The Day the CPK Report Came Out, the Room Fell Silent for Three Seconds
I still remember years ago, when our new process was first introduced to the production line, everyone was full of confidence. However, the first CPK report came out, and it was abysmal, only 1.08, with DPMO soaring to 6210. The entire meeting room instantly fell silent; you could hear a pin drop. The section chief's face turned green, and he immediately asked, "Which station has a problem? Did the mean shift, or did the variance explode?" Everyone exchanged glances because, frankly, at that time, our statistical tools were limited, and many nuances were still unclear.
Where Exactly Is the Problem?
To be frank, many times when we look at data, the first thing we think of is whether the mean has drifted. For example, when measuring wafer thickness, after measuring hundreds of wafers, if the average is 775um, you might initially think there's no problem, as it's within specifications. However, if some machines produce wafers with thickness highly concentrated around 770um, while others are scattered between 760um and 780um, even though the average is the same, the latter has much greater variance, making it easier for defective products to appear. Therefore, the key is that we must not only look at the mean but also the variance!
Variance is an indicator of data dispersion. For two processes, even if their means are the same, if one process's yield fluctuates like a roller coaster, while the other is as stable as a highway, you would naturally choose the stable one. So, how do we know if the variances of different data sets are "similar"? This is where Levene's Test comes in handy.
How Is It Actually Done?
In essence, Levene's Test helps you answer the question: "Do the variances of these data sets truly come from the same population?" Or, "Do their variances have significant differences?"
For example, we have three machines (A, B, C) performing the same process. We want to know if the thickness variances of the products produced by these three machines are the same.
- Collect Data: Randomly sample 30 wafers from each machine and measure their thickness.
- Calculate Test Statistic: This part is usually done by software. You just input the three data sets into Minitab or R, select Levene's Test, and it will output a P-value.
- Interpret Results:
* If the P-value is less than 0.05, then we "reject the null hypothesis." This means that at least one machine's variance is significantly different from the others. At this point, you need to investigate further to find out which machine is causing the issue.
So, the main point is that Levene's Test gives you a P-value, and by checking if it's less than 0.05, you can determine if the variances are "homogeneous."
The Most Common Pitfall
The most common pitfall I've encountered is that after performing Levene's Test and finding a small P-value, people rush to adjust machine parameters. As a result, the problem isn't solved; instead, even previously normal machines become messed up.
Frankly speaking, Levene's Test tells you "there's a problem," but not "where the problem is." A P-value less than 0.05 only indicates a difference between variances; it doesn't mean the one with larger variance is necessarily "bad." Sometimes, machine A's variance is extremely small, while machine B's variance is larger but still within acceptable limits. As a result, you only see the difference but fail to clarify whether it truly requires intervention.
In other words, when you find non-homogeneous variances, the next step should be to visualize the data, for example, by drawing a Box Plot, to see which group has the widest spread. Only then should you conduct a more in-depth analysis of that specific data set to identify the root cause. Don't just make arbitrary adjustments as soon as you see a small P-value!
One Thing You Can Do Today
Take the data you have on hand and run Levene's Test for homogeneity of variances, to see if the P-values are all in line.