## Statistical Process Control

Read this chapter on the basics of statistical process control (SPC). SPC is a standard tool for monitoring whether a process is performing as expected and, if not, where problems occur. While reading, consider how this kind of tool factors in process capacity management.

### Sample Size and Subgrouping

There are a few key conditions that must be met when constructing control charts:

• The initial predictions for the process must be made while the process is assumed to be stable. Because future process quality will be compared to these predictions, they must be based off of a data set that is taken while the operation is running properly.
• Multiple subsets of data must be collected, where a subset is simply a set of n measurements taken over a specific time range. The number of subsets is represented as $k$. A subset average, subset standard deviation, and subset range will be computed for each subset.
• From these subsets, a grand average, an average standard deviation, and an average range are calculated. The grand average is the average of all subset averages. The average standard deviation is simply the average of subset standard deviations. The average range is simply the average of subset ranges.

The upper and lower control limits for the process can then be determined from this data.

• Future data taken to determine process stability can be of any size. This is because any point taken should fall within the statistical predictions. It is assumed that the first occurrence of a point not falling within the predicted limits shows that the system must be unstable since it has changed from the predictive model.
• The subsets are defined, based on the data and the process. For example, if you were using a pH sensor, the sensor would most likely output pages of data daily. If you know that your sensor has the tendency to drift every day, you might select a 30 minute subset of data. If it drifts monthly you might set your subset to be 24 hours or 12 hours.
• Finally, the population size, $N$ is assumed to be infinite. Alternatively, if the population is finite but the sample size is less than 5% of the population size, we can still approximate the population to be near infinite. That is, $n/N < = 0.05$ where $n$ is the sample size and $N$ is the population size.