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Statistical Analysis6 min read

Bootstrap: Statistical Inference for Small Samples

This practical article addresses a common manufacturing challenge: diagnosing process issues and interpreting Cpk with limited sample data. It explains how to accurately identify process problems from small datasets, demonstrating that reliable solutions can be derived even when sample sizes are insufficient for traditional statistical methods.

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

Do you remember? A few years ago, our production line introduced a new process, causing everyone a lot of trouble. One day, the PM suddenly rushed over, his face ashen, saying, "The yield has dropped to 92%, much lower than the expected 95%!" My heart sank, knowing overtime was inevitable. To find the problem, we pulled a batch of only 10 samples for analysis. The results showed a Cpk value of only 1.08, and DPMO surged to 6210. As soon as the report was released, the conference room instantly fell silent, everyone exchanged glances, because these numbers were so different from what we usually saw, making it impossible to determine the root cause of the problem.

What Was the Problem? Your Sample Size Was Not Large Enough!

To be honest, in semiconductor fabs, it's often impossible to use hundreds or thousands of wafers for experiments. Just the cost and time alone would have the boss pinning you to the wall. At this point, you might only have ten or twenty samples, yet you must infer the overall process status from this limited data. How do you know if that Cpk of 1.08 from 10 wafers truly reflects the overall yield, or if you just happened to pick a worse batch? This is the most frustrating aspect of "small sample statistical inference."

In other words, when your sample size is not large enough, traditional statistical methods will yield inaccurate conclusions due to excessive error. You calculate an average from those 10 pieces; how "credible" is it? Would you really dare to report this number to your boss and then decide whether to shut down production? Of course not!

How to Do It in Practice? Use "Bootstrap" to Generate More Samples!

This is where a technique called "Bootstrap" comes in handy. It sounds fancy, but essentially, it means "generating your own samples." How exactly is it done?

  1. Copy, Paste, Recombine: Imagine you have the data from those 10 wafers. You treat these 10 data points as a "population." From these 10 data points, you randomly draw 10 data points "with replacement" to form a new sample. Note that because it's sampling with replacement, some wafer data might appear multiple times in the new sample, while other data might not be drawn at all.

  1. Repeat Millions of Times: You repeat step 1 thousands or tens of thousands of times. Each time, you get a new "pseudo-sample" consisting of 10 data points.

  1. Calculate Statistics: For each "pseudo-sample," you calculate the statistic of interest, such as the mean, standard deviation, or Cpk.

  1. Establish a Distribution: This way, you will obtain thousands or tens of thousands of Cpk values. These Cpk values form a "simulated distribution." This distribution allows you to better understand the possible range of the true Cpk.

For example, the Cpk of our 10 wafers was 1.08. Through the Bootstrap method, we might simulate 10,000 Cpk values and then find that these Cpk values have a 95% probability of falling between 1.0 and 1.25. This allows you to say with more confidence: "Although I only have 10 data points, based on the current results, I expect the overall Cpk to have a 95% chance of falling within this range." Isn't this a more solid basis for judgment than just looking at a single point of 1.08?

The Most Common Pitfall: Mistaking "Random" for "Haphazard"

I remember once, a new rookie engineer was very excited after learning the Bootstrap method. He directly copied and pasted those 10 data points 10 times, turning them into 100 data points, and then used them as a sample for analysis. He thought this was "generating more samples." I almost fell off my chair when I saw his report!

Frankly, this completely missed the point. The essence of the Bootstrap method lies in "random sampling with replacement," which is necessary to simulate the inherent uncertainty of the data itself. Simply copying data is equivalent to repeatedly amplifying the same errors, providing no new information whatsoever. Your "new samples" are actually just rigid repetitions of the original 10 data points, and the calculated statistics will be identical to your initial results, rendering them utterly useless! Therefore, always remember the keywords "random" and "with replacement."

One Thing You Can Do Today

Open your Excel or R/Python, use your small dataset, and run a Bootstrap analysis to see how your statistics are distributed.

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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