By Issa Bass
 

Control Charts

Control charts were developed as a monitoring tool for SPC by Dr. Shewhart; they are among the most important tools in the analysis of production process variations. The purpose of using control charts is

• to help prevent the process from going out of control

The control charts help detect the assignable causes of variation on time so that appropriate actions can be taken to bring the process back in control.

• To keep from making adjustments when they are not needed.

Most production processes allow operators a certain level of leeway to make adjustments on the machines that they are using when it is necessary. Yet over adjusting machines can have negative impacts on the output. Control charts can indicate when the adjustments are necessary and when they are not.

• To determine the natural range (control limits) of a process and to compare it to its specified limits.

If the range of the control limits is wider than the one of the specified limits, the production process will need to be adjusted.

• to inform about the process capabilities and stability
• The process capability refers to its ability to constantly deliver products that are within the specified limits and the stability refers to the quality auditor's ability to predict the process trends based on past experience. A long term analysis of the control charts can help monitor the machine long term capabilities. Machine wear out will reflect on the production output
• to fulfill the need of a constant process monitoring

Samples need to be taken on a regular basis and tested to make sure that the quality of the products sent to the customers meets their expectations.

How to build a control chart

The control charts we are addressing are created for a production process in progress. Samples are taken from the production lines at given time intervals and tested to determine whether they are in conformance with the specifications and their level of conformance are plotted on the charts and monitored.

Let's consider Y, a sample statistic that measures a Critical-To-Quality characteristic of a product (length, color, or thickness…), with a mean  and a standard deviation . The Upper Control Limit (UCL), the Center Line (CL) and the Lower Control Limit (LCL) for the control chart will be given as follow:

Where k is the distance between the center line and the control limits.

Example: Consider the length as being the critical characteristic of manufactured bolts. The mean length of the bolts is 17 inches with a known standard deviation of 0.01. A sample of 5 bolts is taken every half an hour for testing and the mean of the sample is computed and plotted on the chart. That control chart will be called  control chart because it plots the means of the samples.

Based on the Central Limit theorem, we can determine the sample standard deviation and the mean.

The mean will still be the same as the population's mean, 17.

For three sigma control limits, we will have:

Control limits on a control chart should be readjusted every time a significant shift in the process occurs.

A typical control chart is made up of at least four lines: a vertical line that measures the levels of the samples' means, the two outmost horizontal lines represent the UCL and the LCL and the Center Line represents the mean. If all the points plot in between the UCL and the LCL in a random manner , the process is considered to be in control.

What is meant by in control process is not a total absence of variation but instead, when the variations are present, they exhibit a random pattern, they are not outside the control limits and based on past experience, they can be predicted and are strictly due to common causes. The control charts are an effective tool for detecting the special causes of variation

The following chart depicts a process in control and within the specified limits. The Normal curve on the left side shows the specified (desired) limits of the production process while the right chart is the control chart. The specification limits determine whether the products meet the customers' expectations while the control limits determine whether the process is under statistical control. These two charts are completely separate entities. There is no statistical relationship between the specification limits and the control limits.

If some points are outside the control limits, this will indicate that the process is out of control and corrective actions need to be taken.

Let's note that a process with all the points in between the control limits is not necessarily synonymous with acceptable process. A process can be in control with a high variability or too many of the plotted points are too close to one control limit and away from the target.

The following chart is a good example of an out of control process with all the points plotted within the control limits.

In this example, points A, B, C, D, E and F are all well within the limits but they do not behave randomly, they exhibit a run up pattern, in other words they follow a steady (increasing) trend. The causes of this run up pattern need to be investigated because it might be the result of a problem with the process.

The interpretation of the control charts patterns is not easy and requires experience and know-how.

The Western Electric (WECO) Rules

Western Electric put out a handbook in 1956 to determine the rules for interpreting the process patterns. These rules are based on the probability for the points to plot at specified areas of the control charts.

A process is said to be out of control if the following occur:

• A single point falls outside the 3 Sigma limit
• Two out of three successive points fall beyond 2 sigma limit
• Four out of Five successive points fall within 1 sigma or beyond from the mean
• Eight successive points fall on one side of the center line

The WECO rules are very good guidelines for interpreting the charts but they need to be used with caution because they add sensitivity to the trends of the mean.

When the process is out of control, production is stopped and corrective actions taken. The corrective actions start with the determination of category of variation. The causes of variation can be random or assignable. If the causes of variation are solely due to chance, they are called chance causes (By Dr Schewart) or Common causes (by Dr Deming). Not all variations are due to chance, some of them can be traced to specific causes that are not part of the process, in that case, the variations are said to be due to assignable causes (by Schewart) or special causes (by Deming). Finding and correcting special causes of variation are easier than correction common cause since the common causes are inherent to the process.

Types of control charts

Control chart are classified according to whether they monitor attribute data, variable data or multivariate data.

Attribute control charts

Attribute characteristics resemble binary data they can only take one of two given forms. In quality control, the most common attribute characteristics used are “conforming” or “not conforming”, “good” or “bad”.  Attribute data need to be transformed into discrete data to be meaningful.

The types of charts used for attribute data are:

• The p -chart
• The np -chart
• The C -chart
• The U -chart

The p -chart

The p -chart is used when dealing with ratios, proportions or percentages of conforming or non conforming parts in a given sample. A good example for a p -chart is the inspection of products on a production line. They are either conforming or nonconforming. The probability distribution used in this context is the Binomial distribution with p  representing the non-conforming proportion and q (which is equal to1 – p ) representing the proportion of conforming items. Since the products are only inspected once, the experiments are independent from one another.

The first step when creating a p -chart is to calculate the proportion of nonconformity for each sample.

 Where m  represents the number of nonconforming items, b  is the number of items in the sample and p  is the proportion of nonconformity.

 Where is the mean proportion, k  is the number of samples audited and  is the kth  proportion obtained.

The control limits of a p -chart are:

And represents the center line.

np - chart

The np  chart is one of the easiest to build. While the p -chart tracks the proportion of non-conformities per sample, the np chart plots the number of non-conforming items per sample.

The audit process of the samples follows a binomial distribution, in other words, the expected outcome is “good” or “bad”, and therefore the mean number of success is np. 

The control limits for an np chart are as follow:

The c -chart

The c -chart monitors the process variations due to the fluctuations of defects per item or group of items. The c -chart is useful for the process engineer to know not just how many items are not conforming but how many defects there are per item. Knowing how many defects there are on a given part produced on a line might in some cases be as important as knowing how many parts are defective. Here, non-conformance must be distinguished from defective items since there can be several non-conformances on a single defective item.

The probability for a nonconformance to be found on an item, in this case follows a Poisson distribution.

If the sample size does not change and the defects on the items are fairly easy to count, the c -chart becomes an effective tool to monitor the quality of the production process.

If C is the average nonconformity, the UCL and the LCL limits will be given as follow for a k -sigma control chart:

Where c  is the center line and k is the sigma level intended. The most commonly used sigma level is 3.

u- chart

One of the premises for a c -chart was that the sample sizes had to be the same. The sample sizes can vary when the u -chart is being used to monitor the quality of the production process and the u -chart does not require any limit to the number of potential defects. Furthermore, for a p -chart or an np -chart the number of non-conformances cannot exceed the number of items on a sample but for a u -chart, it is conceivable since what is being addressed is not the number of defective items but the number of defects on the sample.

The first step in creating a u -chart is to calculate the number of defects per unit for each sample.

Where u  represents the average defect per sample, c  is the total number of defects and n  is the sample size.

Once all the averages are determined, a distribution of the means is created and the next step will be to find the mean of the distribution, in other word, the grand mean.

 Where k  is the number of samples

The control limits are determined based on  and the mean of the samples n.

Variable Control Charts

 Control charts monitor not only the means of the samples for Critical-To-Quality characteristics but also the variability of those characteristics. When the characteristics are measured as variable data, the , the S and R charts are used. 

These control charts are used more often and they are more efficient in providing feedback about the process performance.

The principle underlying the building of the control charts for variables is the same as the one of the attribute control charts. The whole idea is to determine the mean, the standard deviation and the distance between the mean and the control limits based on the standard deviation.

But since we do not know what the process population mean and standard deviation are, we cannot just plug numbers to these formulas to obtain a control chart. The standard deviation and the mean must be determined from sample statistics. The first chart we will use will be the S  chart to determine whether the process is stable or not.

 and S control Charts

The S  chart is used to determine if there is a significant level of variability in the process, so it plots the standard deviations of the samples taken at regular intervals. A strong variation in the data plots will indicate that the process is very unstable.

Since , the population's variance is unknown, it needs to be estimated using the samples' variance .

    Therefore .      for a number of k  samples.

But using S  as an estimator for would lead to a biased result. Instead,  is used, where  is a constant that depends only on the sample size n.

If   then .

The mean expected of the standard deviation (which is also the Center Line) will be  and the standard deviation of S is .

So the control limits will be as follow:

These equations can be simplified using and

Therefore,

Similarly

These equations can be simplified

Therefore,

and R charts

The building of chart follows the same principle as for the one of attribute control charts with the difference that quantitative measurements are considered for the Critical-To-Quality characteristics instead of qualitative attributes. Samples are taken and measurements of the means  for each sample derived and plotted on the chart.

The center line (CL) is determined by averaging the s.

 Where n represents the number of samples.

The next step will be to determine the Upper Control Limit ( ) and the lower Control Limit ( ). We have determined k  to be equal to 3, the only remaining variable of this equation is  which can be determined in several ways. One way to do it would be through the use of the standard error estimate  and another one would be the use of the mean range.

There is a special relationship between the mean range and the standard deviation for normally distributed data.

 Where the constant is function of n.

Standard error based chart

 The standard error based chart is straight forward. Based on the Central Limit theorem, the standard deviation used for the Control Limits is nothing but the standard deviation of the process divided by the square root of the number of samples and we obtain:

Since the process standard deviation is not known, in theory these formulas make sense but in actuality impractical. The alternative to that is the use of the mean range.

Mean Range based  control charts

When the sample sizes are relatively small , the variations within sample are likely to be small so the range (the difference between the highest and the lowest observe values) can be used en lieu et place of the standard deviation when constructing a control chart.

 or    Where R  is called the relative range.

The mean range is

Where is the range of the kth  sample.

Therefore the estimator of  is  and the estimator of so

The formulas for the control limits become

R chart

For an R  chart, the center line will be and the estimator of sigma is given as . Since , we can replace with its value and therefore obtain .

Let    and  

Therefore the control limits become:

Example

Moving Range

When individual (samples composed of a single item) CTQ characteristics are collected, moving range control charts can be used to monitor process quality. The variability of the process is measured in terms of the distribution of the absolute values of the difference of every two successive observations.

Let be the ith  observation, the moving average range MR will be:

 and the mean MR will be

The standard deviation S is obtained by dividing by the constant .  Since the moving range only involve two observations, n  will be equal to two and therefore for this case,  will always be 1.128.

Since is 1.128, these equations can be simplified

Example: The following data represents diameter measurements of samples of bolts taken from a production line. Find the Control limits.

Therefore, the control limits will be:

About the author
Issa Bass is the managing editor of SixSigmaFirst. He can be reached at issa@sixsigmafirst.com

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