# On properly setting up a control chart

Let me present to you the way I have being 'doing' control charts up until now.

I have a series of $k$ samples of size $n$. I calculate the the sample means $\bar{x}_i$ and the grand mean $\bar{\bar{x}}$ --- the mean of the sample means (or the mean of all the $k\times n$ data points). I also calculate the sample ranges $w_i$ as well as the grand range $\bar{w}$.

Now I use this data to produce what is an approximate 95% confidence interval with AKA inner control limits:

$$\bar{\bar{x}}\pm A'_{0.025}\bar{w},$$

where $A'_{0.025}$ is read from a table and depends on the sample size $n$. Similarly, I produce a 99.8% confidence interval with AKA outer control limits: $$\bar{\bar{x}}\pm A_{0.001}'\bar{w},$$ where again $A'$ depends on $n$ and is read from a table.

Now I set up my control chart with the grand mean and both pairs of inner and outer limits and I plot the sample means and depending on where the sample means lie we make decisions.

The rules we have declared are 1. one sample outside the outer control limits and we have a problem 2. two consecutive samples outside the inner control limits and we have a problem

Now I have read some answers to other questions and I want to say that I have no problem making assumptions about my data (strictly speaking this is a math class).

However I have a number of questions and problems with the way I have being doing things.

1. Do I understand out of (statistical) control? If not is there a difference between being out of statistical control and out of 'quality' control. --- my understanding is that a process is out of control when a change affects the statistics of the process. This change in turn could be something we can affect or not although I am not concerned: all I will say is that the process should be stopped and examined if it is out of control. This leads onto my next question.
2. We use the possibly-out-of-control data to decide what is normal --- --- my understanding is that under the assumption that the population mean is equal to the grand mean we should see our sample means within the control limits with the relevant probability. However, this process could be 'out of control' as I understand it. This could mean two things (a) the population mean is changing or (b) the population mean has changed. In both scenarios I see the out of control data feeding back and essentially saying what is normal both in scenario (a) where the range $\bar{w}$ is potentially big or worse (b) where the grand mean is not estimating the population mean. I would prefer that the the control chart parameters are based on historical or aspirational data.
3. Is it not backwards? --- surely what should happen is that there is a priori a control chart and a worker does the sample, reacts in 'real time' to adverse findings and then stops the process --- or is this more quality control/process capability.
4. In the field? --- how are control charts actually done in the field (at a low level, neglecting trend, stratification, etc). The people learning about this are training to be civil engineers --- although I also do it with Industrial Measurement Control classes (who I am sure see more of it in their other classes beyond maths).
5. In other learning centres? --- is this still taught? What approach is typically taken? Again I am not going to be going into trends, cycles, etc.
6. Should I just teach quality control --- or does nobody do this? --- would it not be a control chart if I worked off historical or aspirational data.

Thank you very much for your help on this, I appreciate it. Not happy teaching it just by plug and chug --- it has to make sense the way we are 'doing' it and at the moment it doesn't.

• It would be incorrect to refer to the intervals as "confidence intervals" (even approximately). They are control limits. You can connect them to interval estimation by characterizing them as approximate prediction limits; indeed, prediction limits are used for quality control purposes. So, if you are using confidence limit tables to compute the $A'_\alpha$, you need to revisit that.
– whuber
Commented Dec 13, 2013 at 16:34
• Under the regime below they certainly are not confidence intervals. Precisely because they were approximate confidence intervals under the regime explained above I knew we were doing it wrong. Thank you for your comment. Commented Dec 13, 2013 at 16:46