Quality Control in Statistics: Monitoring Product Mean with X and CUSUM Charts, Study notes of Statistics

Download Quality Control in Statistics: Monitoring Product Mean with X and CUSUM Charts and more Study notes Statistics in PDF only on Docsity! STAT 401 Ch 16 Quality Control In this chapter we describe some graphical methods that help monitor the quality of manufactured products. We will concen- trate on methods for monitoring quality when quality is quan- tified in terms of the mean (average) value of some trait (char- acteristic of the product). We will describe the X̄-chart and the CUSUM chart. X̄-chart Think of the simple context where we want to monitor the mean strength of plastic sheets. Suppose we have in mind some target value µ0. If the production process shifts (for some rea- son) and the plastic sheets produced have more (or less) average strength we want to interrupt and make adjustments. To deter- mine if and when the production process shifts, samples of size n are taken at time points t1 t2 t3 


. Let X̄1 X̄2 X̄3 


denote the sample means at times t1 t2 t3 


. The X̄-charts plots the points  t1 X̄1   t2 X̄2   t3 X̄3  


if a point falls above the UCL (=Upper Control Limit) or below the LCL (=Lower Control Limit) line the process is declared out of control and corrective measures are taken. 1 Performance Characteristics Similarly to hypothesis testing, we can commit two types of errors: It is possible for a point  ti X̄i  to fall outside the control limits when in fact the true mean equals the target value µ0, and it is possible for a point to fall within the contrail limits when in fact the true mean has shifted away from µ0. In quality contrail it is customary to use UCL µ0
3 σ  n LCL µ0  3 σ n in which case (assuming normality) 2 Unknown parameters The previous discussion of X̄-charts assumed for simplic- ity that both the target value µ0 and the standard deviation σ are known. In practice both σ and µ0 may be unknown. The target value µ0 is typically unknown when the process is first subjected to quality control. It is often the case that the man- agement wants to keep the process at its present state (i.e. sales may be good), but the mean value of the primary quality char- acteristic is unknown. Both µ0 and σ are estimated using k samples gathered when the process is assumed to be in control. Let X̄1 X̄2 


X̄k de- note the k sample means. It is recommended that k 20, with sample size n 3 4 5 or 6. Then use µ̂ ¯̄X 1 k k ∑ i
1 X̄i instead of µ0. Instead of σ we use a bias-adjusted average of the sample standard deviations. Recall that the sample standard deviation S is a biased estimate of σ. If the sample has come from a normal population, it can be shown that E  S  anσ where an 2Γ  n
2  n  1Γ  n  1 2  (see p.167 for definition of the gamma function). As the fol- lowing table indicates, an  1 for large values of n: 5 n 3 4 5 6 7 8 an for n 4 886 921 .940 .952 .959 .965 Let now S̄ 1 k ∑ki
1 Si be the average of the standard devia- tions computed from each of the k samples. Then, if the sample size for each sample is n, it follows (by the properties of expec- tation) that E  S̄  anσ or E S̄ an  σ if all samples are normal with the same variance σ2. Then in- stead of σ use its unbiased estimator S̄
an. Using ¯̄X and S̄
an instead of µ0 and σ, the control limits are LCL ¯̄X  3 S̄ an  n UCL ¯̄X
3 S̄ an  n
Remark 1: Recomputing Control Limits. The control lim- its using S̄
an and S̄
an are based on the assumption that the k sample: (from which X̄1 


X̄k and S1 


Sk were computed) were taken when the process is in control. It is possible, how- ever, that one or more of X̄1 


X̄k falls outside the control lim- its. If this is the case, (and if an assignable cause can be found) it is recommended that control limits be recalculate without the corresponding samples. 6 Remark 2: Supplemental Rules. The large β-values (correspondingly large ARL values) for small deviations from the target value µ0 led to additional rules for declaring the process out of contrail. The following supplemen- tal rules were proposed by the Western Electric Co. Additional ones are implemented by Minitab. 1. Two out of three successive X̄i fall outside the 2σ
 n limits on the same side. 2. Four out of five successive X̄i fall outside the 1σ
 n limits on the same side. 3. Eight successive X̄i fall on the same side. CUSUM Chart The CUmulative SUM procedure is also designed to im- prove the performance of the X̄-chart. The basic idea is that the X̄-chart uses information only from the current sample, ig- noring information contained in previous samples. However, a small deviation from the target is more easily detected by com- bining information in all samples. Thus the CUSUM chart plots t1 S1   t2 S2  


where S1 X̄1  µ0 S2 S1
 X̄2  µ0  S3 S2
 X̄3  µ0  


Si Si  1
 X̄i  µ0  


7 Illustration of a CUSUM Chart (V-mask) as Minitab produces it Given a sequence of k sample means (here k 20), you can ask Minitab to superimpose a V-mask at any sample mean. The above V-mask corresponds to the 15th inspection point. In practice a new V-mask is constructed for each new sample mean as soon as it becomes available. 10 A wood products company manufactures charcoal briquettes for barbecues. It packages these briquettes in bags of various sizes, the largest of which is supposed to contain 40 lb. Table 16.4 displays the weights of bags from 16 different samples, each of size n 4. The first 10 of these were drawn from a normal distribution with µ µ0 40 and σ
5. Starting with the eleventh sample, the mean has shifted upward to µ 40
3. Table 16.4 Observations, χs, and cumulative sums for Example 16.8 Sample Number Observations χ Σ
χt  40  1 40.77 39.95 40.86 39.21 40.20 .20 2 38.94 39.70 40.37 39.88 39.72 -.08 3 40.43 40.27 40.91 40.05 40.42 .34 4 39.55 40.10 39.39 40.89 39.98 .32 5 41.91 39.07 39.85 40.32 40.06 .38 6 39.06 39.90 39.84 40.22 39.76 .14 7 39.63 39.42 40.04 39.50 39.65 -.21 8 41.95 40.74 40.43 39.40 40.41 .20 9 40.28 40.89 39.61 40.48 40.32 .52 10 39.28 40.49 38.88 40.72 39.84 .36 11 40.57 40.04 40.85 40.51 40.49 .85 12 39.90 40.67 40.51 40.53 40.40 1.25 13 40.70 40.54 40.73 40.45 40.61 1.86 14 39.58 40.90 39.62 39.83 39.98 1.84 15 40.16 40.69 40.37 39.69 40.23 2.07 16 40.46 40.21 40.09 40.58 40.34 2.41 11 Figure 16.8 X̄ control chart for the data of Example 16.8 12 Sol. Here µ0 40 but for the computational form we must also know k and h. As always k ∆ 2
15, and the h is found from the Kemp nomogram. However now we have different information: we know that n 4, we know ∆
3 and we know the desired ARL as shift ∆
3 (which is 7). But from this info we can find k l σ
 n
15
5
 4
6 Then we can connect k
6 with the out-of-control ARL=7 to find h 3
8 from which we find h σ n h
95. (We also find an in-control ARL of 500.) Now we are ready to proceed with the computational form of CUSUM: d0 0, 15 d1 max 0 d0
 X̄1  40
15 
max 0 0
 40
20  40
15 

05 d2 max 0 d1
 X̄2  40
15 
max 0 
05
 39
72  40
15 
0 d3 max 0 d2
 X̄3  40
15 
max 0 0
 40
42  40
15 

27 etc. To illustrate the calculations for the ei, we have e0 0 e1 max 0 e0   X̄1  39
85 
max 0 0   40
20  39
85 
0 e2 max 0 e1   X̄2  39
85 
max 0 0   39
72  39
85 

13 etc. The first out-of-control signal comes at the 13th sample and is 16 due to d13 1
17 being greater than h
95. We conculate that the mean has shifted to a higher value which is indeed the case. 17