The following problem enables students to see how the central limit theorem is applied in a manufacturing setting.

A manufacturer of industrial sewing machines is struggling with a problem involving "cam rollers," a particular precision ground part. In a production process such as this, we need to monitor the extent to which the product meets specifications. Statistics can be used to provide on-line quality control procedures to monitor an on-going production process and address the problem of deviations from target specifications. We extract samples of a certain size from the ongoing production process. We then produce line charts of the means (averages) in those samples, and consider their closeness to target specifications. If a trend emerges in those lines, or if samples fall outside pre-specified limits, then we declare the process to be out of control and take action to find the cause of the problem. These types of charts are sometimes also referred to as Shewhart control charts (named after W. A. Shewhart, who is generally credited as being the first to introduce these methods).

For example, an "X-bar chart" contains upper and lower control limits that indicate when the process is out of control. The method for constructing the control limits is a straightforward application of the concept of the sampling distribution and the characteristics of the normal distribution. In particular, if the mean (and variance) of the process does not change (i.e., the process is "in control"), the central limit theorem tells us that successive sample means will be distributed normally around the population mean. Moreover, we also know that the distribution of sample means will have a standard deviation of SD (the standard deviation of individual data measurements) divided by the square root of n (the size of the samples). Thus, for example, approximately 68% of the sample means will fall within approximately ±1×SD/√(n) of the center line, approximately 95% of the sample means will fall within approximately ±2×SD/√(n) of the center line, etc. By setting the control limits a particular distance from the center line, we can thus assess the probability that an "in control" process will go outside these control limits. Then, by making this probability very small (which determines how far from the center line to put the control limits), any time the process does go outside the limits, chances are it has gone out of control.

For example, consider sampling five rollers from each production batch once each day over a 30 day period, and measuring the diameters to the nearest 0.000001 inch. We can then use statistical software to produce the following "X-bar chart" from the data:

- The horizontal axis represents the different samples, while the vertical axis represents the means of the five roller diameters in each sample. The central horizontal line represents the desired target diameter of the rollers in inches. Which statistical population parameter is this analogous to?
- In addition to the center line, the chart includes two additional dashed horizontal lines to represent upper and lower control limits (UCL, LCL, respectively). In particular, these control limits are at ±3×SD/√(n) from the center line (this is indicated by "Sigma level: 3" below the chart). Thus, in order to calculate where the control limits should be drawn on the chart, SD needs to be estimated. How might this have been done in this case?
- For this control chart, what must the value of SD have been?
- The individual points in the chart, representing the samples, are connected by a solid line. If this line moves outside the upper or lower control limits (dashed lines), then a quality problem may potentially exist. What percentage of sample means should fall within the limits when the process is in control?
- Does it seem likely that the process went out of control during this 30-day period, and if so, when did it go out of control?

*Last updated: April, 2012*

*© 2012, Iain Pardoe*