Shewhart Charts

While working for Western Electric in the 1920s, Dr. Shewhart developed a general, practical approach to statistical monitoring of manufacturing processes. He advised managers on implementing these within Western Electric and later published his work in Shewhart (Montgomery, 1931). All Shewhart control charts share several characteristics in common. First, the horizontal axis represents time or lot sequence, but they all have different vertical axes, depending upon the chart time.

Next, all Shewhart control charts have a center line that is drawn parallel to the time axis. This represents the mean of the process, but the value of the process mean can vary depending upon which data are first used to design the chart. In some cases it will be the mean of the data plotted, in others it could be the mean calculated from a much larger number of measurements on the historical operation of the process.

Lastly, all Shewhart control charts have lines drawn to represent either the upper or lower control limits. In most cases both control lines are present, in others where the data have a natural bound, such as zero, only one of these control limits might be drawn.

Shewhart control charts are also broadly classified into two groups: variable and attribute data charts. Variable control charts are used when the quality of interest is a continuous variable, such as the diameter of a valve. If w is a continuous measure of a quality of interest, with mean μw and within-sample standard deviation σw, then the center line is at μw and the upper and lower controls limits are at μw ± kσw. Typically k =3 and the charts are called 3-sigma control charts.

Attribute control charts are used when qualities, not quantities are measured. For example, items may be characterized as conforming or nonconforming to a specification. Items may also be characterized as defective or nondefective. Examples of attributes include the number of failures in a manufacturing run or the number of defects on a computer chip wafer.

P-charts and np-charts are used for plotting the percentage or number of defective items in a sample, respectively. Inspectors, for example, might inspect 100 items in a subgroup and record the number of items with one or more defects. These can be plotted using either the number of defects or the percentage of defects found in each subgroup. In JMSL, class PChart produces p-charts p-charts for plotting the percentage of nonconforming items, and class NpChart produces an np-chart np-chart for the number of nonconforming items. In general, when the subgroup sample sizes are unequal a p-chart is used to adjust for unequal samples; otherwise an np-chart is used to display the number of defective items.

In some cases, instead of recording the number of defective items, the total number of defects of all types might be recorded. For example, the packaging for a semiconductor might be inspected for solder defects such as bridging, insufficient solder, bent leads, etc. The total number of all such defects might be recorded for a sample. Theoretically, there is no upper bound on the total number of defects found in a sample. These are typically plotted using a c-chart or u-chart, depending upon whether the sample size or area is constant from one subgroup to another. In JMSL, c-charts are implemented in class CChart and u-charts are implemented in class UChart.

If a single item can have multiple defects then a CChart or UChart is used, depending upon also, whether the area inspected for defect(s) is consistent or varying. An example of multiple defects per item would be the count of the number of scratches on mirrors. If all samples have an equal opportunity for defects use CChart, otherwise use UChart. So to monitor the number of scratches on mirrors use CChart when all mirrors being made are the same size and use UChart for mirrors being made that are sized differently.

In JMSL the ShewhartControlChart class is the base of a number of classes; it is not usually used by itself. Most of the charts in this chapter extend ShewhartControlChart.