Sample Sizes and A2, A3, B4, d2 Factor Parameters


kristof nuyts


I'm a student Industrial Engineer Elektromechanics in Belgium. For a thesis I have to make a study about Statistical Process Control.

Recently I've encountered a small problem. Suppose we have an on-line process and every second we get the value of a measurement. We calculate the moving average and plot it on a chart. The problem is which samplesize I have to take. Usually people take a sample size of 2 but is there a formula to calculate the sample size?

I have noticed that in some books the parameters A2, A3, B4, d2,... are just written down for sample sizes from 2 to 25. Do these parameters exist for larger sample size or isn't it useful to use these. In other words, is it still useful for plotting the points on a chart with a sample size of (for example) 28.


Kristof N.


Fully vaccinated are you?
If you get 1 measurement every 1 second, how many pieces are going through in 1 second?


Kristof - I am not a statistician so this is only information from my limited knowledge. At least we will get the discussion going.

Assumption - you are running an X-bar & R chart.

It depends on why you are using the chart as to the large sample sizes being useful. If you are trying to control process variability for example, you wouldn't want a large sample size. There is a number of 12 somewhere in the back of my head as the point where the range efficiency drops rapidly.

For large sample sizes you would probably want to run an X-bar & s chart.


Rick Goodson


Let me follow up on Dave's comments.

There are three components of variation in a part, within-piece, piece-to-piece, and time-to-time variation. These are composed of process, material, operator, and miscellaneous components. There are typically two schemes for selecting subgroup samples:

1) Select subgroup samples from product produced at one instant of time or as close to one instant as possible - instant time method.

2) Select product produced over a period of time so that it represents all products - period of time method.

The instant time method will have a minimum of within part variation and a maximum of variation among subgroups. Period of time will have a maximum of within part variation while a minimum of subgroup to subgroup variation. Instant time is the most common used method.

Deciding on the subgroup size is more of a management than a technical decision. As the subgroup size increases, the control limits become closer to the central value which makes the subgroup more sensitive to small variations in the process average. Also, as the subgroup size increases the inspection/test cost increases. When the subgroup size exceeds 10, a sigma chart is used instead of the range chart. Sigma is more sensitive but more difficult to calculate. Range is normally used simply for calculating simplicity.

There is no rule for frequency of subgroups. The inconveniences of factory layout and the cost involved in taking the subgroups have to be balanced against the value of more frequent information. A rule of thumb is to used a variables sampling plan (such as the old MIL-STD-414) to determine the sample size based on daily production. For example, if you produce 3,201 to 8,000 units per day you would take 60 samples per day. If the subgroup size is 5, then the number of subgroups per day would be 60 divided by 5 or 12 subgroups.

With regard to the A2, A3, B4, d2 factors, you need to understand the derivation of the factors to understand why the tables stop at 25. The formula substitutions do not work well in the text based forum format otherwise I would try to show it to you. If you can get hold of a copy of Acheson Duncan's book, Quality Control and Industrial Statistics you can find the background material. Never the less, for the A2 factor for a sample larger than 25 the formula is 3 multiplied times d2 divided by the squart root of the sample size (n). The d2 factor for samples larger than 25 is equal to R divided by sigma. My formula come from an American Society for Testing and Materials standard, ASTM-STP-15D.

Hope this helps.



kristof nuyts

Thanks to all of you for the answers.

With regard to the A2, A3, B4, d2 factors, Rick wrote me to get a copy of Acheson Duncan's book, Quality Control and Industrial Statistics. Too bad I'm just a (poor) student working on a thesis who hasn't got the money for ordering it.

Maybe someone knows a website where I can find the formulas for all these factors.



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