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HAPTER -There is no rule for the frequency of taking subgroups, but the frequency shoul

often enough to detect process changes. The inconveniences of the factory or officeout and the cost of taking subgroups must be balanced with the value of the datatained. general, it is best to sample quite often at the beginning and reducesampling frequency when the data permit.

The precontrol rule for the frequency of sampling could also be used. It is bon how often the process is adjusted. If the process is adjusted every hour, then spling should occur every 10 minutes; if the process is adjusted every 2 hours,sampling should occur every 20 minutes; if the process is adjusted every 3 hothen sampling should occur every 30 minutes; and so forth.

Data CollectionAssuming that the quality characteristic and the plan for the rational subgroup have bselected, a team member such as a technician can be assigned the task of collectingdata as part of his normal duties. The first-line supervisor and the operator should bformed of the technician s activities; however, no charts or data are posted at the wcenter at this time.

Because of difficulty in the assembq of a gear hub to a shaft using a key and keywthe project team recommends using X and charts. The quality characteristic isshaft keyway depth of 6.35 mm (0.250 in.). Using a rational subgroup of four, a teccian obtains five subgroups per day for five days. The samples are measured, thegroup average and range are calculated, and the results are recorded on the formshown in Table 18-4. Additional recorded information includes the date, time, andcomments pertaining to the process. For simplicity, individual measurements are cofrom 6.00 mm. Thus, the first measurement of 6.35 is recorded as 35.

It is necessary to collect a minimum of 25 subgroups of data. A fewer number ofgroups would not provide a sufficient amount of data for the accurate computation ocontrol limits, and a larger number of subgroups would delay the introduction of the ctrol chart.

Trial Central Lines and Control LimitsThe central lines for the X and charts are obtained using the equations

r. jgwhere X average of the subgroup averages (read X double bar )

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STATISTICAL PROCESS CONTROLala on the Depth of the Keyway millimeters-- MEASUREMENTSverageangeNumber Dateime34 XRComment1 7/23505027 6.36.08 11306761 6.40.10 1:454046 6.36.06 3.459489 6.65.10New temporary208440 6.39.10perator

17 7/29251094 6.36.12 11008488 6.42.30Dama;;ed oil line:355178 6.38.06 3:156558 6.51.11ad material/30358057 6.40.08 10209250 6.39.07 11:352996 6.39.06 2:003658 6.38.08 4259834 6.41.06 160.25.19

Trial control limits for the charts are established at J from the central linshown by the equations

DCL = X 3cr-x xLCLI = X - crx

DCl = R crRlCl = 3cr

where DCl:= upper control limitLCL = lower control limitx = population standard deviation of the subgroup averages

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CHAPTER 18

In p~ctice, the calculations are simplified by using the product of th~ average of trange R and a ~~ctor Az to replace the three standard deviations AzR = 30 x in tequation for the X chart. For the R chart, the range is used to estimate the standard dviation of the range. Therefore, the derived equations are

UCLx = X + AzR- -LCLx = X - AzR UCLR = D4RLCLR = D4Rwhere Az D3 and i>4 are factors that vary with the subgroup size and are found in Apendix A. For the X chart, the upper and lower control limits are symmetrical about tcentral line. Theoretically, the control limits for an R chart should also be symmetricabout the central line. But, for this situation to occur, with subgroup sizes of six or lethe lower control limit would need to have a negative value. Because a negative ranis impossible, the lower control limit is located at zero by assigning to D~ the valuezero for subgroups of six or less.

When the subgroup size is seven or more, the lower control limit is greater than zeand symmetrical about the central line. However, when the R chart is posted at the wocenter, it may be more practical to keep the lower control limit at zero. This practieliminates the difficulty of explaining to the operator that points below the lower cotrollimit on the R chart are the result of exceptionally good performance rather than poperformance. However, quality personnel should keep their own charts with the lowcontrol limit in its proper location, and any out-of-controllow points should be invesgated to determine the reason for the exceptionally good performance. Because sugroup sizes of seven or more are uncommon, the situation occurs infrequently.

In order to illustrate the calculations necessary to obtain the trial control limits athe central line, the data concerning the depth of the shaft keyway will be used. FroTable 18-4, LX = 160.25, LR = 2.19, and g = 25; thus, the central lines are

x = L X g = 160.25/25 = 6.41 mmR = L Rig = 2.19/25 = 0.0876 mm

From Appendix Table A, the values for the factors for a subgroup size n of fourAz = 0.729, D3 = 0, and D4 = 2.282. Trial control limits for the X chart are

UCLx = X + AzR= 6.41 + 0.729) 0.0876)= 6.47 mmTrial control limits for the R chart are

UCLR = D4R= 2.282) 0.0876)= 0.20 mm

LCL.x = X - AzR= 6.41 - 0.729) 0.0876)= 6.35 mm

LCLR = D3R= 0) 0.0876)=Omm

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STATISTICAL PROCESS CONTROL

Control charts - Depth of keyway

6.47S sI I .A.6.41

/ \..\ XJ:)e 6.35

ssI 0.09 OJ:)=~ 0LCL

1 I I I 1 I I I t I I I I ] I I I I I I I I I5 10 15 20 25Subgroup number

Figure 18 17 X and R Charts for Preliminary Data with Trial Control Limits

Figure 18-17 shows the central lines and the trial control limits for X and charthe preliminary data.

Revised Central Lines and Control LimitsRevised central lines and control limits are established by discarding out-of-copoints with assignable causes and recalculating the central lines and control limits

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APTER 8

The X chart can now be analyzed. Subgroups 4 and 20 had an assignable cauwhereas the out-of-control condition for subgroup 16 did not. is assumed that sgroup 16 s out-of-control state is due to a chance cause and is part of the natural vation of the process.

The recalculated values are Xo = 6.40 mm and o = 0.079. They are shownFigure 18-=-18.For illustrative purposes the trial values are also shown. The limitsboth the X and charts became narrower as was expected. No change occurredLCLR because the subgroup size is less than 7. The figure also illustrates a simpcharting technique in that lines are not drawn between the points. Also Xo and ostandard or reference values are used to designate the central lines.

The preliminary data for the initial 25 subgroups are not plotted with the revised Ctrollimits. These revised control limits are for reporting the results for future subgrouTo make effective use of the control chart during production it should be displayed iconspicuous place where it can be seen by operators and supervisors.

Control charts - Depth of keyway

6.47E EI18: 6.41

..l~ 6.35

0.20

LCLx

UCLx = 6.46

Xo = 6.40

LCLx =6.34

UCLR =0.18

EEI0.09 r-:

/) ~o L _______ LCLR Ro = 0.08

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STATISTICAL PROCESS CONTROL 50~

Before proceeding to the action step, some final CClmments are appropriate. Firmany analysts eliminate this step in the procedure because it appears to be somewhatdundant. However, by discarding out-of-control points with assignable causes, the cetral line and control limits are more representative of the process. If this step is tcomplicated for operating personnel, its elimination would not affect the next step.-Second, the central line X I for the X chart is frequently based on the specificationIn such a case, the procedure is used only to obtain R If. in our example problem, tnominal value of the characteristic is 6.38 mm, then Xo is set to that value and the uper and lower control limits are

UCLx = Xo z~= 6.38 0.729) 0.079)= 6.44 mm

LCL = Xo zRo= 6.38 ..c. 0.729) 0.079)= 6.32 mm

The central line and control limits for the R chart do not change. This modificatiocan be taken only if the process is adjustable. If the process is not adjustable, then toriginal calculations must be used.

Third, it follows that adjustments to the process should be made while taking data.is not necessary to run nonconforming material while collecting data, because we aprimarily intere~ted in obtaining Ro which is not affected by the process setting. Thedependence of Xo and Ro provides the rationale for this concept.

Fourth, the process determines the central line and control limits. They are not etablished by design, manufacturing, marketing, or any other department, except forwhen the process is adjustable.

When control charts are first introduced at a work center, an improvement in the proceperformance usually occurs. This initial improvement is especially noticeable when tprocess is dependent on the skill of the operator. Posting a quality control chart appeato be a psychological signal to the operator to improve performance. Most workers wato produce a quality product or service: therefore, when management shows an interein the quality, the operator responds.

Figure 18-19 illustrates the initial improvement that occurred after the introductioof the X and R charts in January. Owing to space limitations, only a representative number of subgroups for each month are shown in the figure. During January the subgrouranges had less variation and tended to be centered at a slightly lower point. A reductioin the range variation occurred also.

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APTER 18

Control charts Depth of keyway6.46 1-__ ----------

l

-

EE

I18 6.40

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At the end of June, the periodic evaluation of the past performance showed the nto revise the central lines and the control limits. The performance for the month of Jand subsequent months showed a natural pattern of variation and no quality improment. At that point, no further quality improvement would be possible without a sstantial investment in new equipment or equipment modification.

Dr. Deming has stated that if he were a banker, he would not lend money to anganization unless statistical methods were used to prove that the money was necessaThis is precisely what the control chart can achieve, provided that all personnel usechart as a method of quality improvement rather than a monitoring function.

When the objective for initiating the charts has been achieved, their use should becontinued or the frequency of inspection be substantially reduced to a monitoring actby the operator. The median chart is an excellent chart for the monitoring activity.forts should then be directed toward the improvement of some other ql 1Jity charactistic. If a project team was involved, it should be recognized and rev.arded forperformance and disbanded.

The U.S. Postal Service at Royal Oak, Michigan used a variables control chart toduce nonconformance in a sorting operation from 32% to less than 6%. Th:, activitysulted in an annual savings of 700,000 and earned the responsible team the 1RIT/USA Today Quality Cup for government.

ate of ontrolWhen the assignable causes have been eliminated from the process to the extent thatpoints plotted on the control chart remain within the control limits, the process isstate of control. No higher degree of uniformity can be attained with the existing proceHowever, greater uniformity can be attained through a change in the basic processsulting from quality improvement ideas.

When a process is in control, there occurs a natural pattern of variation, whichillustrated by the control chart in Figure 18-20. This natural pattern of variation

- - - LCLFigure 18 2 atural Pattern of Variation of a ontrol hart

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HAPTER

(1) about 34 of the plotted points on an imaginary band between one standard devia_tion on both s-ides of the central line, (2) about.l~.5 of the pl~tted points in an imag_inary band between one and two standard deviatIOns on both sides of the central lineand (3) about 2.5 of Ihe plotted points in an imaginary band between two and thre~standard deviations on both sides of the central line. The points are located back andforth across the central line in a random manner, with no points beyond the controllim_its. The natural pattern of the points, or subgroup average values, forms its OWnfrequency distribution. If all the points were stacked up at one end, they would form anormal curve.When a process is in control, only chance causes of variation are present. Small vari

ations in machine performance, operator performance, and material characteristics areexpected and are considered to be part of a stable process.When a process is in control, certain practical advantages accrue to the producer andconsumer:I. Individual units of the product will be more uniform, or, stated another way, therewill be less variation.,2. Because the product is more uniform, fewer samples are needed to judge the qual

ity. Therefore, the cost of inspection can be reduced to a minimum. This advantage is extremely important when 100 conformance to specifications is nOI essential.3. The process capability, or spread of the process, is easily attained from 60. With

a knowledge of the process capability, a number of reliable decisions relative to specifications can be made, such as the product specifications; the amount of rework or scrapwhen there is insufficient tolerance; and whether to produce the product to tight specifications and permit interchangeability of components or to produce the product to loosespecifications and use selective matching of components.4. The percentage of product that falls within any pair of values can be predicted with

the highest degree of assurance. For example, this advantage can be very importantwhen adjusting filling machines to obtain different percentage of items below, between,or above particular values.5. It permits the customer to use the supplier s data and, therefore, to test only a few

subgroups as a check on the supplier s records. The and charts are used as statistical evidence of process control.6. The operator is performing atisfactorily from a quality viewpoint. Further

improvement in the process can be achieved only by changing the input factors:materials, equipment, environment, and operators. These changes require action bymanagement.When only chance causes of variation are present, the process is stable and pre

dictable over time, as shown in Figure l8-2l(a). We know that future variation as shown

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STATISTICAL PROCESS CONTROL 5

Size~

Prediction

(a) Only chance causes of variation present

/-, Jf/ y// // / -~--7---::: -Prediction

Size ~ (b) Assignable causes of variation presenFigure 18-21 Stable and Unstable Variation

Out of Control ProcessFigure 18-2 I(b) illustrates the effect of assignable causes of variation over time. The unatural, unstable nature of the variation makes it impossible to predict future variationThe assignable causes must be found and corrected before a natural stable process cacontinue.

The term out of control is usually thought of as being undesirable; however, there asituations where this condition is desirable. It is best to think of the term out of controas a change in the process due to an assignable cause.

A process can also be considered out of control even when the points fall inside tha limits. This situation, as shown in Figure 18-22, occurs when unnatural runs of varation are present in the process. It is not natural for seven or more consecutive pointsbe above or below the central line as shown at (a). Another unnatural run occurs at (bwhere six points in a row are steadily increasing or decreasing. At (c), the space is dvided into four equal bands of 1.5a. The process is out of control when there are twsuccessive points at 1.5a beyond.44 For more information, see A. M. Hurwitz and M. Mather, A Very Simple Set of Process Control RulesQualiry ngineering 5, no. I (1992-1993): 21-29.

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I CHAPTER 8

Process out of controlCL ~-------

XoCL fLCL ~-------

VCL

LCL

--- .-- ~~----

(a) Seven consecutive (b) Six consecutivepoints above points increasing oror below decreasing

Figure 8 22 Some Unnatural Runs Process Out of ontrol

(c)Two consecutivepoints in outerquarter

There are some common questions to ask when investigating an out-of-control proce1. Are there differences in the measurement accuracy of the instruments used?2. Are there differences in the methods used by different operators?3. Is the process affected by the environment? If so, have there been any change4. Is the process affected by tool wear?5. Were any untrained workers involved in the process?6. Has there been any change in the source of the raw materials?7. Is the process affected by operator fatigue?8. Has there been any change in maintenance procedures?9. Is the equipment being adjusted too frequently?

10. Did samples come from different shifts, operators, or machines?It is advisable to develop a checklist for each process using these common question

as a guide.

ess p bilityControl limits are established as a function of the averages-in other words, controllimits are for averages. Specifications, on the other hand, are the permissible variation in tsize of the part and are, therefore, for individual values. The specification or toleranlimits are established by design engineers to meet a particular function. Figure 18-shows that the location of the specifications is optional and is not related to any of tother features in the figure. The control limits, process spread (process capability), dtribution of averages, and distribution of individual values are interdependent. They adetermined by the process, whereas the specificationS: have an optional location. Contr

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Upper specification(optional location) -------------------- USL

Distributionof averages--f-T-(---Control I - -

limi I -ts /+ 3

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APTER 8

LSLTolerance USL

10

Capability

Nominalo

13.09 13.15 13.21 13.27Hole location-mm

Figure 8 24 Relationship of Process Capability to Tolerance

Remember that this technique does not give the true process capability and should beused only if circumstances require its use. Also, more than 25 subgroups can be used toimprove accuracy.

The relationship of process capability and specifications is shown in Figure 18-24Tolerance is the difference between the upper specification limit USL and the lowerspecification limit LSL . Process capability and the tolerance are combined to form acapability index, defined as

USL - LSL6cr

where USL - LSL upper specification - lower specification, or tolerance p capability index6cr process capability

If the capability index is greater than 1.00, the process is capable of meeting the specifications; if the index is less than 1.00, the process is not capable of meeting the specifications. Because processes are continually shifting back and forth, a pvalue of 1.33has become a de facto standard, and some organizations are using a 2..00 value. Usingthe capability index concept, we can measure quality, provided the process is centered.The larger the capability index, the better the quality. We should strive to make the capability index as large as possible. This result is accomplished by having realistic spec

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Min { USL - X) or X - LSL)}C =pk 3cr

A Cpk value of 1.00 is the de facto standard, with some organizations using a value1.33. Figure 18-25 illustrates Cp and Cpk values for processes that are centered and aloff center by lcr.

Case I Cp = USL - LSL)/6 = 8a a = 1.33 a

LSL

Cp = 1.33Cpk = 1.33

USL LSL

Cp = 1.33Cpk = 1.00

Case II Cp = USL - LSL)/6 = a a = 1.00

a

Cp = 1.00Cpk = 1.00

LSL

Cp = 1.00Cpk = 0.67

aCase III Cp = USL - LSL)/6 = 4a/fu = 0.67

6

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CHAPTER 8

Comments concerning Cp and Cpk are as follows:1. The pvalue does not change as the process center changes. Cp = Cpk when the process is centered.3 Cpk is always equal to or less than Cpo4. A pkvalue greater than 1.00 indicates the process conforms to specificatio5. A pkvalue less than 1.00 indicates that the process does not conform to sfications.6. A pvalue less than 1.00 indicates that the process is not capable.7. A pkvalue of zero indicates the average is eqUid to one of the specificlimits.8. A negative pkvalue indicates that the average is outside the specifications

Quality professionals will use these eight items to improve the process. For exaif a lvalue is less than one, then corrective action must occur. Initially 100 intion is necessary to eliminate noncomformities. One solution would be to increastolerance of the specifications. Another would be to work on the process to reducstandard deviation or variability.

ifferent Control Charts for VariablesAlthough most of the quality control activity for variables is concerned with the XR charts, there are other charts that find application in some situations. These chardescribed in Table 18-5.

ontrol Charts for AttributesAn attribute, as defined in quality, refers to those quality characteristics that c' .ospecifications or do not conform to specifications. There are two types:

1. Where measurements are not possible, for example, visually inspected itemsas color, missing parts, scratches, and damage.

2. Where measurements can be made but are not made because of time, cost, orIn other words, although the diameter of a hole can be measured with an insidecrometer, it may be more convenient to use a go-no go gauge and determine if itforms or does not conform to specifications.

Where an attribute does not conform to specifications, various descriptive term

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Commentsentral Limits

TABLE 18-5ifferent ontrol harts for Variables

CentralLineype

X and

Moving average,MX and movingrange, MR

X and moving R

Median and Range

x5

xR

xR

UCLj = X + A 5LCLj = X A 5UCLs = 845LCLs= 835UCLj = X + Ai?LCLj = X AllUCLR = oi?LCLR = oilUCLx = X + 2.660 RLCLx = X 2.660 RUCLR = 3.276 RLCLR = 0) RUCLMd = MdMd + As RMdLCLMd = MdMd As RMdUCLR = D RMdLCLR = Ds RMd

Use when more sensitivity isdesired than R; whenn > 10; and when data arecollected automatically.Use when only one observation is possible at a time.Data needn t be normal.

Use when only one observation is possible at a timeand the data are normal.Equations are based on amoving range of two.Use when process is in amaintenance mode. Benefitsare less arithmetic andsimplicity.

or state that occurs with a severity sufficient to cause an associated product or servicenot to meet a specification requirement. The definition of a defect is similar except it isconcerned with satisfying intended normal or reasonably foreseeable usage require-ments. Defect is appropriate for use when evaluation is in terms of Ilsage and noncon-formity is appropriate for conformance to specifications.

The term nonconforming unit is used to describe a unit of product or service con-taining at least one nonconformity. Defective is analogous to defect and is appropriatefor use when a unit of product or service is evaluated in terms of usage rather than con-formance to specifications.

In this section we are using the terms nonconformity and nonconforming unit Thispractice avoids the confusion and misunderstanding that occurs with defect and defectivein product liability lawsuits.

Variable control charts are an excellent means for controlling quality and subsequentlyimproving it; however they do have limitations. One obvious limitation is that thesecharts cannot be used for quality characteristics that are attributes. The converse is nottrue because a variable can be changed to an attribute by stating that it conforms or doesnot conform to specifications. In other words nonconformities such as missing parts in-correct color and so on are not measurable and a variable control chart is not applicable.

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