Hi. In this lecture we're gonna do a little sort of bonus. I want to talk about the normal distribution again and I want to talk about it in the context of a business practice that has to do with quality control that's known as six Sigma. Six Sigma was a process evolved by Motorola, you know, quite a while or couple of decades ago. In an effort to sort of making production processes more predictable so that we have fewer quality errors. So to understand our works, let us go back and remind ourselves of what Sigma is and then we can understand what six Sigma is. [inaudible] we had a normal distribution, right, we had a mean. You know, we have these Standard Deviations, these Sigmas, one Standard Deviation, two Standard Deviations and so on, right? And then we had a 68 percent of the time. Right that outcome will lie within one stimulation in 95 percent of the time. It will lie within it two standard aviations. So what would lie, how often would we lie within six standard deviations? If I went out here, way out here to six standard deviations. I guess that's even further out. How often would I be inside that? Well the answer's, the only time I would fall outside of it would be 3.4 in a million. Okay? So that means that there's almost no way that I'm gonna be way over here outside of Six, you know, Six Sigma too big, or Six Sigma too small. And so that's gonna be the core idea. Let me explain the idea in the context of an example, and then take it to the production, how it's used in production. So here's an example. Let's go back to the grocery store. So suppose I own a grocery store and I sell bananas. And, on average, I sell 500 bananas a day. You know, I keep, I've kept track of my data, it's a normal distribution, and the standard deviation's ten. So what I wanted to be the case is that if I have any sort of, you know, data within Six Sigma. I'm not gonna run out of bananas. Well, this is easy to solve, all right? Because sigma is equal to ten, right? So that means that Six Sigma. Is gonna be 60. So, if I wanna be within any event within six sigma, I'm still gonna be okay. All I need to do, right, is have 560 bananas on hand, pounds of bananas on hand. And then even if I get a four sigma event, a five sigma event, a 5.8 sigma event, I'm gonna be fine. I'm not gonna run out of bananas. So that the, the idea, right? You want it to be that, even if you get a six sigma event, things are gonna be okay. Okay? So let's see how this works. For production, so suppose I'm making some metal part and this metal part has to be between 500 and 560 mm so this is the range, anything in this range is okay but if I'm outside this range then the part's not going to work. I could be making phones, I could be making car doors, whatever. Now suppose it's the case that what causes the door to be a little thicker or a little thinner than we want is just a bunch of random things being added up so I've got. A normal distribution. Well I should be able to make my production process so I get the mean right in the center of that, right. So we've got 530 which is right in the center. And now I want it to be the case that if I have a six sigma standard deviation, I'm still going to be okay. Well, this isn't very hard to figure out, right. So we can just say here's my distribution, 530s the mean. And I'm gonna have a bell curve. It's not a very good bell curve. [laugh] But I want it to be the case that anything within six sigmas is okay. So, 560 to 500 have to be that's gotta be my six sigma range. So this is gonna be plus six and this is gonna be minus six. Okay. So six sigma is 30 above the mean. Right. This is 560. Minus 530. Equals 30. That means I just want Six Sigma to equal 30. So, if Six Sigma equals 30. That means sigma equals five. So what does that mean? That means if I'm running this company, if I'm sort of making these metal parts, I want it to be the case that my standard deviation. When I, you know, keep track of the standard deviation of my parts, I wanna get that all the way down to five. And if I get that down to five, then if I have any event less than six sigma, the part's still gonna work. Now how do I get it down to five? That's not easy, right, you've got to do continuous quality improvement. So the real management practice was not just computing standard deviations and figuring out what the six is, it was doing all that really hard work that makes it so that sigma falls down to five. So it could be that initially your sigma might have been 30 or twenty or something like that and the idea through continuous improvement as you drive your sigma down so that sigma gets small enough so that even if something really bad happens the process still works and the part still functions and you don't have to do some sort of massive recall. Okay, so that's six Sigma thinking. What six Sigma basically tells us is we can use this idea, right, this model of sort of normal distributions with standard aviations to inform how we, you know, run our production processes so we can figure out, like you know what. We're just making too many mistakes. And if we make mistakes at this level, we're constantly gonna have parts not work. Where as if we can reduce our variation. By reducing our variation, the process is almost always going to work. Our parts will fit in whatever part they've got to fit into. Alright, so that's at least another example how we can use this aggregation things, these techniques, these tools we're using in ways we might never have expected when we first came up with them. Okay. Thank you.