WEBVTT 1 00:00:01.410 --> 00:00:03.150 So, hello, students. 2 00:00:03.150 --> 00:00:07.680 We are going to now go to Problem 1B. 3 00:00:07.680 --> 00:00:09.900 And here, we are going to learn how to calculate 4 00:00:09.900 --> 00:00:11.280 the standard deviation. 5 00:00:11.280 --> 00:00:13.530 So again, first thing I suggest my students to do 6 00:00:13.530 --> 00:00:14.880 is write down the formula. 7 00:00:14.880 --> 00:00:18.660 So what is the formula to calculate the standard deviation? 8 00:00:18.660 --> 00:00:20.043 It is basically, 9 00:00:22.950 --> 00:00:24.990 looks a little intimidating, but it's not, 10 00:00:24.990 --> 00:00:29.103 and I'll show you how to do this in a very simple way. 11 00:00:30.810 --> 00:00:34.473 So, this is our formula to calculate standard deviation. 12 00:00:36.390 --> 00:00:39.000 The best way to do this is to go step by step, 13 00:00:39.000 --> 00:00:41.970 and I'll show you how you can even double check 14 00:00:41.970 --> 00:00:45.150 whether you're doing it correctly or not in the process. 15 00:00:45.150 --> 00:00:47.070 So, I'm going to put a table here. 16 00:00:47.070 --> 00:00:50.670 X sub i is basically our X values. 17 00:00:50.670 --> 00:00:55.200 The middle column here is X sub i minus X bar, 18 00:00:55.200 --> 00:01:00.000 which is our sample mean, which we have already calculated. 19 00:01:00.000 --> 00:01:04.030 And the third column is X sub i minus X bar 20 00:01:05.400 --> 00:01:06.233 and whole squared. 21 00:01:06.233 --> 00:01:09.123 So basically, we are going to whole square the difference. 22 00:01:10.500 --> 00:01:13.230 So again, here, we are going to write the numbers, 23 00:01:13.230 --> 00:01:15.810 the ordered set, which I showed you before, 24 00:01:15.810 --> 00:01:18.483 147, 175, 25 00:01:19.740 --> 00:01:22.593 180, 185, 26 00:01:24.750 --> 00:01:27.063 194, 196, 27 00:01:29.103 --> 00:01:30.543 223, 225, 28 00:01:34.992 --> 00:01:35.825 and 240. 29 00:01:38.430 --> 00:01:42.070 So, the middle column is basically the difference between 30 00:01:43.500 --> 00:01:45.120 this number and the mean. 31 00:01:45.120 --> 00:01:47.740 So for the first one, it will be 49.1. 32 00:01:56.970 --> 00:01:59.673 Second one is negative 21.1. 33 00:02:00.810 --> 00:02:04.323 Third one is 16.1, again, negative. 34 00:02:07.080 --> 00:02:10.143 Fourth one is minus 11.1. 35 00:02:14.400 --> 00:02:17.970 Fifth one is, again, negative 2.1. 36 00:02:17.970 --> 00:02:22.053 Sixth one is negative 0.1. 37 00:02:30.780 --> 00:02:32.900 Now this is going to become positive 26.9. 38 00:02:38.160 --> 00:02:40.740 And this will also be negative 28.9. 39 00:02:44.280 --> 00:02:47.130 And the last one is also gonna be a positive number, 40 00:02:47.130 --> 00:02:49.860 which is 43.9. 41 00:02:49.860 --> 00:02:51.090 Now what I want you to do 42 00:02:51.090 --> 00:02:54.060 is double check whether you are doing this correctly or not, 43 00:02:54.060 --> 00:02:57.450 is to add all the positive and negative values 44 00:02:57.450 --> 00:03:00.720 and see if you come up with zero. 45 00:03:00.720 --> 00:03:03.210 So we are gonna add these numbers, 46 00:03:03.210 --> 00:03:05.640 and add all of these numbers. 47 00:03:05.640 --> 00:03:09.427 So when we do that, what we come up with is negative 99.6 48 00:03:12.840 --> 00:03:16.620 and a positive 99.1. 49 00:03:16.620 --> 00:03:18.600 So, this is what? 50 00:03:18.600 --> 00:03:22.770 It still is a negative 0.5, which is very close to zero, 51 00:03:22.770 --> 00:03:26.340 and the only reason we are getting a negative 0.5 52 00:03:26.340 --> 00:03:27.990 is because we have a decimal here, 53 00:03:27.990 --> 00:03:29.280 if we had a whole number, 54 00:03:29.280 --> 00:03:31.440 this would end up actually being zero, 55 00:03:31.440 --> 00:03:35.250 and that tells you that you have done this correctly. 56 00:03:35.250 --> 00:03:37.800 Then the third column basically is 57 00:03:37.800 --> 00:03:40.350 to whole square the difference. 58 00:03:40.350 --> 00:03:43.473 And when we do that, these are the values we get. 59 00:04:21.158 --> 00:04:24.630 And again, anytime we whole square a negative number, 60 00:04:24.630 --> 00:04:26.313 it always becomes positive. 61 00:04:53.190 --> 00:04:54.390 So what we have done 62 00:04:54.390 --> 00:04:56.550 is we have taken the differences 63 00:04:56.550 --> 00:04:59.340 and we have whole squared them, 64 00:04:59.340 --> 00:05:04.200 which means multiplying the number by itself, 65 00:05:04.200 --> 00:05:06.933 and these are the values we have obtained. 66 00:05:08.114 --> 00:05:08.947 Okay. 67 00:05:09.900 --> 00:05:11.700 I already performed the calculation, 68 00:05:11.700 --> 00:05:14.100 so it would not take a humongous amount of time 69 00:05:14.100 --> 00:05:15.690 for us to go through this, 70 00:05:15.690 --> 00:05:17.790 but again, I strongly encourage you 71 00:05:17.790 --> 00:05:19.440 to do the calculation yourself, 72 00:05:19.440 --> 00:05:22.650 just to make sure you understand the step by step process. 73 00:05:22.650 --> 00:05:24.480 And of course, if you have any questions, 74 00:05:24.480 --> 00:05:25.560 please let me know. 75 00:05:25.560 --> 00:05:26.670 So what we are going to do 76 00:05:26.670 --> 00:05:29.190 is right now sum up the third column, 77 00:05:29.190 --> 00:05:31.493 and that will be 6,728.9. 78 00:05:36.330 --> 00:05:37.830 So as you can see right now, 79 00:05:37.830 --> 00:05:41.550 we have calculated our numerator, 80 00:05:41.550 --> 00:05:44.280 which is kind of the most, I would say, 81 00:05:44.280 --> 00:05:46.560 intimidating part of this formula. 82 00:05:46.560 --> 00:05:50.490 And the sample size is still what? 83 00:05:50.490 --> 00:05:51.323 Nine. 84 00:05:51.323 --> 00:05:54.360 So we deduct one from nine, we get eight. 85 00:05:54.360 --> 00:05:57.170 So we put that here, 6,728.9, 86 00:06:00.930 --> 00:06:02.733 we divide that by 8. 87 00:06:04.080 --> 00:06:04.913 We get, 88 00:06:13.860 --> 00:06:15.123 there's a decimal here, 89 00:06:16.830 --> 00:06:19.770 and when we do the square root, 90 00:06:19.770 --> 00:06:24.770 we get 29.0019. 91 00:06:24.810 --> 00:06:26.070 Given the decimals, 92 00:06:26.070 --> 00:06:30.120 the situation here, if we just round it up, 93 00:06:30.120 --> 00:06:31.620 we will get 29, 94 00:06:31.620 --> 00:06:36.620 because it is too far out to really use the decimals here, 95 00:06:37.110 --> 00:06:38.662 because it would be what? 96 00:06:38.662 --> 00:06:43.470 29.002, so we will just use 29. 97 00:06:43.470 --> 00:06:47.160 So, basically, 29 is our standard deviation. 98 00:06:47.160 --> 00:06:50.040 Again, as I say, it looks kind of intimidating 99 00:06:50.040 --> 00:06:51.420 at the beginning, but it's not. 100 00:06:51.420 --> 00:06:54.210 As long as you follow the step by step process, 101 00:06:54.210 --> 00:06:56.370 it is actually quite simple. 102 00:06:56.370 --> 00:06:57.570 So the first thing, again, 103 00:06:57.570 --> 00:07:00.120 is to create that ordered set 104 00:07:00.120 --> 00:07:03.090 and put all your ordered numbers here 105 00:07:03.090 --> 00:07:05.910 from smallest to largest, 106 00:07:05.910 --> 00:07:07.770 and then you create the second column 107 00:07:07.770 --> 00:07:11.250 where you are deducting your X value, 108 00:07:11.250 --> 00:07:13.650 basically your mean from your X value. 109 00:07:13.650 --> 00:07:15.480 And then what you want to do 110 00:07:15.480 --> 00:07:18.090 is calculate all the negative numbers, 111 00:07:18.090 --> 00:07:19.410 sum them up, 112 00:07:19.410 --> 00:07:22.380 sum up all your positive numbers and look at the difference. 113 00:07:22.380 --> 00:07:25.050 If the difference is zero, or very close to zero, 114 00:07:25.050 --> 00:07:28.560 you know you have done the right thing. 115 00:07:28.560 --> 00:07:31.170 And again, please keep in mind this will only work 116 00:07:31.170 --> 00:07:33.660 if you have the ordered set done properly. 117 00:07:33.660 --> 00:07:35.910 Okay? If you have not done the order set properly, 118 00:07:35.910 --> 00:07:37.290 it's going to be very difficult 119 00:07:37.290 --> 00:07:40.230 to sum up the negatives on the positives, 120 00:07:40.230 --> 00:07:41.340 not that you cannot do it, 121 00:07:41.340 --> 00:07:42.840 but it'll be much more challenging. 122 00:07:42.840 --> 00:07:46.950 So again, strongly recommend, please do an ordered set. 123 00:07:46.950 --> 00:07:49.170 And then, the third column is basically 124 00:07:49.170 --> 00:07:50.160 doing the whole square. 125 00:07:50.160 --> 00:07:53.340 And as you can see, I did all of those. 126 00:07:53.340 --> 00:07:57.000 Again, please feel free to let me know 127 00:07:57.000 --> 00:07:58.020 if you have any questions, 128 00:07:58.020 --> 00:08:00.570 I'm here to help, as I've said many times, 129 00:08:00.570 --> 00:08:03.480 and I really want you to understand the process. 130 00:08:03.480 --> 00:08:04.313 Okay? 131 00:08:04.313 --> 00:08:06.960 I know we can use statistical software packages, 132 00:08:06.960 --> 00:08:09.360 and they do calculate these things for us, 133 00:08:09.360 --> 00:08:10.320 but at the same time, 134 00:08:10.320 --> 00:08:13.890 we do need to understand the process ourself, how it works, 135 00:08:13.890 --> 00:08:17.010 so if the computer or the statistical software package 136 00:08:17.010 --> 00:08:20.437 gives us a weird answer, we will be like, 137 00:08:20.437 --> 00:08:21.480 "This is not feasible." 138 00:08:21.480 --> 00:08:24.000 So we do need to understand the process 139 00:08:24.000 --> 00:08:26.159 a little bit ourself, how it works. 140 00:08:26.159 --> 00:08:28.440 So, that's about it for 1B, 141 00:08:28.440 --> 00:08:31.113 and now we are going to move to 1C.