Why control charts "work"
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Plus or minus "3 sigma" limits are typical

The control limits as pictured in the graph are probability limits. They were determined so that, if chance causes alone were present, the probability of a point falling above the upper limit would be one out of a thousand, and similarly, a point falling below the lower limit would be one out of a thousand. We would be searching for an assignable cause if a point would fall outside these limits. Where we put these limit will determine the risk of undertaking such a search when in reality there is no assignable cause for variation. 

Since two out of a thousand is a very small risk, the 0.001 limits may be said to give practical assurances that, if a point falls outside these limits, the variation was caused be an assignable cause. It must be noted that two out of one thousand is a purely arbitrary number. There is no reason why it could have been set to one out a hundred or even larger. The decision would depend on the amount of risk the management of the quality control program is willing to take. In general (in the world of quality control) it is customary to use limits that approximate the 0.002 standard. 

If the system of chance causes generates a variation in X that follows the normal distribution, the 0.001 probability limits are very close to the 3s limits. From normal tables we glean that the 3s in one direction is 0.00135 or in both directions 0.0027. For normal deviations, therefore, the 3s limits are the practical equivalent of 0.001 probability limits. 

In the U.S., whether X is normally distributed or not, it is an acceptable practice to base the control limits upon a multiple of the standard deviation. Usually this multiple is 3 and thus the limits are called 3 sigma limits. This term is used whether the standard deviation is the universe or population parameter, or some estimate thereof, or simply a "standard value" for control chart purposes. It should be inferred from the context what standard deviation is involved. (Note that In the U.K. statisticians generally prefer to adhere to probability limits.)

If the underlying distribution is skewed, say in the positive direction the 3-sigma limit will fall short of the upper 0.001 limit, while the lower 3-sigma limit will fall below the 0.001 limit. This situation means that the risk of looking for assignable causes of positive variation when none exists, will be greater than one out of a thousand. But the risk of searching for an assignable cause of negative variation, when none exists, will be reduced. The net result, however, will be an increase in the risk of a chance variation beyond the control limits.  How much this risk will be increased will depend on the degree of skewness. 

If variation in quality follows a Poisson distribution, for example,  for which np = .8,  the risk of exceeding the upper limit by chance would be raised by the use of 3-sigma limits from 0.001 to 0.009 and the lower limit reduces from 0.001 to 0. For a Poisson distribution the mean and variance both equal np. Hence the upper 3-sigma limit is 0.8 + 3 sqr(.8) = 3.48 and he lower limit = 0. For np = .8 the probability of getting more than 3 successes = 0.009. 

Does this mean that when all points fall within the limits, the process is in control? Not necessarily. If the plot looks non-random, that is, if the points exhibit some form of systematic behavior, there is still something wrong. For example, if the first 25 of 30 points fall above the center line and the last 5 fall below the center line, we would wish to know why this is so. Statistical methods to detect sequences or nonrandom patterns can be applied to the interpretation  of control charts. To be sure, in control implies that all points are between the control limits and they form a random pattern.