Shewhart's theory of variability refers to the normal variability, or noise, that occurs within a system. He determined that in any engineering or manufacturing process there will be variability caused by either chance causes or assignable causes. Chance causes of variability refer to the normal noise within a system and it is caused by small and, in most cases, excusable sources. This type of variability within a process is generally unavoidable. However, assignable causes of variability are often avoidable and they deserve special attention when they occur. This is especially important when this type of variability occurs often in a system because generally it means that products are of inconsistent quality. Shewhart described his theory of variability in the following equation:
where xt is the measurement, m0 is the mean of the sample, and et is the random noise of the sample at time t. Using this relatively simple equation, Shewhart was able to develop charts that would plot this variability within a system.
However, given the equation, it is impossible to determine whether a signal is random or assignable in nature. Therefore, Shewhart needed more information to monitor a system successfully. The et term carries with it a mean of zero and a particular variance of sO2. When an unstable signal occurs that throws the system out of control, these assumptions are not true. Therefore either the mean of the normal noise is shifted or the variance at the time of instability is not equal to the constant variance of the system [Montgomery, 1992]. Using this information, Shewhart was able to develop three new equations using a statistic, w. The equations are as follows:
where mw is the mean of the sample, L is the width of the control limits, and sw is the standard deviation of the sample. UCL refers to an upper limit of the process variable. Similarly, LCL refers to a lower limit of the process variable. If the value of the variable crosses either of these control limits, the process is considered out-of-control for that point. CL is the centerline of the process and although it does not have to be plotted on a chart, it does give the user a reference to the mean of the sample and how far the data deviates from this mean.
With these new values, it is possible to make charts of the data measurements. A process is considered stable or in control if the data remains between the two control limits. The system goes out of control, or unstable, once data proceeds outside the control limits.
In the figure above is shown a Shewhart control chart. This particular chart contains 47 data points and the control limits were set with L being equal to 3. In other words, the control limits were set 3 standard deviations from the mean of the data set. This example shows data that is in control by the standards set by the user. If the user wanted a more conservative chart, he or she would have to set the control limits closer to the mean. To create a Shewhart chart of your own, click on the demonstration button below:
This section gives a good idea of the concepts behind control charts and their importance in determining the stability of a system. It is possible to get much more accurate charts than the charts proposed by Dr. Shewhart. For a given measurement, the Shewhart chart determines its stability based strictly upon the stability of its preceding measurement, but not on the sequence of measurements made before the given measurement. Therefore, the Shewhart chart does have a couple of inconsistencies that may make it somewhat inaccurate. The next section deals more with the uses of control charts and how inconsistencies in Shewhart analysis have given way to more advanced and accurate models for univariate control charts.