In 1988, creative EU bureaucrats in Brussels created EU Resolution 1677-88. The resolution said that for a cucumber, in order to be of the highest quality standard, its curvature should not exceed ten millimeters. With a cucumber an eleven millimeter curvature, it was no longer in the highest quality level. Nine millimeter, however, was fine. Now this little anecdote clearly tells us something about the creativity of politicians to come up with stupid laws. But more to the point of this session, it reminds us that often the matter of good or bad is not as binary as it sounds. Oftentimes, there's an underlying parameter that can be measured, be it a geometry, a weight, or a temperature, and this underlying measurement has to be compared with some specification. This is the basic idea behind Six Sigma. Now I have to confess that I found it was below my academic dignity of going to the local market and purchasing 50 cucumbers only to take then out a ruler and measure their curvature. Instead to illustrate the concept of Six Sigma I did something much sweeter. Sweeter, though potentially unhealthier. I went to the local grocery store and bought 50 bags of M&Ms. I then took out a high resolution scale to measure the weight of these 50 bags. Here is the data that I got. Take a look at this data. For example, we can compute the average of the weights in the sample. This was about 50 gram and I have to confess was significantly higher than what the company has labeled on the bag. That was about 47. You also see that the standard deviation is about 1.1. To get a visual of the distribution we can use the histogram analysis in Excel. For that, go to Data Analysis, click on Histogram, enter the data here as the Input Range, enter the data. As the Bin Range enter a number that is slightly below the lowest value and the number that is slightly above the highest value, and then click on okay. Now you get a chart that looks like this. This is a histogram of the weights and, surprise surprise, it's interesting to see how the laws of statistics kick in. The distribution here is almost normal-like, with most of the values being here somewhere in the middle around the mean. And some of the bags being either extremely light or extremely heavy. We will use that statistical distribution in our calculations in a moment to figure out how many defects there will be in a very large sample. Now how should we define an M&M bag as defective? Having eaten them all I have to confess that all of them were quite yummy, and so I have a hard time speaking about defect. But lets say for the sake of argument that we have a specification that a bag should have at least forty eight grams of chocolate in there. We refer to this number as the LSL which stands for the lower specification limit. Similarly we speak of the USL, the upper specification level, as the number that if exceeded we would call the bag defective, so lets say for the sake of argument the bag is defective if they have more than 52 grams of chocolate in there. With these two numbers I can compute a number that is known as the capability score. The capability score, also known as the CP score, looks at the ratio between at the ratio between the weights of the specification level relative to six times the standard deviation in the process. This number here tells us the capability of the process. Notice that the way that we can increase process capability is by either making the specifications more forgiving. It's the higher capability if would assume the specifications were 47 to 53. Or maybe using the standard variation. Both of these will reduce the likelihood of defect. Let me illustrate the object of capability support on the following slide. Just as a reminder, this is the definition of the capability score. The upper specifications of it minus the lower specifications of it, divided by 6 times the standard deviation. Now imagine two distributions. One of them has a density function that has a little larger variance. And you notice here, how you can go three standard deviations from the mean of the distribution before you hit the specification limit. Now on the lower case you see the distribution that has a lower variance that you need to go six standard deviations before you're going to incur a defect. Now clearly defects are less likely in the lower case. There's simply less probability mass at the tails here. So this suggests that we can compute or we can translate the capability scores of the distribution into the probability of defects. Let me illustrate this calculation by going back into our spreadsheet. So how likely is it going to be that we encounter bag of M&Ms that are heavier than 52 grams? So what's the probability that the bag is too heavy? I can get to this by using the normal distribution function in Excel and looking at the 52 grams relative to a distribution with 50 as a mean and 1.1 as the standard deviations. That probability is 96% that it stays below this, or the probability is to heavy, it simply one minus that, which is 3.4%. Now next, ask ourselves what's probability that this is too small of a bag, or that this is too light? Well for that I have to look at the normal distribution. This time with the lower specification number, 50 and 1.1 is the standard deviation. And this is equals 3.3%. Now for a defect I either need to have the bag be too heavy or too light, and so the sum of those two is simply the probability of a defect. Now I can take this number and I can multiply this with, say, a million units in a production run to get a number that is known as the ppm, the parts per million. So we have 67,818 defective per million parts. So you notice that the capability score for around 0.6 as we just saw in the M&M example. This equating to a defect probability of around 0.067%, or put differently, 67,000 defects in a million parts. Now, in this table we show the relationship between the capability score and defect probabilities. For example, at a capability score of 1, we can go three sigmas from the mean to either side of the specification levels. And we have a defect probability of 0.027. Put differently, you're going to have 2,700 defects per million parts. No where did these numbers come from? Now let's first look at a three sigma process. What's the defect probability? Well I have to go three standard deviations before I hit the specification element. Let's assume the underlying distribution is a standard normal distribution, which is having a mean of zero and a standard deviation of one. So for a defect, I have to hit the number 3. So I'm looking at the normal distribution, mean 0, standard deviation of 1, which is the value of 99.865%. So 1 minus that probability, gets me the probability of the part being too large. This is also the probability of the part being too small. So, assuming here's symmetry, I can just double this number and that gives you the 0.027 that you all saw on the earlier table. Now let's move this further and look at the six sigma process. With six sigma, you have to go six standard deviations and you notice that this number here becomes ridiculously small. It's a little hard to interpret here so let's not look at it as a probability but as a defect per million parts. So we have to multiply this with one million and we see that the number here is roughly 0.002. In other words we're going to have 2 defects per billion parts. So a quality target is typically expressed in defect probabilities, or parts per million. We see in this table that can be matched to a capability score. This allows me to ask myself for a given specification limit, what is the amount of variability in the process that I can tolerate before violating my quality goal. Let me go back to the example of the M&Ms. So we said we had the USL-LSL/6sigma and that had to be, if I'm aiming for 6 sigma operation, that would have to equate to 2. Now, in our example, the difference here between the USL and the LSL was 4 and so that gives me an equation I can solve. 4/6sigma=2. And so in other words sigma is equal to one-third. So just to gain some confidence in our calculations, let's go back to our sales spreadsheet and take the current comparably observed standard deviation of 1.1 and replace it by our new goal of a standard deviation of one-third. Of the numbers we compute and you see that the parts per million go down to 0.002 which is the two parts per billion that we talked about. In 2009 the EU bureaucrats finally decided to cancel the resolution 1677-88. That makes you hopeful that one day they can also resolve the Euro crisis. But more to the point of this session, we saw that variation exists almost everywhere. Even in a highly industrialized product, such as packaged M&M chocolate, we saw a considerable amount of variation from one package to the other. Would you really call the package a defect just because it has one extra gram of chocolate in that? That is a matter of the product specification. We saw how the CP score measures the process capability by really looking at the variation that you have in the process relative to the allowable variation that you get out of the specifications. We also saw that it's the capability score of something that tells you how many defects you're likely to make in a thousand or in a million parts. And that's a really good metric that you want to track over time to see how you're getting better, or across your suppliers to see who's giving you the highest quality product.
