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ISU STAT 496 - Control Charts Construction

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Control Chart ConstructionW. Robert StephensonDepartment of StatisticsIowa State University1 Charts for Measurements• Rational subgroupsObserved measurements are grouped together into rational subgroups. For each sub-group, the subgroup Mean (average) is calculated as well as a subgroup Range (dif-ference between the largest and smallest values in a subgroup.) The subgroup rangesprovide information about short-term (within subgroup) variability. The subgroupmeans, when viewed over time, provide information about longer-term (between sub-group) variability. Control limits are established based on the short-term variability(the ranges) and the average of the subgroup means. These are sometimes referred toas “retrospective” limits. TheX − R control chart pair provides a pictorial represen-tation of the short- and long-term variability in the process.The following notation will be helpful in developing formulas for the control limits.Data: Xtiis the ithmeasurement in the tthsubgroup. There are n measurementsin each subgroup1and m subgroups.Range: Rt=maximum(Xti) − minimum(Xti)fort=1, 2, 3, ..., mAverage Range:R=Rt mMean: Xt=ni=1XtinAverage Mean: X=Xt mEstimate of short term standard deviation: ˆσ=Rd2.Coefficients, such as d2and others used on the following page, can be found in Ta-ble 1 on page 2 of this handout. These coefficients are based on the assumption thatmeasurements are normally distributed.1The number of measurements in each subgroup does not have to be the same, but it is simpler to havea constant subgroup size.1–Range(R)Chart:Plot:Rtfor t=1, 2, 3, ..., m.UCL: D2Rd2=D4RCL:RLCL: D1Rd2=D3R– Subgroup Mean (X)Chart:Plot:Xtfor t=1, 2, 3, ..., m.UCL: X +3Rd2 √n=X + A2RCL:XLCL:X − 3Rd2 √n=X − A2RTable 1: Control Chart CoefficientsSubgroupSizend2D1D2D3D4A22 1.128 0 3.686 0 3.267 1.8803 1.693 0 4.358 0 2.575 1.0234 2.059 0 4.698 0 2.282 0.7295 2.326 0 4.918 0 2.115 0.5776 2.534 0 5.078 0 2.004 0.4837 2.704 0.205 5.203 0.076 1.924 0.4198 2.847 0.387 5.307 0.136 1.864 0.3739 2.970 0.546 5.394 0.184 1.816 0.33710 3.078 0.687 5.469 0.223 1.777 0.30811 3.173 0.812 5.534 0.256 1.744 0.28512 3.258 0.924 5.592 0.284 1.716 0.26613 3.336 1.026 5.646 0.308 1.692 0.24914 3.407 1.121 5.693 0.329 1.671 0.23515 3.472 1.207 5.737 0.348 1.652 0.22320 3.735 1.548 5.922 0.414 1.586 0.18025 3.931 1.804 6.058 0.459 1.541 0.1532• Rational Subgroups (Cont.)The subgroup range is not the only way to measure short-term variability. Anothercommon measure is the subgroup Standard Deviation. The subgroup standard de-viation replaces the subgroup range in the calculation of control limits. Appropriatecoefficients can be found in Table 2 on page 4 of this handout. Again, the coefficientsare based on the assumption that measurements are normally distributed.The following notation will be helpful in developing formulas for the control limits.Data: Xtiis the ithmeasurement in the tthsubgroup. There are n measurementsin each subgroup and m subgroups.Standard Deviation:st=ni=1Xti− Xt 2(n − 1)orst=ni=1X2ti−ni=1Xti2n(n − 1)for t=1, 2, 3, ..., m.Average Standard Deviation:s=stmMean: Xt=ni=1XtinAverage Mean: X=Xt mEstimate of short term standard deviation: ˜σ=sc4.22There are other ways to estimate the short-term standa rd deviation based on the subgroup standarddeviations. One alternative uses the square root of the pooled variance, sp. Provided subgroup sizes areequal: sp=s2tm.3– Standard Deviation (s)Chart:Plot:stfor t=1, 2, 3, ..., m.UCL: s +3sc41 − c24= B4sCL:sUCL:s − 3sc41 − c24= B3s– Subgroup Mean (X)Chart:Plot:Xtfor t=1, 2, 3, ..., m.UCL: X +3sc4 √n= X + A3sCL:XLCL:X − 3sc4 √n= X −A3sTable 2: More Control Chart CoefficientsSubgroupSizenc4B3B4B5B6A32 0.7979 0 3.267 0 2.606 2.6593 0.8862 0 2.568 0 2.276 1.9544 0.9213 0 2.266 0 2.088 1.6285 0.9400 0 2.089 0 1.964 1.4276 0.9515 0.030 1.970 0.029 1.874 1.2877 0.9594 0.118 1.882 0.113 1.806 1.1828 0.9650 0.185 1.815 0.179 1.751 1.0999 0.9693 0.239 1.761 0.232 1.707 1.03210 0.9727 0.284 1.716 0.276 1.669 0.97511 0.9754 0.321 1.679 0.313 1.637 0.92712 0.9776 0.354 1.646 0.346 1.610 0.88613 0.9794 0.382 1.618 0.374 1.585 0.85014 0.9810 0.406 1.594 0.399 1.563 0.81715 0.9823 0.428 1.572 0.421 1.544 0.78920 0.9869 0.510 1.490 0.504 1.470 0.68025 0.9896 0.565 1.435 0.559 1.420 0.6064• Individual MeasurementsOccasionally measurements are taken in such a way that no rational subgrouping ispossible. In this case, each individual measurement becomes its own subgroup of size1. With subgroups of size 1, short-term variability must be quantified differently. TheMoving Range, the absolute difference of successive measurements, provides informa-tion about short-term variability provided successive measurements are not separatedtoo much in time. These moving ranges are treated like ranges from subgroups of size2, i.e. d2=1.128.The following notation will be helpful in developing formulas for the control limits.Data: Xtis the measurement at time t.Moving Range: MRt=|Xt− Xt−1| for t=2, 3, ..., mAverage Moving Range:MR=MRt m−1Average Value: X=Xt mEstimate of short term standard deviation: ˆσ=MR1.128.–MovingRange(MR)Chart:Plot:MRtfor t=2, 3, ..., m.UCL: 3.267MRCL:MRLCL: 0– Individual Measurement (X)Chart:Plot:Xtfor t=1, 2, 3, ..., m.UCL: X +2.660MRCL:XLCL:X − 2.660MR52 Charts for Counts and Proportions• Number of defective items in a subgroup of size nWhen items in subgroups of size n are classified as either “good”, i.e. conforming,nondefective, in-spec, etc. or “bad”, i.e. nonconforming, defective, out-of-spec, etc.,control charts for the Number of Defective Items (or the Fraction Defective)ineach subgroup can be constructed. The control limits are based on the normal approx-imation to the binomial distribution.The following notation will be helpful in developing formulas for the control limits.Data: Each subgroup consists of n items.3Xtis the Number of Defective Itemsin the tthsubgroup.Fraction Defective:ˆpt=Xtnfor t=1, 2, 3, ..., m.Average Fraction Defective:p =Xtmn=ˆptmEstimate of the standard deviation of the fraction defective:p(1−p)nEstimate of the standard deviation of the number of defective items:np(1 − p)– p Chart for fraction defective:Plot:ˆptfor t=1, 2, 3,


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