00:33:25,740 --> 00:33:32,570 them. For sure it is hard to tell this direction 412 00:33:32,570 --> 00:33:36,130 if the data is large. Because maybe you will find 413 00:33:36,130 --> 00:33:40,250 as X increases for one point, Y maybe decreases 414 00:33:40,250 --> 00:33:43,610 for that point. So it depends on the data you 415 00:33:43,610 --> 00:33:49,010 have. Anyway, let's see how can we compute R. I 416 00:33:49,010 --> 00:33:53,770 will use Excel to show how can we do these 417 00:33:53,770 --> 00:33:54,550 calculations. 418 00:34:02,110 --> 00:34:06,530 The screen is clear. But give me the data of X and 419 00:34:06,530 --> 00:34:06,750 Y. 420 00:34:10,710 --> 00:34:14,310 X is 64. 68. 421 00:34:18,910 --> 00:34:26,830 68. 78. There is one 68. 78. 74. 422 00:34:31,120 --> 00:34:37,600 Seventy-four. Seventy-five. Seventy-six. 423 00:34:38,360 --> 00:34:42,240 Seventy-seven. Seventy-five. So that's the values 424 00:34:42,240 --> 00:34:49,440 of X, Y values. Seventy. Seventy-five. Seventy 425 00:34:49,440 --> 00:34:49,800 -seven. 426 00:35:17,230 --> 00:35:23,730 So first we have to compute it. x times y so 427 00:35:23,730 --> 00:35:28,270 that's as x times 428 00:35:28,270 --> 00:35:38,230 the value of y so 46 times 65 equals 4160 x 429 00:35:38,230 --> 00:35:46,050 squared so this value squared for y squared 65 430 00:35:48,660 --> 00:35:53,700 Square and we have to do this one for the rest of 431 00:35:53,700 --> 00:36:03,160 the data So 432 00:36:03,160 --> 00:36:07,600 that's the sum of XY sum X squared and Y squared 433 00:36:07,600 --> 00:36:13,480 now the summation So 434 00:36:13,480 --> 00:36:17,180 that's the sum of X and Y 435 00:36:20,380 --> 00:36:27,040 We have to compute the mean of x and y. So that is 436 00:36:27,040 --> 00:36:31,380 this sum divided by n, where n is 10 in this case. 437 00:36:34,600 --> 00:36:36,100 So this is the first step. 438 00:36:41,820 --> 00:36:48,900 Let's see how can we compute R. R, we have sum of 439 00:36:48,900 --> 00:37:00,780 x, y. minus n is 10 times x bar times y bar. This 440 00:37:00,780 --> 00:37:03,100 is the first quantity. The other one is square 441 00:37:03,100 --> 00:37:08,580 root of sum 442 00:37:08,580 --> 00:37:15,420 x squared minus n x bar squared. 443 00:37:18,830 --> 00:37:24,770 times some y squared minus n times y bar squared. 444 00:37:28,930 --> 00:37:34,090 And we have to find the square root of this value. 445 00:37:34,210 --> 00:37:40,810 So square root, that will give this result. So now 446 00:37:40,810 --> 00:37:46,990 R equals this value divided by 447 00:37:49,670 --> 00:37:54,890 155 and round always to two decimal places will 448 00:37:54,890 --> 00:38:05,590 give 87 so r is 87 so first step we have x and y 449 00:38:05,590 --> 00:38:12,470 compute xy x squared y squared sum of these all of 450 00:38:12,470 --> 00:38:18,100 these then x bar y bar values are given Then just 451 00:38:18,100 --> 00:38:20,820 use the formula you have, we'll get R to be at 452 00:38:20,820 --> 00:38:31,540 seven. So in this case, if we just go back to 453 00:38:31,540 --> 00:38:33,400 the slide we have here. 454 00:38:36,440 --> 00:38:41,380 As we mentioned, father's height is the 455 00:38:41,380 --> 00:38:45,640 explanatory variable. Son's height is the response 456 00:38:45,640 --> 00:38:46,060 variable. 457 00:38:49,190 --> 00:38:52,810 And that simple calculation gives summation of xi, 458 00:38:54,050 --> 00:38:57,810 summation of yi, summation x squared, y squared, 459 00:38:57,970 --> 00:39:02,690 and some xy. And finally, we'll get that result, 460 00:39:02,850 --> 00:39:07,850 87%. Now, the sign is positive. That means there 461 00:39:07,850 --> 00:39:13,960 exists positive. And 0.87 is close to 1. That 462 00:39:13,960 --> 00:39:17,320 means there exists strong positive relationship 463 00:39:17,320 --> 00:39:22,480 between father's and son's height. I think the 464 00:39:22,480 --> 00:39:25,060 calculation is straightforward. 465 00:39:27,280 --> 00:39:33,280 Now, for this example, the data are given in 466 00:39:33,280 --> 00:39:37,460 inches. I mean father's and son's height in inch. 467 00:39:38,730 --> 00:39:41,050 Suppose we want to convert from inch to 468 00:39:41,050 --> 00:39:44,750 centimeter, so we have to multiply by 2. Do you 469 00:39:44,750 --> 00:39:52,050 think in this case, R will change? So if we add or 470 00:39:52,050 --> 00:39:59,910 multiply or divide, R will not change? I mean, if 471 00:39:59,910 --> 00:40:06,880 we have X values, And we divide or multiply X, I 472 00:40:06,880 --> 00:40:09,460 mean each value of X, by a number, by a fixed 473 00:40:09,460 --> 00:40:12,600 value. For example, suppose here we multiplied 474 00:40:12,600 --> 00:40:19,460 each value by 2.5 for X. Also multiply Y by the 475 00:40:19,460 --> 00:40:24,520 same value, 2.5. Y will be the same. In addition 476 00:40:24,520 --> 00:40:28,920 to that, if we multiply X by 2.5, for example, and 477 00:40:28,920 --> 00:40:34,960 Y by 5, also R will not change. But you have to be 478 00:40:34,960 --> 00:40:39,400 careful. We multiply each value of x by the same 479 00:40:39,400 --> 00:40:45,700 number. And each value of y by the same number, 480 00:40:45,820 --> 00:40:49,640 that number may be different from x. So I mean 481 00:40:49,640 --> 00:40:56,540 multiply x by 2.5 and y by minus 1 or plus 2 or 482 00:40:56,540 --> 00:41:01,000 whatever you have. But if it's negative, then 483 00:41:01,000 --> 00:41:05,640 we'll get negative answer. I mean if Y is 484 00:41:05,640 --> 00:41:08,060 positive, for example, and we multiply each value 485 00:41:08,060 --> 00:41:13,000 Y by minus one, that will give negative sign. But 486 00:41:13,000 --> 00:41:17,640 here I meant if we multiply this value by positive 487 00:41:17,640 --> 00:41:21,320 sign, plus two, plus three, and let's see how can 488 00:41:21,320 --> 00:41:22,540 we do that by Excel. 489 00:41:26,320 --> 00:41:31,480 Now this is the data we have. I just make copy. 490 00:41:37,730 --> 00:41:45,190 I will multiply each value X by 2.5. And I will do 491 00:41:45,190 --> 00:41:49,590 the same for Y 492 00:41:49,590 --> 00:41:57,190 value. I will replace this data by the new one. 493 00:41:58,070 --> 00:42:00,410 For sure the calculations will, the computations 494 00:42:00,410 --> 00:42:09,740 here will change, but R will stay the same. So 495 00:42:09,740 --> 00:42:14,620 here we multiply each x by 2.5 and the same for y. 496 00:42:15,540 --> 00:42:19,400 The calculations here are different. We have 497 00:42:19,400 --> 00:42:22,960 different sum, different sum of x, sum of y and so 498 00:42:22,960 --> 00:42:31,040 on, but are the same. Let's see if we multiply 499 00:42:31,040 --> 00:42:38,880 just x by 2.5 and y the same. 500 00:42:41,840 --> 00:42:49,360 So we multiplied x by 2.5 and we keep it make 501 00:42:49,360 --> 00:42:57,840 sense? Now let's see how outliers will affect the 502 00:42:57,840 --> 00:43:03,260 value of R. Let's say if we change one point in 503 00:43:03,260 --> 00:43:08,480 the data set support. I just changed 64. 504 00:43:13,750 --> 00:43:24,350 for example if just by typo and just enter 6 so it 505 00:43:24,350 --> 00:43:33,510 was 87 it becomes 0.7 so there is a big difference 506 00:43:33,510 --> 00:43:38,670 between 0.87 and 0.7 and just we change one value 507 00:43:38,670 --> 00:43:45,920 now suppose the other one is zero 82. The other is 508 00:43:45,920 --> 00:43:48,260 2, for example. 1. 509 00:43:53,380 --> 00:43:59,200 I just changed half of the data. Now R was 87, it 510 00:43:59,200 --> 00:44:02,920 becomes 59. That means these outliers, these 511 00:44:02,920 --> 00:44:06,180 values for sure are outliers and they fit the 512 00:44:06,180 --> 00:44:07,060 correlation coefficient. 513 00:44:11,110 --> 00:44:14,970 Now let's see if we just change this 1 to be 200. 514 00:44:15,870 --> 00:44:20,430 It will go from 50 to up to 63. That means any 515 00:44:20,430 --> 00:44:26,010 changes in x or y values will change the y. But if 516 00:44:26,010 --> 00:44:30,070 we add or multiply all the values by a constant, r 517 00:44:30,070 --> 00:44:31,170 will stay the same. 518 00:44:35,250 --> 00:44:43,590 Any questions? That's the end of chapter 3. I will 519 00:44:43,590 --> 00:44:48,990 move quickly to the practice problems we have. And 520 00:44:48,990 --> 00:44:55,270 we posted the practice in the course website.