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# Tests for Special Causes

Nelson Rules is a set of rules to detect the presence of special causes or non-random behaviour in the process. These are commonly used to check for process stability in Statistical Process Control (SPC). They were first published by Lloyd S Nelson in 1984.

Western Electric Rules is a set of rules to detect the presence of special causes or non-random behaviour in the process. They were codified by a specially-appointed committee of the manufacturing division of the Western Electric Company in 1956.

Westgard Rules is a set of statistical patterns, each being unlikely to occur by chance variability i.e. they are indicative of special causes. They are a set of modified Western Electric rules, developed by James Westgard.

Montgomery Rules is a set of rules to detect the presence of special causes or non-random behaviour in the process. These are commonly used to check for process stability in Statistical Process Control (SPC). They were first published by Douglas C Montgomery.

An application-oriented question on the topic along with responses can be seen below. The best answer was provided by Gulshan Kumar on 25th Nov 2022.

Applause for all the respondents - Rahul Arora, Gulshan Kumar, Himanshu Sharma.

## Question

Q 524. What is the difference between the various set of tests for detection of special causes - Nelson, Montgomery, Western Electric, Westgard? What is the criteria to use these different set of rules? Illustrate your answers with examples.

Note for website visitors - Two questions are asked every week on this platform. One on Tuesday and the other on Friday.

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Special Cause Variation may exist in a process that appears to be in control i.e within upper & lower control limits & it may also be reflected in trends or behaviors that appear non-normal. There are various tests for finding potential special cause variation.

Most of these tests divide the control region of the chart into 3 zones which are usually 1 sigma apart from each other. These zones are as shown below:-

Now Let us explore some of the most common rules for testing special causes in control charts  :-

Western Electric rules for control charts :

Western Electric rules are the original set of four control chart rules used to conduct stability analysis. These rules are used with all the control charts & are commonly known as WECO rules. These rules are as stated below :-

• 1 One point above UCL (Upper Control Limit) or below LCL (Lower Control Limit)
• 2 Two points above or below 2 sigma zone i.e. zone which is usually 2σ above or below mea
• 3 Four out of five points above or below 1 sigma i.e. zone which is usually 1σ above or below mean
• 4 Eight points in a row above or below the central mean line

Below are the examples demonstrating the WECO rules:-

Nelson rules for control charts :

Nelson rules were developed in the 1950s & can be used with any control chart & they are the extension of the WECO rules. These rules are as stated below :-

• One point above UCL (Upper Control Limit) or below LCL (Lower Control Limit)
• 2 Two points above or below 2 sigma zone i.e. zone which is usually 2σ above or below mean
• 3 Four out of five points above or below 1 sigma i.e. zone which is usually 1σ above or below mean
• 4 Eight points in a row above or below the central mean line
• 5 Six points in a row ascending or descending (trend)
• 6 Fifteen points in a row between -1σ & 1σ
• 7 Fourteen points in a row alternating up & down
• 8 Eight points in a row above 1σ or below -1σ

Below are some of the examples showcasing some of the above rules :-

Westgard rules for control charts :

Westgard rules include the WECO rules-1, 2, 4 & rule-5 of the Nelson rules. These rules are used with levey jenning charts in laboratories. These rules are as stated below :-

• 1 One point above UCL (Upper Control Limit) or below LCL (Lower Control Limit)
• 2 Two points above or below 2 sigma zone i.e. zone which is usually 2σ above or below mean
• 4 Eight points in a row above or below the central mean line
• 5 A trend of seven points in a row increasing or decreasing
• 9 Four points above or below 1 sigma or two points in a row with one above & one below 2 sigma

Here the rule-9 is unique & is not included in other control chart rule sets.

Let us now summarize the differences of different rules :-

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Dr. Walter Shewhart of Bell labs developed a theory of Statistical process Control in 1924. Dr. Shewhart presented his theories in a series of lectures  that were published in a book. Economic control of Quality manufacturing Product 1931. SPC came into wide use during 1940 as a result of war production effort. Dr Shewhart think that All process display variation

There are two component of variation

1-    A steady component that is inherent that Dr. Shewhart attributed to chance and undiscoverable causes, currently called random variation.

2-    An intermittent component attributed to assignable causes that could be economically discovered and removed.

Variation of a particular process characteristic can be quantified by sampling the characteristic and estimating the parameters of distribution.

Changes in the distribution can be revealed by plotting the parameters versus time.

Dr. Walter Shewhart define 8 rule to detect variation and out of control condition of a process. Tests 1, 5, 6, 2 defined by the Western Electric CO (1958) as the original 4 rules. Test 2 modified to from 8 to 9 points and after modification test 1-8 are defined by Lloyd S. Nelson (1984). These 8 rule to detect and control variation has been defined in different book by different author and one of the popular books by by Douglas C. Montgomery, 2012.

Below is the Nelson 8 rule to detect and control variation :

1-    1 point above UCL or Below LCL, Indicates Large change in process

2-    2 Point above/ below 2 σ Indicates Major special cause variation

3-    4 out of 5 point above/ below 1σ Zone Indicates A small shift – special cause

4-    8/9 point in a row above/below centre line Indicates Process Mean shift

5-    6 point a row increasing or decreasing (trend) Indicates A trends up or drift

6-    15 point in a row +/- 1 σ – near centre line Indicates Old or incorrect limit, within subgroup variation is large compare to between subgroup variations.

7-    14 point in a row alternating up and down Indicates alternating pattern due to over adjustment, shift to shift or machine to machine variation.

8-    8 point in a row above 1σ and below 1σ Indicates Mixture pattern due to over control or 2 different processes on same chart.

The 4 Western Electric rules that indicate an out of control process condition. The position of data point and their trends compared to the centre line value (mean) and control limit indicate out of control or non-random signal in data present. Four Western Electric rules are:

a.    1 point above UCL or Below LCL

b.    2 out of 3 point in a row falling in 3σ or above zone

c.    4 out of 5 point above/ below 1σ zone

d.    9 Point in a row one side of centre line.

The Westgard rules are a set of modified Western Electric rule, developed by Jemes Westgard Mainly focus on faulty accuracy or precision of the measurement system. They are used for laboratory quality control, in runs consisting of measurement of multiple samples and published in his book seminars on quality control. Below are the rules:

1-    1 Point above or below 2σ Indicates inaccuracy or imprecision

2- 1 Point above or below 3σ Indicates inaccuracy or imprecision

3-    2 Consecutive point above or below 2σ same side of centre line Indicates inaccuracy or imprecision

4-    2 measurement in same run have 4σ difference such as 2σ above mean and 2σ below mean Indicates imprecision

5-    4 Consecutive point above 1σ same side of mean Indicates Inaccuracy

6-    10 Consecutive measurement same side of mean Indicates Inaccuracy in measurement

Conclusion: So first decide objective to use control chart and decide which control chart is usefull and what data needs to collected. Ensure to train people on control chart concept and make them understood about importance and use of control chart. Many time we experience no enough care being taken in data collection as not following define intervals, in correct measurement that may result in wrong decision based up on data. To find out causes closely observe Man, Machine, Material, Method, Environment & Process parameter etc. Control chart will indicate you problem which can give direction to investigate further and arrive at root cause.

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Very interesting question and all the responses are detailed. The best answer has been provided by Gulshan.

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