X-bar R charts with varying subgroup size

Question

I am using control limits to check if a process is going out of control or not and detect the mean shifts over time.

I am aware of how to apply the control limits, but not really sure about the statistical background behind it.

Now that I have set the background, I'll come to the question: I have a process for which a lot of variables are measured, and every product that goes through the process is measured for these variables (since it is an automated testing).

1. So, now since I'm measuring all the products that pass through the process, am I looking at the population or the sample? 2. If it is a population, then the question is should I sample or not? I know X bar and R charts are based on sampling. But is that designed keeping in mind the computational and operational complexities of measuring each and every product and calculating? or is there a more statistical reason behind it? 3. Can I use in someway, all the observation in an Xbar R Chart? without sampling (cause if I choose a rational subgroup as a day, the subgroup sizes are variable and that won't fit in to the standard calculations 4. If it is a sample, then is my only option I-MR charts? sampling a sample doesn't make a lot of sense to me.

Edit:- After discussion with whuber, I have decided to rephrase my question.

Since I am measuring all the products that pass through my process, I have varying sub group size. I know I-MR charts is an option. But what about X-bar R charts

Is it possible to have the Xbar-R chart even with varying subgroup size?

• Is the constraint for not using variable subgroup sizes in Xbar-R chart only the sigma estimator?
• If we find a way around the sigma estimator and use the actual SD for each subgroup and then combine them to get overall SD, would that be a statistical blunder?

Any help is hugely appreciated. Even if you guys can direct me to the relevant literature, I can try and read up on the same.

Answer 1

After some research in to the topic, I have stumbled upon two journals which address this point. 1. Control Charts for Measurements with Varying Sample Sizes (Burr, Irving W.(1969, ASQC)) 2. Standardization of Shewhart Control Charts (Nelson, Lloyd S.(1989, ASQC))

An excerpt from the Nelson (1989) below:

When subgroup sizes differ there are three approaches usually recommended.
1. Draw the actual control limits for each subgroup separately.
2. Use the average of the subgroup sizes and calculate limits based on this >average size, and calculate the exact limit whenever doubt exists.
3. Standardize the statistic to be plotted and plot the results on a chart with >a centerline of zero and limits at ±3.

He goes on to explain why 1 and 2 are not the ideal way to do:

The first alternative may yield not only a messy chart but also one to which runs tests cannot be applied—specifically the trend and zigzag tests

The second alternative can also have the kind of problem just described. Further, one must be ready to calculate the exact limit when the approximate one is called into question

The third alternative, standardization, yields a neat chart for which interpretation is not a problem. The centerline is always at zero (although it is desirable to indicate on the chart the value of the mean of the original data), and because the vertical scale is a “sigma scale,” the zones for carrying out tests for special causes are always at ±1, ±2, and ±3.

The formulae for implementing the Standardized approach is in the below Table.

For more details, please refer to the mentioned research papers I hope this will come in useful for someone who stumbles across the same scenario as mine.