To review the approach, the raw data is plotted in two ways. The actual points are plotted in one chart, and the moving ranges (differences between points i and i-1) are plotted in another chart. A horizontal line is drawn on each chart at the mean of the data. Control limits are calculated using the moving range as a measure of variability instead of standard deviation. The upper control limit (UCL) of the moving range chart is calculated as 3.27 times the mean of the moving range, and this is plotted on the moving range chart. The upper and lower control limits (UCL and LCL) of the individual values is given my the mean of the individual values ± 2.66 times the mean of the moving range, and these are plotted on the individuals chart.

If there is a trend in the data, the moving range will be smaller than the standard deviation, because the basis for determining variability is difference from the previous point, rather than from the mean of all points.

Individuals-Moving Range Analysis

This table shows the individual values and moving ranges for the three main browsers. The means and control limits are computed below the table of values, and values in the table are colored red if they lie outside the control limits. The values show browser usage each month by percent of visits to my site.

There is a lot of red (i.e., out of statistical control) in the IE and Chrome individual values, notably at the beginning and end, indicating a trend from start to finish. Firefox shows only one red point, and there’s no obvious trend. The only red value in the moving range data is a single point for IE.

The data is plotted in the following I-MR charts. The Y axis ranges are the same for all browsers for easy comparison. The trends for Internet Explorer and Chrome are rather obvious when the new control limits are plotted.

For IE (above) the upper and lower control limits calculated using mean and standard deviation were 67.9% and 53.5%, much further apart than those in the I-MR chart; in fact, those limits fall outside the Y axis scale of the I-MR chart. For Chrome (below), the Mean-SD upper control limit is 10.4%, which also falls outside the corresponding I-MR chart. Both calculations for Chrome’s LCL are below zero; since this makes no physical sense, zero is used.

The Firefox control limits based on mean and SD are further apart than the I-MR limits, by one percentage point (32.4% and 27.0%), but would still be visible in this chart.

I-MR Analysis for Sparse Data

The conclusion from my earlier post was that three points over 18 months is insufficient data to judge whether there was a trend in the browser usage percentages. This conclusion holds when the more rigorous I-MR evaluation is carried out. If we perform the above analysis on four points, one point every six months, the I-MR calculations and charts show the processes are in control, and cannot be attributed to changing patterns of usage.

The moving range values are much larger than for monthly data points, since six months of changes are lumped into one point. As a result, the control limits are pushed far enough away from the means that there are no out-of-control points.

The data is plotted in the following I-MR charts. The Y axis ranges are the same for all browsers for easy comparison. Although we “see” trends for Internet Explorer and Chrome, since there are no points outside the control limits and not enough points to invoke the special Western Electric rules, we cannot conclude there is any variation not attributable to random fluctuations.

Peltier Technical Services, Inc., Copyright © 2010.
 
Licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.
 

In fact, in my last engineering job before becoming a full-time Excel jockey, I was frustrated by many operational features in the manufacturing facility where I worked. As I reread Wheeler’s book, I could relate many observed behaviors in the plant with examples in the book. Most examples, of course, showed misuse of reporting as well as optimization of separate departments to the detriment of the business as a whole. My employer talked the talk of SPC and Six Sigma, but nobody with any authority understood statistics or processes, so the company walked a random walk through the jargon of process control.

Types of Control Chart

A control chart, or run chart, is essentially a time series that shows variation in a process output over a period of time. The time may be defined by an actual date or time, or by a number indicating how many times the process has been carried out. Control charts were developed in the 1920s by Walter Shewhart while he was working at Bell Labs. Shewhart was investigating ways to improve product reliability by reducing and controlling manufacturing variability, and the control chart was Shewhart’s means of distinguishing random variability inherent in a process from “special causes” extrinsic to the process.

Wheeler’s book demonstrates statistical process control using the simplest type of control chart. This is called an Individuals chart, so named because it is based on individual measured values. The moving range (difference between sequential measurements) is used to calculate control limits for the chart. Individuals charts are also called XMR (or XmR) charts, to denote the individual X values and the moving ranges.

Individuals (XMR) chart, comprised of X chart (top) and R chart (bottom)
from Introduction to Control Charts (Run Charts)

The Individuals chart is used when individual values or rates are collected periodically. Examples of this type of data may be daily scrap rates in a production line, or a company’s monthly sales. Depending on the type of data being tracked, and on how it’s collected, there are actually several different types of control charts.

Other types of control charts are better suited to data which is collected in groups. While the Individuals chart plots the individual measured values and uses the moving range to provide statistical context, XbarR and XbarS charts are used when a batch of measurements comprise each sample. The average measurement from the group (Xbar) is plotted, while either the range (XbarR, for small batches) or sample standard deviation (XbarS, for larger sample sizes) of each sample is used for the calculation of control limits. I have seen one reference to the use of medians rather than means in these charts, but these variants must be rare.

If counts are measured (for example, the number of defective parts in a batch), several other control chart types have been developed to display the run chart behavior. These are known as P, C, nP, and u charts.

Selecting a Control Chart

To help sort out the different types of control charts, Wheeler has laid out the requirements of each in the form of a flow diagram. The only version I have of Wheeler’s diagram is a second-hand distorted GIF file, so I’ve redrawn the flow chart in both vertical and horizontal orientations. This is a handy reference if you are building your own control charts, but an SPC software package will automatically select the appropriate control chart based on the data provided.

Vertical Flow Diagram for Selection of Proper Control Chart
Link to Image File – Link to PDF File

Horizontal Flow Diagram for Selection of Proper Control Chart
Link to Image File – Link to PDF File

In a series of posts I will show how to create each type of control chart in my favorite statistical process control software platform, and I will present real-world examples of each.

Interpreting Control Charts

I will also cover interpretation of control charts. A process is in control when its variability is described by the natural statistical variation defined by measurements taken over a period of time. When the process goes “out of control” (or “out of statistical control”), it indicates a possible change to the process, because the statistical rules determining the variability in the process have changed.

The natural process limits in a control chart are constructed to contain variations within 3 standard deviations above and below the mean, or 99.7% of naturally occurring variations. Any measurement outside this range is out of control, and special causes for this variation should be identified.

Obviously, a process is out of control when a measurement falls outside of the upper or lower control limits drawn on the control chart. The probability that this may occur naturally is 0.3%, sufficiently rare to warrant examination of the process when this occurs.

Other patterns in the data also merit special investigation. For example, too many consecutive measurements above or below the mean may indicate loss of statistical control of a process. Too many consecutive measurements that trend in one direction, or too many consecutive alternating measurements, may signal a process shift. A change in the process may be indicated by having too much, or not enough, variation within the ± 3 sigma limits.

Not enough variation sounds like a good thing, doesn’t it? Well, it probably is. Not all process change is necessarily bad. But if this change is not understood, then it cannot be reproduced, measured, and controlled. Thus, even if the change results in less variability, it is not a good change.

A number of rules have been established that define patterns that may indicate change in the underlying process. I will cover these special rules in a post in this series on the control chart. Software packages that produce run charts have these rules built in, but it is good to understand their basis.

Peltier Technical Services, Inc., Copyright © 2010.
 
Licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.
 

Jorge Camoes suggests in Validate Your Tables – US Census Bureau Example that we make a quick chart to make sure the data “looks” correct. Jorge was looking at some data from the US Census Bureau, and it didn’t seem correct. So he made a simple plot.

As Jorge says, “Can you see [the error] now? Of course you can. Try not to see it…”

Jorge’s simple example is a great illustration. The following is another great example of how much easier it is to see patterns in charts than in tabulated data. This example is taken from Understanding Variation: The Key to Managing Chaos, by Donald J. Wheeler.

This small, 136-page book is very useful because of its down-to-earth approach to SPC and its reliance on real business and manufacturing examples. I’ve cited other examples from this book in my post Introducing Control Charts (Run Charts).

Here is the weekly production data for a plant, as presented in Understanding Variation, and unfortunately as it would be presented in so many formal and informal reports in business and industry.

Do you see any patterns? Sure. Right. The first step towards clarity is to put the data into a single list (CSV file), and make a simple line chart. This type of chart, which shows sequential values along a line chart, is called a control chart or run chart and it is a basic component of Statistical Process Control.

As Jorge might say, “Can you see [the pattern] now? Of course you can. Try not to see it…”. When you look at the complete picture, and include dates (CSV file), you can even form a hypothesis about why the pattern exists.

I’ve made up the dates to go with the production figures, but only to help the illustration. Anyone who has worked in a production environment, in fact, any environment in which there are requirements due at regular intervals, knows this general pattern. Every week, some of the monthly quota is met, but in the last week of the month, there is a flurry of activity needed to actually meet the monthly quota. This is simple human nature of course, the same characteristic that leads to an all-nighter whenever a term paper is due.

While employed as a scientist/engineer for a large manufacturing corporation, I pulled a stint as trainer for my company’s Total Quality program. This role occupied half of my time for three or four years. A large component of our Total Quality initiative was related to Statistical Process Control (SPC). And of all the volumes of course materials used in this training, the most useful was a small, 136-page book called Understanding Variation.

Understanding Variation: The Key to Managing Chaos
Donald J. Wheeler

What made the book so useful was its down-to-earth approach to SPC and its reliance on real business and manufacturing examples.

After discussing the randomness but orderliness of data, the book introduces time series charts. Basically, a record of performance is put on paper, in graphical form, so a quick glance can show whether things are on track. Once you have such a chart, you need a system that helps indicate how much variation is acceptable, and how much means the process is out of control.

Let’s take a look at some sample data. I’ve generated a couple of months of normally distributed process data, which looks like this:

The gray vertical lines separate adjacent weeks. Without any special training, one could look at this data, see that it varies, and say that “some days are better than others”. But we have no way to characterize the variation. Walter Shewhart first applied some simple but effective numerical techniques to this type of data, and he called the result “control charts”. While the techniques were relatively simple to apply, they were based on solid statistical methods.

The first step is to add a line to the chart above representing the mean of the data:

We still can’t say anything more insightful than the data is distributed about the mean. In order to judge the variability in the data, we need first to quantify the variability. This can be done by analyzing a moving range chart, which is an effective measure of variation in the data, and simpler to compute than the standard deviation (at least in the days prior to desktop computers). Essentially, for a given day, we plot the absolute difference between that day’s value and the previous day’s value.

We can add a line for the mean as we did with the chart of raw values:

We still don’t know how much variability is “acceptable”, that is, attributable to common variation in the process. This is where some calculations are required. When using the moving range to determine “typical” variation, the maximum acceptable value of the moving range is 3.27 times the mean of the moving range. For this data, the mean of the moving range is 3.2, so the maximum acceptable moving range, or the Upper Control Limit for Ranges, is 10.5.

The variability in moving range is well under the Upper Control Limit (UCL). We can use the moving range to compute the control limits on the individual values data: the mean of the moving range is multiplied by 2.66, then added to and subtracted from the process data mean (this is roughly equivalent to using 3 times the standard deviation of the individual values data). Our Upper and Lower Control Limits (UCL and LCL) are thus 28.1 and 11.1.

In conjunction, these two graphs comprise an XMR chart, because it analyzes the X (individual values) and MR (moving range) data.

A quick glance at these charts shows that the process data falls well within the upper and lower control limits, that is, it is operating within the bounds of random variation, and we can say that the process is “In Control”. In truth, there are additional tests for randomness, because any patterns in the data even within the control limits means the process may be “Out of Control”. A detailed treatment of these additional tests is beyond the scope of this introduction.

In addition to the XMR methodology, there are several other types of control charts, which are used for different types of data and data collection. I plan to describe them in a future post.

Further Reading

ISO 9001 – Introduction to SPC

Control Charts on Wikipedia

Interpreting Control Charts

Selecting the Right Control Chart

Excel Control Charts on the Peltier Tech Web Site

Run Chart with Mean and Standard Deviation Lines

This example shows how to build a run chart with a mean and lines for one standard deviation above and below the mean. For an “official” run chart calculated based on mean and standard deviation, the upper and lower control limits are usually set at ±3 SD. It is a simple matter to adjust the example to show any control limits you have calculated.

Dynamic Control Chart

This links to a zipped workbook that uses VBA code to adjust the chart when the data changes. The positions of the lines are adjusted for changes in the data, and the points are formatted so that points below the mean are filled with blue and those above white, while diamonds are used for points within one SD of the mean, circles for two SD, and triangles for three SD; points outside of the control limits are red squares.

Historical note: This was the first chart I ever formatted point-by-point with VBA. Back in Excel 97 on an underpowered Windows 95 laptop, it was very slow, taking almost a minute to process the chart as seen above. It happens in a flash in Excel 2003 on a computer with enough juice to support Windows XP.

Commercial Statistical Process Control Software

If you have a need for extensive Statistical Process Control capabilities, then you are not likely to want to build an entire solution yourself. There are a number of commercial packages that provide SPC functionality for Excel. There is one in particular which I will mention here: SPC for Excel. Written and supported by a client of mine, Business Process Improvement, or BPI Consulting, SPC for Excel is a turnkey SPC system in the form of an Excel add-in. SPC for Excel is compatible with Excel versions 2000 through 2007; this reduced view of its Excel 2007 ribbon tab shows the capabilities of SPC for Excel.

There are a lot of terms that mean SPC, or that are a part of SPC. I plan to address some of these topics in more detail in future posts.

• Process Control
• Process Capability
• Quality Assurance
• Quality Control
• Total Quality Management
• Reliability Engineering
• Six Sigma
• Lean Manufacturing
• Continuous Improvement

Statistical tools commonly used in SPC include the following (with links to pages on the Peltier Tech web site):

• Run Charts (Control Charts)
• Box and Whisker Charts
• Probability Plots
• Histograms
• Pareto Charts
• Scatter Plots
• Regression Analysis
• Analysis of Variance

I will discuss several of these charting techniques in future posts.

Here are a few online resources you can visit to learn more about SPC.

• National Institute of Standards and Technology
  NIST/SEMATECH e-Handbook of Statistical Methods
  Process Control Techniques
  
• Business Process Improvement
  SPC Articles and Newsletter
  SPC for Excel Software
  
• Wikipedia
  Statistical Process Control on Wikipedia
  

Stay tuned for my follow-up posts on SPC. I will build a running index on this page as I add topics.