WEBVTT 00:00:00.000 --> 00:00:02.460 The following content is provided under a Creative 00:00:02.460 --> 00:00:03.740 Commons license. 00:00:03.740 --> 00:00:06.060 Your support will help MIT OpenCourseWare 00:00:06.060 --> 00:00:10.090 continue to offer high quality educational resources for free. 00:00:10.090 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.560 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.560 --> 00:00:17.904 at ocw.mit.edu. 00:00:22.070 --> 00:00:25.490 DUANE BONING: So let me begin by just giving you 00:00:25.490 --> 00:00:27.620 a little bit of a preview of what's 00:00:27.620 --> 00:00:29.450 coming up over the next couple of weeks 00:00:29.450 --> 00:00:32.210 in terms of the calendar. 00:00:32.210 --> 00:00:35.180 The problem set that's due today you'll notice 00:00:35.180 --> 00:00:37.520 is perhaps a little bit long. 00:00:37.520 --> 00:00:39.380 But the good news is the next problem 00:00:39.380 --> 00:00:42.170 set you'll look at and go, is that all? 00:00:42.170 --> 00:00:44.360 Because it will be a very short. 00:00:44.360 --> 00:00:47.780 Part of the reason is the problem set that goes out today 00:00:47.780 --> 00:00:51.120 will be due a week from today, next Tuesday. 00:00:51.120 --> 00:00:55.000 But a week from Thursday will be our first quiz. 00:00:55.000 --> 00:00:57.230 And so the upcoming problem set is 00:00:57.230 --> 00:01:02.120 meant to give you some time to go back and re-integrate 00:01:02.120 --> 00:01:06.110 and re-digest the material that we've covered so far. 00:01:06.110 --> 00:01:09.800 The material that is on this problem set that's going out, 00:01:09.800 --> 00:01:13.292 it's going to be a couple more, a few more problems still 00:01:13.292 --> 00:01:15.500 related to control charting, some of the things we'll 00:01:15.500 --> 00:01:16.940 talk about today. 00:01:16.940 --> 00:01:19.170 And that will be covered on the quiz. 00:01:19.170 --> 00:01:22.520 So the quiz will be statistical basics 00:01:22.520 --> 00:01:27.830 up through statistical process control and control charting. 00:01:27.830 --> 00:01:29.930 That'll be a week from Thursday. 00:01:29.930 --> 00:01:33.890 This Thursday I'll be giving a lecture that shifts 00:01:33.890 --> 00:01:35.780 a little bit to yield modeling. 00:01:35.780 --> 00:01:41.570 And that will not be on either this problem set or this quiz. 00:01:41.570 --> 00:01:45.740 So that's the plan. 00:01:45.740 --> 00:01:47.520 We'll have more details on the quiz 00:01:47.520 --> 00:01:51.110 and I think even some example quizzes that you 00:01:51.110 --> 00:01:54.050 can look at from the past. 00:01:54.050 --> 00:01:59.950 But other quick warnings or previews 00:01:59.950 --> 00:02:04.900 on the quiz itself is the plan is to make that closed book. 00:02:04.900 --> 00:02:07.750 But you will be able to bring a page of notes in. 00:02:07.750 --> 00:02:12.880 So you can have a single page of cheat sheet or notes, formulas, 00:02:12.880 --> 00:02:15.655 examples, whatever you can fit for quiz one 00:02:15.655 --> 00:02:20.200 on the one side of one page. 00:02:20.200 --> 00:02:22.720 And also bring a calculator. 00:02:22.720 --> 00:02:28.120 We will provide for you any statistics tables 00:02:28.120 --> 00:02:32.320 that might be needed, typically out of the back of Montgomery 00:02:32.320 --> 00:02:34.540 or [INAUDIBLE]. 00:02:34.540 --> 00:02:36.730 So we'll have those if you need them. 00:02:36.730 --> 00:02:37.510 Yes? 00:02:37.510 --> 00:02:41.662 AUDIENCE: So [INAUDIBLE] 00:02:41.662 --> 00:02:43.370 DUANE BONING: We're not giving you those. 00:02:43.370 --> 00:02:46.600 So those should go on your one page of notes 00:02:46.600 --> 00:02:48.390 that you can bring in. 00:02:48.390 --> 00:02:50.817 Or you can just remember them if you're really good. 00:02:50.817 --> 00:02:52.650 AUDIENCE: But you'll give us like a Z table? 00:02:52.650 --> 00:02:54.275 DUANE BONING: We'll give you a Z table. 00:02:54.275 --> 00:03:00.270 We'll give you, if you need it, t, Chi squared. 00:03:00.270 --> 00:03:01.817 We'll give you those tables. 00:03:01.817 --> 00:03:04.883 AUDIENCE: Is there a gamma table? 00:03:04.883 --> 00:03:06.800 DUANE BONING: I'm sure there is a gamma table. 00:03:06.800 --> 00:03:09.838 I'm not sure whether or not it's in the-- 00:03:09.838 --> 00:03:13.430 AUDIENCE: [INAUDIBLE] gamma of something-- 00:03:13.430 --> 00:03:14.970 DUANE BONING: It arises in-- 00:03:14.970 --> 00:03:17.066 AUDIENCE: [INAUDIBLE] 00:03:19.815 --> 00:03:21.440 DUANE BONING: And that might be a clue. 00:03:21.440 --> 00:03:26.375 If you don't have a gamma table, don't try to use the gamma. 00:03:26.375 --> 00:03:28.850 AUDIENCE: [INAUDIBLE] 00:03:28.850 --> 00:03:30.768 Can you put a sample exam so that we know-- 00:03:30.768 --> 00:03:31.560 DUANE BONING: Yeah. 00:03:31.560 --> 00:03:37.920 We'll try to dig up an example exam so you can take a look. 00:03:37.920 --> 00:03:39.060 OK. 00:03:39.060 --> 00:03:40.950 So any other questions on the quiz? 00:03:40.950 --> 00:03:45.180 Any questions from Singapore on the upcoming quiz? 00:03:45.180 --> 00:03:49.140 You guys look so eager for it. 00:03:49.140 --> 00:03:49.800 OK. 00:03:49.800 --> 00:03:52.140 So today what I want to do is pick up a little bit 00:03:52.140 --> 00:03:54.790 where Dave Hardt left off. 00:03:54.790 --> 00:03:58.590 So the plan here today will be a very brief review 00:03:58.590 --> 00:03:59.970 on control charting. 00:03:59.970 --> 00:04:02.880 The basic idea I want to be able to hit, now 00:04:02.880 --> 00:04:04.810 that you've seen what he described 00:04:04.810 --> 00:04:07.890 and what you've been doing hopefully on the problem 00:04:07.890 --> 00:04:10.050 set that's due at the end of the day, 00:04:10.050 --> 00:04:14.100 is reinforce the relationship here 00:04:14.100 --> 00:04:18.329 between hypothesis tests and control chart. 00:04:18.329 --> 00:04:20.730 And so I'm going to just remind us 00:04:20.730 --> 00:04:22.800 a little bit about hypothesis tests 00:04:22.800 --> 00:04:25.290 and these different types of errors. 00:04:25.290 --> 00:04:30.690 The probability of false alarm, probability of a misalarm, 00:04:30.690 --> 00:04:33.720 missed alarm or beta, and how the sample 00:04:33.720 --> 00:04:35.880 size plays with that. 00:04:35.880 --> 00:04:39.230 Because when you design control charts, 00:04:39.230 --> 00:04:44.240 alpha, recall, basically sets your upper control limits 00:04:44.240 --> 00:04:46.800 and lower control limits. 00:04:46.800 --> 00:04:52.880 Beta falls out, beta being related 00:04:52.880 --> 00:04:55.700 to essentially the probability, again, 00:04:55.700 --> 00:05:00.200 of missing an alarm when the process has, in fact, shifted. 00:05:00.200 --> 00:05:04.680 And how do you change or affect beta? 00:05:04.680 --> 00:05:06.430 By modifying the sample size. 00:05:06.430 --> 00:05:08.310 So I want to talk a little bit about those. 00:05:08.310 --> 00:05:11.320 I think you might have touched on them briefly. 00:05:11.320 --> 00:05:15.950 But I want to touch on those and also 00:05:15.950 --> 00:05:17.600 remind us of the definition or give you 00:05:17.600 --> 00:05:19.830 the definition of average run length. 00:05:19.830 --> 00:05:22.820 And I think if you've gotten to that point in the problem set, 00:05:22.820 --> 00:05:26.900 you've already read about that as well. 00:05:26.900 --> 00:05:34.740 So let me start, again, with the basic hypothesis test. 00:05:34.740 --> 00:05:38.760 Because remember, all that a control chart is 00:05:38.760 --> 00:05:42.000 is a running hypothesis test where every new sample comes 00:05:42.000 --> 00:05:45.360 in, you're doing a hypothesis or you're 00:05:45.360 --> 00:05:47.190 asking the test on the hypothesis, 00:05:47.190 --> 00:05:56.040 has the process shifted or is it still my nominal distribution? 00:05:56.040 --> 00:06:03.880 So the basic situation with the hypothesis test, if you recall, 00:06:03.880 --> 00:06:09.800 what we have is our assumed distribution. 00:06:09.800 --> 00:06:14.240 I'm going to assume here that it's a Gaussian. 00:06:14.240 --> 00:06:16.570 We'll call that h0. 00:06:16.570 --> 00:06:20.470 It's got some mu, some true process mean. 00:06:20.470 --> 00:06:24.280 And what we said is we pick control limits, 00:06:24.280 --> 00:06:28.600 or in the hypothesis test case acceptance regions. 00:06:28.600 --> 00:06:32.110 But I'm going to use the control chart terminology, since that's 00:06:32.110 --> 00:06:33.670 what we're dealing with here. 00:06:33.670 --> 00:06:36.490 We pick where these limits are based 00:06:36.490 --> 00:06:41.870 on a probability of false alarm. 00:06:41.870 --> 00:06:43.460 Seems a little weird, right? 00:06:43.460 --> 00:06:48.440 But basically what we do is we assign 00:06:48.440 --> 00:06:56.950 if alpha is our false alarm rate, we put alpha over 2 00:06:56.950 --> 00:06:58.360 in each of the two tails. 00:06:58.360 --> 00:07:03.480 Now, that's based on this key idea 00:07:03.480 --> 00:07:06.720 that if we're in control, most of the time 00:07:06.720 --> 00:07:08.670 the data will be in here. 00:07:08.670 --> 00:07:10.440 Every now and then, just by random 00:07:10.440 --> 00:07:14.730 chance alone, even when nothing has changed, we're still in h0, 00:07:14.730 --> 00:07:18.590 I do get an unusual point out in the tails. 00:07:18.590 --> 00:07:26.120 And if we do 3 sigma, we saw that's about 3 in 1,000 points 00:07:26.120 --> 00:07:28.545 will fall way outside in the tails. 00:07:28.545 --> 00:07:30.170 So when we get a point out in the tail, 00:07:30.170 --> 00:07:35.350 we say something might be unusual going on. 00:07:35.350 --> 00:07:39.550 Either something has changed in the process or I got unlucky 00:07:39.550 --> 00:07:42.680 and I just pulled a sample from the tails. 00:07:42.680 --> 00:07:45.260 So in fact, there is, in a process 00:07:45.260 --> 00:07:48.700 that's always in control, there is this false alarm rate. 00:07:48.700 --> 00:07:53.890 Now, another way of talking about this false alarm rate 00:07:53.890 --> 00:07:57.460 is also the inverse of the false alarm 00:07:57.460 --> 00:08:02.020 rate, which is how many runs does it take on average 00:08:02.020 --> 00:08:03.730 before you get a false alarm? 00:08:03.730 --> 00:08:08.560 And this is referred to as the average run length. 00:08:08.560 --> 00:08:15.190 So here sliding back to the slides. 00:08:15.190 --> 00:08:18.380 Can we put the PowerPoint slides on both screens? 00:08:18.380 --> 00:08:18.880 Great. 00:08:18.880 --> 00:08:19.713 Thank you very much. 00:08:22.270 --> 00:08:26.260 So looking back here, this is our nominal probability. 00:08:26.260 --> 00:08:28.270 Again, 3 out of 1,000. 00:08:28.270 --> 00:08:33.580 But now if I turn it around and ask the question, how often 00:08:33.580 --> 00:08:39.700 or what is the average number of runs that we have to go? 00:08:39.700 --> 00:08:43.690 Well, it's just 1 over that probability 00:08:43.690 --> 00:08:50.680 on any one run of having the point fall in the tails. 00:08:50.680 --> 00:08:53.920 So that's the definition of our average run length. 00:08:53.920 --> 00:08:58.090 And actually, a notation that I like in Montgomery 00:08:58.090 --> 00:09:01.900 is we'll define this as average run length sub-zero. 00:09:01.900 --> 00:09:05.170 What that does is remind me that that's the average run 00:09:05.170 --> 00:09:09.850 length associated with the h0, the default probability 00:09:09.850 --> 00:09:11.006 distribution. 00:09:13.680 --> 00:09:15.120 So that's the alpha part. 00:09:15.120 --> 00:09:17.360 Where does beta come in? 00:09:17.360 --> 00:09:21.550 Beta comes in when we actually have a shift. 00:09:21.550 --> 00:09:26.500 So the whole idea is now imagine that I've got 00:09:26.500 --> 00:09:30.160 a shift in the distribution. 00:09:30.160 --> 00:09:35.215 I've had a shift from mu to mu plus some delta. 00:09:38.550 --> 00:09:41.060 I'm going to label that the alternative hypothesis. 00:09:41.060 --> 00:09:43.170 So the idea is for a given delta, 00:09:43.170 --> 00:09:46.860 I'm now actually drawing parts from a different distribution, 00:09:46.860 --> 00:09:50.030 my h1 distribution. 00:09:50.030 --> 00:09:52.250 I've already defined the acceptance region 00:09:52.250 --> 00:09:54.800 based on my acceptable average run 00:09:54.800 --> 00:09:58.570 length or my acceptable false alarm rate. 00:09:58.570 --> 00:10:01.620 By the way, there's nothing magic about plus minus 3 sigma. 00:10:01.620 --> 00:10:06.030 If you don't want to respond once out of three times out 00:10:06.030 --> 00:10:11.340 of every 1,000 where about once every 333 runs, 00:10:11.340 --> 00:10:13.560 you don't want to respond to a false alarm rate, 00:10:13.560 --> 00:10:19.530 it doesn't cost you that much to have a false alarm or it costs 00:10:19.530 --> 00:10:22.260 a lot to have a false alarm, but it doesn't cost you 00:10:22.260 --> 00:10:25.290 that much to have a bad part, then maybe you only want 00:10:25.290 --> 00:10:28.200 to respond 1/10 as much. 00:10:28.200 --> 00:10:30.780 In which case, you would change your control limits, 00:10:30.780 --> 00:10:35.360 move them out a little bit so that you set it 00:10:35.360 --> 00:10:37.490 based on the average-- 00:10:37.490 --> 00:10:41.270 set based on the average run length or the probability 00:10:41.270 --> 00:10:43.530 that you can accept out of the tail. 00:10:43.530 --> 00:10:44.780 So that's set. 00:10:44.780 --> 00:10:48.830 But now we said there is also the type two 00:10:48.830 --> 00:10:59.210 error or the beta error, which is the probability of a missed 00:10:59.210 --> 00:11:00.020 alarm. 00:11:00.020 --> 00:11:04.490 And that's set by those control limits. 00:11:04.490 --> 00:11:07.490 So now the probability that I would 00:11:07.490 --> 00:11:12.710 have thought it was a good part even though an alarm occurred 00:11:12.710 --> 00:11:15.590 is just the other side of that same control limit 00:11:15.590 --> 00:11:19.670 but now drawn from the h1 distribution. 00:11:19.670 --> 00:11:23.570 I really have to shift, but the point 00:11:23.570 --> 00:11:26.760 fell in this part of the tail of the h1 distribution. 00:11:26.760 --> 00:11:28.280 And I declared it good. 00:11:28.280 --> 00:11:31.820 I said there's no alarm. 00:11:31.820 --> 00:11:37.640 And we saw last time basically, or a couple of classes 00:11:37.640 --> 00:11:42.050 ago, this is also it's just the tail there 00:11:42.050 --> 00:11:43.070 in that distribution. 00:11:43.070 --> 00:11:53.960 It's UCL minus delta over sigma over square root of n. 00:11:53.960 --> 00:11:56.510 By the way, this is just this part of the tail. 00:11:56.510 --> 00:12:00.200 To be really careful, I should also subtract off 00:12:00.200 --> 00:12:03.770 this tail, this part right there, 00:12:03.770 --> 00:12:06.020 because I got something way out there, 00:12:06.020 --> 00:12:08.600 I would call it a failed part. 00:12:08.600 --> 00:12:12.480 And so my beta is slightly smaller than this. 00:12:12.480 --> 00:12:14.930 But as I've drawn it here, I'm really only worried 00:12:14.930 --> 00:12:16.460 about this part of the tail. 00:12:16.460 --> 00:12:19.280 But there is the other part of a lower control 00:12:19.280 --> 00:12:23.240 limit plus delta minus. 00:12:23.240 --> 00:12:23.740 Excuse me. 00:12:23.740 --> 00:12:25.390 It's still a lower control limit still 00:12:25.390 --> 00:12:30.250 minus delta over single route in. 00:12:30.250 --> 00:12:32.300 OK. 00:12:32.300 --> 00:12:32.900 Great. 00:12:32.900 --> 00:12:37.300 There is also another definition of average run 00:12:37.300 --> 00:12:41.310 length associated with this. 00:12:41.310 --> 00:12:43.500 And the question you would pose here 00:12:43.500 --> 00:12:51.660 is what is the average run length to detect shift 00:12:51.660 --> 00:12:53.100 when it actually occurred? 00:13:03.170 --> 00:13:05.440 It's a little tricky here, because there's actually 00:13:05.440 --> 00:13:06.700 two steps. 00:13:06.700 --> 00:13:13.080 You can imagine this would be the probability of not 00:13:13.080 --> 00:13:16.050 missing an alarm and then inverting it, 00:13:16.050 --> 00:13:19.440 just like we did here. 00:13:19.440 --> 00:13:23.220 So this little tail here is the probability of missing alarm. 00:13:23.220 --> 00:13:25.740 There is also a definition referred 00:13:25.740 --> 00:13:31.530 to as the power of a test, which is 1 minus beta. 00:13:34.290 --> 00:13:36.690 And that's basically the probability 00:13:36.690 --> 00:13:39.060 of catching an alarm on any one run. 00:13:39.060 --> 00:13:43.060 And that's just, it's everything to the right of this. 00:13:43.060 --> 00:13:47.310 And now if I draw from h1 but not in the region 00:13:47.310 --> 00:13:51.630 where I would have declared it h0, that's 1 minus beta. 00:13:51.630 --> 00:14:03.000 An average run length sub 1 is defined as 1 over 1 minus beta. 00:14:03.000 --> 00:14:05.210 So this is the average number of runs 00:14:05.210 --> 00:14:08.900 it takes to detect the shift assuming it actually occurred. 00:14:11.470 --> 00:14:12.200 OK? 00:14:12.200 --> 00:14:14.660 Everybody happy with this so far? 00:14:14.660 --> 00:14:17.100 Is it starting to fit together? 00:14:17.100 --> 00:14:20.850 Now how do I change-- 00:14:20.850 --> 00:14:23.390 well, why would I change alpha? 00:14:23.390 --> 00:14:26.390 I change alpha based on a decision 00:14:26.390 --> 00:14:31.730 outside on acceptance of my acceptability of making 00:14:31.730 --> 00:14:32.690 a false alarm. 00:14:32.690 --> 00:14:33.800 How do I change beta? 00:14:36.770 --> 00:14:41.990 What if I say it's taking me too long to detect a shift? 00:14:41.990 --> 00:14:45.490 I want it to take less time. 00:14:45.490 --> 00:14:46.270 Change n. 00:14:46.270 --> 00:14:49.390 So what changes when I change n? 00:14:49.390 --> 00:14:51.250 Does alpha change when I change n? 00:14:54.350 --> 00:14:56.790 No. 00:14:56.790 --> 00:15:00.390 Alpha is set a priori based on my false alarm rate. 00:15:00.390 --> 00:15:02.460 When I change beta-- 00:15:02.460 --> 00:15:04.980 or when I change n, I do change beta. 00:15:04.980 --> 00:15:06.540 Do I change the control limits? 00:15:11.240 --> 00:15:12.740 I do change the control limits. 00:15:12.740 --> 00:15:17.550 I change the control limits because my sample size changed 00:15:17.550 --> 00:15:19.990 and I move them to maintain the same alpha. 00:15:19.990 --> 00:15:23.410 So let me draw the picture. 00:15:23.410 --> 00:15:25.150 Let's see. 00:15:25.150 --> 00:15:29.540 If I draw down here, can people in Singapore still see this? 00:15:29.540 --> 00:15:30.800 OK, great. 00:15:30.800 --> 00:15:34.760 So if I increase n, what's going to happen 00:15:34.760 --> 00:15:39.170 is I have a sample size that tightens by my sampling 00:15:39.170 --> 00:15:42.860 distribution for x bar. 00:15:42.860 --> 00:15:45.680 Remember, h0 is really an x bar based on. 00:15:45.680 --> 00:15:49.070 And so that distribution will have the same mean, 00:15:49.070 --> 00:15:50.910 but it will get tighter. 00:15:50.910 --> 00:15:52.670 So now I have an h0. 00:15:52.670 --> 00:15:56.180 But I'm still setting my control limits 00:15:56.180 --> 00:15:57.785 based on an acceptable alpha. 00:16:03.230 --> 00:16:05.510 But the control limits will get tighter, 00:16:05.510 --> 00:16:08.960 because I've got a tighter distribution. 00:16:08.960 --> 00:16:11.840 I still have the same alpha in the tail. 00:16:11.840 --> 00:16:13.460 And what's cool here is I'm still 00:16:13.460 --> 00:16:17.240 detecting the same mean shift. 00:16:17.240 --> 00:16:18.560 Trying to draw down here. 00:16:18.560 --> 00:16:20.360 Boom, boom, boom. 00:16:20.360 --> 00:16:25.650 So now I've got a tighter distribution here as well. 00:16:25.650 --> 00:16:27.050 Here's my h1. 00:16:27.050 --> 00:16:29.690 I've got a tighter distribution on h1 00:16:29.690 --> 00:16:32.420 and I've still got just a little bit of tiny stuff 00:16:32.420 --> 00:16:34.100 over here on the tail. 00:16:34.100 --> 00:16:36.530 And that's now my beta, and it's much smaller 00:16:36.530 --> 00:16:37.760 if I've increased n. 00:16:41.260 --> 00:16:45.160 So I can basically pick my sample size 00:16:45.160 --> 00:16:48.580 to give me the acceptable false-- 00:16:48.580 --> 00:16:52.510 or the acceptable missed alarm rate or the acceptable average 00:16:52.510 --> 00:16:56.080 run length to detect the shift. 00:16:56.080 --> 00:16:59.650 And there are tables in both Montgomery 00:16:59.650 --> 00:17:02.890 and [INAUDIBLE] that are referred 00:17:02.890 --> 00:17:08.950 to as operating curves or OC's that basically give you 00:17:08.950 --> 00:17:15.400 the relationship between sample size, size of shift, 00:17:15.400 --> 00:17:22.000 and the beta or average run length one that you can accept. 00:17:22.000 --> 00:17:23.770 You could go in and calculate them. 00:17:23.770 --> 00:17:30.550 Turns out this is actually a really nasty calculation to do, 00:17:30.550 --> 00:17:32.800 because there's no close-- 00:17:32.800 --> 00:17:35.860 if you need to find the n to give you a certain beta, 00:17:35.860 --> 00:17:37.750 you have to solve it iteratively. 00:17:37.750 --> 00:17:40.450 There's no closed form for that. 00:17:40.450 --> 00:17:44.290 Q sum are the cumulative probability 00:17:44.290 --> 00:17:47.080 that's associated with beta. 00:17:47.080 --> 00:17:49.580 So there's no formula for it. 00:17:49.580 --> 00:17:51.610 And these tables, basically, or charts 00:17:51.610 --> 00:17:55.330 have basically calculated that for you. 00:17:55.330 --> 00:17:58.300 You can set up Excel or whatever to do that, 00:17:58.300 --> 00:18:01.090 to solve it iteratively if you want. 00:18:01.090 --> 00:18:01.670 Yeah? 00:18:01.670 --> 00:18:04.770 AUDIENCE: [INAUDIBLE] 00:18:13.420 --> 00:18:15.400 DUANE BONING: So the question was, 00:18:15.400 --> 00:18:18.820 we're changing alpha or changing sample size. 00:18:18.820 --> 00:18:20.290 The distribution gets tighter. 00:18:20.290 --> 00:18:20.905 Why doesn't-- 00:18:24.380 --> 00:18:26.140 AUDIENCE: Yeah, why doesn't alpha? 00:18:26.140 --> 00:18:29.770 DUANE BONING: Why doesn't alpha change? 00:18:29.770 --> 00:18:33.460 If we kept the same lower control limit and upper control 00:18:33.460 --> 00:18:36.070 limit and I tightened this. 00:18:36.070 --> 00:18:39.340 So if I kept these the same but the distribution tightened, 00:18:39.340 --> 00:18:41.620 then alpha would change. 00:18:41.620 --> 00:18:45.400 What's interesting about classic control chart 00:18:45.400 --> 00:18:49.240 design is the upper control limit and lower control limit 00:18:49.240 --> 00:18:51.320 are based on the statistics. 00:18:51.320 --> 00:18:54.070 So they are based on alpha. 00:18:54.070 --> 00:19:00.650 We nailed down alpha first and then set the upper 00:19:00.650 --> 00:19:02.570 and lower control limits. 00:19:02.570 --> 00:19:04.070 And there's a really important thing 00:19:04.070 --> 00:19:07.730 that we'll get to, actually, in just a second about process 00:19:07.730 --> 00:19:13.400 capability where then we start talking about upper 00:19:13.400 --> 00:19:21.710 and lower spec limits, which is related to the notion of where 00:19:21.710 --> 00:19:25.070 the parts property really needs to be 00:19:25.070 --> 00:19:27.540 for it to function correctly. 00:19:27.540 --> 00:19:30.800 And those are very different than control limits 00:19:30.800 --> 00:19:32.750 but often confused. 00:19:32.750 --> 00:19:35.870 And so nominally, your spec limits 00:19:35.870 --> 00:19:40.130 wouldn't actually change if you're changing sample size. 00:19:40.130 --> 00:19:43.100 The part still needs to function correctly 00:19:43.100 --> 00:19:45.980 within a certain band. 00:19:48.510 --> 00:19:51.930 So in that case, you change sample size. 00:19:51.930 --> 00:19:53.640 Your spec limits would not. 00:19:53.640 --> 00:19:57.510 But what people often do is erroneously 00:19:57.510 --> 00:20:00.360 set the control limits to equal the spec limits. 00:20:00.360 --> 00:20:02.280 And that's wrong. 00:20:02.280 --> 00:20:05.190 What you want to do is set the control limits based purely 00:20:05.190 --> 00:20:08.850 on the statistics based on your false alarm rate. 00:20:08.850 --> 00:20:14.070 Chance of observing by chance or by random probability, 00:20:14.070 --> 00:20:17.102 an unusual event. 00:20:17.102 --> 00:20:20.752 AUDIENCE: [INAUDIBLE] 00:20:22.050 --> 00:20:24.090 For example, in the problem set, there 00:20:24.090 --> 00:20:26.082 was a question [INAUDIBLE]. 00:20:40.253 --> 00:20:41.670 DUANE BONING: I guess I don't want 00:20:41.670 --> 00:20:44.820 to say too much about the problem set right now. 00:20:44.820 --> 00:20:51.570 But you can often relate things back to the underlying 00:20:51.570 --> 00:20:52.570 probabilities. 00:20:52.570 --> 00:20:54.780 And what you should usually do is relate them 00:20:54.780 --> 00:20:56.700 back to the underlying probabilities 00:20:56.700 --> 00:21:02.250 and set up the hypothesis test based on shifts, 00:21:02.250 --> 00:21:07.560 questions about there can be two sided shifts, one sided shifts. 00:21:07.560 --> 00:21:09.550 Both things could shift by a certain amount, 00:21:09.550 --> 00:21:11.258 and it might be just worried about what's 00:21:11.258 --> 00:21:12.960 happening in the tail. 00:21:12.960 --> 00:21:14.590 There's other hypotheses. 00:21:14.590 --> 00:21:18.180 These are all for changes in the mean. 00:21:18.180 --> 00:21:21.690 If I'm looking at some other distribution, 00:21:21.690 --> 00:21:24.840 like changes in the variance, I could set up 00:21:24.840 --> 00:21:27.840 a similar hypothesis test and perhaps other probability 00:21:27.840 --> 00:21:30.120 distributions apply. 00:21:30.120 --> 00:21:32.520 Turns out with the S chart, we're 00:21:32.520 --> 00:21:36.900 really tracking the mean of standard deviation 00:21:36.900 --> 00:21:39.640 or the mean of the sample standard deviation. 00:21:39.640 --> 00:21:42.000 And in those cases, we typically still 00:21:42.000 --> 00:21:44.370 are assuming that the distribution 00:21:44.370 --> 00:21:47.700 of standard deviation for large samples 00:21:47.700 --> 00:21:51.570 is Gaussian within correction factors to make 00:21:51.570 --> 00:21:54.750 it come close in setting the control limits. 00:21:54.750 --> 00:21:59.650 Things like C4's and those sorts of things. 00:21:59.650 --> 00:22:04.080 But going back, it's not always exactly the same set up. 00:22:04.080 --> 00:22:06.420 But you can almost always reason it out 00:22:06.420 --> 00:22:09.060 by going back and setting up what 00:22:09.060 --> 00:22:12.540 is my normal situation, what is the thing I'm trying to detect, 00:22:12.540 --> 00:22:14.970 and then look at the probabilities associated 00:22:14.970 --> 00:22:17.515 with those cases. 00:22:17.515 --> 00:22:19.140 AUDIENCE: I have a question, Professor. 00:22:19.140 --> 00:22:20.307 DUANE BONING: Yes, question. 00:22:22.350 --> 00:22:24.420 AUDIENCE: If I give you a control chart, 00:22:24.420 --> 00:22:28.770 you can't really tell whether it's a false alarm 00:22:28.770 --> 00:22:32.032 or it's a true alarm, can you? 00:22:32.032 --> 00:22:33.240 DUANE BONING: That's correct. 00:22:37.660 --> 00:22:39.820 AUDIENCE: Like how do you use a control chart then? 00:22:39.820 --> 00:22:41.620 Because if I gave a control chart, 00:22:41.620 --> 00:22:45.270 it could be with alpha probability of a false alarm 00:22:45.270 --> 00:22:50.720 or with 1 minus beta probability of a true alarm. 00:22:50.720 --> 00:22:52.720 [INAUDIBLE] 00:22:52.720 --> 00:22:57.280 DUANE BONING: This is a fundamental important point 00:22:57.280 --> 00:23:01.030 about how control charting works, which goes back 00:23:01.030 --> 00:23:04.000 to the idea that if nothing changes, you do 00:23:04.000 --> 00:23:05.530 have false alarms. 00:23:05.530 --> 00:23:08.890 And if something changes, you've got a real alarm. 00:23:08.890 --> 00:23:10.390 All you've got on the control chart 00:23:10.390 --> 00:23:14.860 is a point that's lying outside of your normal bounds. 00:23:14.860 --> 00:23:20.090 All by itself from the control chart, you can't tell. 00:23:20.090 --> 00:23:22.080 You are exactly right. 00:23:22.080 --> 00:23:26.090 So in SPC methodology, what you are supposed 00:23:26.090 --> 00:23:30.110 to do any time you see an alarm is 00:23:30.110 --> 00:23:33.590 go and look at the equipment, investigate the problem, 00:23:33.590 --> 00:23:36.800 and decide was this a false alarm? 00:23:36.800 --> 00:23:39.800 In other words, is the process still operating normally? 00:23:39.800 --> 00:23:43.710 Or has something important changed? 00:23:43.710 --> 00:23:48.330 Now, it turns out that in an awful lot of factories, 00:23:48.330 --> 00:23:54.460 people have adapted to this notion of false alarm 00:23:54.460 --> 00:23:59.290 in a very qualitative way, which is they see one alarm point. 00:23:59.290 --> 00:24:00.400 They look at it. 00:24:00.400 --> 00:24:01.870 They don't investigate the process. 00:24:01.870 --> 00:24:04.330 They ignore it. 00:24:04.330 --> 00:24:05.530 And then they wait. 00:24:05.530 --> 00:24:07.340 They do the next point. 00:24:07.340 --> 00:24:09.670 If it's also an alarm, then they say, 00:24:09.670 --> 00:24:12.160 ah, I think something may really be wrong 00:24:12.160 --> 00:24:14.400 and then they investigate. 00:24:14.400 --> 00:24:16.315 What are they doing there? 00:24:16.315 --> 00:24:17.940 How could you change your control chart 00:24:17.940 --> 00:24:19.590 to actually do what they are doing? 00:24:24.010 --> 00:24:26.920 What essentially they're doing is saying, 00:24:26.920 --> 00:24:30.670 I don't believe the evidence. 00:24:30.670 --> 00:24:35.710 The 3 in 1,000 chance of seeing one point outside of a control 00:24:35.710 --> 00:24:39.520 limit is not a strong enough evidence for me. 00:24:39.520 --> 00:24:41.020 What they're really doing is looking 00:24:41.020 --> 00:24:47.470 for a smaller probability of two points in a row 00:24:47.470 --> 00:24:49.930 being outside of the control chart 00:24:49.930 --> 00:24:52.130 as much stronger evidence. 00:24:52.130 --> 00:24:56.380 So in this case, roughly it's twice as infrequent. 00:24:56.380 --> 00:24:58.990 Or not twice, but the product. 00:25:03.370 --> 00:25:05.890 The probability of seeing both of those points outside 00:25:05.890 --> 00:25:09.460 of the control limit, and again, not alarming on the first one 00:25:09.460 --> 00:25:12.400 is the square of that. 00:25:12.400 --> 00:25:14.770 So that's a very infrequent thing. 00:25:14.770 --> 00:25:17.740 Then they say I think there's really something wrong. 00:25:17.740 --> 00:25:21.230 Now, you could actually have adapted your control chart 00:25:21.230 --> 00:25:25.490 either by having it only signal alarms 00:25:25.490 --> 00:25:30.170 once you've got two alarm points or changing the control 00:25:30.170 --> 00:25:35.310 limits so that you basically move the control limits out 00:25:35.310 --> 00:25:36.580 even further. 00:25:36.580 --> 00:25:41.370 So you've got to really have an alpha over 2 that's that small. 00:25:41.370 --> 00:25:45.960 So maybe you go to 4, plus or minus 4 sigma or 4.5 sigma, 00:25:45.960 --> 00:25:52.665 something like that, in order to have that much evidence. 00:25:52.665 --> 00:25:56.930 AUDIENCE: What steps are efficient to find [INAUDIBLE]?? 00:25:56.930 --> 00:25:58.160 DUANE BONING: I'm sorry? 00:25:58.160 --> 00:25:59.635 What steps? 00:25:59.635 --> 00:26:00.260 AUDIENCE: Yeah. 00:26:00.260 --> 00:26:01.885 There is a huge machine, and everything 00:26:01.885 --> 00:26:03.980 can be somewhat changed. 00:26:03.980 --> 00:26:05.590 I don't know where I should start. 00:26:05.590 --> 00:26:07.760 DUANE BONING: Well, I think the steps 00:26:07.760 --> 00:26:09.320 that you use to actually investigate 00:26:09.320 --> 00:26:11.870 depend completely on the process, 00:26:11.870 --> 00:26:14.090 but there's a few rules of thumb. 00:26:14.090 --> 00:26:18.080 Many of you have probably had actual process experience 00:26:18.080 --> 00:26:20.360 on different tools, different manufacturing lines. 00:26:20.360 --> 00:26:22.250 What are some things that you would do? 00:26:22.250 --> 00:26:25.520 I'll tell you the number one thing I would do. 00:26:25.520 --> 00:26:27.290 AUDIENCE: Check the screen? 00:26:27.290 --> 00:26:28.700 DUANE BONING: Look at the screen. 00:26:28.700 --> 00:26:29.570 Pull the power plug! 00:26:29.570 --> 00:26:30.290 Plug it back in! 00:26:30.290 --> 00:26:30.790 No. 00:26:33.090 --> 00:26:33.916 Yes? 00:26:33.916 --> 00:26:34.790 AUDIENCE: Check the measurement? 00:26:34.790 --> 00:26:35.540 DUANE BONING: Yes. 00:26:35.540 --> 00:26:37.290 That's exactly what I would do. 00:26:37.290 --> 00:26:40.350 I would always doubt my measurement equipment first. 00:26:40.350 --> 00:26:42.050 They also have failures. 00:26:42.050 --> 00:26:45.230 Rather than pull the process, turn the process off 00:26:45.230 --> 00:26:48.290 and panic with the actual equipment, 00:26:48.290 --> 00:26:51.420 I'd want to verify the measurement first. 00:26:51.420 --> 00:26:54.980 So I take it off, do an independent measurement 00:26:54.980 --> 00:26:57.240 of those parts, and try to understand it. 00:26:57.240 --> 00:27:02.720 So let's say that indeed I see, yes, that part looks like it's 00:27:02.720 --> 00:27:04.880 outside of the control limits. 00:27:04.880 --> 00:27:06.377 What's another thing it might do? 00:27:09.059 --> 00:27:11.790 Do you look just at that one single point? 00:27:11.790 --> 00:27:13.593 What other data do you have? 00:27:13.593 --> 00:27:17.043 AUDIENCE: You can look at the [INAUDIBLE].. 00:27:17.043 --> 00:27:17.960 DUANE BONING: Exactly. 00:27:17.960 --> 00:27:20.190 So now you can look back at the control chart 00:27:20.190 --> 00:27:23.370 data that has been plotted either manually in old systems 00:27:23.370 --> 00:27:25.740 or in the computer system. 00:27:25.740 --> 00:27:28.020 Track back historically and start to look, 00:27:28.020 --> 00:27:32.790 is there some other evidence that wasn't triggering an alarm 00:27:32.790 --> 00:27:35.050 but that might give you a sense? 00:27:35.050 --> 00:27:39.300 And a telltale one might be, in fact, a slow drift 00:27:39.300 --> 00:27:40.620 in the points. 00:27:40.620 --> 00:27:43.190 Now, some people might set up the WECO rules, 00:27:43.190 --> 00:27:46.930 the Western Electric rules, to also watch for drifts. 00:27:46.930 --> 00:27:49.230 But that's pretty rare, frankly. 00:27:49.230 --> 00:27:52.950 People tend to just use the upper and lower control 00:27:52.950 --> 00:27:57.010 limits in a single point outside of them. 00:27:57.010 --> 00:28:00.210 So you might look back at the rest of the data 00:28:00.210 --> 00:28:02.970 and try to see is there any additional evidence. 00:28:02.970 --> 00:28:06.173 AUDIENCE: The trends can be incorporated into [INAUDIBLE].. 00:28:06.173 --> 00:28:07.090 DUANE BONING: Exactly. 00:28:07.090 --> 00:28:09.090 So we could also have-- 00:28:09.090 --> 00:28:12.330 the point was trends could be incorporated in the control 00:28:12.330 --> 00:28:15.780 chart as well, either with this and additional rules 00:28:15.780 --> 00:28:19.220 or, in fact, other control charting 00:28:19.220 --> 00:28:22.310 approaches we'll talk about later that 00:28:22.310 --> 00:28:28.040 are designed to track or detect very subtle trends. 00:28:28.040 --> 00:28:30.800 But again, the point would be often 00:28:30.800 --> 00:28:34.730 you are trying to debug the process and look back. 00:28:34.730 --> 00:28:38.000 So you are a good pattern detector, 00:28:38.000 --> 00:28:41.420 and you might detect a pattern. 00:28:44.030 --> 00:28:46.070 Failing that, I think then you start 00:28:46.070 --> 00:28:53.240 looking around to see if maybe there's not a systematic trend. 00:28:53.240 --> 00:28:55.940 What you're hypothesizing, in fact, 00:28:55.940 --> 00:28:58.280 is that there was a shift, not a trend. 00:28:58.280 --> 00:29:02.110 And you start talking to people, looking at the equipment, 00:29:02.110 --> 00:29:03.680 talking with the operators, and ask, 00:29:03.680 --> 00:29:05.570 did something unusual happen? 00:29:05.570 --> 00:29:08.960 The whole idea is you've detected an unusual event. 00:29:08.960 --> 00:29:14.150 Might that be correlated with something else in the operation 00:29:14.150 --> 00:29:15.440 of the machine changing? 00:29:15.440 --> 00:29:17.780 So you start investigating and looking for things 00:29:17.780 --> 00:29:22.250 that might explain a shift. 00:29:22.250 --> 00:29:23.720 So you go to talk to the operator, 00:29:23.720 --> 00:29:25.460 and it's a new operator. 00:29:25.460 --> 00:29:28.150 You've never seen that operator before. 00:29:28.150 --> 00:29:29.210 Say when did you start? 00:29:29.210 --> 00:29:30.815 They say, an hour ago. 00:29:30.815 --> 00:29:31.315 Aha. 00:29:34.250 --> 00:29:37.017 And then it gets progressively more complicated. 00:29:37.017 --> 00:29:38.600 You're going to have to dive into more 00:29:38.600 --> 00:29:41.480 details on either the process, the feedstock, all 00:29:41.480 --> 00:29:46.100 of those sources of variability that we talked about earlier. 00:29:46.100 --> 00:29:48.150 Good questions. 00:29:48.150 --> 00:29:50.090 OK. 00:29:50.090 --> 00:29:53.185 So I've sort of flown by these. 00:29:53.185 --> 00:29:54.560 These are actually slides, again, 00:29:54.560 --> 00:30:00.820 that were pulled out of last Thursday's lecture 00:30:00.820 --> 00:30:03.370 but just illustrate this notion of 00:30:03.370 --> 00:30:07.510 and give some example numbers of exactly what we're 00:30:07.510 --> 00:30:10.210 talking about here with average run length. 00:30:10.210 --> 00:30:14.590 The example plotted here is the same chart but drawn 00:30:14.590 --> 00:30:18.370 a little bit differently for kind of smaller shifts. 00:30:18.370 --> 00:30:24.790 Here we have a delta u that moves the probability 00:30:24.790 --> 00:30:27.440 under the assumption of a shift. 00:30:27.440 --> 00:30:31.970 And this pe down here is basically 1 minus beta. 00:30:31.970 --> 00:30:35.380 So this is the probability of detecting 00:30:35.380 --> 00:30:42.640 the true shift under the control limits again 00:30:42.640 --> 00:30:46.310 that were set based on h0. 00:30:46.310 --> 00:30:48.700 And so here is an example where, again, the false alarm 00:30:48.700 --> 00:30:54.550 rate was 3 out of 1,000 with this small one sigma shift. 00:30:54.550 --> 00:30:58.480 Now your tail probability there only becomes 24 out 00:30:58.480 --> 00:31:01.780 of 1,000 or an average run length, 00:31:01.780 --> 00:31:06.100 if I invert that, of about 42 samples 00:31:06.100 --> 00:31:11.980 are required before you can detect that small one sigma 00:31:11.980 --> 00:31:12.550 shift. 00:31:12.550 --> 00:31:16.450 Now, one sigma shift doesn't seem all that small to me. 00:31:16.450 --> 00:31:23.910 If that's too long to detect a shift, then what do you do? 00:31:26.580 --> 00:31:27.630 Increase sample size. 00:31:27.630 --> 00:31:28.530 That's exactly right. 00:31:32.730 --> 00:31:34.830 And here is simply the definition. 00:31:34.830 --> 00:31:38.640 What's going on there if we increase the sample size is 00:31:38.640 --> 00:31:44.280 the standard deviation, again, of our charts, our x bar chart. 00:31:44.280 --> 00:31:48.690 h0 and h1 are shrinking. 00:31:48.690 --> 00:31:56.260 That moves our three sigma limits closer together 00:31:56.260 --> 00:31:58.810 so that you get smaller probabilities. 00:31:58.810 --> 00:31:59.566 Yeah? 00:31:59.566 --> 00:32:01.190 AUDIENCE: Even if we increase the n, 00:32:01.190 --> 00:32:07.290 still we have to get more samples [INAUDIBLE].. 00:32:07.290 --> 00:32:08.040 DUANE BONING: Yes. 00:32:08.040 --> 00:32:13.020 So the question was to increase, then, 00:32:13.020 --> 00:32:15.160 you got to take more samples. 00:32:15.160 --> 00:32:18.180 So there is a little bit of this two level 00:32:18.180 --> 00:32:21.120 hierarchy in sampling assumed here 00:32:21.120 --> 00:32:22.350 when I'm talking about this. 00:32:22.350 --> 00:32:28.560 We'll also talk about ways to deal with more continuously 00:32:28.560 --> 00:32:31.530 sampled, even driving in smaller, not larger, 00:32:31.530 --> 00:32:34.710 so that you can improve detectability. 00:32:34.710 --> 00:32:38.970 But the two level hierarchy that I'm talking about here 00:32:38.970 --> 00:32:46.200 is basically this notion that in many lines, 00:32:46.200 --> 00:32:49.690 you've got lots and lots of parts coming off very quickly. 00:32:49.690 --> 00:32:54.540 So in time, you've got parts coming very rapidly. 00:32:54.540 --> 00:32:56.730 And the basic picture here is you're 00:32:56.730 --> 00:33:01.230 grabbing a sample of size n at some point in time. 00:33:01.230 --> 00:33:03.870 Maybe it's a four point sample. 00:33:03.870 --> 00:33:08.150 And then much later, I'm grabbing another four. 00:33:08.150 --> 00:33:11.200 So I am sampling from a what's pictured 00:33:11.200 --> 00:33:13.770 as a much larger group. 00:33:13.770 --> 00:33:18.420 And so the idea here is I actually 00:33:18.420 --> 00:33:21.060 have more than one degree of freedom. 00:33:21.060 --> 00:33:24.030 We've been talking about I can change just the sample size, 00:33:24.030 --> 00:33:28.080 but I might still in time only be sampling-- 00:33:31.020 --> 00:33:33.880 I could have actually the same time period. 00:33:33.880 --> 00:33:36.090 So I'm not actually changing if I just 00:33:36.090 --> 00:33:38.070 modify my sample size here. 00:33:38.070 --> 00:33:42.000 I might actually not change the frequency at which 00:33:42.000 --> 00:33:45.840 I draw my size and sample. 00:33:45.840 --> 00:33:48.810 I might do this once an hour. 00:33:48.810 --> 00:33:51.390 And once every hour, I draw a sample of size four. 00:33:55.640 --> 00:33:57.980 So there's an additional degree of freedom 00:33:57.980 --> 00:34:02.840 that's not in here, which is I could also shorten my sample 00:34:02.840 --> 00:34:05.190 time, if you will. 00:34:05.190 --> 00:34:11.159 Because then it takes fewer minutes of clock time 00:34:11.159 --> 00:34:16.239 to get to my 42 runs that I sample. 00:34:16.239 --> 00:34:18.960 So I haven't really talked about sort of the trade off 00:34:18.960 --> 00:34:20.130 between these two. 00:34:20.130 --> 00:34:22.469 But this is the assumed model. 00:34:22.469 --> 00:34:24.600 And you could also do things, and I 00:34:24.600 --> 00:34:28.260 think what was kind of implicit in your question, 00:34:28.260 --> 00:34:31.409 is I could continuously sample everything. 00:34:31.409 --> 00:34:34.260 And these days of automated inspection equipment 00:34:34.260 --> 00:34:37.770 on the equipment, on the manufacturing equipment, 00:34:37.770 --> 00:34:41.670 you're very often sampling and measuring virtually every part. 00:34:41.670 --> 00:34:48.000 In which case, it's sort of like every part gets measured, 00:34:48.000 --> 00:34:50.639 in which case I have a very interesting trade 00:34:50.639 --> 00:34:55.300 off between n, which is exactly the point you were making. 00:34:55.300 --> 00:34:59.340 If I increase n, then it takes me 00:34:59.340 --> 00:35:05.930 longer to get more runs that would form each point 00:35:05.930 --> 00:35:06.870 on the control chart. 00:35:06.870 --> 00:35:08.787 So there is a little bit of a trade off there. 00:35:11.340 --> 00:35:13.200 Is that clear? 00:35:13.200 --> 00:35:14.050 Almost? 00:35:14.050 --> 00:35:14.970 AUDIENCE: Almost. 00:35:14.970 --> 00:35:18.240 Because why we don't simple increase-- 00:35:18.240 --> 00:35:22.180 because [INAUDIBLE] 00:35:24.270 --> 00:35:28.350 So why we just don't increase the coefficient? 00:35:28.350 --> 00:35:31.350 So it's more sensitive to any change. 00:35:31.350 --> 00:35:34.090 Why we should increase the x? 00:35:34.090 --> 00:35:35.340 DUANE BONING: Well, let's see. 00:35:35.340 --> 00:35:38.490 If you increase, go from 3 sigma to 5 sigma, 00:35:38.490 --> 00:35:42.270 you increase the control limit and you make it less sensitive. 00:35:42.270 --> 00:35:45.070 You have to have even more extreme low points. 00:35:45.070 --> 00:35:49.590 So in that case, you decrease the sensitivity of the chart 00:35:49.590 --> 00:35:52.630 to false alarms. 00:35:52.630 --> 00:35:55.110 So in that case, you would want to-- 00:35:57.630 --> 00:36:00.390 so alpha and beta are different. 00:36:00.390 --> 00:36:03.360 When you play with them a little bit on the problem set, 00:36:03.360 --> 00:36:05.580 it'll reinforce, I think, a little bit of this. 00:36:11.160 --> 00:36:17.320 So again, this is basically the example we just talked about. 00:36:17.320 --> 00:36:17.820 OK. 00:36:17.820 --> 00:36:21.270 So I want to shift to kind of this point 00:36:21.270 --> 00:36:24.510 that I was making earlier about the difference between control 00:36:24.510 --> 00:36:27.390 limits and spec limits. 00:36:27.390 --> 00:36:32.130 But I'm going to get to it through this notion of process 00:36:32.130 --> 00:36:34.260 capability. 00:36:34.260 --> 00:36:38.820 And CP and CPK, if some of you have run into that 00:36:38.820 --> 00:36:42.270 or played with that yet on the problem set. 00:36:42.270 --> 00:36:43.950 The point here is that once we've 00:36:43.950 --> 00:36:49.200 got a distribution like this, based on historical data 00:36:49.200 --> 00:36:51.960 that we use to build our control chart, 00:36:51.960 --> 00:36:55.750 we basically have a model of our process. 00:36:55.750 --> 00:36:57.670 It's a very simple statistical model 00:36:57.670 --> 00:37:00.020 that says it has a particular mean 00:37:00.020 --> 00:37:02.920 and it has a particular variance in the normal distribution 00:37:02.920 --> 00:37:04.000 case. 00:37:04.000 --> 00:37:06.820 And an important thing that we can do with that model 00:37:06.820 --> 00:37:09.100 is answer a couple of key questions 00:37:09.100 --> 00:37:11.320 that really come up a lot. 00:37:11.320 --> 00:37:12.325 How good is the process? 00:37:14.900 --> 00:37:16.280 Is the mean where we want it? 00:37:16.280 --> 00:37:20.170 Is the variance where we want it? 00:37:20.170 --> 00:37:22.390 And in particular, how good is it 00:37:22.390 --> 00:37:26.800 with respect to our requirements for manufacturing 00:37:26.800 --> 00:37:30.220 a particular part or product? 00:37:30.220 --> 00:37:32.720 This is the notion of process capability. 00:37:32.720 --> 00:37:36.580 And what this also pulls in, then, for the first time, 00:37:36.580 --> 00:37:39.430 unlike all of this, which is just probabilities 00:37:39.430 --> 00:37:44.701 on a normally operating or a not normally operating process, 00:37:44.701 --> 00:37:47.620 process capability pulls in for the first time 00:37:47.620 --> 00:37:52.410 the notion of design specifications. 00:37:52.410 --> 00:37:55.710 Different than control limits. 00:37:55.710 --> 00:37:58.140 So the basic way to characterize your process 00:37:58.140 --> 00:38:01.920 is you have to assure yourself that the process is indeed 00:38:01.920 --> 00:38:04.030 in control. 00:38:04.030 --> 00:38:05.490 It's operating normally. 00:38:05.490 --> 00:38:09.630 No common cause events. 00:38:09.630 --> 00:38:13.080 When that's the case for a normally distributed process, 00:38:13.080 --> 00:38:17.430 it's characterized entirely by mean and variance 00:38:17.430 --> 00:38:20.490 or on our control chart by x bar and s. 00:38:20.490 --> 00:38:24.210 So indeed, if the control limits have been set correctly 00:38:24.210 --> 00:38:28.290 in relation to known sample sizes 00:38:28.290 --> 00:38:31.410 and known probabilities of error and so on, 00:38:31.410 --> 00:38:33.300 then x bar and s tells you everything 00:38:33.300 --> 00:38:38.070 about the underlying process mean and variance. 00:38:38.070 --> 00:38:39.900 And now we can start to use those 00:38:39.900 --> 00:38:43.140 to look at where the process sits 00:38:43.140 --> 00:38:45.240 and those statistics sit with respect 00:38:45.240 --> 00:38:49.700 to tolerances or quality loss functions. 00:38:49.700 --> 00:38:52.460 Quality loss functions may be a little less familiar to you. 00:38:52.460 --> 00:38:55.670 Everybody has an intuitive idea of tolerance. 00:38:55.670 --> 00:38:57.590 So let's talk about that first. 00:38:57.590 --> 00:39:00.350 Basic idea of a tolerance on, say, a part measurement 00:39:00.350 --> 00:39:02.710 is that there is an upper and lower limit. 00:39:02.710 --> 00:39:05.690 So let's say it's some characteristic dimension. 00:39:05.690 --> 00:39:09.050 We have a nominal dimension or a target dimension. 00:39:09.050 --> 00:39:11.000 I'll label that x star. 00:39:11.000 --> 00:39:14.270 And then we have an upper spec limit and a lower spec limit 00:39:14.270 --> 00:39:17.600 that we simply say, as long as the part 00:39:17.600 --> 00:39:20.360 dimension is within that range, we're 00:39:20.360 --> 00:39:22.280 going to call it a good part. 00:39:22.280 --> 00:39:25.850 And it meets maybe the design requirements. 00:39:25.850 --> 00:39:30.380 It meets our consumer's requirements, what have you. 00:39:30.380 --> 00:39:33.410 And what we're saying is in essence, 00:39:33.410 --> 00:39:36.440 anything, anywhere within those spec limits 00:39:36.440 --> 00:39:41.420 is equally good and equally shippable. 00:39:41.420 --> 00:39:44.150 So there's no notion of better or worse. 00:39:44.150 --> 00:39:49.160 It's simply on off decision, it's acceptable or not. 00:39:49.160 --> 00:39:51.440 So that's a spec limit or not. 00:39:51.440 --> 00:39:53.840 Now, the idea of a quality loss function 00:39:53.840 --> 00:39:57.830 is a beautiful idea that actually 00:39:57.830 --> 00:40:02.780 conforms a little bit better to what we really care about. 00:40:02.780 --> 00:40:05.150 And one could think of a quality loss function 00:40:05.150 --> 00:40:09.470 as this penalty function for any deviation 00:40:09.470 --> 00:40:12.710 away from our target x star. 00:40:12.710 --> 00:40:15.560 Now graphically, a typical quality loss function 00:40:15.560 --> 00:40:22.610 that is used is a quadratic cost for these deviations. 00:40:22.610 --> 00:40:30.670 Where the further I get away from x bar, 00:40:30.670 --> 00:40:32.200 I penalize more and more. 00:40:32.200 --> 00:40:33.790 Not just even in a linear fashion 00:40:33.790 --> 00:40:35.470 but in a square fashion. 00:40:35.470 --> 00:40:41.090 And then I have some normalized or some constant out front. 00:40:41.090 --> 00:40:43.840 And what this basically says is you really 00:40:43.840 --> 00:40:46.120 want to be on target. 00:40:46.120 --> 00:40:49.270 So as many of your parts are right close to the target, 00:40:49.270 --> 00:40:52.210 the lower your loss function is. 00:40:52.210 --> 00:40:54.910 And then the further away you get, 00:40:54.910 --> 00:40:56.590 the worse and worse things are. 00:40:56.590 --> 00:41:01.160 So this really drives you to do two things. 00:41:01.160 --> 00:41:02.990 To minimize quality loss function, 00:41:02.990 --> 00:41:05.660 what do you need to do? 00:41:05.660 --> 00:41:08.870 Want to target your process, get it means centered. 00:41:08.870 --> 00:41:11.600 And second, you want to reduce the deviation, 00:41:11.600 --> 00:41:14.550 reduce the variance in your process. 00:41:14.550 --> 00:41:16.640 So with one quality loss function, 00:41:16.640 --> 00:41:21.420 you're really driving the process towards both goals. 00:41:21.420 --> 00:41:26.960 Now, there are questions about how do you calibrate this? 00:41:26.960 --> 00:41:30.380 And there's some rules of thumb, for example, people 00:41:30.380 --> 00:41:35.160 take in upper and lower spec limits, 00:41:35.160 --> 00:41:42.440 you can start to sort of relate these to some cost associated 00:41:42.440 --> 00:41:47.480 with if a part actually goes out of spec and either 00:41:47.480 --> 00:41:52.130 you shift it or not, what is the cost of throwing away 00:41:52.130 --> 00:41:52.910 that part? 00:41:52.910 --> 00:41:55.310 That might give you a point on the curve 00:41:55.310 --> 00:41:59.360 and establish an actual dollar value, for example, 00:41:59.360 --> 00:42:01.340 right at those spec limits. 00:42:01.340 --> 00:42:07.520 But it actually isn't that important what the number is. 00:42:07.520 --> 00:42:10.910 It's really that you've got this quadratic dependence which 00:42:10.910 --> 00:42:18.680 drives you towards minimizing both being off target 00:42:18.680 --> 00:42:20.480 and the variance. 00:42:20.480 --> 00:42:23.690 So here's an example on the use of tolerances. 00:42:23.690 --> 00:42:25.463 And now pulling in both our distributions 00:42:25.463 --> 00:42:26.255 and control charts. 00:42:29.930 --> 00:42:34.600 So here we're assuming the process is normal. 00:42:34.600 --> 00:42:40.000 And what I've done here is drawn, first off, 00:42:40.000 --> 00:42:46.720 our in control process with its mean 00:42:46.720 --> 00:42:50.140 and its plus minus 3 sigma points. 00:42:50.140 --> 00:42:54.550 Now, notice here I've superimposed x star. 00:42:54.550 --> 00:42:56.830 So this is the desired target. 00:42:56.830 --> 00:43:00.100 And notice we're off target a little bit. 00:43:00.100 --> 00:43:02.860 There's a deviation between where our process mean really 00:43:02.860 --> 00:43:04.750 sits and our target. 00:43:04.750 --> 00:43:07.405 And I've also drawn these spec limits. 00:43:10.480 --> 00:43:15.160 So now you can start to evaluate various questions like, well, 00:43:15.160 --> 00:43:18.670 what's the probability not of a false alarm, 00:43:18.670 --> 00:43:23.980 but the probability of a nonconforming part? 00:43:23.980 --> 00:43:24.830 And that's easy. 00:43:24.830 --> 00:43:28.490 You're just looking at the tails. 00:43:28.490 --> 00:43:30.460 So I can actually go in and say, OK, 00:43:30.460 --> 00:43:33.070 the probability of a nonconforming part 00:43:33.070 --> 00:43:36.730 up on this side is just that tail to the right 00:43:36.730 --> 00:43:41.190 based on the actual performance I need. 00:43:41.190 --> 00:43:46.620 You can also start to ask questions of, well, 00:43:46.620 --> 00:43:49.920 how does my probability of getting nonconforming parts 00:43:49.920 --> 00:43:54.025 change as a function of how far off target I am? 00:43:54.025 --> 00:43:55.650 That's just shifting the thing and it's 00:43:55.650 --> 00:43:57.250 moving those tails around. 00:43:57.250 --> 00:43:59.220 So you can evaluate both of these. 00:43:59.220 --> 00:44:03.660 And essentially, a very nice, canonical way 00:44:03.660 --> 00:44:12.570 of talking about how far apart your spec limits 00:44:12.570 --> 00:44:14.580 are compared to your distribution, 00:44:14.580 --> 00:44:18.090 how good your underlying process is 00:44:18.090 --> 00:44:20.370 compared to your spec limits. 00:44:20.370 --> 00:44:24.300 Is this notion of process capability, or CP, and then 00:44:24.300 --> 00:44:26.530 CPK, as we'll see in a minute, also 00:44:26.530 --> 00:44:30.430 pulls in the idea of the mean difference. 00:44:30.430 --> 00:44:35.700 So first off, definition of CP or process capability ratio 00:44:35.700 --> 00:44:40.670 is simply the distance between your upper and spec limits 00:44:40.670 --> 00:44:46.260 and six sigma of your underlying process. 00:44:46.260 --> 00:44:49.860 Now, you start to hear the six sigma kinds of terminology. 00:44:49.860 --> 00:44:51.913 Yeah? 00:44:51.913 --> 00:44:52.830 AUDIENCE: [INAUDIBLE]. 00:44:52.830 --> 00:44:55.605 For example, if you have 30 or 40 steps, 00:44:55.605 --> 00:44:59.580 and each unit [INAUDIBLE] has its own [INAUDIBLE],, 00:44:59.580 --> 00:45:04.570 is there a system [INAUDIBLE]? 00:45:04.570 --> 00:45:08.465 And the second question was, for [INAUDIBLE],, for example, 00:45:08.465 --> 00:45:13.920 or for [INAUDIBLE],, you have thousands of parameters, 00:45:13.920 --> 00:45:16.170 each of which has a spec. 00:45:16.170 --> 00:45:20.910 So what's the quality loss function? 00:45:20.910 --> 00:45:27.420 Each of them is at some point [INAUDIBLE] 00:45:27.420 --> 00:45:31.680 all of them as something [INAUDIBLE].. 00:45:31.680 --> 00:45:33.687 Otherwise, you're not s 00:45:33.687 --> 00:45:36.270 DUANE BONING: So you're asking, really, I think two questions. 00:45:36.270 --> 00:45:38.520 I'll give you a really quick answer right now and then 00:45:38.520 --> 00:45:40.600 we'll defer some of those. 00:45:40.600 --> 00:45:44.010 So your first question is, for example, in semiconductor 00:45:44.010 --> 00:45:46.380 manufacturing or in many processes, 00:45:46.380 --> 00:45:48.750 we have multiple process steps. 00:45:48.750 --> 00:45:50.670 And what you can often do is have 00:45:50.670 --> 00:45:53.700 control charts for each individual one of those. 00:45:53.700 --> 00:45:56.010 You can talk about the process capability 00:45:56.010 --> 00:45:57.910 of each one of those. 00:45:57.910 --> 00:46:01.770 And yes, indeed, there is also very often 00:46:01.770 --> 00:46:03.570 the question about sort of quality roll 00:46:03.570 --> 00:46:10.530 up of how quality loss or deviations either add 00:46:10.530 --> 00:46:16.720 up or compound themselves across multiple steps. 00:46:16.720 --> 00:46:18.070 So we'll talk about that. 00:46:18.070 --> 00:46:21.030 But the first quick answer there is 00:46:21.030 --> 00:46:24.730 if each of those process steps are independent, 00:46:24.730 --> 00:46:29.400 then it's simply an additive function. 00:46:29.400 --> 00:46:32.820 Where it gets interesting is if those two process steps are not 00:46:32.820 --> 00:46:33.580 independent. 00:46:33.580 --> 00:46:35.580 So for example, the second process step 00:46:35.580 --> 00:46:38.770 may magnify or shrink or, in fact, 00:46:38.770 --> 00:46:42.000 even be designed to compensate for deviations 00:46:42.000 --> 00:46:44.010 in the first process step, in which case 00:46:44.010 --> 00:46:47.140 there is intentional correlation between the two. 00:46:47.140 --> 00:46:49.710 So it's not always a trivial question 00:46:49.710 --> 00:46:54.420 of how multiple steps interact to aggregate. 00:46:54.420 --> 00:46:56.560 The second question is a little different, 00:46:56.560 --> 00:47:02.040 which is especially in today's highly instrumented processes, 00:47:02.040 --> 00:47:05.730 there are a zillion parameters that you can measure. 00:47:05.730 --> 00:47:07.680 There's a zillion control charts. 00:47:07.680 --> 00:47:09.720 And what we've been talking about so far 00:47:09.720 --> 00:47:12.840 is a single parameter control chart. 00:47:12.840 --> 00:47:14.790 And there's some very important corrections 00:47:14.790 --> 00:47:19.650 one has to make when you've got many, many control charts 00:47:19.650 --> 00:47:21.870 and many, many parameters. 00:47:21.870 --> 00:47:24.120 And just to give you the intuition, 00:47:24.120 --> 00:47:28.680 if the probability for a single control chart with plus 00:47:28.680 --> 00:47:36.450 minus 3 sigma control limits of a false alarm is 3 out of 1,000 00:47:36.450 --> 00:47:40.860 or an average run length of 333 and you have 1,000 00:47:40.860 --> 00:47:44.220 independent control charts, how often do you think you're 00:47:44.220 --> 00:47:47.050 going to see a false alarm? 00:47:47.050 --> 00:47:49.270 All the time. 00:47:49.270 --> 00:47:52.990 So the quick answer there is you actually 00:47:52.990 --> 00:47:57.070 have to think about an aggregate false alarm rate, not 00:47:57.070 --> 00:47:59.500 an individual alpha on every chart. 00:47:59.500 --> 00:48:01.480 And you really should make an adjustment. 00:48:01.480 --> 00:48:03.340 Basically spread your control limits 00:48:03.340 --> 00:48:06.940 on every one of your independent individual charts 00:48:06.940 --> 00:48:10.510 much wider so that you don't drive the engineer 00:48:10.510 --> 00:48:13.580 nuts chasing false alarms. 00:48:13.580 --> 00:48:17.950 So we'll, I think, talk about multivariate situations 00:48:17.950 --> 00:48:20.170 also a little bit later. 00:48:20.170 --> 00:48:22.880 Very good questions. 00:48:22.880 --> 00:48:25.280 So the key idea here on the process capability 00:48:25.280 --> 00:48:27.950 is simply talking about the spread 00:48:27.950 --> 00:48:30.170 in your process compared to your spec limits 00:48:30.170 --> 00:48:32.090 and how good that is. 00:48:32.090 --> 00:48:37.190 Notice that if I'm off target, this definition CP 00:48:37.190 --> 00:48:38.580 doesn't change. 00:48:38.580 --> 00:48:41.060 This is purely talking about the inherent variance, 00:48:41.060 --> 00:48:43.280 the inherent spread in your process 00:48:43.280 --> 00:48:45.650 compared to your spec limits. 00:48:45.650 --> 00:48:50.400 Your inherent six sigma spread compared to your spec limits. 00:48:50.400 --> 00:48:55.260 If you also want to worry about mean, and you often do, 00:48:55.260 --> 00:48:58.800 another measure is the CPK. 00:48:58.800 --> 00:49:02.490 And what that looks at is the shortest distance either 00:49:02.490 --> 00:49:05.880 on the right or the left of the chart between your spec limit 00:49:05.880 --> 00:49:07.500 and your mean. 00:49:07.500 --> 00:49:10.530 How many sigma apart is your mean 00:49:10.530 --> 00:49:12.360 from one of the spec limits? 00:49:12.360 --> 00:49:18.240 And divide that by 3 sigma, 3 sigma on either side. 00:49:18.240 --> 00:49:25.745 And so now you're penalized for being offset from your target. 00:49:25.745 --> 00:49:27.120 So let's look at this graphically 00:49:27.120 --> 00:49:29.250 and get a little bit of a feel here. 00:49:29.250 --> 00:49:31.180 And here I'm just using tolerance limits. 00:49:31.180 --> 00:49:34.480 I'm not talking quality loss at this point. 00:49:34.480 --> 00:49:37.200 CP, CPK are actually just defined 00:49:37.200 --> 00:49:41.330 for pure plus minus tolerance limits. 00:49:41.330 --> 00:49:46.830 So what's drawn here is a set of spec limits that go-- 00:49:46.830 --> 00:49:49.620 these are sigma markings. 00:49:49.620 --> 00:49:56.040 So if I look at, first off, CP and CPK, my lower spec limit, 00:49:56.040 --> 00:50:00.420 upper spec limit, those are six sigma apart 00:50:00.420 --> 00:50:03.330 compared to the variance of my underlying process. 00:50:03.330 --> 00:50:05.550 That's just a CP of 1. 00:50:05.550 --> 00:50:09.360 So qualitatively, CP of 1 you should mesh in your mind 00:50:09.360 --> 00:50:14.250 just says my spread is such that my plus minus 3 sigma tails 00:50:14.250 --> 00:50:21.050 are right at the spec limits if I'm perfectly centered. 00:50:21.050 --> 00:50:23.600 And as pictured here, I am perfectly centered. 00:50:23.600 --> 00:50:26.030 So my CPK is also 1. 00:50:26.030 --> 00:50:29.270 In other words, my minimum distance between mu 00:50:29.270 --> 00:50:32.750 and my upper spec limit or mu and my lower spec limit 00:50:32.750 --> 00:50:35.750 is both three sigma. 00:50:35.750 --> 00:50:39.800 So the minimum of those two is still 1. 00:50:39.800 --> 00:50:43.490 So my CPK is 1. 00:50:43.490 --> 00:50:46.610 Now let's consider a slightly different picture. 00:50:46.610 --> 00:50:53.530 Here I have my spec limits different from my-- 00:50:53.530 --> 00:50:58.610 or basically my x bar or x star here and my mu 00:50:58.610 --> 00:50:59.810 are very different. 00:50:59.810 --> 00:51:05.090 My process spread is exactly the same as before. 00:51:05.090 --> 00:51:08.360 I still have an upper spec limit minus lower spec limit 00:51:08.360 --> 00:51:10.010 equal to 6 sigma. 00:51:10.010 --> 00:51:14.900 So my CP, in some sense, my inherent capability 00:51:14.900 --> 00:51:17.570 of the process in terms of spread 00:51:17.570 --> 00:51:20.180 is still exactly the same. 00:51:20.180 --> 00:51:22.940 But now because I'm off centered, 00:51:22.940 --> 00:51:28.790 the distance from mu to the lower spec limit is 0. 00:51:28.790 --> 00:51:32.840 So my CPK is 0 in this case. 00:51:32.840 --> 00:51:36.500 So qualitatively, these are very different. 00:51:36.500 --> 00:51:39.230 Here's another situation. 00:51:39.230 --> 00:51:41.310 What's changed here? 00:51:41.310 --> 00:51:44.150 Now in this case, my spread is much smaller. 00:51:44.150 --> 00:51:46.250 I've got a tighter distribution. 00:51:46.250 --> 00:51:49.010 And in fact, my upper spec limit to lower spec limit 00:51:49.010 --> 00:51:50.570 is basically 12 sigma. 00:51:53.180 --> 00:51:55.970 That's great, right? 00:51:55.970 --> 00:51:58.350 So that's a CP of 2. 00:51:58.350 --> 00:52:00.620 Now, I'm still off centered. 00:52:00.620 --> 00:52:02.060 I'm mistargeted. 00:52:02.060 --> 00:52:05.030 And so in this case my minimum distance 00:52:05.030 --> 00:52:07.700 from mean to the lower spec limit, 00:52:07.700 --> 00:52:09.560 that is the smallest distance. 00:52:09.560 --> 00:52:11.600 And that's still three sigma. 00:52:11.600 --> 00:52:16.250 So when I divide that distance there, I get CPK of 1. 00:52:21.450 --> 00:52:23.190 And here's one last example. 00:52:23.190 --> 00:52:25.020 In this case, now I've re-centered, 00:52:25.020 --> 00:52:31.110 got my CPK up to my full theoretical possible CP of 2 00:52:31.110 --> 00:52:34.440 by putting it back on the target. 00:52:34.440 --> 00:52:36.030 Putting my mean back on the target. 00:52:39.430 --> 00:52:45.400 So one of the things you use these CPs and CPKs for 00:52:45.400 --> 00:52:48.310 is in manufacturing. 00:52:48.310 --> 00:52:50.230 And you also can use it in design 00:52:50.230 --> 00:52:52.870 to get a quick feel for the relationship 00:52:52.870 --> 00:52:56.320 between your manufacturing capability, the inherent spread 00:52:56.320 --> 00:53:03.250 and mean to target centering, compared to your design specs. 00:53:03.250 --> 00:53:08.680 So you can characterize your CP and CPK and then evaluate, 00:53:08.680 --> 00:53:13.480 well, if I make some change in my design spec, what 00:53:13.480 --> 00:53:15.640 probabilities would that do? 00:53:15.640 --> 00:53:18.940 If I tightened up my spec limits and start to say, 00:53:18.940 --> 00:53:22.667 if you just even know just a CP and CPK, you can back out, 00:53:22.667 --> 00:53:25.600 because that's all in terms of normalized standard deviation, 00:53:25.600 --> 00:53:27.880 you can back probabilities associated 00:53:27.880 --> 00:53:30.940 with the number of parts that are not going to conform 00:53:30.940 --> 00:53:32.950 to those specifications. 00:53:32.950 --> 00:53:36.760 So if I purely tell you CP and CPK, 00:53:36.760 --> 00:53:38.920 you know probabilities, fractions 00:53:38.920 --> 00:53:41.500 of nonconforming parts. 00:53:41.500 --> 00:53:43.960 And then you can start to also compare 00:53:43.960 --> 00:53:48.550 how much would it buy me to move my mean in terms of fraction 00:53:48.550 --> 00:53:50.620 of nonconforming parts. 00:53:50.620 --> 00:53:52.300 How much would it buy me? 00:53:52.300 --> 00:53:55.090 It might cost a lot of energy and time 00:53:55.090 --> 00:53:57.310 to move the mean a little bit closer to the center 00:53:57.310 --> 00:53:58.630 of my spec limits. 00:53:58.630 --> 00:54:00.310 Is it worth it or not? 00:54:00.310 --> 00:54:04.170 Or would my time be better spent shrinking my variance 00:54:04.170 --> 00:54:06.850 if I were looking for possible process improvements? 00:54:06.850 --> 00:54:10.150 You can start to make those comparisons. 00:54:10.150 --> 00:54:12.340 And the last thing I want to do with the CP and CPK 00:54:12.340 --> 00:54:15.550 is give you a little bit of qualitative feel for these. 00:54:15.550 --> 00:54:21.670 Because what is a good value for CP and CPK? 00:54:21.670 --> 00:54:24.880 Well, the larger the better. 00:54:24.880 --> 00:54:29.710 A CP or CPK of 0, that's not good. 00:54:29.710 --> 00:54:32.410 When you start getting in the ranges of two, 00:54:32.410 --> 00:54:35.110 that's often what is talked about in manufacturing 00:54:35.110 --> 00:54:36.880 as being a goal. 00:54:36.880 --> 00:54:38.890 But it really depends on your process 00:54:38.890 --> 00:54:43.570 and what the costs of nonconforming parts are. 00:54:43.570 --> 00:54:47.590 But just to give you a feel, then, for these probabilities 00:54:47.590 --> 00:54:51.040 of nonconforming parts and what kind of CPs or CPKs 00:54:51.040 --> 00:54:56.710 those correspond to, just recognize that a CPK of 1 00:54:56.710 --> 00:55:03.430 corresponds to 3 sigma tail right at the spec limit. 00:55:03.430 --> 00:55:11.050 And being a 3 sigma tail off of whatever my mean is. 00:55:11.050 --> 00:55:12.310 So I might be off target. 00:55:12.310 --> 00:55:18.580 But 3 sigma out from the process mean, that's 3 sigma. 00:55:18.580 --> 00:55:21.130 Think of this is the number of sigmas. 00:55:21.130 --> 00:55:24.130 And that corresponds to about 1 in 1,000 parts 00:55:24.130 --> 00:55:25.540 falling outside of our tail. 00:55:30.620 --> 00:55:33.713 In fact, you can ask it upper spec limit, lower spec limit. 00:55:33.713 --> 00:55:35.880 And we actually know the number much more carefully. 00:55:35.880 --> 00:55:41.120 It's really 3 out of 1,000, those probabilities. 00:55:41.120 --> 00:55:44.900 If I go to a CPK of about 1.33, that corresponds roughly 00:55:44.900 --> 00:55:46.801 to a Z of 4. 00:55:46.801 --> 00:55:47.810 4 sigmas. 00:55:47.810 --> 00:55:51.020 I'm four sigmas away from the spec limit. 00:55:51.020 --> 00:55:56.240 You can see I get about 100x reduction in the number 00:55:56.240 --> 00:55:58.520 of nonconforming parts. 00:55:58.520 --> 00:56:00.920 If I can go another standard deviation, 00:56:00.920 --> 00:56:06.060 or a CPK of about 1.67, I get another factor of 100. 00:56:06.060 --> 00:56:09.440 And similarly, I get about another factor of 100 00:56:09.440 --> 00:56:12.500 if I can get to a CPK of 2. 00:56:12.500 --> 00:56:20.190 CPK of 2 means my mean is 6 sigma away 00:56:20.190 --> 00:56:26.360 from either my minimum distance from my upper 00:56:26.360 --> 00:56:27.980 or lower spec limit. 00:56:27.980 --> 00:56:30.610 And that's a very small number. 00:56:30.610 --> 00:56:36.140 That's really 6 sigma away or about one part per billion. 00:56:36.140 --> 00:56:39.050 10 to the minus 9. 00:56:39.050 --> 00:56:43.430 So why do you want something like a CP of 2 or a CPK of 2? 00:56:43.430 --> 00:56:49.060 Because one part per billion seems awfully small. 00:56:49.060 --> 00:56:52.180 And the reason, really, is so that you are 00:56:52.180 --> 00:56:56.230 robust to small mean shifts. 00:56:56.230 --> 00:57:02.280 To get down to even if you've got that small mistargeting 00:57:02.280 --> 00:57:06.540 that you've still got enough room in an inherent process 00:57:06.540 --> 00:57:10.470 with, say, a CP of 2 or CPK of 2 to be 00:57:10.470 --> 00:57:17.790 able to have a very small number of nonconforming parts. 00:57:17.790 --> 00:57:19.780 So whoops, what happened here? 00:57:19.780 --> 00:57:25.320 So here's the picture where my CPK and CP are both 2. 00:57:25.320 --> 00:57:26.880 Perfectly mean centered. 00:57:26.880 --> 00:57:31.950 Again, that's the 6 sigma distance here between the tail. 00:57:31.950 --> 00:57:35.790 And we said the probability of having way 00:57:35.790 --> 00:57:43.170 out here in the tail, that's my fraction of nonconforming. 00:57:43.170 --> 00:57:45.640 And it's about one part per billion, 00:57:45.640 --> 00:57:48.640 maybe two parts per billion. 00:57:48.640 --> 00:57:51.990 But now let's say that I actually have a mean shift. 00:57:51.990 --> 00:57:53.730 That was a little weird. 00:57:53.730 --> 00:57:55.920 Where I've actually shifted by 2 sigma. 00:57:58.830 --> 00:58:01.410 And recall we said back with control charting, 00:58:01.410 --> 00:58:03.570 detecting small mean shifts at 1 or 2 sigma 00:58:03.570 --> 00:58:05.670 are actually kind of hard. 00:58:05.670 --> 00:58:07.590 You actually have to derive sample size 00:58:07.590 --> 00:58:12.630 way high to have a very rapid detection. 00:58:12.630 --> 00:58:14.850 And what we'd like to do in this case now, 00:58:14.850 --> 00:58:18.660 I've got still the same inherent variability or a CP of 2, 00:58:18.660 --> 00:58:21.480 but that 2 sigma shift still leaves 00:58:21.480 --> 00:58:27.210 me 4 sigma before my tail starts impinging 00:58:27.210 --> 00:58:29.580 very much on my spec limit. 00:58:29.580 --> 00:58:34.260 And so now I'm here at about 31 or 32 parts 00:58:34.260 --> 00:58:36.780 per million nonconforming. 00:58:36.780 --> 00:58:40.380 So that's kind of the drive towards six sigma 00:58:40.380 --> 00:58:45.240 is that it allows you to accommodate or have 00:58:45.240 --> 00:58:49.050 a process that is capable enough that it can accommodate both 00:58:49.050 --> 00:58:54.040 the variance and target deviations. 00:58:54.040 --> 00:59:00.740 Both of them are hard to maintain as small as possible. 00:59:00.740 --> 00:59:03.010 Now, one can also go back and, if you're 00:59:03.010 --> 00:59:05.200 given the quality loss function, you 00:59:05.200 --> 00:59:11.950 can also ask if you have the probability. 00:59:11.950 --> 00:59:13.510 So think of this as the PDF. 00:59:13.510 --> 00:59:16.250 If I have the probability distribution function, 00:59:16.250 --> 00:59:22.270 I've got my underlying process with some mean and variance. 00:59:22.270 --> 00:59:25.690 You can also do that integral and ask questions 00:59:25.690 --> 00:59:30.070 about what is my expected quality loss 00:59:30.070 --> 00:59:35.530 and do the same kinds of things for mean deviations or variance 00:59:35.530 --> 00:59:36.580 changes. 00:59:36.580 --> 00:59:40.360 You just have to go in and essentially do the calculation. 00:59:40.360 --> 00:59:42.130 One thing that's nice is if you look 00:59:42.130 --> 00:59:46.055 at the expectation of that quality loss function 00:59:46.055 --> 00:59:47.680 and just go through and do a little bit 00:59:47.680 --> 00:59:53.240 of the basic expectation math on that where you split out, 00:59:53.240 --> 00:59:57.370 factor out x minus x bar, do that squaring 00:59:57.370 --> 01:00:02.770 and then relate components of the expansion, what 01:00:02.770 --> 01:00:06.760 you get is one component that is clearly related 01:00:06.760 --> 01:00:13.660 to your variance in the process and another component that 01:00:13.660 --> 01:00:17.560 is clearly related to your mistargeting or deviation 01:00:17.560 --> 01:00:19.010 from the mean. 01:00:19.010 --> 01:00:22.120 So in one quality loss function, again, you 01:00:22.120 --> 01:00:25.900 can see and relate in the normal distribution sense 01:00:25.900 --> 01:00:34.510 how both variance and how target deviation contributes 01:00:34.510 --> 01:00:35.530 to quality loss. 01:00:38.620 --> 01:00:41.680 OK, so just to recap a little bit 01:00:41.680 --> 01:00:45.610 on process capability, what you've got 01:00:45.610 --> 01:00:47.230 is some characterization. 01:00:47.230 --> 01:00:50.740 You've got a model of your real process. 01:00:50.740 --> 01:00:53.110 Where it's really operating when in control 01:00:53.110 --> 01:00:55.550 in the best situation. 01:00:55.550 --> 01:00:56.980 You know its mean. 01:00:56.980 --> 01:00:58.510 You know its variance. 01:00:58.510 --> 01:01:02.140 You've got the requirements, the spec limits. 01:01:02.140 --> 01:01:06.130 You don't use the spec limits to set your control limits. 01:01:06.130 --> 01:01:10.210 Control limits are based on detection of unusual events. 01:01:10.210 --> 01:01:12.850 And that's based purely on your model. 01:01:12.850 --> 01:01:16.540 You might superimpose your spec limits 01:01:16.540 --> 01:01:18.580 to be able to quickly calculate and think 01:01:18.580 --> 01:01:25.060 about what the probability of nonconforming parts are. 01:01:25.060 --> 01:01:27.730 But those are different things. 01:01:27.730 --> 01:01:30.340 To relate the two, a good way of thinking about it 01:01:30.340 --> 01:01:35.770 is the CP and CPK or an expected loss function. 01:01:35.770 --> 01:01:39.250 What's nice about CP and CPK is we basically normalized 01:01:39.250 --> 01:01:44.100 away so that if I came up to you and told you 01:01:44.100 --> 01:01:52.150 my process has a CP of 3 and a CPK of 2.5, you'd go, 01:01:52.150 --> 01:01:55.430 wow, I'm impressed. 01:01:55.430 --> 01:02:00.410 That's a pretty good process or awfully loose spec limits. 01:02:00.410 --> 01:02:01.503 One of the two. 01:02:01.503 --> 01:02:02.420 You don't really know. 01:02:04.970 --> 01:02:09.170 AUDIENCE: If each step is six sigma [INAUDIBLE] but there 01:02:09.170 --> 01:02:17.695 is no-- and let's say, the real determinant [INAUDIBLE] 01:02:17.695 --> 01:02:19.320 you could have six sigma for everything 01:02:19.320 --> 01:02:22.870 and you could still have zero, because [INAUDIBLE].. 01:02:22.870 --> 01:02:26.740 Is there a CP and CPK for [INAUDIBLE]?? 01:02:26.740 --> 01:02:29.920 DUANE BONING: I'm not quite sure what you mean by tracking. 01:02:29.920 --> 01:02:33.460 If by that you mean being on target. 01:02:33.460 --> 01:02:37.090 AUDIENCE: Yeah, let's say [INAUDIBLE] 01:02:37.090 --> 01:02:41.290 horizontal and vertical [INAUDIBLE].. 01:02:41.290 --> 01:02:43.720 And each step by itself would to be six sigma. 01:02:43.720 --> 01:02:46.960 And if the tracking is off, what would 01:02:46.960 --> 01:02:48.907 be within that [INAUDIBLE]? 01:02:48.907 --> 01:02:51.490 DUANE BONING: So that's, again, another multivariate question, 01:02:51.490 --> 01:02:54.640 looking at how important it is for two parameters 01:02:54.640 --> 01:02:58.150 to be close to each other, even if one or the other 01:02:58.150 --> 01:02:59.260 could deviate. 01:02:59.260 --> 01:03:00.430 It's OK if they both. 01:03:00.430 --> 01:03:01.570 AUDIENCE: [INAUDIBLE] 01:03:01.570 --> 01:03:03.070 DUANE BONING: And so what I would do 01:03:03.070 --> 01:03:07.270 is if the importance is the difference between those two 01:03:07.270 --> 01:03:10.720 dimensions, that's what I would characterize 01:03:10.720 --> 01:03:14.300 and that's what I would chart, put a plot on, 01:03:14.300 --> 01:03:16.660 and that's what I would set my spec limits on. 01:03:16.660 --> 01:03:19.780 I might also still care about, say, channel length. 01:03:19.780 --> 01:03:21.792 And so I might have another chart on that too. 01:03:21.792 --> 01:03:23.500 I have to be careful, because they're not 01:03:23.500 --> 01:03:24.940 completely independent. 01:03:24.940 --> 01:03:32.110 But what you really want to do is get your statistical process 01:03:32.110 --> 01:03:37.540 control apparatus targeted at or applied to the things that 01:03:37.540 --> 01:03:38.650 are most critical to you. 01:03:38.650 --> 01:03:40.510 And so in your scenario, the tracking 01:03:40.510 --> 01:03:41.990 between those two parameters. 01:03:41.990 --> 01:03:44.905 That's what I would monitor. 01:03:44.905 --> 01:03:45.405 OK. 01:03:47.970 --> 01:03:52.320 So this is just a recap of the x bar chart with a little-- 01:03:52.320 --> 01:03:53.940 that's kind of ugly. 01:03:53.940 --> 01:03:55.510 A little example. 01:03:55.510 --> 01:03:59.318 I wonder if I can just get rid of all those. 01:03:59.318 --> 01:04:01.040 Let me just try to go forward here. 01:04:04.960 --> 01:04:06.940 I think this is trying to reveal a data point 01:04:06.940 --> 01:04:10.400 one at a time, which we don't really want to do. 01:04:10.400 --> 01:04:12.610 So here's an example. 01:04:12.610 --> 01:04:18.590 Based on what you've seen about an x bar chart, 01:04:18.590 --> 01:04:22.820 and assuming here we've really set plus minus 3 sigma control 01:04:22.820 --> 01:04:26.180 limits on this chart, if you were wandering, 01:04:26.180 --> 01:04:30.170 moving along, and just triggering just 01:04:30.170 --> 01:04:33.710 on, say, extremal points, what do you think? 01:04:33.710 --> 01:04:35.780 Was this an alarm? 01:04:39.750 --> 01:04:42.270 Well, sure. 01:04:42.270 --> 01:04:43.410 How about that one? 01:04:43.410 --> 01:04:44.730 That was an alarm. 01:04:44.730 --> 01:04:46.710 Do you think they were real? 01:04:46.710 --> 01:04:48.000 Was it a false alarm? 01:04:48.000 --> 01:04:50.190 A real alarm? 01:04:50.190 --> 01:04:51.570 Why do you think it was real? 01:04:51.570 --> 01:04:54.720 AUDIENCE: Because [INAUDIBLE]. 01:04:54.720 --> 01:04:58.240 And those two are [INAUDIBLE] as well. 01:04:58.240 --> 01:05:03.610 So I expect to have more points [INAUDIBLE].. 01:05:03.610 --> 01:05:06.130 DUANE BONING: So the real answer is we don't know, 01:05:06.130 --> 01:05:07.930 but you're doing exactly the right thing 01:05:07.930 --> 01:05:09.580 going back to some of the discussion we 01:05:09.580 --> 01:05:12.700 had earlier of looking for additional evidence. 01:05:12.700 --> 01:05:16.330 So your evidence here was saying if I got this alarm, 01:05:16.330 --> 01:05:21.310 I might look back and say one of the previous points, hmm, 01:05:21.310 --> 01:05:24.550 there's an awful lot of them to one side of the mean. 01:05:24.550 --> 01:05:26.950 And in fact, maybe adrift, although at this point 01:05:26.950 --> 01:05:30.010 sort of looks like it's counter to the drift. 01:05:30.010 --> 01:05:33.820 So there's a little counter evidence. 01:05:33.820 --> 01:05:35.860 By the time I get to this point, now 01:05:35.860 --> 01:05:38.770 I'm starting to get really convinced that something 01:05:38.770 --> 01:05:41.930 is weird. 01:05:41.930 --> 01:05:43.600 Because now look at this. 01:05:43.600 --> 01:05:48.010 What's the probability of having all of those points? 01:05:48.010 --> 01:05:54.230 Almost all except for that one or two to one side of the mean. 01:05:54.230 --> 01:05:57.140 Well, you could actually go and calculate that probability. 01:05:57.140 --> 01:05:59.980 Sounds like a problem set problem. 01:05:59.980 --> 01:06:03.100 But qualitatively, you know something 01:06:03.100 --> 01:06:06.850 looks like it's off target. 01:06:06.850 --> 01:06:11.070 So this is the kind of thinking that you would want to do. 01:06:11.070 --> 01:06:15.480 Now, this is a little tricky. 01:06:15.480 --> 01:06:19.900 If you were just eyeballing it, what do you think a CP or a CPK 01:06:19.900 --> 01:06:21.535 would be for this process? 01:06:24.780 --> 01:06:25.470 I have no idea. 01:06:25.470 --> 01:06:26.762 I haven't done the calculation. 01:06:26.762 --> 01:06:31.050 You could go and do the calculation based on what? 01:06:31.050 --> 01:06:33.090 What would you do to actually formally try 01:06:33.090 --> 01:06:37.053 to calculate a CP or a CPK for this process? 01:06:37.053 --> 01:06:37.988 AUDIENCE: [INAUDIBLE] 01:06:37.988 --> 01:06:38.780 DUANE BONING: Yeah. 01:06:38.780 --> 01:06:41.900 You'd calculate the mean and the standard deviation. 01:06:41.900 --> 01:06:44.480 First off, you have the true mean 01:06:44.480 --> 01:06:48.710 and you have the true control limits. 01:06:48.710 --> 01:06:51.200 Could you calculate a CP or CPK based 01:06:51.200 --> 01:06:52.640 on mean and standard deviation? 01:06:52.640 --> 01:06:53.848 I can give you those numbers. 01:06:58.819 --> 01:07:01.362 AUDIENCE: Because you need upper and lower spec limits. 01:07:01.362 --> 01:07:02.820 DUANE BONING: You need spec limits. 01:07:02.820 --> 01:07:05.790 Control chart by itself never gives you spec limits. 01:07:05.790 --> 01:07:07.350 So you don't know. 01:07:07.350 --> 01:07:10.800 It may well be I've got really tight spec limits. 01:07:14.430 --> 01:07:17.665 My parts really need to operate down in here. 01:07:17.665 --> 01:07:19.290 By the way, you also have to be careful 01:07:19.290 --> 01:07:23.170 because if these are based on n samples, 01:07:23.170 --> 01:07:26.350 the sample distribution is tighter than the underlying 01:07:26.350 --> 01:07:28.580 process distribution. 01:07:28.580 --> 01:07:31.570 So if you see spec limits also on your control chart, 01:07:31.570 --> 01:07:32.950 it's kind of a warning. 01:07:32.950 --> 01:07:36.820 Because usually a control chart is based on sampling. 01:07:36.820 --> 01:07:41.230 So by itself, you can't really know what your CP and CPK are. 01:07:44.120 --> 01:07:46.740 And I would be cautious about assuming 01:07:46.740 --> 01:07:52.260 your spec limits are anywhere near your control limit. 01:07:52.260 --> 01:07:55.140 Don't assume your spec limits are plus minus 3 sigma. 01:07:55.140 --> 01:07:57.450 You're really falling in the trap 01:07:57.450 --> 01:08:01.140 of sort of basically assuming a CP 01:08:01.140 --> 01:08:05.663 or a CPK of 1 process, which is not often the case. 01:08:05.663 --> 01:08:09.540 AUDIENCE: [INAUDIBLE] outside your control limits? 01:08:09.540 --> 01:08:10.900 DUANE BONING: Absolutely. 01:08:10.900 --> 01:08:11.430 Why? 01:08:11.430 --> 01:08:15.880 Here you're having a huge fraction of nonconforming. 01:08:15.880 --> 01:08:16.380 Yeah. 01:08:16.380 --> 01:08:18.345 So that's my point. 01:08:18.345 --> 01:08:19.890 Your spec limits may be very tight 01:08:19.890 --> 01:08:21.720 and your process would be horrible. 01:08:21.720 --> 01:08:23.744 This is not a capable process. 01:08:23.744 --> 01:08:25.800 So normally yes. 01:08:25.800 --> 01:08:31.569 What you normally want to do is trigger on alarms 01:08:31.569 --> 01:08:35.470 much before you have parts that are falling outside of spec so 01:08:35.470 --> 01:08:38.350 that you can react to drifts in the process 01:08:38.350 --> 01:08:41.710 before they cause you disaster. 01:08:41.710 --> 01:08:43.710 Absolutely. 01:08:43.710 --> 01:08:44.640 That was implicit. 01:08:44.640 --> 01:08:46.439 I'm glad you made it explicit. 01:08:46.439 --> 01:08:49.010 Yes, yes. 01:08:49.010 --> 01:08:50.600 OK. 01:08:50.600 --> 01:08:53.180 So I think we've talked about a lot of these. 01:08:53.180 --> 01:08:55.729 What I'd like to do is give you a little glimpse 01:08:55.729 --> 01:09:01.640 of some additional control chart ideas that are beyond the x bar 01:09:01.640 --> 01:09:04.100 s or x bar range chart. 01:09:04.100 --> 01:09:06.640 There's some very good things about the x bar chart. 01:09:06.640 --> 01:09:11.479 It's relatively simple, fairly transparent, meaning you 01:09:11.479 --> 01:09:13.265 sort of know what each data point means. 01:09:13.265 --> 01:09:16.010 You know what an alarm means. 01:09:16.010 --> 01:09:19.700 It relates very closely to assumptions of normality. 01:09:19.700 --> 01:09:24.590 And in fact, by the act of sampling, 01:09:24.590 --> 01:09:27.649 you're enforcing some of these assumptions. 01:09:27.649 --> 01:09:30.470 By sampling within larger than 1, 01:09:30.470 --> 01:09:32.899 generally, you've got the central limit theorem 01:09:32.899 --> 01:09:34.010 on your side. 01:09:34.010 --> 01:09:39.140 So x bar really trends towards a normal distribution. 01:09:39.140 --> 01:09:41.340 And we'll talk about this a little bit. 01:09:41.340 --> 01:09:45.050 If I've got long times between samples, 01:09:45.050 --> 01:09:49.310 if there's very, very short time scale 01:09:49.310 --> 01:09:51.710 correlation between sample points, 01:09:51.710 --> 01:09:54.740 these samples are truly independent measures 01:09:54.740 --> 01:09:56.190 of the process. 01:09:56.190 --> 01:09:59.930 So as long as I've got reasonably long sampling times, 01:09:59.930 --> 01:10:02.330 each of my samples are an independent draw 01:10:02.330 --> 01:10:05.450 from the process. 01:10:05.450 --> 01:10:10.130 The problems are, yes, sample size, I need n more than one. 01:10:10.130 --> 01:10:12.980 And very often I may be measuring every one. 01:10:12.980 --> 01:10:16.730 So I'd like to get to some ideas or control charts that 01:10:16.730 --> 01:10:20.480 might be able to either react faster, a shorter average run 01:10:20.480 --> 01:10:23.000 length, or be able to utilize the fact that you've 01:10:23.000 --> 01:10:28.860 got more data and I can have this faster time response. 01:10:28.860 --> 01:10:31.740 So in fact, let's think real briefly about the limit. 01:10:31.740 --> 01:10:36.020 What if your sample size is just one? 01:10:36.020 --> 01:10:37.910 Could you go to that limit? 01:10:37.910 --> 01:10:41.930 There's a couple of tricky points here. 01:10:41.930 --> 01:10:44.160 The good thing is you have a lot of data. 01:10:44.160 --> 01:10:46.860 I can look at every part and respond quickly. 01:10:46.860 --> 01:10:50.880 But now a lot of these assumptions on the statistics 01:10:50.880 --> 01:10:54.870 or the calculation of these basic statistics 01:10:54.870 --> 01:10:56.490 don't have meaning anymore. 01:10:56.490 --> 01:11:01.540 For a sample of size one, what is the average? 01:11:01.540 --> 01:11:03.910 Well, OK, it's just that value. 01:11:03.910 --> 01:11:08.260 What's the standard deviation? 01:11:08.260 --> 01:11:09.610 No, it's undefined. 01:11:09.610 --> 01:11:12.755 1 over n minus 1 in the standard. 01:11:12.755 --> 01:11:13.880 You don't have enough data. 01:11:13.880 --> 01:11:17.320 You have no estimate of a variance 01:11:17.320 --> 01:11:19.810 from that sample size of one. 01:11:19.810 --> 01:11:22.870 So an approach here is essentially 01:11:22.870 --> 01:11:27.400 use kind of a variant referred to as a running control chart 01:11:27.400 --> 01:11:29.030 or a running average. 01:11:29.030 --> 01:11:31.630 So you look back not just on the new sample 01:11:31.630 --> 01:11:34.540 but some number of previous data points and kind of 01:11:34.540 --> 01:11:39.670 have a window that moves along or some calculated average 01:11:39.670 --> 01:11:43.210 of past history where I might weight my most 01:11:43.210 --> 01:11:46.630 recent measurement either equally or more 01:11:46.630 --> 01:11:49.150 importantly than my past measurements 01:11:49.150 --> 01:11:54.520 and use that as an estimate for the process average 01:11:54.520 --> 01:11:56.880 and the process variance. 01:11:56.880 --> 01:11:59.260 AUDIENCE: Does that violate the independent assumption? 01:11:59.260 --> 01:12:00.430 DUANE BONING: Absolutely. 01:12:00.430 --> 01:12:04.090 It does violate the independent assumption. 01:12:04.090 --> 01:12:06.800 If, however, they truly are independent, 01:12:06.800 --> 01:12:09.550 you can still formulate statistics and probabilities 01:12:09.550 --> 01:12:10.450 associated with that. 01:12:10.450 --> 01:12:16.650 In other words, if your process data was independent. 01:12:16.650 --> 01:12:21.190 What you've got is from any one point in time, 01:12:21.190 --> 01:12:22.800 I can calculate the probabilities 01:12:22.800 --> 01:12:24.060 associated with that. 01:12:24.060 --> 01:12:26.460 But if I then calculated that for my next point 01:12:26.460 --> 01:12:28.260 in time using some of the same data, 01:12:28.260 --> 01:12:31.650 I would never use some of the WECO rules 01:12:31.650 --> 01:12:35.707 like the probability of two points in a row falling above, 01:12:35.707 --> 01:12:37.290 because they're no longer independent. 01:12:37.290 --> 01:12:41.530 I can't just multiply those two probabilities. 01:12:41.530 --> 01:12:43.320 So it would get really messy trying 01:12:43.320 --> 01:12:48.330 to calculate sequence probabilities 01:12:48.330 --> 01:12:51.480 using a running average. 01:12:51.480 --> 01:12:53.190 Now, that's different for we'll see 01:12:53.190 --> 01:12:55.590 this notion of a cumulative sum chart 01:12:55.590 --> 01:12:58.320 where actually you want to use the fact that there 01:12:58.320 --> 01:13:02.790 is dependence to aggregate deviations 01:13:02.790 --> 01:13:04.780 and detect them even faster. 01:13:04.780 --> 01:13:09.340 So you use that very notion of lack of independence. 01:13:09.340 --> 01:13:11.820 So let's talk about these sort of running average or an n 01:13:11.820 --> 01:13:13.770 equals 1 kind of design. 01:13:13.770 --> 01:13:16.210 What you'd like to do is have increased sensitivity. 01:13:16.210 --> 01:13:19.680 You want to detect small changes, especially 01:13:19.680 --> 01:13:22.320 small delta changes. 01:13:22.320 --> 01:13:24.730 So small shifts are one class. 01:13:24.730 --> 01:13:30.570 There's also cases where you want to detect a small shift. 01:13:30.570 --> 01:13:33.420 We'll come to that chart a little bit later. 01:13:33.420 --> 01:13:36.420 You could also have the desire to not 01:13:36.420 --> 01:13:37.810 have too many false alarms. 01:13:37.810 --> 01:13:40.130 So you still want noise rejection. 01:13:40.130 --> 01:13:41.880 And one of the problems with some of these 01:13:41.880 --> 01:13:48.060 are you might, in fact, actually have higher variance 01:13:48.060 --> 01:13:49.030 estimators. 01:13:49.030 --> 01:13:51.210 And you've got to be very careful, because you 01:13:51.210 --> 01:13:56.440 can get larger false alarm rates unless you compensate for that. 01:13:56.440 --> 01:13:59.670 But the basic idea, and this almost 01:13:59.670 --> 01:14:01.970 goes back to your earlier question, 01:14:01.970 --> 01:14:05.640 imagine, first off, what we're doing with just a regular x bar 01:14:05.640 --> 01:14:08.970 chart but where we're measuring every part 01:14:08.970 --> 01:14:11.710 and then just aggregating in samples of different size. 01:14:11.710 --> 01:14:13.650 So what I've got here is the run data. 01:14:13.650 --> 01:14:16.020 And then in the little purple pictures 01:14:16.020 --> 01:14:21.280 is an x bar calculated for each group of four sample points. 01:14:21.280 --> 01:14:23.680 So every four I calculate the average, 01:14:23.680 --> 01:14:27.250 then I move on to the next four and calculate the average. 01:14:27.250 --> 01:14:31.440 And so in a typical sort of x bar chart with sample size n 01:14:31.440 --> 01:14:39.420 equals 4 where my sample time is also n equals 4, 01:14:39.420 --> 01:14:43.920 I've got a smoother curve here with the x bar. 01:14:43.920 --> 01:14:46.330 And we didn't explicitly talk about it, 01:14:46.330 --> 01:14:48.840 but one of the advantages of the x bar 01:14:48.840 --> 01:14:53.310 chart, what we talked about was the squeezing 01:14:53.310 --> 01:14:55.000 of the distribution down. 01:14:55.000 --> 01:14:59.730 Another way of looking at it is we're filtering high frequency 01:14:59.730 --> 01:15:06.570 deviations down to try to detect a smoother average that's 01:15:06.570 --> 01:15:10.530 more indicative of the true mean of the process. 01:15:10.530 --> 01:15:13.710 I'm really not trying to detect variances. 01:15:13.710 --> 01:15:16.170 I'd just as soon have one signal that just told me 01:15:16.170 --> 01:15:18.570 how good my mean is. 01:15:18.570 --> 01:15:22.660 And so an x bar really does filtering of your data. 01:15:25.390 --> 01:15:29.050 So if you think about it from the filtering perspective, 01:15:29.050 --> 01:15:30.460 what does the filtering do? 01:15:30.460 --> 01:15:32.290 It reduces some of these sharp peaks. 01:15:35.380 --> 01:15:38.500 It aggregates or hides some of this intermediate data, 01:15:38.500 --> 01:15:40.450 reduces this high frequency content 01:15:40.450 --> 01:15:45.580 to try to get to the low frequency, either drift 01:15:45.580 --> 01:15:49.690 or a mean shift a single mean shift that's applying 01:15:49.690 --> 01:15:52.730 to many data points together. 01:15:52.730 --> 01:15:57.710 And so you can also think about these as filtering operations. 01:15:57.710 --> 01:15:59.770 And any time you've got filtering, 01:15:59.770 --> 01:16:02.140 you do get at some of these questions 01:16:02.140 --> 01:16:07.480 we were kind of asking about about how related 01:16:07.480 --> 01:16:12.890 or how independent are samples at different points in time. 01:16:12.890 --> 01:16:15.460 So one can actually go in and have 01:16:15.460 --> 01:16:17.650 a measure of this independence. 01:16:17.650 --> 01:16:22.390 And if the process itself within a sample, the points 01:16:22.390 --> 01:16:24.250 are highly correlated, you actually 01:16:24.250 --> 01:16:27.670 have to explicitly think about this and for control charting 01:16:27.670 --> 01:16:30.970 spread those data points out. 01:16:30.970 --> 01:16:34.750 There is an autocorrelation function 01:16:34.750 --> 01:16:39.520 that's rxx as a function of the time separation 01:16:39.520 --> 01:16:42.400 or the number of sample points between them 01:16:42.400 --> 01:16:44.650 that has this definition here. 01:16:44.650 --> 01:16:49.600 If you think about it in terms of the time 01:16:49.600 --> 01:16:51.700 for an autocorrelated function, that 01:16:51.700 --> 01:16:56.770 is in fact inherent in a linear first order system, 01:16:56.770 --> 01:17:00.580 the autocorrelation function drops off fairly rapidly 01:17:00.580 --> 01:17:04.420 from 0 time gap. 01:17:04.420 --> 01:17:09.040 So if I sample at this point in time and I go some time away, 01:17:09.040 --> 01:17:15.430 the similarity of those sample points decays away. 01:17:15.430 --> 01:17:19.960 If it were truly uncorrelated, each and every new data point 01:17:19.960 --> 01:17:22.370 is completely independent of the other. 01:17:22.370 --> 01:17:24.280 And the point is simply here that if you're 01:17:24.280 --> 01:17:26.770 sampling from some physical processes, 01:17:26.770 --> 01:17:29.110 there may be an inherent correlation that 01:17:29.110 --> 01:17:31.100 isn't the one you're trying to get at. 01:17:31.100 --> 01:17:34.480 You're trying to detect long term trends to not 01:17:34.480 --> 01:17:37.900 detect very small shifts. 01:17:37.900 --> 01:17:40.840 And so you can play around with the sample 01:17:40.840 --> 01:17:46.810 time compared to the inherent correlation of the process. 01:17:46.810 --> 01:17:48.940 All this is saying is you want to pick your sample 01:17:48.940 --> 01:17:55.060 time ultimately such that if I pick this sample here and then 01:17:55.060 --> 01:18:00.700 my sample time some point later, this underlying short term 01:18:00.700 --> 01:18:03.760 correlation is not what I'm detecting. 01:18:03.760 --> 01:18:08.950 I'm detecting again a big mean shift. 01:18:08.950 --> 01:18:10.960 So we've been talking about this. 01:18:10.960 --> 01:18:12.580 Again, what you're looking for here 01:18:12.580 --> 01:18:16.850 is maybe pulling your small sample out, using, in fact, 01:18:16.850 --> 01:18:18.520 some of the correlation there to be 01:18:18.520 --> 01:18:21.940 able to say what the best estimate of the mean 01:18:21.940 --> 01:18:26.290 is right at that point in time and then have uncorrelated 01:18:26.290 --> 01:18:27.310 samples around that. 01:18:30.480 --> 01:18:37.530 OK, well what I think I'll do is probably stop now. 01:18:37.530 --> 01:18:39.600 I think I'll pick up next time, defer 01:18:39.600 --> 01:18:42.780 or wait a little bit on diving into the yield 01:18:42.780 --> 01:18:45.882 until maybe mid time Thursday. 01:18:45.882 --> 01:18:47.340 Because I do want you to get a feel 01:18:47.340 --> 01:18:49.410 for these alternative charts. 01:18:49.410 --> 01:18:52.170 So the charts we'll talk about next time 01:18:52.170 --> 01:18:55.920 include a moving average chart. 01:18:55.920 --> 01:18:59.760 Essentially you aggregate your data and use 01:18:59.760 --> 01:19:04.740 that to form estimates of local means and variances. 01:19:04.740 --> 01:19:06.870 And then we'll look at other clever ways 01:19:06.870 --> 01:19:11.370 of using more recent data and weight towards those. 01:19:11.370 --> 01:19:13.560 So we'll talk about an exponentially weighted 01:19:13.560 --> 01:19:17.825 moving average control chart and what that idea is. 01:19:17.825 --> 01:19:20.070 And then finally, we'll also give you 01:19:20.070 --> 01:19:24.420 a little bit of a feel for a cumulative sum chart where 01:19:24.420 --> 01:19:27.510 we aggregate deviations from the mean, 01:19:27.510 --> 01:19:30.750 because we're really looking for was there a small change 01:19:30.750 --> 01:19:34.140 or is there a small drift going on in the process? 01:19:34.140 --> 01:19:35.970 And how do we increase the sensitivity 01:19:35.970 --> 01:19:40.750 to detect that small shift as rapidly as possible? 01:19:40.750 --> 01:19:42.960 So we'll talk about charts that are really 01:19:42.960 --> 01:19:49.552 meant to squeeze that average run length for detection down. 01:19:49.552 --> 01:19:50.760 So we'll see you on Thursday. 01:19:53.310 --> 01:19:55.860 And again, while you're working on today's problem set, 01:19:55.860 --> 01:19:58.380 if you want to feel good, take a glance 01:19:58.380 --> 01:20:00.615 as soon as it's released at next week's problem set. 01:20:00.615 --> 01:20:02.941 It will make you feel much better.