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Statistical Analysis6 min read

Testing the Normality Assumption: Shapiro-Wilk vs Anderson-Darling

This article discusses a scenario where initial factory yield was poor, and a CPK report's data severely mismatched the DPMO, leaving everyone perplexed. It highlights that many statistical metrics, such as CPK, rely on the crucial assumption of data normality, and if this assumption is violated, calculated values are invalid. Therefore, prior to analysis, it is essential to perform a normality test, such as the Shapiro-Wilk test, to ensure data validity and prevent erroneous conclusions.

That Day the CPK Report Came Out, the Entire Room Fell Silent for Three Seconds

I still remember many years ago, we had a new process in our factory, and everyone was confidently saying the yield would skyrocket. However, when the first CPK report came out, the data showed 1.08, with a DPMO of 6210. The meeting room fell silent for three seconds, and the air instantly condensed. The boss's face turned green because, theoretically, such a CPK value simply couldn't correspond to such a poor DPMO. Everyone exchanged bewildered glances, not knowing what the problem was. I thought to myself, could it be "that thing" again?

Where's the Problem? Is Your Data "Normal"?

In essence, statistical metrics like CPK and DPMO all share a major prerequisite: your data must be "normally distributed." What does this mean? It means that if your process data is plotted as a histogram, it should resemble a bell-shaped curve, high in the middle and low on both sides. Many statistical tools assume your data is normal; if it isn't, the calculated numbers might be completely inaccurate, just like the situation we encountered that day.

So, the key is that before starting any analysis, you must first test whether your data truly conforms to a normal distribution. This is where the Shapiro-Wilk test and the Anderson-Darling test come into play. Frankly, both of these help you determine "whether your data looks like a bell curve."

How to Do It in Practice? Just Look at the P-Value!

When you run a normality test in Minitab or JMP, you will see a P-value. This P-value is the key to making a judgment.

  1. Shapiro-Wilk Test:

* This is the most commonly used test, particularly suitable for small samples (usually less than 50 data points).

* If the P-value > 0.05 (or your set significance level alpha), then we "cannot reject" the assumption that your data is normally distributed. In other words, you can safely assume it is normal.

* If the P-value <= 0.05, it indicates that your data is very likely not normally distributed. At this point, you need to be careful!

  1. Anderson-Darling Test:

* This test is more sensitive to the tails of the distribution (i.e., extreme values).

* Again, look at the P-value: P-value > 0.05 indicates the data might be normal; P-value <= 0.05 indicates it might not be.

* Its statistical power is typically slightly higher than Shapiro-Wilk, especially when dealing with large samples, making it more likely to detect non-normal conditions.

So, the key is that when you see a P-value less than 0.05, it indicates that your data is very likely not normal. If you then use CPK formulas to calculate, the results will be distorted. That day's CPK of 1.08 and DPMO of 6210 was precisely because the data was fundamentally not normally distributed, leading to an "inflated" calculated CPK value.

The Most Common Pitfall: Forcing CPK Without Checking the P-Value

The biggest pitfall I encountered was when I first joined the factory. A senior colleague taught me to run CPK but didn't specifically emphasize performing a normality test first. At that time, I took the data, threw it directly into the software for calculation, and submitted the report after seeing decent CPK numbers. However, the boss noticed the DPMO figures were off and called me in for questioning. Only then did I discover that my data was a bimodal distribution, completely non-normal! I was severely reprimanded then, and it was only then that I realized how crucial this preliminary step is. Often, when we see a CPK report where the DPMO and CPK logic doesn't align, there's an 80% chance that the normality test was not passed.

One Thing You Can Do Today

Before your next statistical analysis, first run a Shapiro-Wilk or Anderson-Darling test on your data in Minitab or JMP, and check the P-value!

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