The meaningful and correct application of control charts can be deceptively difficult. At first glance, introducing a control chart to the shop floor seems easy enough. Pick an important parameter to measure, determine a sampling strategy, teach someone how to plot the points and voila--a control chart is born. The results: audit scores are higher, Cpk values get reported and customers are happy to see statistical process control (SPC) activity. Using control charts might even result in cost savings. However, do the alarms generated by control charts provide justification to perform process interrogation? Are the sources of variation captured within each subgroup useful for identifying true process changes? These are just some of the questions that must be answered before the chart can be considered a successful process improvement tool.

Not to diminish the importance of passing audits and making customers happy, but a control chart's true function is to provide real-time feedback needed to control and improve processes. The enthusiasm enjoyed from the benefits listed above will fade if the data displayed on the charts do not help front line operators make better process decisions. Operators will reject SPC if they are required to feed data into a control chart that provides no useful information. Introducing useless statistical tools on the shop floor can negate improvement activities and jeopardize an SPC program's effectiveness.

Of the many SPC implementation gaffs that occur in the workplace, two mistakes are widespread. Unfortunately, these mistakes are not only the most common, but they are the most severe and pose the greatest risks to an SPC program.

**SPC application mistake #1**

The first mistake is incorrect subgrouping. A fundamental assumption for any traditional Shewhart control chart is that the individual data values within a subgroup are independent, meaning that one measurement in a subgroup should not be influenced by another. For example, an aerosol can whose height is a critical dimension is measured in three locations (A, B, and C) that are 120 degrees apart. These multiple checks are performed to ascertain a can's height uniformity. Because the three values come from the same can, each value within the subgroup size of three is dependent on, or related to, the other heights. That is, if the can is extra tall, all three height measurements will be high; if the can is short, all three will be low.

A common approach for monitoring can height is to measure the three heights, then plot the average and range of these three readings on an -X and R chart. The average and range values, provided in the table, Can Height Dataset (p. 35), contain a sample data set of ten subgroups gathered with this strategy, including the -X and range calculations. These are graphed in the -X and range chart, Incorrect Method (p. 35), and are in a state of statistical control, which indicates a consistent amount of within-subgroup variability. Because the range chart is in control, the Cp value can be calculated, which is 3.83. However, the -X chart is out-of-control, with seven of the ten plot points exceeding its control limits. One explanation for this is that the control limits are calculated incorrectly, but checking the math proves that the control limits are calculated correctly. Therefore, what could cause so many of the subgroup averages to be out of control? In addition, the high Cp value of 3.83 is questionable, but a check of the math verifies that it too has been calculated correctly. What is the problem?

Let's explore the facts.

- The ranges represent within-can variation.

- The -X chart is based on can-to-can variation. Each -X value represents the average height of an individual can.

- The control limits are computed from the average range (---X+ or - A2-R).

The sampling procedure violates the assumption of data independence because the three values making up the subgroup are from the same part. However, appropriate subgrouping practice requires each measurement within a subgroup to be independent. If the subgroup were composed of one height measurement from three different cans, each individual measurement would not be influenced by any of the others and would be independent subgroup averages, they are treated as individuals on this chart because each average is the best one number representation of height for an individual can.

The 3-D chart, titled the Correct Method (p. 37), is constructed for the can height measurements. The R(within) chart is identical to the Range chart, titled the Incorrect Method, but it is more correctly called a Range chart for within-piece variation. The control limits are calculated in the conventional manner, D3-R and D4-R, where -R is the average of the within-can ranges. The subgroup size used to determine the D3 and D4 factors is based on the number of measurements per piece, which is three in this example. Control limits for the moving range chart are computed using the standard method where the D3 and D4 factors are based on n=2, minus the consecutive averages used to calculate the moving range.

The -X portion of this 3-D chart is treated as an Individual X (IX) chart, with each plot point representing the average height of a single can. However, the control limits for this special version of an IX chart are computed using the centerline of the moving-range chart ( -X + or - A2 ---M---R ). Note that the within piece variability from the R(within) chart is not mistakenly used for calculating the control limits for the IX chart. Instead, the correct estimate of can-to-can variability is used, which is indicated by the ---M---R.

With the correct control limits, the 3-D chart now indicates stability for three different statistics, time-to-time (IX chart), can-to-can (MR chart) and within-can (R(within) chart). Now that control has been demonstrated, process capability can be estimated. The within-can Cpk is 2.87. The can-to-can capability, 0.77. Finally, the combined, or pooled, Cpk is 0.74.

The original, and incorrect, analysis led to the belief that the process Cp was 3.83, but this 3-D analysis correctly reports the Cp as 0.77.

**SPC application mistake #2**

The second mistake is also related to incorrect subgrouping. Sometimes several process streams are combined in the same subgroup. For example, one process stream might be considered a machine or assembly line. Mixing process streams is not advisable because the standard deviations among the processes are more than likely different. The end result is control limits that do not correctly represent any individual process stream, and, thus, there is an inability to suitably interpret process signals on the control chart. This error can mask statistical signals and seriously degrade problem solving and process improvement.

Another example of this mistake might include mixing data from multiple spindles within the same subgroup or creating a subgroup whose injection mold data includes measurements across multiple cavities. To illustrate this mistake, consider a machine with eight filling heads, where each head is controlled by a separate pump. Because the heads are controlled separately, each can be considered a separate process worthy of its own individual control chart.

The data in the table, Bottle Fill Volume (p. 39), represents fill volumes for bottles filled by each head. A common strategy is to combine one fill volume from each head into a subgroup of size eight, then calculate the average and sample standard deviation, s.

Because of its incorrect subgrouping strategy, the resulting -X and s chart, titled Incorrect Method-Traditional Xbar/Sigma Chart (p. 37), appears to be in control. All that can be said about this process is that bottles are being consistently filled with liquid, but this conclusion may be invalid because all eight heads are averaged into one value. If any of the heads were "misbehaving," the -X chart would probably mask the problem. Although combining the eight fill heads into a single subgroup seems convenient, it will seriously undermine the ability to problem solve and improve filling performance. Why? Because the differences between the heads are lost in the average and range.

To avoid the subgrouping mistakes associated with combining all eight heads into a single subgroup, plot each head on its own chart. This is statistically correct, but not very practical because this solution requires eight control charts to manage this one filling machine, as is seen in the graphic, One Chart Per Head (p. 38). Imagine trying to control a filling machine with 20 or 30 heads.

Another option might be to plot all process streams on a single chart, resulting in 8 separate data lines. However, displaying all data stream lines results in a "spaghetti chart" where the highs and lows are indistinct and confusing, as is seen in the graphic, Spaghetti Chart (p. 38). The confusion would grow as the number of heads increases.

**Solving mistake #2**

The solution to this problem is a group chart that provides the simplicity of one chart, but maintains a separate analysis of each fill head. Instead of averaging the fill heads, or plotting each on its own chart, treat all eight individual values as a group representing fill volumes at a given period of time. From this group, only the minimum (min) and maximum (max) fill volumes are highlighted and plotted, as seen in the table, Minimum and Maximum Fill Volumes (p. 39). Then each min and max plot point on the chart is labeled with a number representing the associated fill head. The resulting chart is called a Group IX-MR chart.

The group chart is intentionally absent of control limits because sometimes group charts are used for plotting dependent data streams. As illustrated in the mistake, subgrouping dependent data results in incorrect control limits. The power of a group chart is two-fold: one, to clearly and distinctly illustrate the extremes in a data set; and two, to present the data to users so that opportunities for improvement are clearly indicated.

A group chart can be considered a running Pareto chart of variables data--the most important information is displayed to the user which facilitates the identification of improvement actions.

**Interpreting the group chart**

When interpreting group charts, look for runs in the min and max positions, as shown in the Correct Method Group IX-MR chart (p. 40). Notice that the min plot points on the IX chart are predominately represented by heads 2 and 6, which are consistently filling less volume than the other heads. Looking at the moving range chart, the majority of the max plot points are from head 6. Therefore, the variability from head 6 is consistently greater than the other heads. The distance between the IX chart's min and max lines displays the variation within a single rotation through all eight heads. That is, the closer the lines, the more consistent the fill volume across all heads.

There are several different variations of a group chart, but they all are designed to plot multiple process or data streams on a single chart. With just a single group chart, one can answer many questions about a process and promptly uncover tremendous amounts of information. Conversely, users armed only with traditional SPC methods would need to closely examine each of several different charts, while simultaneously juggling and comparing those charts and their center lines. They would also likely require the aid of off-line analysis by a statistician.

**SPC solves problems**

With SPC's ability to put tremendous amounts of information and problem-solving knowledge into the hands of operators, it is surprising how many companies have not adopted SPC or have abandoned its use. Surprising, that is, unless the situations that bedevil companies trying to leverage the power of SPC are analyzed. Upon closer inspection, it is likely that there are frustrated operators, misinformed managers and a lack of support for statistical methods on the manufacturing shop floor. Why? Because there is a general lack of understanding of correct subgrouping conventions and a belief that SPC "just doesn't work" in their particular situations. More likely, the lack of success could be attributed to an incorrect subgrouping strategy as well as the use of the wrong statistical tool.

For the most part, users are familiar with the standard -X and R chart. However, the 3-D chart is a superior tool for evaluating three important sources of variation: time-to-time, piece-to-piece, and within-piece. Additionally, the Group Chart is a clever analysis tool that allows users to visualize the individual performances of multiple process streams on a single chart. With these tools in the hands of operators and support personnel, SPC has the opportunity to become not only less troublesome, but infinitely more capable of providing the critical real-time information required for sustaining a successful SPC system.

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