STATISTICAL INFERENCE

STATISTICAL CONTROL CHARTS

Control Charts For Variables Data

1. In order to get natural limits for data from a process, you need to estimate the variability of the distribution of the data (if it exists).
2. The occurrence of special causes will disturb the process and your estimate of variability.
3. Observations taken close together in time (or space) are likely to differ only because of common cause variation, since special causes tend to occur more rarely, at irregular intervals.
4. The process variance reflects only common cause variation. It is a measure of the variability of the process when it is operating 'normally', without the influence of extraneous special causes. You should estimate the process variance from a sample of observations taken close together. But if the observations are too close together they may be highly correlated, and we could get an underestimate of common cause variability. Thus there is some skill involved in choosing this group of observations, which we call a rational subgroup (following Shewhart's terminology).
5. The averages of the rational subgroups will have a more normal-like distribution than the raw data because of the Central Limit Theorem. They will have a standard error of where s is the process (common cause) standard deviation and n is the subgroup size .
6. Therefore if you can estimate the standard deviation s of the process the averages of subgroups of size n will be within one, two and three of the process mean roughly 68%, 95%, and 99.7% of the time.
7. You can estimate s and hence get a standard error for the subgroup averages, by averaging all of the estimates of s from different subgroups. You can estimate the process mean by averaging all of the averages from different subgroups.
8. If any subgroup average deviates too far from the overall average (by more than 3 standard errors, say) then that subgroup is likely deviating due to a special cause and you should seek to uncover the root cause and remove it. The best way to do this is through teamwork - form a team of appropriate people to improve the process.

Rational Subgroups

It is important that we try to choose our subgroups in such a way that common cause variability is represented in within subgroup variation, but special causes occur between subgroups. (Shewhart called this a rational choice, and coined the term 'rational subgroups' which has remained in vogue.)

It is important to think carefully about the sampling scheme, and use it to decide what variation is represented by differences within the subgroup, and what variation causes differences between subgroups. You may sample differently depending on the purposes of the study.

Example - Variation In Percent Solids Test

The team took a sample, split it into three parts, and performed the test which included heating in an oven on each subsample. They did this for seventeen samples. The natural rational subgroup is the triplicate sample. Since this is a measurement study, you want to eliminate as much variability of the product as possible: all of the samples came from the same source. (The team wanted to use a control chart to signal variation in the measurement process, not in the material being measured.) Further, triplicates are just a splitting of one sample. Differences within the triplicates (within-subgroup variation) are likely to be due to differences in handling and weighing the three parts, and differences within the oven, if they were not placed close together. Differences between subgroups would be due to different oven runs, different samples (designed to be small), degradation of materials or drift in equipment over time, change in sample preparation method, etc. If the purpose of the study had been to investigate the manufacturing process, the samples would have been taken from the process, not from a single source, and they would have been spread out enough in time or space so that "normal" variation in product would have occurred within a subgroup.

Arrange the subgroups so that variation which is considered normal process variation will have an opportunity to occur within the subgroup, while special causes which you want to detect are likely to occur between sampling the subgroups.

How large should rational subgroups be? Walter Shewhart found that subgroups of four or five worked well in a variety of situations; these tended to be applications involving discrete items which occurred frequently. Data which occurs infrequently (chemical batches, accounting data, administrative data), motivate charts of individual values - see below. When subgroups are large, nonnormality is not a problem, and also smaller special causes can be detected - if they persist long enough to alter substantially the mean of at least one subgroup. When individual observations are charted, large but transient special causes are easier to detect, but ensuring approximate normality of individual values is important.

Note: Control charts can be constructed for the process mean (x-bar charts) and the process variability (R and S charts which plot the range and standard deviation). The emphasis in this course will be to construct and interpret x-bar charts.