Quality Control

Probability · Axiom Academy

Using the Central Limit Theorem to ensure product quality Imagine you manage a factory producing precision bolts. Your production line manufactures 50,000 bolts per day, and each bolt must have a diameter between 9.9mm and 10.1mm to meet specifications. The problem: You can't measure every single bolt—it would take too long and be too expensive. But if you let defective bolts slip through, customers will receive faulty products. This is where statistical quality control saves the day. By sampling a small number of bolts and using the Central Limit Theorem, you can make reliable decisions about the entire production run. Sampling from the Production Line Every hour, the quality control team randomly selects and measures 25 bolts. Let's simulate what happens over a typical 8-hour shift. Production Sample (n = 25 bolts per hour): Key observation: Notice how the sample defect rate varies from sample to sample, even though the true defect rate stays constant. This is sampling variability—and it's exactly what the CLT helps us understand. Quality engineers use control charts to monitor production over time. The chart plots sample means and shows control limits based on the expected variation from the CLT. How it works: The CLT tells us that sample means will follow a normal distribution centered at the true mean. We set control limits at μ ± 3σ/√n, which should contain 99.7% of all sample means if the process is working correctly.

This is the written version of the interactive lesson above. See the full Probability course.