# Quality control - non-normal distribution

Suppose we have a production line, which makes widgets. Assume also that we know distribution of widget's length ~ $N(\mu, \sigma)$ We measure widget's length and if it is within $\mu+/- 2 \sigma$ we let it pass, otherwise, we consider the part defective. This is reasonable approach when distribution is normal, but what should we do if distribution is not normal, suppose distribution is so bad that it does not even have density function.

Similar, if we measure width and length of a widget, in normal case we can get an elliptical region for quality control, but what do we do in non-normal case.

So far I found that it is common to transform distribution to normal. Are there any other approaches?

Am I over-complicating the situation?

• +1. But I am wondering what is so bad about distributions without density functions? They crop up all the time--any distribution for counts is an example--and often are easy to work with. I would also question your assumption that a $\pm 2\sigma$ limit is appropriate. That would have a terrible ARL and therefore be useful chiefly in situations where the process variation cannot be improved. – whuber Jun 11 '15 at 21:45
• The underlying question "what makes a part defective?" isn't a statistical one. You would need to consider that question in formulating a policy about classifying parts as defective. – Glen_b -Reinstate Monica Jun 12 '15 at 4:10
• I agree, my question is not well posed. I guess what I am interested in is the fact that for normal distribution we can make statements like 95% of observations will lay with in $\mu \pm 3 \sigma$, while for some other distributions this statement is not always as meaningful. There is some sort of sense of direction present in normal distribution. – user1700890 Jun 12 '15 at 13:45

It looks like you are confounding Capability and Statistical Process Control.

Shewhart Charts or Control Charts typically plot what is assumed to be Gaussian data with control limits of $\pm3s$. Control limits are not used to declare a part defective, they are used to help determine when a process is in "statistical control." In fact, if a chart was plotted of your process as described, the chart would detect tampering by identifying that no points of data were beyond $\pm2s$. The process should then be examined to discover the source of the tampering.

Capability is a measure of process performance vs. specification levels or tolerances. In this instance, a part must meet the requirement or it is declared defective.

Processes can be capable but out of control; they can also be incapable but in control.

For capability, the National Institute of Science and Technology recommends the use of $C_{npk}$, the non-parametric $C_{pk}$, for non-normal data. This modified index is defined as: $$C_{npk}=\min \left\{ \frac{USL-\tilde{x}}{X_{99.5\%}-\tilde{x}},\frac{\tilde{x}-LSL}{\tilde{x}-X_{.5\%}}\right\}$$ Where $X_{99.5\%}$, $\tilde{x}$, and $X_{.5\%}$ are the 99.5th, 50th, and .5th percentiles of the data based on the underlying distribution, respectively.

For control charts, you need to either transform the data with a process such as Box-Cox (the common approach), or create a chart with lines based on $X_{99.5\%}$, $\tilde{x}$, and $X_{.5\%}$ of the distribution the data is from (which is equivalent to $\overline{x}\pm3s$ in a normal distribution (much less common approach).