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Rare Events and X-mR Charts

Rare Events and X-mR Charts

May 2007

In this issue:


Last month’s publication reviewed what happens to control charts when there is excessive round-off in the data (chunky data) for variable control charts, such as the individuals control chart (X-mR) and the Xbar-R control chart. This month’s publication examines what happens with attributes control charts (p, np, c and u charts) when the average value is too low, less than 1.0. In this case, the charts are not helpful. For example, you might be monitoring the number of lost time accidents or the number of environmental spills. These events (we hope) happen rarely. This newsletter examines how to handle this type of data. It involves converting the count data to rates and using the individuals control chart. We will use the c control chart and counting data as the example.


c Control Charts

sample c chart

The c control chart is used to monitor the variation in counting type data (see our July 2004 newsletter). c control charts are used to look at variation in counting type attributes data. They are used to determine the variation in the number of defects in a constant subgroup size. Subgroup size usually refers to the area being examined. For example, a c chart can be used to monitor the number of injuries in a plant. In this case, the plant is the subgroup. Since the plant doesn’t change size very often, it is a subgroup of constant size.

To use the c chart, the opportunities for defects to occur in the subgroup must be very large, but the number that actually occurs must be small. For example, the opportunity for injuries to occur in a plant is very large, but the number that actually occurs is small.

Suppose we have the following data for plant injuries per month. Remember you need a good operational definition of what constitutes an injury.

Jan-05: 2
Feb-05: 0
Mar-05: 3
Apr-05: 1
May-05: 0
Jun-05: 6
Jul-05: 2
Aug-05: 3
Sep-05: 3
Oct-05: 0
Nov-05: 0
Dec-05: 3
Jan-06: 4
Feb-06: 2

This data can be used to make a classical c control chart on plant injuries. This chart is shown above. The plant averages about 2 injuries per month. The process is in statistical control (see our April 2004 newsletter on how to interpret control charts). This means that the plant can expect to have between 0 and 6 injuries per month with a long term average of 2 as long as the process stays the same. There is only common cause of variation present. To improve the process, it must be fundamentally changed.

Rare Events

Now consider a plant that has excellent safety. It is a rare event when someone gets hurt. The data below represents this plant’s safety record based on number of injuries per month:

Jan-05: 0
Feb-05: 0
Mar-05: 0
Apr-05: 1
May-05: 0
Jun-05: 0
Jul-05: 1
Aug-05: 0
Sep-05: 0
Oct-05: 1
Nov-05: 0
Dec-05: 0
Jan-06: 0
Feb-06: 1
Mar-06: 0
Apr-06: 0
May-06: 0
Jun-06: 0
Jul-06: 1
Aug-06: 0
Sep-06: 0
Oct-06: 1
Nov-06: 0
Dec-06: 0

A c control chart on this data is shown in the figure below.

c control chart for the above data

Note that the average number of injuries per month is 0.25. The upper control limit is 1.75. The upper control limit is seven times the size of the average. It would take two injuries for the chart to show a special cause of variation (out of control point). The chart is not too useful. A different approach is shown below.

Converting Counts to Rates

The data can be useful if it is converted to rates. The rate we are looking at is the number of injuries per year. Below are the dates of the injuries for the data above.


The first step is to determine the days between injuries. For example, there are 91 days between 4/5/2005 and 7/6/2005. To determine the yearly rate, you calculate the following:

(1 injury/91 days)*(365 days/year) = 4.1 injuries per year

This is an instantaneous injury rate. This rate can also be calculated for the other days between injuries. This gives the following data for injuries per year:

4/5/2005: –
7/6/2005: 4.1
10/3/2005: 4.2
2/22/2006: 2.6
7/15/2006: 2.6
10/14/2006: 4.1

This data can then be used to construct a X-mR based on the injuries per year (see our October 2006 newsletter for information on how to construct an individuals chart). The X chart for this data is shown in the figure below. The average injury rate is 3.5 injuries per year. This chart is in statistical control. The rate of yearly injuries does not appear to be changing – which means you are not decreasing the injury rate.

x rate sample

The moving range chart is shown in the figure below. The average moving range is 0.8. This chart is also in statistical control. This means that the difference in rates between consecutive injuries is predictable and consistent.

r rate sample


The idea is to now use this chart to monitor the number of injuries. For example, suppose another accident happens on 1/2/2007. The days from the last accident is 78 which gives a yearly rate of 4.7. This is within the control limits on the X chart. Now, the next accident occurs on 2/4/2007. The days from the last accident (1/2/2007) is 32 days which gives a yearly rate of 11.4. This is clearly beyond the upper control limit and signals a special cause of variation. There has been an increase in the yearly injury rate.

Note that we could have plotted the days between injuries. The X chart and moving range chart for the days between injuries are shown below and give similar information. Note that the value of 32 days is now below the lower control limit.

A rule of thumb is that if you want to detect a deterioration in a process (such as safety), use the instantaneous rates. A point above the upper control limit will show that the process is deteriorating. If you want to detect an improvement in a process (such as safety), use the time between events. A time above the upper control limit will indicate an improvement.


x days chart
r days chart


This month’s publication examined how to use control charts with rare events – like the time between lost time accidents or environmental spills.  Instead of using the classical c control chart for the number of the defects, the data are converted to rates or time between events.  The data are then plotted on the individuals control chart.

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Thanks so much for reading our SPC Knowledge Base. We hope you find it informative and useful. Happy charting and may the data always support your position.


Dr. Bill McNeese
BPI Consulting, LLC

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