How to Choose the Right Control Chart for Statistical Process Control

    For characteristics which are evaluated with variable data, AIAG’s SPC manual recommended 4 types of control charts:

1. Average and Range Chart (X ̅-R Chart)
2. Average and Standard Deviation Chart (X ̅-s Chart)
3. Median and Rang Chart (X ̃-R Chart)
4. Individuals and Moving Range Chart (X-MR Chart)

    Arising from the 4 options, a question that many SPC beginners may have is: which control chart should I use for my process? The answer can be found in P.177 of the 4th edition of the SPC manual, but it is very brief. Below is a more detailed explanation about the selection procedure of the right control chart:

    First, let’s take a look at the X ̅-R Chart, which is probably the most popular chart. For this chart, the control limits of X ̅ and R are calculated with the following equations:

where R ̅ is the average of ranges of individual subgroup. From the above equations, it can be seen that when R ̅ is 0,and. The control limits make no sense in such case. That means if one needs to use X ̅-R chart, there must exist detectable within-subgroup variation. The above equations also show that the width between UCL and LCL of X ̅  is. In order not to see too many undesired out of control signals, the X ̅-R Chart should only be used when the between-subgroup variation is insignificant compared with within-subgroup variation. Otherwise, the future data point of X ̅ may easily go out the control limits due to big between-subgroup variation (Note: generally, big between-subgroup variation, compared with within-subgroup variation, is not accepted as it indicates the existence of special causes. But if both Cpk and Ppk are significantly larger than the requirement, e.g. Cpk=8, Ppk=5, it may be accepted). As a summary, to use X ̅-R chart, the within-subgroup variation must not be 0 and the between-subgroup variation must be insignificant compared with within-subgroup variation.

    Here is an example when the X ̅-R chart should not be used: The process is to produce a chemical solution by adding solvent into the distilled water. It is produced one bath a time, and one bath is one lot. The monitored characteristic is the composition of the solution. As the solution is homogenous, taking several samples from one bath as one subgroup gives no within-subgroup variation. On the other hand, the difference between different lots may be more significant, as the amount of solvent added to each bath may differ. So using X ̅-R chart is inappropriate for this process.

  In the above case, one should use the X-MR chart. The X-MR chart requires no subgroups (or you may consider that the subgroup size is 1). Individual sample is taken, and the difference between consecutive samples is calculated to estimate the process variation and the control limits, as in the following equations:

where (MR) ̅ is the average of the differences between all consecutive samples.

    X ̅-s chart is close to X ̅-R chart. The key difference is that the process variation in this chart is estimated with the average standard deviations, instead of average ranges, of the subgroups, and the control limits are calculated with the following equations:

where s ̅ is the average of the standard deviations of individual subgroup. Similar to X ̅-R Chart, X ̅-s chart should only be used when within-subgroup variation is detectable, and the between-subgroup variation is insignificant compared with within-subgroup variation.

Since X ̅-s chart is very close to X ̅-R chart, when should they be used respectively? In fact, X ̅-s is always preferred over X ̅-R chart, as the process variation σ ̂_X ̅  is better estimated with average standard deviation of subgroups than average range of subgroups. One may consider using X ̅-R chart only if the SPC cannot be done with computer and hence calculation of standard deviation is inconvenient. But if the subgroup size is 9 or more, the estimation of σ ̂_X ̅  with average range of subgroups becomes too poor, and X ̅-s chart should be used in this case, even the calculation of standard deviation is difficult.

    X ̃-R chart is also close to X ̅-R chart. But in this chart, median, instead of average, of each subgroup is monitored. Comparing X ̃-R chart with X ̅-R chart, the latter is preferred, since for process control, one cares about the location of the process average, which is better estimated with X ̅ than X ̃. X ̃-R chart should be used only if the calculation of subgroup average is not convenient, e.g. when doing SPC on a paper.

    As a summary, when choosing the right control charts, one needs to follow the below priority:

1. Average and Standard Deviation Chart (X ̅-s Chart)
2. Average and Range Chart (X ̅-R Chart)
3. Median and Rang Chart (X ̃-R Chart)
4. Individuals and Moving Range Chart (X-MR Chart)

i.e. X ̅-s chart should be always considered first, but if the subgroup size is less than 9 and the calculation of standard deviation is not convenient, X ̅-R chart can be used as an alternative. And if the calculation of subgroup average is also not convenient, then X ̃-R chart can be used instead of X ̅-R chart. All the above control charts require that subgroups with detectable within-subgroup variation can be obtained, and the between-subgroup variation is insignificant compared with within-subgroup variation. If it’s not possible to obtain such subgroups, one can then use X-MR chart.