In the last session, we saw how even a very industrialized product, such as M&M's chocolate can have considerable variation from one product to the next. So when you're on the line if you encounter some variation, That the chocolate in this package was heavier than the chocolate in this package. How do you react? How do you know whether this was just part of the normal way of doing business or whether there was cause for concern? In this session, we will introduce the concept of control charts. Control charts will help us to distinguish between what we would call as normal and abnormal variation. Control charts are part of a broader toolbox known as statistical process control. And we formalize the idea of normal and abnormal variation with the following example. Now, I know that it was the last five to six week, you had to suffer through my crappy handwriting, and so, that example will resonate with you. On this page here, I've written three rows of the letter R. For the first row, I use my better hand, If there is such a better thing in my handwriting such as a better hand. I've used my right hand to write eight times if letter R. Now, you notice not every R is identical. There's some small variation from one letter to the other, but they look pretty similar. We refer to the variation from one letter to the other as a common cause variation and the level of common cause variation here is low. Now look at the second row. This row was written with my left hand and I really can't write with my left hand. Now, you notice that the variation from one letter to the other is bigger as you go from say, the seventh to the eighth letter. There's a high level of variation, But it is still a common cause variation. They all kind of look similar in pattern. Now, look at the last row. In the last row, You notice a big jump in the types of letters as you go from the fourth letter to the fifth letter. The reason for that, I wrote the first letters with my right hand, and then switched hands to write the last four set of letters. This is called an assignable cause. There's something in the underlying process generating the letters that changed. Our role in statistical process control is to measure the amount of common cost variation, exactly using the tools that we saw in the last session talking about six sigma, But then also being aware and recognizing if an assignable cost variation happens. So, how can I distinguish between an assignable cause variation and a common cause variation in my process? Let's revisit our example of the M&M bags. Imagine I would go to the grocery store everyday and take a sample of five bags of M&M's. I would then take the average of these bags and I would plot them on a chart that roughly looks like this. Now, I know that the bag's weight will follow some underlying distribution. If I have a weight that is equal to 50 or close to 50, I know that this behaved according to our sample mean that we have computed before. However, if I assigned the bags that is particularly heavy or particularly light, I should be concerned that there might be some assignable cause going on. Now, in our calculations, we're going to set our control limit as three standard deviations above the mean, then we're going to set the lower control limit at three standard deviations below the limit. I can then ask myself, well, with what probability will a sample average of five bags fall within this range and outside of this range? Now, statistics tells us that this band, six standard deviation wide, explains for 99.7% of the observed cases. But differently, if I observe an average of a sample of five that is outside this band, I can say with a 0.3% confidence that some assignable cause just happened in the system. So let's revisit our M&M's data. Suppose that the sample that we discussed in the last session was obtained as follow. I went to the store on ten subsequent days and everyday I got five bags. Now, let's do some calculations. First of all, we can compute the averages of some weights that I obtain on any different day. Now, how do I interpret that number? This is an average of a five numbers. I know that by the laws of statistics that the standard deviation of a sample of five will have a standard deviation of, the standard deviation of the overall population divided by the square root Of five. So, if I want to figure out the line for the control line that is three standard deviations below that, I have to basically look at the mean of 50 in this overall population, And I have to subtract three times the standard deviation of the samples that I'm likely to draw on a particular day. Similarly, so this is a lower control limit. The upper control limit is computed simply by taking the mean and adding three times the estimate of the standard deviation. So, this is the mean, This is the lower control limit, And this is the outer control limit. Now, of course, the mean changes from day to day, as you can see here in the chart. Now, when we plot this information. You see the basic idea of the control chart. You see how the sample mean is changing from day to today and how all of these changes are inside the control limits. Now, please separate in your mind the idea of specification limits that we talked about early on and the control limits. The control limits only tell us to what extent the process is behaving according to it's normal deviation. It doesn't say if anything whether the parts produced are defective or not. It's just looking for a sign of a cause variation. If I now go to the store in the,, on the eleventh day, and I'm going to draw a sample with an average that is either above the green line or below the red line, I know there wasn't a sign of a cause that occurred. Based on our discussion of six sigma in the previous session and the current session's discussion of control charts, we can summarize the idea of statistical process control. Statistical process control is basically a never-ending circle. You start by collecting data about your process and measuring the current capability. You then use control charts to see if the current process performance is in line with the empirical regularities that you've seen in the process. You're looking for assignable cause variation. When the sign of a cross-variatoon pops up, you try to figure out why it happened. The control chart's give you the trigger event, They tell you that something has happened, you then have to identify the root cause. We're talking about root cause problem solving in one of the coming sessions. Once we find, identify the root cause, be it a machine or an operator, we try to eliminate the sign of the root cause variation. This gets you to, back to the starting point. As you're doing this, hopefully, you're able to reduce the variation in the process and also increase the capability score of your operation. In this session, we saw how we can use the laws of statistics to permanently and continuously test the hypothesis that the variation that we saw in the process was an abnormal variation due to some assignable cause or whether it was just a normal way of doing business. Control charts are a very powerful tool of keeping on top of your process. Between you and me, let me confess, that I'm even using a control chart to map out my own weight on every morning in the week. Now, one nice side benefit of the control chart is just tracking the data in it by itself, even if you don't use the control limits, But just plotting the mean over time can be very motivational and very visual. Oftentimes, really just seeing the data, If you think about a diet, but also about a factory performance, seeing the data is extremely powerful. But there are many other forms of control charts beyond the X pi charts that we saw in this session. However, I think with the intuition that you got in this session, you have the basic ideas behind this broader set of tools known as statistical process control.
