Statit Support: Criteria for Lack of Control

Robert F. Hart, Ph. D.
Marilyn K. Hart, Ph.D.

A control chart is a statistical tool used to examine the stability of a process. It allows the identification of special causes of variation in the process which can promote process improvement through the removal of unwanted special causes. However, in some cases it is used solely to assess whether a process appears to be in a state of control. This knowledge may be important for several reasons [Hart and Hart, 2002]:

1.	If a state of control exists, future performance may be predicted.
2.	Whether a state of control may be inferred may be important information in its own right, either to quality personnel within the organization or to an outside quality agency.
3.	The process should be in control before a process capability is inferred.
4.	It is beneficial for the process to be in control before using it for standard values.

It must be recognized that all processes cannot be improved at the same time. Determining which processes are out of control may be an important part of a screening activity to determine where to use limited quality improvement resources.

Chart Interpretation

Before any meaningful interpretation of a control chart can be made, the purpose of the chart must be defined: either for the "assessment" of the past process stability or for future improvement of the process. [Hart, et al 2004]

When "assessment" is the control chart purpose, usually only the fundamental criterion for lack of control (points outside the 3-sigma limits) is used. If additional criteria are added, too many stable processes may show additional indications leading to erroneous conclusions of lack of control (instability).

Even without supplemental criteria, this one criterion will often give indications of lack of control (points outside the 3-sigma control limits) even though no special-cause variation exists. The probability of this happening is called the false-alarm risk or alpha risk. For example, with a normally distributed process that has a known mean and standard deviation, an Xbar chart with 24 subgroups has a false alarm risk of 0.063. This may be calculated by noting that the probability of any one point being inside the 3-sigma limits is 0.9973 so the probability of all 24 being inside is 0.9973²⁴ = 0.937. Therefore, the probability of one or more points outside the 3-sigma limits by pure chance is 0.063. (In the usual case when the mean and standard deviation are not known but are estimated from the data, the calculations get much more complicated.) Inclusion of the use of the s chart would drive this false alarm risk even higher, and this is before any supplemental criteria are added. For example, Hilliard and Lasater [1966] wrote of their study where the false alarm risk was 0.25 with 25 subgroups of size four when three criteria were used. In other words, one in every four charts would give an indication of special-cause variation even when none was present!

However, when process improvement is the control chart purpose, the objective in chart interpretation is to use whatever criteria may be helpful to find indications which "suggest existence of nonrandom influences." The criteria most often used are based on the AT & T rules (originally called the Western Electric rules) [Western Electric, 1956]. Some of these rules were not clearly spelled out. For instance, AT&T claimed that trends may be indicated by x’s on one side of the chart followed by x’s on the other but did not specify how many x’s were needed to determine the trend. These rules were then better spelled out by Lloyd Nelson [1984, 1985], so are often called Nelson’s rules. They begin by dividing the area between the centerline and the 3-sigma upper control limit into three zones, each representing one-sigma. From the center line to 1-sigma above the centerline is zone C; from 1-sigma to 2-sigma is zone B; from 2-sigma to the upper control limit of 3-sigma is zone A. The zones are then similar for the lower half of the chart. (C is between centerline and -1-sigma, B is between -1 and -2-sigma, A is between -2 and -3-sigma.) The criteria are then:

A.	1 point above Zone A
B.	1 point below Zone A
C.	2 of 3 successive points in upper Zone A or beyond
D.	2 of 3 successive points in lower Zone A or beyond
E.	4 of 5 successive points in upper Zone B or beyond
F.	4 of 5 successive points in lower Zone B or beyond
G.	8 points in a row above center line
H.	8 points in a row below center line
I.	15 points in a row in Zone C (above and below center)
J.	8 points on both sides of center with 0 in Zone C
K.	14 points in a row alternating up and down
L.	6 points in a row steadily increasing or decreasing

These criteria are based upon the data being fairly normally distributed, so these rules may be used with confidence for Xbar charts but only for individual charts if the data are indeed fairly normally distributed. In reality, many people ignore rules E and F. Rules I, J and K do not occur often and require some comment. Rule I checks for stratification. Suppose that surgeon X always takes a long time to do a procedure; surgeon Y always takes shorter. If a subgroup was always made with one surgery time from X and one from Y, the averages would always be in the middle. But the ranges (or standard deviations) would always be large, so the control limits on the averages would always be wide. Hence, the averages plotted on the control chart would tend to “hug” the centerline giving 15 points in a row in Zone C (above and below center) as stated in rule I. This is an indication of not forming the subgroups correctly. Subgroups should be homogeneous, that is the elements of a subgroup should be as much alike as possible.

On the flip side, rule J checks for mixture. Continuing on this last example of the two surgeons, if some subgroups are from surgeon X and some from surgeon Y, all surgeon X’s average will be above zone C and all of surgeon Y’s will be below zone C. No points will be in zone C. Control charts are most useful when they are applied to a single-stream process.

The classic case of rule K is the use of an egg timer with a bubble on one side of the glass restricting the flow in one direction. The timer is used on one side then the other, alternating between the fast and slow flows. In the 25 years these authors have worked in the area of quality, this rule has never been violated in their work so needless to say, this rule is not broken often.

All of these rules have only a few chances in a thousand to happen by chance for normally distributed data points. For the derivations of these probabilities, see Western Electric [1956].

But what if the plotted points are not normally distributed? Then only some of the criteria or modified criteria are applied. For instance, for R charts, Statit uses:

A.	1 point above Zone A
B.	2 successive points in or above upper Zone A
C.	3 successive points in or above upper Zone B
D.	7 successive points in or above upper Zone C
E.	10 successive points in or below lower Zone C
F.	6 successive points in or below lower Zone B
G.	4 successive points in lower Zone A

For the s chart:

A.	1 point above Zone A
B.	1 point below Zone A

For p, u, and c charts:

A.	1 point above Zone A
B.	1 point below Zone A
C.	9 points in a row above center line
D.	9 points in a row below center line
E.	6 points in a row steadily increasing or decreasing
F.	14 points in a row alternating up and down

Conclusions

When making a control chart, the purpose of the chart must first be established: assessment or improvement. Then the criteria used to determine lack of control can be intelligently determined.

References