If you have worked with count charts with large denominators, you have probably seen control limits that seem too narrow to be of much value. The p-chart is one of the attributes charts with this flaw.
A p-chart counts two things: 1) the number of non-conforming items (the numerator) and 2) the number of items inspected (the denominator). If you look at the glass half-full rather than half-empty, you might count the number of conforming items (rather than non-conforming). In either case, when the denominator is large, a problem may be present.
Consider the following chart which shows a p-chart from a plastic shopping bag manufacturer.
This chart measures the percent of plastic bags that failed a particular test. The bag manufacturer averages about 20,000 bags inspected in each sample and about 600 failures. As you can tell by looking at this chart, the limits seem too tight to be useful. The control limits are considered to be overly-dispersed.
One solution to the problem of overly-dispersed control limits is to display the percentage data on an individuals chart, as the chart below shows.
This chart is created by calculating each sample’s percentage of non-conforming bags and then displaying this calculated percent on an Individuals (X) chart. This technique is widely recommended and, as you can see from the chart above, is more useful than using the traditional p-chart. The primary problem with this approach occurs when the denominator changes by a significant amount in each sample. The control limits do not adequately adjust for excessively small or large numbers in the denominator. So, an individuals chart is better, but what approach is best?
The chart below demonstrates the best way to deal with count data that is overly-dispersed.
This chart type combines advantages of both the traditional p-chart with its changing control limits from subgroup-to-subgroup, and the individuals chart with its more appropriate control limits. This chart type is called a p’chart (p prime). In this example, the p-prime chart exposes three out-of-control points that were missed using the individuals chart.
Note how the control limits seem appropriate and how they adjust periodically. The adjustment comes from the fact that some samples have smaller or larger denominators compared to the average denominator. Thus, the control limits compensate correctly for small and large amounts of data.
The p-prime chart is a must for anyone with overly-dispersed data. If you aren’t sure if you have overly-dispersed data, simply run a test in CHARTrunner SPC software. CHARTrunner version 3.6 has a button called “Test for overdispersion.” With one click, you will know whether or not you should switch your “traditional” attributes chart to one of the prime attributes charts.