Why is not 100% valid for FPC 100% test

It is a common practice to conduct 100% inspection in order to avoid non-conforming products being shipped. Every product produced is inspected and judged to pass or fail. Goods will be shipped and non-conforming items will be left for repair or scrap. It's all simple and straightforward. It seems that 100% inspection is a very effective method, regardless of the errors of the inspectors.

We assume that the characteristic values of the product are normally distributed, denoted by Y, as shown in the figure below.

The qualification standard of the product is known, and the specification values of the upper and lower limits have also been given, such as LSL and USL in the above figure. Products between LSL and USL will be eligible products. But we don't know the actual Y value, we need to use a measurement system to measure it, the measured value of Y is called X.

The measured X value is not equal to Y, because there is a measurement error, which we call E. Because the measurement error is also normally distributed, we obtain the distribution of the measurement value as shown in the figure below, which is similar to the distribution shape of Y, but the variance of X is larger, equal to the variance of Y plus the variance of the measurement error E.

Next we use the bivariate normal distribution to show the correlation between Y and X.

The probabilistic model of this distribution can be represented by a series of ellipses in the XY coordinate plane. The figure below shows one, two, and three standard deviation contours of two bivariate normal distributions, where the intraclass correlation coefficients are set as 0.95 and 0.80.

As can be seen from the above figure, as the measurement error increases, the ellipse of the bivariate normal distribution becomes fatter and the main axis is skewed. What we are interested in is: which range of measurements X corresponds to a qualified Y.

The white range in the figure below indicates the range of qualified products

The icon below shows the range of products shipped after 100% inspection.

At this point, we have clearly seen that the range of products shipped after 100% inspection is different from the range of actual qualified products.

The range of products that pass 100% inspection ≠ the range of qualified products

We classify all products into four categories based on the quality of the products and whether they are shipped or not:

Qualified inspection products (GS: Good and Shipped)

Qualified inspection rejected product (GR: Good and Rejected)

Unqualified products passed the inspection (BS: Bad and Shipped)

Unqualified inspection rejected product (BR: Bad and Rejected)

GS vs BR is the result we want to see, BS will cause trouble for customers and GR will cause trouble for itself.

The following figure shows the position of each category in the bivariate normal distribution.

For the producer, he hopes that the proportion of qualified products shipped (PGS: Proportion of good product that is shipped) is as high as possible:

PGS = GS / (GS + GR1 + GR2)

For the customer, what he wants is that the proportion of bad products that are rejected (PBR: Proportion of bad product that is rejected) is as high as possible:

PBR = (BR1 + BR2) / (BR1 + BR2 + BS1 + BS2)

Unfortunately, PGS and PBR are not linearly related.

Below is an evaluation table presenting the values of PGS and PBR for different levels of measurement error and the proportion of overall nonconforming product.

Interpret the set of data in the bottom right corner of the above table: when the overall product failure rate is 1% and the measurement error is large (ICC = 0.8), the producer will have a 98.6% probability of shipping qualified products , but the proportion of defective products rejected is only 70%, and a large number of defective products will flow to customers.

How to solve this problem? There seem to be only two ways.

The first method: adhere to the 100% test, then improve the measurement system, so that the ellipse in the bivariate normal distribution is flattened like a straight line, then both PGS and PBR will be very high.

The price is: you need a near-perfect measurement system, which often requires a huge investment that doesn't solve the problem, it just leaves it behind.

The second method: Improve the process capability so that the output of the process is within the upper and lower limits of the specification. At this time, 100% inspection is no longer required, which can not only save the investment in upgrading the test system, but also save the cost of inspection, and more. The point is to actually solve the problem.

So far, our conclusions are very clear:

The existence of measurement system error makes 100% inspection unable to have 100% validity;

100% inspection cannot solve quality problems, and it will also require unnecessary equipment investment and personnel inspection costs;

The measurement system should be used to substantially improve the quality and consistency of the production process, not for inspection purposes