How to Derive Statistical Process Control Limits

Learn how to derive statistical process control limits about the process average using the range and moving range.

Statistical process control is a process control method developed by Walter Shewhart. He discovered two kinds of mistakes made in trying to make product uniform. The first mistake assumes a product came from special cause variation when it came from common cause variation. The other mistake assumes a product came from common cause variation when it came from special cause variation. These types of mistakes cost money! If we attribute any outcome to common causes when it came from special causes we risk making poor product. If we attribute any outcome to special causes when it came from common causes then unwanted is process intervention. In either case, the economic loss is large and production of poor product is inevitable. We can reduce and regulate the frequency of either mistake using statistical process control charts and its’ three sigma control limits.

In a continuous production setting we collect samples. Critical feature of the samples are then measured. This process of data collection and measurement continues for the life of the product. We use the results to compute statistics like the average, sample standard deviation or range. These statistics describe the process and plotting them  versus time tells us if the process is stable.

Placing limits of variation about the process mean helps detect process stability or instability. We call these limits statistical control limits or three sigma control limits.  We use the following expression to calculate the mean dispersion about the process mean.

Using this expression, we can derive statistical control limits about the process mean. Shown is the general form of the control limit equation.

In this expression, u is the process mean and σ_X is the dispersion about the process mean. Making the proper substitution we may use the expression as shown.

In this expression, we estimate u using the process average X-bar. To estimate dispersion, σ_X, use the following statistical relationship.

R-bar is the average Range and d2 is a numerical constant and based on the sample size gathered. To calculate the average Range requires we compute the Range for each group of samples collected over time. We call each group of samples a subgroup. To calculate the subgroup Range, take the difference between the highest and lowest value in a subgroup. The average Range is the average of all subgroup Ranges.

We use the R-bar/d2 estimate to calculate statistical control limits for two type of control charts. They are the x-bar and individuals charts.

The expression, in brackets, is the A2 constant. Since n is variable, the A2 constant depends on the subgroup sample size.

Suppose we check a product feature from a process and collect five consecutive samples at various times. In such a case, we may use X-bar and Range charts to watch the process average and dispersion with time. Since n=5 then d2=2.326. Substituting these values we get the A2 constant and simplify the control limit equation.

There are cases when samples cannot form a subgroup. This happens when we collect a single sample with time. In this case, we use Individual and Moving Range charts to check a characteristic of interest when the sample size is n=1. To calculate dispersion, we compute the moving range. The moving range is the difference between two consecutive samples. The average of the Moving Ranges is MR-bar. Since the smallest number of samples that makes up a moving range is n=2 then d2=1.128. Performing the right substitutions we have the following expression. The expression shown, in brackets, is the E2 constant.

As shown, these statistical process control expressions simplify how to compute the control limits about the process average. They are useful when the operators of a process are maintaining manual control charts. Using three sigma control limits, based on common causes, assures we detect special causes. Investigating special cause points beyond three sigma limits leads to a stable and predictable process. Once a process is stable and predictable than costs are predictable. The end-user is the benefactor of a reliable and dependable product.