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Statistical Analysis6 min read

Kruskal-Wallis and Friedman Tests: Non-parametric ANOVA

This article is a true godsend for engineers, addressing a common challenge: how to compare the performance of different equipment or processes using semiconductor manufacturing data like yield and DPMO when it deviates from a normal distribution. It explains why the traditional ANOVA is often unsuitable in such scenarios and introduces non-parametric tests as a powerful alternative. After reading, you will be equipped to select the appropriate statistical method and make reliable data-driven decisions, avoiding misinterpretations of irregular data.

That day, when the CPK report came out, the whole room fell silent for three seconds, and then someone asked, "Can we even compare these?"

I remember two years ago, a new process was being introduced, and we wanted to compare the yield performance of three equipment vendors. It was the Q3 sprint, and everyone was under immense pressure. We ran three batches of products on each of the three machines. The results showed Vendor A's yield at 99.85%, Vendor B's at 99.82%, and Vendor C's at 99.83%. The data looked very close, but Ah-De, from process integration, immediately frowned and asked, "Can we compare these? These yield data are not normally distributed at all; is ANOVA accurate here?" The conference room instantly fell silent. I knew he was referring to "non-parametric tests."

What's the Problem? Why Can't We Directly Use ANOVA?

Simply put, the commonly used ANOVA (Analysis of Variance) has a crucial prerequisite: your data must be "normally distributed" and have "equal variances." However, in semiconductor manufacturing, your data often doesn't meet these conditions. For instance, data like yield or DPMO (Defects Per Million Opportunities) frequently exhibit skewed distributions. Or, if you only have "ranked" data, such as quality scores (1-5 levels), then ANOVA is even less applicable. If you insist on using it, the results might be no better than random guessing, and a beautifully written report would just be self-deception.

Therefore, the key is that when your data doesn't conform to a normal distribution or has unequal variances, we need to find "non-parametric" alternatives. The Kruskal-Wallis test and Friedman test, which we'll discuss today, are non-parametric versions of ANOVA for such situations. Kruskal-Wallis is used to compare median differences among three or more independent samples; Friedman is used to compare median differences among three or more paired samples (or repeated measures).

How Is It Actually Done?

Let's take the example of the equipment vendors mentioned earlier. Suppose you now have three equipment vendors A, B, and C, and each equipment ran five batches of products, with the following DPMO data:

  • Vendor A: 6210, 6350, 6180, 6290, 6320
  • Vendor B: 6400, 6520, 6380, 6450, 6500
  • Vendor C: 6300, 6280, 6330, 6250, 6270

First, you run a normality test in Minitab and find that these DPMO data are not normally distributed at all. In this situation, you cannot use One-way ANOVA; you must switch to the Kruskal-Wallis test.

  1. Data Preparation: Consolidate all DPMO data into one column, and in another column, label which equipment vendor it belongs to (A, B, C).
  2. Perform the Test: In Minitab, you will find `Stat` -> `Nonparametrics` -> `Kruskal-Wallis`.
  3. Interpret the P-value: If the P-value is less than your significance level (typically 0.05), it indicates that the DPMO median of at least one equipment vendor differs significantly from the others. If the P-value is greater than 0.05, it indicates no significant difference between the groups.

In other words, the Kruskal-Wallis test ranks all data together and then determines if there's a difference between groups based on these ranks. It compares medians, not means.

What about Friedman? Imagine you're not comparing different equipment vendors, but rather want to see if there's a difference in cleanliness scores (1-10 points, higher is better) after the "same batch of products" undergoes three different cleaning modes (A, B, C). Because it's the same batch of products, these are "paired samples," and in this case, you should use the Friedman test.

  1. Data Preparation: Place the cleanliness scores for the three cleaning modes into three separate columns, with each row representing a batch of products.
  2. Perform the Test: In Minitab, you will find `Stat` -> `Nonparametrics` -> `Friedman`.
  3. Interpret the P-value: Again, look at the P-value. If it's less than 0.05, it indicates a significant difference in the cleanliness medians among the three cleaning modes.

The Most Common Pitfalls

To be honest, I also fell into traps when I first started learning. The most common mistake is to "reflexively run ANOVA" whenever comparing multiple groups. Once, we were evaluating materials from different suppliers, wanting to see if there was a difference in material surface roughness. The data volume wasn't large, but because it was based on manual visual assessment, it couldn't possibly be normally distributed. Yet, I directly ran ANOVA, and the P-value was small, so I wrote the report with full confidence. Later, when my boss asked for the data distribution plot, I realized it was severely skewed, and my face went pale. My boss simply said, "Next time, check the data's 'appearance' first." After that incident, I learned my lesson. Before any statistical analysis, the first step is always to examine the data distribution plot to determine if it's normal.

Another pitfall is that while non-parametric tests are very useful, their "statistical power" is generally slightly lower than that of parametric tests (like ANOVA). This means that when a true difference exists, non-parametric tests might have a harder time detecting it. Therefore, if your data meets the assumptions of parametric tests, it's generally better to prioritize ANOVA. But if the data does not meet the assumptions, don't hesitate; Kruskal-Wallis or Friedman is your salvation.

One Thing You Can Do Today

Open your latest ANOVA report and examine the raw data distribution plot.

Want to try it yourself?

Every tool mentioned in this article is available on InsightFab — just upload a CSV to analyze.

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