spctoolkit

by Donald J. Wheeler


Charts for Rare Events


Counts of rare events are inherently insensitive and weak.


Your imagination is the only limitation on the use of Shewhart's charts. They are such a basic tool for analyzing and understanding data that they can be used in all kinds of situations-the key is to see how best to apply them. This column will illustrate how to use Shewhart's charts to track rare events.

Department 16 has occasional spills. The most recent spill was on July 13. Spills are not desirable, and everything possible is done to prevent them; yet they have historically averaged about one spill every seven months. Of course, with this average, whenever they have a spill, they are 690 percent above average for that month. When dealing with very small numbers, such as the counts of rare events, a one-unit change can result in a huge percentage difference.

Counts of rare events would commonly be placed on a c-chart. (While the c-chart is a chart for individual values, and while most count data may be charted using an XmR chart, the XmR chart requires an average count that exceeds 1.0. The c-chart does not suffer this restriction.) The average count is found by dividing the total number of spills in a given time period by the length of that time period. During the past 55 months, a total of eight spills occurred, which gives an average count of:

c bar = 8 spills/55 months = 0.145 spills per month

This average will be the central line for the c-chart, and the upper control limit will be computed according to the formula:

UCLc = c bar + 3 (square root of c bar ) = 0.145 + 3 (square root of 0.145 ) = 1.289

The c-chart is shown in Figure 1. In spite of the fact that a single spill is 690 percent above the average, the c-chart does not show any out-of-control points. This is not a problem with the charts but rather a problem with the data. Counts of rare events are inherently insensitive and weak. No matter how these counts are analyzed, there is nothing to discover here.

Yet there are other ways to characterize the spills. Instead of counting the number of spills each year, they could measure the number of days between the spills. The first spill was on Feb. 23, Year One. The second spill was on Jan. 11, Year Two. The elapsed time between these two spills was 322 days. One spill in 322 days is equivalent to a spill rate of 1.13 spills per year.

The third spill was on Sept. 15, Year Two. This is 247 days after the second spill. One spill in 247 days is equivalent to a spill rate of 1.48 spills per year. Continuing in this manner, the remaining five spills are converted into instantaneous spill rates of 1.24, 1.61, 1.64, 2.12 and 3.17 spills per year. These seven spill rates are used to compute six moving ranges and are placed on an XmR chart in Figure 2.

The average spill rate is 1.77 spills per year, and the average moving range is 0.42. These two values result in an upper natural process limit of:

UNPL = 1.77 + 2.66 x 0.42 = 2.89
the lower natural process limit will be:

LNPL = 1.77 ­p; 2.66 x 0.42 = 0.65
and the upper control limit for the moving ranges will be:

UCL = 3.268 x 0.42 = 1.37

This XmR chart for the spill rates is shown in Figure 2.

The last spill results in a point that is above the upper natural process limit, which suggests that there has been an increase in the spill rate. This signal should be investigated, yet it is missed by the c-chart.

In general, counts are weaker than measurements. Counts of rare events are no exception. When possible, it will always be more satisfactory to measure the activity than to merely count events. And, as shown in this example, the times between undesirable rare events are best charted as rates.

About the author

Donald J. Wheeler is an internationally known consulting statistician and the author of Understanding Variation: The Key to Managing Chaos, Advanced Topics in Statistical Process Control and Understanding Statistical Process Control, Second Edition.

© 1996 SPC Press Inc. Telephone (423) 584-5005. E-mail: dwheeler@qualitydigest.com.