Understanding Control Chart Detection Rules

In the realm of statistical analysis in problem solving (especially Six Sigma), Statistical Process Control or Control Charts are a commonly taught and used statistical tool, however we find it can often become overly complex in it’s teachings, or quite often they can be totally misunderstood. The reality of control charts is actually very simple, and I would argue only serves as an all-round benefit when used appropriately.

Firstly, if you want to know the history of Shewhart’s control charts, check out these wonderful articles by Dennis Crommentuijn-Marsh:

• The Birth of the Control Chart – Discovering the Drivers for Shewhart’s Creation
• Celebrating a century of understanding variation

This brief article however intends to focus on the detection rules that have since evolved from Shewhart’s original memo in truly understanding their usage and benefit to you.

Control Charts & Variation

There is always one simple picture that we like to use to demonstrate the fundamental purpose of a control chart in a nutshell…

Like any form of graph or graphic, it is there to visualise something to make a certain piece of information clearly visible and understandable to anyone walking past. The fundamental premise of the control chart is to help visualise when unnatural variation may have occurred in a data set / process that may have a knowable cause. In other words, it helps you see when something changed in the process or to check if you have made a significant change to a process yourself – this change may be desired, or undesired.

The point is, if you see signs of unnatural variation, go and investigate, learn from it, and take action to address it if needed.

It is worth briefly defining what we mean by these types of variation, as through the evolution of statistical process control, inevitably means many people have gotten their hands in the pie and created their own terminology. What we’ll do here is quickly cover the different terminology you might use and what we really mean when we say it:

There are various reasons why these terms came to be, most of them trivial, but that is the way of data! For eases sake, I will clarify the terminology I prefer to use is Natural and Unnatural variation as I think holistically, this is the most applicable definition to any situation and I find makes it easier for many to understand.

The purpose of Shewhart’s control chart is to help distinguish between these two types of variation, in a clear, visual, and convenient way.

The Detection Rules Explained

This is where detection rules come into play, and there has been some small evolution over time. The two most common sets of rules (or guidelines) I see being utilised across industries are the Western Electric rules, and the Nelson rules, and there can be some debate on which one to use. Ignoring other methods that have been developed or utilised, a very oversimplified timeline of these is:

Whilst there is debate on their usage, the reality is that each tried to enhance the previous and are actually all from the same place (each was based off of Shewhart’s fundamental principles), but their usage has since exploded and as such, probably been misunderstood also. Nelson Rule’s were mostly just an attempt of ‘standardising’ practice from the original Western Electric Handbook to try and have more consistent application of them.

Let’s begin to explain how you can use these in the most appropriate way depending on your situation, and explain why using all of them without thought is actually problematic. We’ll try to clearly show what the rules are across both Western Electric and Nelson, but there is only so much I can apologise for in remembering what number is in what system – and the last thing I want to do is create an entirely new numbering system.

Rule	Description	Example	How is it useful	When to use
Shewhart Rule 1 Western Electric Rule 1 Nelson Rule 1	1 point falls outside the control limits (typically ±3σ from the mean).		Useful when monitoring for large shifts in the process as this will filter out the vast majority of natural variation with a very low error rate. This will cover most situations as this is what provides organisations with the ability to see shifts that are “economically” important to respond to. However it is not as good at detecting small shifts in the process.	Use this on all control charts. This will capture the majority of situations.
Western Electric Rule 2 Nelson Rule 5	2 out of 3 consecutive points are beyond 2σ from the mean on the same side.		This rule comes in to be able to detect smaller shifts in the process where rule 1 perhaps cannot. It is able to detect more intermediate size shifts rather than the more large shifts in rule 1. This simply increases the sensitivity of the control chart. This can be very useful where you have a process that requires slightly higher resolution of shifts.	Use this alongside rules 1 for increased sensitivity in intermediate shifts. This can be used on virtually all control charts for increased sensitivity and little increase in false alarms. Can’t be used on Moving Range charts or when there is no clear ‘sequence’ to your data.
Western Electric Rule 3 Nelson Rule 6	4 out of 5 consecutive points are beyond 1σ from the mean on the same side.		Use for identifying even smaller changes than 1 and 2. Best for processes where small, sustained changes can indicate significant impact. This is the next level to rules 1 and 2, where they cannot detect these small shifts, rule 3 possibly can. This simply increases the sensitivity of the control chart even further, but the gain is less significant. This can be useful where you have a process that requires even higher resolution of shifts alongside rules 1 and 2.	Use this alongside rules 1 and Western Electric 2/ Nelson 5 for increased sensitivity in intermediate and smaller shifts. This can be used on virtually all control charts for increased sensitivity and little increase in false alarms. Can’t be used on Moving Range charts or when there is no clear ‘sequence’ to your data.
Western Electric Rule 4 Nelson Rule 2	8 consecutive points are all above or all below the mean. (Nelson = 9 points)		Use to detect longer-term drifts or shifts in the process. This when used with Rules 1, 2, and 3 gives minimal gain. But when used simply with Rule 1, gives the ability to detect intermediate shifts circa 80% of the time. This is useful to supplement rule 1 and fundamentally is very simple to use and see. Using rules 1 and 4 together without any other rules can capture the majority of situations. Nelsons modification is simply more conservative as it significantly reduces the false alarm rate.	Use alongside Rule 1 when you require more sensitivity to intermediate/smaller shifts. Can’t be used on Moving Range charts or when there is no clear ‘sequence’ to your data.
Nelson Rule 3	6 consecutive points show a consistent increasing or decreasing trend.		Use to detect trends in data over a period. You might see this as a trend of deterioration or gradual improvement. This should theoretically increase the sensitivity of the control chart. Avoid in processes known to have natural periodic trends, as it may incorrectly signal a change. This rule by all accounts gives the least benefit for the potentially highest false error rate and so should be used very pragmatically.	Use if you wish to detect increasing or decreasing trends in a data set, but be pragmatic when responding to it. Use only in needed situations (if at all).
Nelson Rule 4	14 consecutive points alternate in direction (up and down).		This rule can indicate issues related to ‘autocorrelated’ data. This rule will often detect when 2 sources of data have been mixed onto the same chart and often is a data collection issue.	Very useful in the setup of a control chart, but is less important in the ongoing running once data collection has been standardised. Generally you could seldom use this rule, due to the fact that the previous rules will likely capture changes first, but this can add some sensitivity if needed with not much increase in false alarms. Nelson recommended using this on all charts for this reason, but it is not essential.
Nelson Rule 7	15 consecutive points are within 1σ of the mean.		This is a diagnostic for stratification. Test 7 reacts when the observations within each subgroup consistently come from multiple sources with different means as this creates a “hugging the centreline” pattern. This means that each subgroup of observations may alternate between different conditions, such as data being collected from 2 similar processes, and this data being put into the same subgroup (mixed together). This can help identify potential data collection errors and sometimes within-subgroup noise.	Very useful in the setup of a control chart, but is less important in the ongoing running once data collection has been standardised. Use this when creating your initial control chart and data collection methods.
Nelson Rule 8	8 consecutive points are all more than 1σ from the mean, on either side.		This is a diagnostic for stratification and like Test 7, however test 8 reacts when the observations within each subgroup consistently come from the same source, but is a different source to other subgroups. This means that each subgroup of observations may have been taken under different conditions, such as data being collected from 1 process, that produces one output, then switched to produce a different output. This data appears consecutive however the reality is subgroup 1 is a different process to subgroup 2. This is simply the inverse of Nelson Rule 7.	Very useful in the setup of a control chart, but is less important in the ongoing running once data collection has been standardised. Use this when creating your initial control chart and data collection methods.

The Problem With These Rules

The simple fact of the matter is that as you add more detection rules, your false alarm rate goes up. If I keep asking a chart to “look for this pattern, now this pattern, oh and this pattern, and yes, that pattern too”, then eventually it will match a pattern and give a signal. This is akin to ‘throwing everything at the wall and seeing what sticks’. Something will eventually stick, you have to balance whether the additional rule is worth the potential response it might create.

This isn’t saying never apply additional rules, but apply them in the situations that you need them and if and when appropriate. If you are ever in doubt of how to appropriately use them, I often like to describe the detection rules for dummies:

1. Rule 1 will capture the vast majority of cases where control charts will be useful to an organisation.
2. If you need more than this, using the 4 Western Electric Zone (Nelson rules 1, 2, 5 and 6) tests give more resolution without significantly increasing false positive rates.
3. Nelson rules may provide the rules to help you in given situations, especially when setting up the control chart.