# Control charts – tools for understanding variation

## Features

**Author:**G Robin Henderson**Date:**18 Dec 2013**Copyright:**Image appears courtesy of iStock Photo. All figures are copyright of G Robin Henderson except sketch by Deming which is used by kind permission of the Deming Foundation and Tom Nolan.

**“The control chart does a marvellous job under a wealth of applications. It works.” ****W Edwards Deming**

**Introduction**

Statistical thinking is an essential requirement for the improvement of processes in all areas of human activity. The statement that “statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write!” is often attributed to the author H G Wells but is believed to be a paraphrased version from Sam Wilks’ 1950 presidential speech to the American Statistical Association. In the second edition of *Statistical Thinking*, Hoerl and Snee (2012) state:

Statistical thinking is a philosophy of learning and action based on these fundamental principles:

1. All work occurs in a system of interconnected processes

2. Variation exists in all processes

3. Understanding and reducing variation are keys to success.

Control charts, or process behaviour charts, are tools for understanding variation. The basic idea of the control chart was introduced in a memo written by Dr Walter Shewhart on 16th May 1924 at the Western Electric Company in the USA (Ryan 2000). However, David Salsburg (2001) in his book The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century suggests that the mathematical formulation of a control chart was first proposed by W S Gosset (“Student” of t-test fame) and that a control chart appeared even earlier in a textbook written by G U Yule. Shewhart distinguished between two types of variation. On the one hand there is chance or common cause variation – random variation that is inherent in the process used to create the product or service. On the other hand there is special cause variation – non-random variation that is extraneous to the process and which may require removal in order to restore process performance to a desired state. Although developed originally as a statistical tool for industrial applications, various forms of control charts have subsequently been developed for applications in a wide variety of fields including healthcare and service industry in general. W Edwards Deming was influenced by Shewhart’s work and became a vigorous advocate of the deployment of statistical tools for quality improvement. Yet today the distinction between common and special cause variation is not known as widely as it deserves to be. A short on-line biography of Shewhart has been created by O’Connor and Robertson (2013).

In one of his famous seminars, Dr Deming cited the example of an 11-year old school student, Patrick Nolan, who learned of the distinction through monitoring the time of his arrival by bus at his school. In essence a control chart consists of a time-ordered plot of the data with horizontal lines that represent the limits of common cause variation. Data points that fall outside the common cause variation band indicate the possible presence of special cause variation. A sketch of Patrick’s chart created by Dr Deming that appeared in T*he New Economics* is displayed in Figure 1.

Figure 1. Sketch by Dr Deming of a control chart of Patrick Nolan’s data.

Special causes were identified corresponding to the two points outside the limits – on one occasion there was a new driver on the route and on the second there was a problem with the door-closing mechanism.

There are many types of control chart. For example, the widely used statistical software package Minitab (http://www.minitab.com/en-us/products/minitab/) offers a menu with 24 types. We consider below the control chart for individual measurements, a type of chart that Donald Wheeler (1993) presents in *Understanding Variation* along with an image of a Swiss Army knife in order to indicate its versatility.

**A control chart for individual measurements**

Consider a manufacturing process for lubricating oil where the target viscosity is 9.0 CSt at 100°C. Viscosity measurements were made on the output at 15-minute intervals during a production run with the first observation being made at 08:00. Figure 2 shows a control chart of the data created once 25 observations were available. All the charts in this section were created using Minitab; details of how to do so may be found in Henderson (2011).

Figure 2. Control chart of the first 25 viscosity measurements.

Essentially it is a run chart of the data “clothed” through the addition of a centre line at the mean of the 25 initial data values (8.98) and lower and upper “three sigma” control limits. The limits are placed at the mean plus / minus three standard deviations i.e. at 7.93 and 10.03. Shewhart (1931) argued from experience that the use of three sigma limits made economic sense. The standard deviation has to be estimated from the data. It is conventional not to estimate the standard deviation through application of the usual formula for sample standard deviation to the set of 25 measurements but rather using a method based on “local” rather than “global” variability. The rationale for this approach is that, if the preliminary data includes any special cause variation that the chart creator is unaware of, then the use of such methods mitigates the detrimental effect such special cause variation can have on the location of the chart limits. Historically the ranges of consecutive pairs of observations, moving ranges, have provided the most widely-used method of estimation for the standard deviation The details of the calculations will be given later – the main point to note is that all the data points in Figure 2 lie between the “tram” lines formed by the chart limits thereby suggesting that only common cause variation is present.

Figure 3. Schematic of a process performing in a stable, predictable manner.

At this stage we are using the control chart to decide whether we have a process that is stable and predictable over time, within limits of variation due to common causes only (Scenario 1 – Figure 3), or a process that is unstable and unpredictable over time, with both common and special cause variation affecting performance, (Scenario 2 – Figure 4). This is referred to as Phase I application of control charts. In the schematics the blue curves represent the underlying statistical distributions that may be considered to yield the observation made at the corresponding point in time. With only common cause variation present we can think of successive observations being yielded by the same distribution all the time. With special cause variation also present we can think of successive observations being yielded by distributions that change with time.

Figure 4. Schematic of a process with unstable, unpredictable performance.

Scrutiny of the chart in Figure 2 reveals no points beyond the chart limits so it was decided to “roll out” the chart with those limits and centre line for further monitoring of the process. This is referred to as Phase II application of control charts. Later in the day with additional data plotted the chart appeared as in Figure 5.

Figure 5. Control chart with a signal providing evidence of special cause variation.

The point on the chart corresponding to the measurement of viscosity made at 18:15 lies above the upper chart limit, thus providing evidence that a special cause may be affecting the process. Subsequent investigation by the process team revealed a clogged filter that was replaced. One could then proceed to continue to monitor viscosity using the chart with the limits established using the first 25 observations. In the case of major changes to the process it might be advisable to start the whole charting process again i.e. to take another series of initial viscosity measurements and to plot an initial chart. If there are no points outside the limits on this new chart then it could be adopted for further routine monitoring.

In his foreword to *Understanding Statistical Process Control* by Wheeler and Chambers (1992), Deming refers to the history of Shewhart’s perception of two types of variation as follows. (The quotation that appears at the beginning of this article is from the same source.)

*How did the problem arise? The management of the Western Electric Company, the Hawthorne Plant, Chicago, sought to achieve uniformity, so that a telephone company that bought their product could depend upon it. The aim was noble. Their methods though were folly. They took action, made some kind of change at every sign of departure from uniformity. They were smart enough and honest enough to observe that their actions only made this worse. They sought help. The problem went to Dr. Shewhart…'*

In addition to a point out of the three sigma limits providing evidence of the presence of special cause variation, three other widely used criteria are:

- 8 points in a row on same side of centre line
- 2 out of 3 points more than 2 standard deviations from centre line (same side)
- 4 out of 5 points more than 1 standard deviation from centre line (same side)

These tests, along with the earlier criterion discussed, are referred to as the Western Electric Company Rules. When the additional three criteria are applied the chart previously displayed in Figure 5 now appears as shown in Figure 6. Note that horizontal lines have been added at two standard deviations either side of the mean.

Figure 6. Alternative evidence of the presence of special cause variation.

The occurrence of two out of three points beyond two standard deviations from the centre line and both above it, provides earlier evidence, by 45 minutes, of the presence of special cause variation affecting the process. The three relevant points are indicated in Figure 6. In employing Shewhart charts a balance has to be struck between having too many detection rules and associated increased risk of false alarm signals of special cause variation and the risk of failing to detect process changes timeously.

The consequence of “making some kind of change at every sign of departure from uniformity” is referred to as tampering. This may be illustrated by simulation for the oil viscosity. Imagine that the process operatives have a control setting for viscosity which is set at the target value of 9.0 and that after the initial 25 observations were made that an operative begins to oversee the process and who adjusts the process on the basis of each new observation as follows. If the viscosity observed is 9.2, for example, reduce the control setting by 0.2 and if the viscosity observed is 8.9, for example, increase the control setting by 0.1. Simulated data for this scenario are displayed in Figure 7.

Figure 7. Control chart of process data with tampering applied to process.

There are no signals providing evidence of special cause variation from this chart. However the control chart of the moving ranges shown in Figure 8 does provide evidence of a process change. In fact it can be shown that the type of tampering simulated increase process variability by 40%. Deming (1986) discusses tampering and describes funnel experiments that may be used to illustrate its consequences. Henderson (2011) presents simulations of the funnel experiments and displays associated individual value control charts.

Figure 8. Moving range chart providing evidence of special cause variation.

**Creation of control charts**

Some argue that when someone is using control charts for the first time that there is merit in plotting the data using pencil and paper and in doing the relevant calculations by hand. There is no doubt that software is invariably used to create charts in practice. In this section the charts were created using the spreadsheet software Microsoft Excel. (The American Society for Quality (http://asq.org/index.aspx) provides a free Microsoft Excel template for the creation of Shewhart mean and range control charts that readers may also find of value).

We consider a control chart of the weekly manhours lost for Department X. The data and formulae for the required calculations in Excel are displayed in Table 1, with the 21 data values in the second column, with header X. Here 20 moving ranges are available. The expected value of the range of random samples of size n from a normal distribution with standard deviation is d2. By regarding consecutive observations as samples of n =2, division of the average moving range by d2, which is 1.128 for n = 2, yields the standard deviation estimate of 6.87 and the control limit values of 61.10 and 102.33. Strictly speaking this estimation procedure should only be used with normally distributed data but it has been shown to be robust for non-normally distributed data.

Table 1. Formulae required to create control chart in Excel.

Having found no evidence of special cause variation affecting manhour losses for department X we can now “roll-out” the chart with the centre line and limits calculated for future monitoring.

Week | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |

Manhours lost | 92 | 93 | 86 | 80 | 93 | 76 | 71 | 89 | 71 |

Table 2. Additional manhours loss data for Weeks 22 to 30.

Following plotting of the additional data in Table 2 the chart after 30 weeks data are to hand is shown in Figure 9. We are now employing the chart in Phase 2 mode i.e. we are looking for any evidence that the process has departed from its state of statistical control i.e. that it has been affected by special cause variation.

Figure 9. Control chart of manhours loss data for the first 30 weeks.

There are no points out of the control limits on the updated chart. Thus there is no evidence of special cause variation – the process continues to be stable and predictable, it continues in what is often referred to as a state of statistical control. Let us now suppose that as a result of a process improvement project changes were made at the end of Week 30 to procedures in Department X with a view to reducing the losses and that the plotting the further data in Table 3 the chart after 40 weeks was as displayed in Figure 10.

Week | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |

Manhours lost | 64 | 58 | 73 | 62 | 62 | 63 | 79 | 58 | 68 | 64 |

Table 3. Additional data on manhours loss following improvement project.

Figure 10. Control chart of manhours loss data for the first 40 weeks.

The two points below the lower chart limit provide evidence that the changes have been effective and that a new stage of process performance has been entered. Hence we can compute a new centre line and limits for the data from week 31 onwards as shown in Figure 11. The new centre line is at the value 65.1 as compared with the original centre line value of 87.1. Thus the charts provide evidence of a decrease in the mean weekly loss of around 12 manhours.

Figure 11. Control chart of manhours loss data for the two stages.

In this second illustration of the use of a control chart for individual measurements the signals of special cause variation were welcome ones. In other situations special cause variation would be unwelcome e.g. in the case of the Viscosity example considered earlier if the process mean departs significantly from the target then steps would need to be taken to shift the mean back to the target value. Were the revised chart for manhours lost to yield signals from points falling above the upper limit, then this would provide evidence that manhour losses had increased again. Action would then be required to seek out and eliminate the special causes in order to get the process “back on track” again. Montgomery (2009) states that states that when used in this manner “the control chart becomes a logbook in which the timing of process interventions and their subsequent effect on process performance is easily seen”.

In this article we have scratched the surface of a vast topic. For readers wishing to learn more, Caulcutt (2004) has published two articles in the magazine *Significance *that are available on-line (http://www.significancemagazine.org/view/index.html) and are highly informative. For technical details on the charts discussed and others the references by Henderson (2011), Montgomery (2009) and Wheeler and Chambers (1993) may be consulted.

**The challenge!**

In *Statistical Method from the Viewpoint of Quality Control*, Shewhart (1939) wrote:

'*The long-range contribution of statistics depends not so much on getting a lot of highly trained statisticians into industry as it does in creating a statistically minded generation of physicists, chemists, engineers and others who will in any way have a hand in developing and directing the production processes of tomorrow.'*

Eighty years have elapsed and the author believes that we have failed to fully respond to Shewhart’s challenge! It should be broadened to include the creation of statistically minded people in all spheres of business and service activity, in politics and in journalism. Everyone should learn about common and special cause variation and be armed with a simple graphical tool that distinguishes between the two – the Shewhart control chart. After all, an 11-year old child could understand it!

__Acknowledgement__

The author wishes to acknowledge the encouragement given by Alison Oliver at Wiley and the most helpful comments from an anonymous reviewer.

__References__

(1) Caulcutt, R. (2004) Managing by fact. *Significance*, 1(1): 36–38.

(2) Caulcutt, R. (2004) Control charts in practice. *Significance*, 1(2): 81–84.

(3) Deming, W.E. (1986) Out of the Crisis. Cambridge: MIT Press.

(4) Deming, W.E. (2000) *The New Economics*, 2nd edn. Cambridge: MIT Press.

(5) Henderson, G.R. (2011) *Six Sigma Quality Improvement with Minitab*, 2nd edn. Chichester, John Wiley & Sons Ltd.

(6) Hoerl, R.W. and Snee, R.D. (2012) *Statistical Thinking: Improving Business Performance*, 2nd edn. Hoboken, NJ, John Wiley & Sons, Inc.

(7) O’Connor, J.J. and Robertson E.F. (2013) The MacTutor History of Mathematics Archive http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Shewhart.html (accessed 26th September 2013)

(8) Montgomery, D.C. (2009) *Introduction to Statistical Quality Control*, 6th edn. Hoboken, NJ: John Wiley & Sons, Inc.

(9) Ryan, T.P. (2000) *Statistical Methods for Quality Improvement*. 2nd edn. New York: John Wiley & Sons, Inc.

(10) Salsburg, D. (2001) The Lady Tasting Tea – How Statistics Revolutionized Science In the Twentieth Century. W.H. Freeman and Company, New York.

(11) Shewhart, W.A. (1931) *Economic Control of Quality of Manufactured Product*. New York: D. Van Nostrand. Also available in a 50th anniversary edition published in 1980 by the American Society for Quality, Milwaukee, WI.

(12) Shewhart, W.A. (1939) *Statistical Method from the viewpoint of Quality Control*. Graduate School of the Department of Agriculture, Washington, D.C.

(13) Wheeler, D.J. (1993) *Understanding Variation – The Key to Managing Chaos*. Knoxville, TN: SPC Press.

(14) Wheeler, D.J. and Chambers, D.S. (1992) *Understanding Statistical Process Control*, 2nd edn. Knoxville, TN: SPC Press.

G Robin Henderson

Halcro Consultancy

halcro.robin@gmail.com

## Connect: