Increasing variability always degrades the performance of a production system. This is the law of variability. This law, as discussed by Hopp and Spearmen, shows that understanding variation is very important for the quality of a process. Therefore, I want to show you how to study variability across groups. I will explain how to perform a test for equal variances. Moreover, I will motivate why this is useful. Let's go back to the example of four machines producing coffee. Together, we investigated that the average moisture content of the coffee beans depended on the machine it was produced on. The data looked like this. We showed that the means of moisture percentage differ across the four machines. But what about the variation within a machine? Machine two has a wide range of moisture contents, while machine four appeared a lot more consistent. For producing consistent coffee, we probably want the machine with the smallest variation. Let's take a closer look at these two machines. The mean level is near equal for machine two and four. However, the variation of the two machines may differ. This is the data of machine two in a time order. And here, we see the data for a machine four in a time order. The moisture content of machine two goes up and down more than that of machine four. If the machine produces very inconsistently, meaning a lot of variation, this might cause problems. It could lead to a lot of scrap and defects which will cost you money. If you want your moisture percentage to be, for instance, lower than 10.5 percent, and there is a lot of variation, like in machine two, a lot of your batches will show a moisture percentage which is higher than this 10.5, and these batches will have to be thrown away or reprocessed. If the variation is smaller, like in machine four, this will happen less often. Therefore, you will probably like to know if one of the machine produces more consistently than the others. The output of the ANOVA analysis already gave us some information on the variation. We see that machine four has a smaller standard deviation and therefore, produces most consistently within the sample at least. However, are these differences between these standard deviation is statistically significant, or is it a coincidence that machine four has the smallest variation? Let us use a statistical test, the test of equality of variances to study this. Now, pause your video, load the data into Minitab before continuing. This is what your data in Minitab should look like. Note that I stacked the four columns with the individual machine data into one column which is moisture and one column which shows the machine. For testing of equal variances, we have to go to the stats menu because this is statistical analysis. On the stat, we go to ANOVA. And there, you can find the test for equal variances here. Now, Minitab asks you, "What is your response?" Well, our response or the y variable or the CTQ is moisture of course. Next, we have to fill in the factors or the influence factors, and that, in this example, is of course machine. Well, that's it. So, okay. Now, Minitab gives us some output. And one of them is this graph, the test of equal variances. And the other output is in the session window, lots of different, what, results. Now, let's study the output. The estimates for the standard deviation in moisture percentage for the machines are given. To check if the differences between these are statistically significant, we look at the p-value, compute it according to Levene's method. In this case, the p-value is relatively large. It shows us that there is a 32.5 percent chance that the difference between the machines is a chance fluctuation, which is much larger than our threshold of five percent. This means that we did not find evidence that there is a difference in variation between machines, and we cannot draw any conclusions on what the machine produces most consistently. Now, suppose that we would have had 50 measurements for each machine instead of 10. This would give us a different dataset. The individual value plot would look like this. The equal variance test now gives us this output. What would you conclude? Do you see a statistical difference between the variances of the machines? We see a very low p-value, which means that there is a statistical difference between the variances of the machine. And we can conclude the machine four produces most consistently. The test for equal variances also has implications for the ANOVA analysis. Remember that when we performed ANOVA analysis, we selected the options and uncheck the box assume equal variances. In this case of our first example, we found no significant difference between the variances of each machine. So, actually, we did not need to un-check this box. It is preferred not to do this because then the p-value would be more precise. Let's summarize. You now know why variances are important and learned how to compare them across groups. You know that knowledge about equality of variances can be used to improve the p-value over and over.
