402

23. - 25. 5. 2012, Brno, Czech Republic, EU

ANALYSIS OF NONRANDOM PATTERNS IN CONTROL CHART AND ITS SOFTWARE

SUPPORT

Darja NOSKIEVIČOVÁ

FMMI, VŠB-TU Ostrava, 17. listopadu 15, 708 33 Ostrava – Poruba, Czech Republic,

[email protected]

Abstract

Identification of assignable causes of the process variability and restriction and elimination of their influence

are the main goals of statistical process control (SPC). Identification of these causes is associated with so

called tests for special causes or runs tests. From the time of formulation of the first set of such rules

(Western Electric rules) several different sets have been created (Nelson rules, AIAG rules, Boeing AQS

rules, Trietsch rules). This paper deals with the comparison analysis of these sets of rules and with the

analysis of the computer support of them. At the end of this paper some recommendations for the correct

use of these rules are formulated.

Keywords:, Common causes, special causes, runs tests, natural pattern, unnatural pattern

1. INTRODUCTION

Statistical process control (SPC) is an approach to process control that has been widely used in any

industrial or non-industrial fields. It is primarily a tool for understanding process variation [1]. SPC is based

on so called Shewhart´s conception of the process variability. This conception distinguishes variability

caused by obviously effected common causes (process is considered to be statistically stable) from

variability caused by abnormal special (assignable) causes (process is considered not to be statistically

stable) using control charts. The main goals of SPC is an identification of abnormal variability caused by

special (assignable) causes with the aim to make the process stable, minimize the process variability,

improve the process performance. To meet these goals SPC must be built as the problem solving instrument

and the sequence of the subprocesses “Out-of control signal revelation – Root cause identification – Action

acceptance – Verification of action” must be the axis of the SPC application. In this paper the stress is put on

the first subprocess applying to so called Shewhart control charts.

Control chart is the main SPC instrument for the analysis of the process variation over time. It is a graphical

depiction of the process variation and its natural and unnatural patterns. Control chart displays a value of the

quality characteristic that has been measured or some sample statistics that has been computed from

measured values in sample versus the sample number or time. Central line in the control chart represents

the average value of the quality characteristic corresponding to the statistically stable process (only natural

pattern is presented and only common causes influence the process). Control limits are set so that when the

process is statistically stable, nearly all the points in control chart fall between them (for mostly applied ±3σ

control limits and assumed normal distribution it is 99,73% points). As long as the points are within the

control limits and they show natural random pattern, the process is considered to be statistically stable. But if

some points are out of the control limits or the points inside the limits show nonrandom (unnatural) pattern it

is assumed that assignable cause of the abnormal process variation is present and must be removed from

the process via searching for it and corrective action or some improvement must be realized.

In general the points plotted on a control chart form an irregular up-and-down pattern – natural or unnatural.

An advantage of using control charts is that it enables observation of the process behaviour over time to

search for occurring of any unnatural pattern [2] as a symptom of the special (assignable) cause and as a

signal for the reduction of the process variation and for improvement.


Natural pattern is such pattern where the points fluctuate randomly. It can be characterized as follows [3]:

most points lie near the central line; only few of the points spread out and reach the control limits; very rarely

some point goes out of the control limit. When some or all these attributes are missing the pattern will be

classified as unnatural. Many types of unnaturalness can be recognized in control charts. But several generic

patterns that can be detected for various processes were defined. In table 1 we can find mostly

recommended (for instance in [3], [4], [5], [6], [7]) and applied nonrandom patterns for Shewhart control

charts with their description and their typical symptom in control chart. To standardize and simplify the

process of the typical unnatural patterns recognition statistical tests (rules) that quantify the length of the

unnatural patterns were formulated.

Table 1 Unnatural pattern description

2. TESTS FOR UNNATURAL PATTERNS RECOGNITION

Tests for unnatural patterns supplement the evidence of the unnatural pattern given by the position of the

points relative to the control limits or central line with evidence based on the statistical theory of runs [8]

except the Shewhart rule (point out of the control limit). They are based on the probability calculations that

tell us the portion of points lying near the central line, near the control limits, etc. We will discuss tests that

cover unnatural patterns no. 1 - 6 (see the former table), i.e. shifts, trends, mixtures, systematic variation and

stratification. Table 2 contains sets of the rules that are the most well – known and applied (Shewhart rule

[9], Western Electric rules [3], Nelson tests [10], ISO 2859 tests [11], special sets formulated by large

companies (Boeing [12], AIAG [13]) and the most new set by Trietsch [2].

Rules 1 - 4 were defined to quickly recognize patterns linked to the shifts in the process. Rule 5 is connected

with the trends, rule 6 and 8 with patterns caused by incorrect sampling and rule 7 with abnormal oscillation.

Historically first rule for the detection of the assignable cause developed W. S. Shewhart – in table 2 it is

Rule 1. Set of Western Electric Rules added to the Shewhart criterion run tests covering such unnatural

patterns as smaller or sustained shifts (called zone tests), mixtures, stratification and systematic variation. To

be able to apply these tests the region between control limits in control chart must be divided into 6 zones

each of 1 sigma width and the location of the predefined sequences of the points in relation to these zones

must be evaluated. As normal distribution of the applied sample statistics is supposed, in zones C 68,27%

values, in zones B 27,18% values and in zones A 4,28% values are expected when process is statistically

No. Unnatural (non-

random) pattern

Pattern Description Symptom in control chart

1 Large shift

(strays, freaks)

Sudden and high change Points near and or beyond control limits

2 Smaller sustained

shift

Sustained smaller change Series of points on the same side of the central

line

3 Trends A continuous changes in one

direction

Steadily increasing or decreasing run of points

4 Stratification Small differences between values in

a long run, absence of points near

the control limits

A long run of points near the central line on the

both sides

5 Mixture Saw-tooth effect, absence of points

near the central line

A run of consecutive points on both sides of

central line, all far from the central line

6 Systematic

Variation

Regular alternation of high and low

values

A long run of consecutive points alternating up

and down

7 Cycle Recurring periodic

movement

Cyclic recurring patterns of points


Table 2 Tests for unnatural patterns

stable.

Rules 2, 3 and 4 were designed for quicker identification of the shifts from the mean as compared to the test

1 (early warning indicators).

As it can be seen in table 2 the most complex as to the rate of covering all eight defined rules are Nelson

rules [10]. In addition to the Westerns Electric set Nelson defined run test for the oscillation and for the

trends. In rule 4 he changed the length of the run from 8 to 9 points. As compared to the Western Electric

rules the sequence of the rules had been changed by Nelson, too. Standard for Shewhart control charts ISO

8250 has wholly copied the Nelson set of rules. Also Trietsch in his publication works with the Nelson rules

but in rules 6, 7, 8 he has changed the length of the runs to reach better statistical properties of these tests.

Tests for unnatural

patterns

Shewhart

Rule

Western

Electric

Rules

Nelson

Tests

ISO

8258

AIAG

Rules

Boeing

ASQ

Rules

Trietsch

Rule 1

One or more points more

than 3 sigma from the

mean (beyond zone A)

1

1 1 1 1 1

Rule 2

2 of 3 consecutive points

between 2 and 3 sigma

from the mean (in zone A

or beyond)

2 5 5 2 5

Rule 3

4 of 5 consecutive points

between 1 and 3 sigma

from the mean (in zone B

or beyond)

3 6 6 3 6

Rule 4

8 consecutive points on

one side of the mean (in

zone C or beyond)

4 2

9 points

2

9 points

2

7 points

4 2

9 points

Rule 5

Six consecutive points

steadily increasing or

decreasing

3 3 3 3

Rule 6

15 consecutive points both

above and below central

line ( in zones C)

7 7 7

13 points

Rule 7

14 consecutive points

alternating up and down

4 4 4

13 points

Rule 8

Eight points in a row on

both sides of the central

line with none in zones C

8 8 8

5 points


Rules of the companies Boeing and AIAG have less complex sets of rules based on the first four Western Electric zone rules.

3. METHODOLOGY FOR APPLICATION OF THE RULES FOR UNNATURAL PATTERN

RECOGNITION

In literature (see for instance [9], [3], [14], [15], [10], [16], [17], [4]) several strategies for the application of the

rules for unnatural pattern recognition from different authors can be found. They have many differences but

the main idea is common: runs test must not be applied routinely; it is really impractical and simultaneous

application of all known rules brings a danger of unacceptable rise of the false alarm.

As it can be found in the mentioned literature some authors differ in the idea what rules to apply routinely.

But predominantly they recommend rule 1 in conjunction with rule 4. For additional sensitizing of the control

charts some of the authors recommend rules 2 and 3. Wheeler recommends using of these rules only when

increasing sensitivity of a control chart is absolutely necessary. From some strategies it can be seen an effort

to apply separately rules for the identification of patterns for shifts and patterns for subgrouping problems

(for instance Wheeler intents [16] that the main use of Rule 6 is at start-up of SPC than in an on-going

control). Some authors put the stress on the phases of the SPC implementation when selecting suitable

rules. Montgomery means that „Sensitizing rules can be helpful when the control chart is first applied and the

focus is on stabilizing an out-of-control process. However, once the process is reasonably stable, the

routine use of these sensitizing rules to detect small shifts or to try to react more quickly to assignable

causes, should be discouraged” ([4], p. 177).

4. TESTS AND TYPES OF CONTROL CHARTS

When selecting suitable rules for the selected control chart it must be considered that ([10], [3], [2], [16.]):

All rules can be applied to x-bar chart and individuals chart supposing normal distribution of the variable;

Rules 1- 4 can be applied to the dispersion charts (R, s) without any modification when sample size is 5

or more (it should lead to the symmetric control limits);

Nonparametric rules (Rule 5 and 7) work sufficiently well for any continuous distribution.

The rules can be applied without any problems to attribute control charts np and c supposing that

normal approximation is valid and control limits are reasonably symmetrical.

The same pays for the control charts p and u with constant control limits.

5. ANALYSIS OF SELECTED SW PRODUCTS SUPPORTING UNNATURAL PATTERNS

RECOGNITION

As it was mentioned in the previous chapter routine application of runs rules can lead to reducing the

effectiveness of SPC implementation. Now days there are many SW products that support this part of SPC.

They enable to increase effectiveness of the problem solving in SPC but at the same time they bring danger

of routine application of the runs rules with its negative effects.

In this charter the analysis of SW (Statgraphics [18], [19], Minitab [20]) used for the education at the

Department of quality management at the University of Ostrava, Czech Republic and really excellent

statistical SW used by many well known Czech universities (Statistica [21]) will be done in connection with

their support to the control chart unnatural pattern recognition and with respect to the problems discussed in

the previous chapter.

In general all analysed SW products cover the set of Nelson rules (the same used in ISO 8258) with some

little departures. All Nelson rules (all tests for assignable causes) including Shewhart criterion 1 (Rule 1)

contains only SW Minitab. The rest of the analysed SW have rule 1 incorporated into the basic control chart


analysis outside the set of runs tests. Statgraphics products have slightly different definition of run in rule 5,

rule 8 (Statgraphics Plus) and Rule 8 (Statgraphics Centurion). Detailed characterization of every analyzed

SW can be found in table 3.

As to the complexity in general it can be stated that all analysed SW cover the set of Nelson rules (the same

used in ISO 8258) with some little departures. Only SW Minitab contains all Nelson rules (tests for

assignable causes) including Shewhart criterion 1 (rule 1). The rest of the analysed SW have rule 1

incorporated into the basic control chart analysis outside the set of runs tests ( for that reason in these SW

the term „runs tests“ is used). From the point of view of the common types of the unnatural patterns defined

in this paper (large shift, smaller sustained shift, trends, stratification, mixture, systematic variation) all

analysed SW cover all these types except Statgraphics Plus where systematic variation is not considered.

All analyzed SW cover all standard Shewhart control charts for variables and for attributes. Statgraphics

products cover in addition ARIMA residuals chart and CusCore chart.

To use the tests correctly the user needs to have at least basic knowledge of the statistical base of the tests.

This request is to a certain extent solved only in SW Statistica.

Different default runs length definition as compared to the Nelson rules can be found in the Statgraphics SW

products in rule 5, rule 8 (Statgraphics Plus) and rule 7 (Statgraphics Centurion).

The analyzed SW also differ in the default rules. SW Statgraphics Plus and Statistica have preset all defined

rules, covering unnatural patterns as all types shifts in the mean, systematic variation, stratification and

mixture; Statgraphics Centurion only first four (according to the numbering in table 2 - rules 4, 2, 3, 5, i.e.

rules covering different types of shifts).WS Minitab has only first four rules ((according to the numbering in

table 2 - rules 1, 4, 5, 7 covering basic types of the mean shift and systematic variation). For less

experienced user setting all the tests as a default can be confusing. Better access is setting the limited

number of rules. It is not good strategy to apply all tests simultaneously – it leads to the higher false alarm

and to more complicated analysis of the process. The limited number of rules as a default is a smaller evil.

In addition every rule is not suitable for each type of the chart. In SW Statgraphics Centurion and Minitab the

user is to some extend protected from incorrect application of runs rules as to the type of control chart by the

limited default set of runs that can be applied to various control charts. But most precisely it is solved in

Minitab where the system itself enables to use only suitable runs tests when it is needed – for R, S, moving

Range, np, p, u, c charts only tests 1, 4, 5 and 7 (numbering according table 2) are allowed.

All the tested SW are flexible as to the choice of the run test and the run length. We can select only test or

tests we need. It offers the possibility to set the unique strategy for the analysis of the process instability

related to the process behaviour. The possibility to change the length of the runs can lead to the selection of

the optimal length connected with the acceptable value of false signal. Except SW Minitab all tested SW

enable to change definition of zones. It offers possibility to adopt the rules to control chart that is not based

on the normal distribution. It is sufficiently flexible for the user but the information how to decide about

suitable tests or the length of runs is not appropriately detailed.

Indication of patterns in control chart is solved in all analyzed SW. SW Statistica offers in addition the

following function: when the out-of-control conditions are investigated and an explanation for them is found,

investigator can assign descriptive labels to those out-of-control samples and explain the causes and actions

that have been taken. Having causes and actions displayed in the chart will document that the centre line

and the control limits of the chart are affected by special cause variation in the process.

As it was mentioned above simultaneous application of more rules leads to rise of false alarm and it can lead

to worse and longer identification of really affected uncommon causes. Warning about simultaneous using

more rules is included only in the adviser of SW Minitab and Statistica. The same conclusion can be done as

to the warning about the necessity to have a deep knowledge about the analyzed process. But without

appropriate knowledge about the process the unnatural patterns analysis can´t offer acceptable results.


Table 3 Multicriteria analysis of the selected SW

Properties/ SW

Statgraphics

Plus V.15

Statgraphics

Centurion

v. XV

Minitab

v. 15

Statistica

v. 16

Term used for unusual pattern Unusual pattern Unusual

sequence

Nonrandom

pattern

Systematic

pattern

Term used for tests Runs chart Runs tests Tests for

special causes

Runs tests

Complexity 6 from 8 7 form 8 8 from 8 7 form 8

Statistical base of tests no no no yes

Coverage of unnatural pattern

Large shift yes yes yes yes

Smaller sustained shift yes yes yes yes

Trends yes yes yes yes

Stratification yes yes yes yes

Mixture yes yes yes yes

Systematic Variation no yes yes yes

Different default runs definition as

compared to Nelson rules

2 2 0 0

Default number of rules 6 all First 4 First 4 (WE) 7 all

Possibility to choice the rules yes yes yes yes

Possibility to change the length of runs yes yes yes yes

Possibility to change definition of zones yes yes no yes

Indication of patterns in control chart yes yes yes yes

Record of assignable causes into

control chart

no no no yes

Record of actions into control chart no no no yes

Warning about simultaneous using more

rules

no no yes yes

Warning about the necessity to have a

deep knowledge about process

no no yes yes

Types of control chart covered (S-

Shewhart, O- other)

S + 0 S +0 S S

The same default set of rules for

different control charts

yes yes yes yes

Information to that control charts are

rules applicable

Yes (no precise) Yes (no precise) partial partial

Defined potential general assignable

causes

no no no yes

References yes yes yes yes

Description of rules yes yes yes yes

Interpretation of rules no no no yes


Description of the set of the rules can be found in each analyzed SW. But the most detailed description

including possible interpretation of the rules and definition of possible assignable causes offers only SW

Statistica.

The former analysis of the properties of the analyzed SW showed that the best solution in the field of runs

tests application offer SW Minitab and Statistica. But they also suffer from some disadvantages that can lead

little experienced user to incorrect results and to complicate procedure of the process control. It can

unfortunately result in ignoring these runs tests in the analysis of the process stability or failing of the whole

SPC application. For that reason in the next charter simple methodology for the correct application of tests

for special casus is formulated.

6. RECOMMENDATIONS FOR THE APPLICATION OF RULES FOR UNNATURAL PATTERNS

RECOGNITION

Before applying runs tests or tests for special causes check the default set of tests for selected

control charts.

Never routinely apply all runs tests or tests for assignable causes even if they are preset in your SW.

Check if chosen control charts have reasonably symmetrical control limits because in such situation

next conclusions are valid:

- all tests (Rule 1 - Rule 8) can be applied to x-bar chart and individuals chart supposing normal

distribution of the variable;

- rules 1- 4 can be applied to the dispersion charts (R, s) without any modification when sample

size is 5 or more;

- the rules can be applied without any problems to control charts np and c supposing that normal

approximation is valid and control limits are reasonably symmetrical.

- The same pays for control chart p and u with constant control.

- Tests 5 and 7 work sufficiently well for any continuous distribution because of their

nonparametric nature.

In the situation when control charts have reasonable symmetrical control limits, select suitable test

according to the following schema:

- Rules 6 and 8 apply at the beginning of the SPC implementation to verify rational subgrouping.

- In the phase of the statistical stability ensuring start with Rule 1 and 4.

- If additional sensitizing of control chart is necessary add Rules 2 and 3 or only one of them.

- When your knowledge of the process has risen during the previous SPC implementation adds

some of the other standard tests or your own fitted test.

- Test 5 apply singularly (its marginal increase in sensitivity to real signals is more than offset by

the greater increase in false alarm ([16], p. 137).

Rules 1 – 4 apply to both halves of the control chart but separately.

When using a couple of control charts (for instance x-bar and R) start with an analysis of the

dispersion control chart.

Keep in mind that where the run test is violated does not always indicate where the process change

had occurred.

7. CONCLUSIONS

Based on the theoretical background made in the introductory chapter this paper dealt with the analysis and

comparison of the several known sets of the rules for the special causes identification. Shewhart rule,

Western Electric Rules, Nelson tests, ISO 2859 tests, special sets formulated by large companies (Boeing,

GE) and the most new set by Trietsch were compared. In the light of the conclusions of these analyses the

next multi-criteria analysis of the selected SW product with the focus on the tests for the special causes


identification were done with the aim to reveal the advantages and disadvantages of every discussed SW

from the point of view of the support for meeting the main goal of SPC. Based on this analysis several

recommendations for the correct and effective application of these tests were then formulated.

ACKNOWLEDGEMENT

This paper was elaborated in the frame of the specific research project No.

SP2012/42, which has been solved at the Faculty of Metallurgy and Materials

Engineering, VŠB-TU Ostrava with the support of Ministry of Education, Youth

and Sports, Czech Republic.

REFERENCES

[1] STAPENHURST, T. Mastering Statistical Process Control. Oxford: Elsevier, 2005, 460 p. ISBN 0750665297.

[2] TRIETSCH, D. (1999). Statistical quality control. A loss minimization approach. London: World Scientific. 387 p.

ISBN 9810230311.

[3] WESTERN ELECTRIC COMPANY, INC. Statistical Quality Control Handbook, 2nd

ed. Easton: Mack Printing Co., 1958. 328 p.

[4] MONTGOMERY, D.C. Introduction to Statistical Quality Control. New York: J. Wiley Sons, 2001. 796 s.

ISBN 0-471-31648-2.

[5] GRIFFITH, G.K. Statistical Process Control Methods for Long and Short Runs. Milwaukee, Wisconsin: ASQC Quality Press,

1996. 250 p. ISBN 0-87389-345-X.

[6] EVANS, J. R., LINDSAY, W.M. Managing for Quality and Performance Excellence. 7th ed. Mason, OH: Thomson, 2008. 783 p.

ISBN 10:0-324-64686-0.

[7] BESTERFIELD, D. H. Quality Control. 8th

ed. New Jersey: Prentice Hall, 2009, 552 s. ISBN 978 0135 000 953.

[8] GRANT, E. L., LEAWENHWORTH, R.S. Statistical Quality Control. New York: McGraw Hill, 1996. 764 p. ISBN 0-07-024162-7.

[9] SHEWHART, W. A. 1931 Economic Control of Quality of Manufactured Product. New York: Van Nostrand, 1931. 501 p. ISBN

0-873890-76-0. (republished by Milwaukee: ASQ, 1980).

[10] NELSON, L. S. 1984, The Shewhart Control Chart – Tests For Special Causes. Journal Of Quality Technology,1984, V.16, No.4,

pp. 337-339.

[11] ČSN ISO 8258 Shewhartovy regulační diagramy. Praha: ČSNI, 1994.

[12] D1-9000 ASQ Boeing Company, 2000.

[13] AIAG SPC-3, 2005.

[14] BURR, I., W.: Statistical Quality Control Methods. New York: Marcel Dekker Inc., 1976. 532 p. ISBN 9780824763442.

[15] OTT, E. R. Analysis of Means – A Graphical Procedure. Journal of Quality Technology. V.15, pp. 10-18. 1983.

[16] WHEELER, D. Advanced Topics in Statistical Process Control. Knoxville: SPC Press, Inc., 2004. ISBN 0-945320-63-9.

[17] MITRA, A. Fundamentals of Quality Control and Improvement. New York: MacMillan Publishing Company, 1993, 651 s. ISBN

0-02-381791-7.

[18] STATGRAPHICS PLUS v. 5.1. Manugistics Inc., 2000.

[19] STATGRAPHICS CENTURION,v XV. StatPoint Technologies, 2006.

[20] MINITAB v. 16. Minitab Inc., 2010.

[21] STATISTICA v. 10, StatSoft, Inc., 2010.

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