In the last session we saw how even a very industrialized product, such as M&M chocolate, can have considerable variation from one product to the next. So when you're on the line and 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 that there was the cause for concern? This session will introduce a concept of controlled chance. Controlled chance will help us to distinguish between what we would call as normal and abnormal variation. Controlled chance a part of a broader toolbox known as statistical process control. Let me formalize the idea of normal and abnormal variation with the following example. Now I now that over the last few weeks you've had to suffer through my crappy hand writing, and so this example will resonate with you. On this page here I've written three rows of the letter r. For the first row, I used my better hand, if there is such a thing in my handwriting such as a better hand. I used my right hand to write eight times the 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'll 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 I switched hands to write the last four sets 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 Cause Variation exactly using the tools that we saw in the last section talking about six sigma. But then also being aware and recognizing if an Assignable Cause 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 every day and take a sample of five bags of M&Ms. 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 bags weight would follow some underlying distribution. If I have a weight that is equal to 50 or close to 50, I know that this behaves according to our sample mean that we have computed before. However, if I find a bag 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 are going to set the control limit as three standard deviations above the mean. And we're going to set the lower control limit at three standard deviations below the mean. And can then ask myself with what probability was sample average of five bags fall within this range and outside of this range. Now statistics tells us, that this band, six standard deviations wide, explains for 99.7% of the observed cases. Put differently, if I observe an average of a sample of five that is outside this band, I can say it with 0.3% confidence, that some assignable cause just happened in this system. So let's revisit our M&M data. Suppose that the samples that we discussed in the last session was obtained as follows: I went to the store on ten subsequent days and every day I got five bags. Now let's do some calculations. First of all, we can compute the averages of the weight that I obtained on any different day. Now how do we interpret that number? This is an average over five numbers. I know that, by the laws of statistics, the standard deviation of the sample of five will have a 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 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 upper control limit. Now of course the mean changes from day to day, as you see here on the chart. Now when we plot this information, you see the basic idea of a control chart. You see how the sample mean is changing from day to day and how all of these changes are inside the control limit. Now please separate in your mind the idea of the specification limits that we talked about earlier on and the control limits. The control limits only tell us to what extent the process is behaving according to its normal variation. It doesn't say anything whether the past produce is defective or not, it's just looking for an assignable cause variation. If I now go to the store on an 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 was an assignable 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 this assignable cause variation pops up, you try to figure out why it happened. The control charts give you the trigger event. They tell you that something has happened, you then have to identify the root cause. We'll talking about root cause problem solving in one of the coming sessions. Once you find the root cause, be it a machine or an operator, you try to eliminate the assignable cause variation. This gets you back to the starting point. As you are doing this, hopefully you are 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 a variation that we saw in the process was an abnormal variation due to some assignable cause, or whether it was just the 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 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 limit, but just plotting the mean over time can be very motivational and very visual. Oftentimes, really, just seeing that data, if you think about a diet, but also about a factory performance, seeing that data is extremely powerful. Now there are many other forms of control charts beyond the X bar 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.
