What is a control chart that monitors the proportion of defects in a sample?

Índice Show

Variable Control Charts:
X bar control chart.
Range “R” control chart.
Standard Deviation “S” control chart.
Attribute Control Charts:
“u” and “c” control charts.
“p” and “np” control charts.
Pre-control Charts.

The Information you need, provided in an easy to understand visual format.

There are various types of control charts which are broadly similar and have been developed to suit particular characteristics of the quality attribute being analyzed. Two broad categories of chart exist, which are based on if the data being monitored is “variable” or “attribute” in nature.

Variable Control Charts:

X bar control chart.

This type of chart graphs the means (or averages) of a set of samples, plotted in order to monitor the mean of a variable, for example the length of steel rods, the weight of bags of compound, the intensity of laser beams, etc.. In constructing this chart, samples of process outputs are taken at regular intervals, the means of each set of samples are calculated and graphed onto the X bar control chart. This chart can then be utilized to determine the actual process mean, versus a nominal process mean and will demonstrate if the mean output of the process is changing over time.

Range “R” control chart.

This type of chart demonstrates the variability within a process. It is suited to processes where the sample sizes are relatively small, for example <10. Sets of sample data are recorded from a process for the particular quality characteristic being monitored. For each set of date the difference between the smallest and largest readings are recorded. This is the range “R” of the set of data. The ranges are now recorded onto a control chart. The center line is the averages of all the ranges.

Standard Deviation “S” control chart.

The “S” chart can be applied when monitoring variable data. It is suited to situations where there are large numbers of samples being recorded. The “S” relates to the standard deviation within the sample sets and is a better indication of variation within a large set versus the range calculation. An advantage of using the standard deviation is that all data within a set are utilized to determine the variation, rather than just the minimum and maximum values.

Develop your understanding of Control Charts, Process Capability, Process Improvement, etc. …

Attribute Control Charts:

Attribute control charts are utilized when monitoring count data. There are two categories of count data, namely data which arises from “pass/fail” type measurements, and data which arises where a count in the form of 1,2,3,4,…. arises. Depending on which form of data is being recorded, differing forms of control charts should be applied.

“u” and “c” control charts.

The “u” and “c” control charts are applied when monitoring and controlling count data in the form of 1,2,3, …. i.e. specific numbers. An example of such data is the number of defects in a batch of raw material, or the number of defects identified within a finished product. The c chart is used where there can be a number of defects per sample unit and the number of samples per sampling period remains constant. In the u chart, again similar to the c chart, the number of defects per sample unit can be recorded, however, with the u chart, the number of samples per sampling period may vary.

“p” and “np” control charts.

P charts are utilized where there is a pass / fail determination on a unit inspected. The p chart will show if the proportion defective within a process changes over the sampling period (the p indicates the portion of successes). In the p chart the sample size can vary over time. A similar chart to the p chart is the np chart. However, with the np chart the sample size needs to stay constant over the sampling period. An advantage of the np chart is that the number non-conforming is recorded onto the control rather than the fraction non conforming. Some process operators are more comfortable plotting the number rather than the fraction of non-conformances.

Pre-control Charts.

Where a process is confirmed as being within statistical control, a pre-control chart can be utilized to check individual measurements against allowable specifications. Pre-control charts are simpler to use than standard control charts, are more visual and provide immediate “call to actions” for process operators. If however a process is not statistically “capable” i.e. having a Cpk of at least 1, pre-control can result in excessive process stoppages.

Develop your understanding of Control Charts, Process Capability, Process Improvement, etc. …

Image by Analytics Association of the Philippines on LinkedIn

Quality control charts represent a great tool for engineers to monitor if a process is under statistical control. They help visualize variation, find and correct problems when they occur, predict expected ranges of outcomes and analyze patterns of process variation from special or common causes. Quality control charts are often used in Lean Six Sigma projects and DMAIC projects under the control phase and are considered as one of the seven basic quality tools for process improvement.

However, how can we determine the right quality control chart to use for monitoring a process? The following decision tree can be used to identify which is the correct quality control chart to use based on the given data:

Quality Control Charts Decision Tree

For the following examples, we will be focusing on quality control charts for discrete data that consider one defect per unit (i.e. defective or not defective unit), for when the sample size is constant and for when it is not.

The p-chart is a quality control chart used to monitor the proportion of nonconforming units in different samples of size n; it is based on the binomial distribution where each unit has only two possibilities (i.e. defective or not defective). The y-axis shows the proportion of nonconforming units while the x-axis shows the sample group. Let’s take a look at the R code using the qcc package to generate a p-chart.

p-chart R codep-chart example using qcc R package

The p-chart generated by R provides significant information for its interpretation, including the samples (Number of groups), both control limits (UCL and LCL), the overall proportion mean (Center) the standard deviation (StdDev), and most importantly, the points beyond the control limits and the violating runs. Engineers must take a special look at these points in order to identify and assign causes attributed to changes in the system that led to nonconforming units.

np-chart

The np-chart is a quality control chart used to monitor the count of nonconforming units in fixed samples of size n. The y-axis shows the total count of nonconforming units while the x-axis shows the sample group. Let’s take a look at the R code using the qcc package to generate a np-chart.

np-chart R codenp-chart example using qcc R package

The np-chart generated by R also provides significant information for its interpretation, just as the p-chart generated above. In the same way, engineers must take a special look to points beyond the control limits and to violating runs in order to identify and assign causes attributed to changes on the system that led to nonconforming units.

We have gone through one of the many industrial engineering applications that R and the qcc package have to offer. As you might have noticed, just with few lines of code we were able to construct quality control charts and get significant information to be used during Lean Six Sigma and DMAIC projects for process improvement. Once again, I invite you to continue discovering the amazing stuff you can perform using R as an industrial engineer.

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If you found this article useful, feel welcome to download my personal code on GitHub. You can also email me directly at and find me on LinkedIn. Interested in learning more about data analytics, data science and machine learning applications in the engineering field? Explore my previous articles by visiting my Medium profile. Thanks for reading.

- Robert