Last year I wrote an Introduction to Control Charts (Run Charts). I referenced a favorite book of mine, Understanding Variation: The Key to Managing Chaos, by Donald Wheeler of SPC Press, an expert in Statistical Process Control. This book is a quick read, and it’s a great introduction to control charts, written clearly using layman’s terms, with a number of very good examples to illustrate their use.
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.
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.