Control Charts are the last of the elementary seven tools of QC. Because of its wide use and misuse, a good portion of this section will be devoted to it.

In the minds of many, quality professionals and nonprofessionals alike, the control chart is synonymous with statistical process control. For a number of years it held center stage in the discipline that used to be called statistical quality control. Developed by Shewhart, the control chart’s main function, past and present, is to maintain a process under control, once it’s inherent variation has been reduced through the design of experiments. Quoting Dr. Shewhart, “an adequate science of control for management should take into account the fact that measurements of phenomena in both social and natural science for the most part obey neither deterministic or statistical laws, until assignable causes of variability have been found and removed.”

Gradual change in a critical adjustment for condition in a process is expected to produce a gradual change in the data pattern. An abrupt change in the process is expected to produce an abrupt change in the data pattern. Civil processes are often ingested without reference to any data. The data from even the simplest process will provide unsuspected information on its behavior. In order to benefit from data coming from either regularly, or in special studies from a temperamental process, it is important to follow one important and basic rule: “plot the data in a time sequence”. Once this is done, too important analysis methods can be employed to diagnose the behavior of time sequence data. These are (1) use of run criteria and (2) control charts with control limits and various other criteria that will signal the presence of assignable causes.

Run criteria are used to test the hypothesis that the data represent random variation from stable sources. A simple way to describe will run is when someone repeatedly tosses a coin and produces a run of six heads in succession. If this happens we realize it’s very unusual. Similarly for any process we are studying we sample that process for an attribute of interest to us and we plotted in sequence above or below the average of all the readings we have. What we will get is a median line with one half the points above and half below. The order in which points of fallen above and below the median however may not be random. There may be runs of points above or below the median and the question is depending upon the run length of her run above or below the median what does that tell us about the process stability.

In the “Example of Production Data for Product Weight” figure we see that the data for product weight has been plotted in a sequential manner, that the median product weight has been calculated and drawn on the figure, and that there are Runs of data above and below this median going from left to right of seven, three, three, one, one, two, one, and six, respectively. In this case each of the 24 points on the plot (n) is an average of 4 sub-group samples (k) taken at the same time.

Too many Runs above and below the median indicate the following possible engineering reasons. (1) samples are being drawn alternately from two different populations resulting in a saw-tooth effect. These effects occur regularly in portions of sets of data. Their explanation is often found to be two different sources, that being analysts, machines, raw materials which enter the process alternately were nearly alternately. (2) there may be three or four different sources which enter the process in a cyclic manner. Too few Runs are quite common. Their explanations include: (a) a general shift in the process average, (b) an abrupt shift in the process average, (c) is slow cyclic change in the averages.

The total number of Runs in the Product Weight data is eight. The average expected number, and the standard deviation of the sampling distribution, are given by the following formulas.

Average Number of Runs = (n + 2) / 2 = 13 = m+1 {used in the formula below}

Standard Deviation of Runs =square root of ( (m * (m-1) ) / ( (2 * m) – 1 ) =

The expected Number of Runs of Exactly Length “s” derived from the table in the “Runs Above and Below the Median of Length “s” figure. When running these calculations for the run lengths observed in the Example of Production Data for Product Weight study, one would find that there were too few short runs and too many long runs observed. In particular, the two long Runs of 6 and 7 suggest that the data are not random. When looking at the figure itself we see that in there appears to be an overall increase in filling weight during the study. Thus using run criteria in seeing whether run links are the result of a random or nonrandom process is important hints on whether or not there is a large assignable cause that could be found and controlled to greatly improve the process.

This can be seen from the above example there are criteria which indicate the presence of assignable causes by unusual runs in the set of points comprised of subgroup samples. The method works even if the subgroup size is one, all the way up to very large subgroups.

The ultimate value of using run criteria can be summarized by saying “if the distribution of Runs are not as shown in the “Run Criteria” below, there is a process that can greatly benefit from finding assignable causes”. This is important in production scale-up runs from R&D and Business Development projects. It is critical to know when the new process is stable and when it is not.

Run Criteria
1. The total number of Runs about the median is = (n+2) / 2
2. The Run length of expected Runs, as shown in the “Runs Above and Below the Median of Length “s” figure, are inconsistent with those Run length frequencies observed
3. A Run of length greater than six is evidence of an assignable cause warranting investigation.
4. A long Run-up or Run-down usually indicates a gradual shift in the process average. A Run-up or Run-down of length five or six is usually longer than expected.

The Shewhart control chart is a well-known and powerful method of checking on the stability of a process. It was conceived as a device to help in production with routine hour by hour adjustments and its value in this regard is unequaled. The control chart provides a graphical time sequence of data from the process itself. This permits a run analysis to study the historical behavior patterns. Further, the control chart provides additional signals to the current behavior of the process such as the upper and lower control limits which define the maximum expected variation of the process.

The control chart is a method of studying your process from a sequence of small random samples from the process. The basic idea of the procedures to collect 3 to 5 samples at each point of regular time intervals. Sample sizes of four or five are usually best. Sometimes it’s more expedient to use sample sizes of 12 or three. Sample sizes of larger than six or seven or not recommended. Quality characteristic of each unit of the sample is then measured and the measurements. Their range is plotted on a Range or “R” control chart and their their average recorded on a second control chart.

In starting control charts it is necessary to collect some data to provide the preliminary information regarding the central lines on the average value and ranges of the values. It is usually recommended to collect the data over 20 to 25 time intervals before generating a plot. A lower number can be used but it is not recommended to drop below 14.

Step one is to plot the values on a time axis and compute the average and range of each sample in the time sequence.
Step two is to draw the average of the average values and average of the range values of the population on the chart as horizontal lines.
Step three is to compute the Upper and Lower Control Limits on the range values. These are obtained using the “Factors Used to Calculate the UCL/LCL of the Range” figure. Note that “n” in this figure is the number of samples taken at each point in time that were averaged to obtain the single “time” point. The UCL and LCL are drawn on the range chart as horizontal lines. The UCL is D4 * Range Average. The LCL is D3 * Range Average
Step four is to compute the three Sigma control limits on the average and draw them as lines on the chart as well. The upper control limit is the average plus 3 times the standard deviation of the point average values. The lower control limit is the average minus 3 times the standard deviation of the point average values.