Chapter 6 Variables Control Charts. Statistical Quality Control (D. C. Montgomery)

Transcription

1 Chapter 6 Variables Control Charts 許 湘 伶 Statistical Quality Control (D. C. Montgomery)

2 Introduction I Variable: a numerical measurement A single measurable quality characteristic, such as a dimension( 尺 寸 ), weight, or volume, is called a variable. deal with a quality characteristics, necessary to monitor both 1. the mean value of the quality characteristic: x control chart 2. variability: a control chart for the standard deviation: s control chart a control chart for the range: R control chart (widely used)

3 Introduction II Separate x and R charts are maintained for each quality characteristic of interest. Important: maintain( 使 繼 續 ) control over both the process mean and process variability

4 Statistical Basis of the Charts I Assumptions a quality characteristics is normally distributed with µ and σ Size n: x i N (µ, σ 2 ), i = 1,..., n Average: n i=1 x = x i n N (µ, σ 2 x), σ x = σ x The probability is 1 α that any sample mean will fall between (µ, σ: known) ] [ ] σ σ µ Zα/2 σ x, µ + Z α/2 σ x = [µ Z α/2 n, µ + Z α/2 n

5 Statistical Basis of the Charts II If the underlying distribution is nonnormal: the central limit theorem We usually will not know µ and σ 怎 麼 做? Estimated from preliminary samples or subgroups taken when the process is thought to be in control

6 Statistical Basis of the Charts III The best estimator of µ: m samples: m = each sample containing n observations: n = 4, 5, 6 The grand average: x = mi=1 x i m (the center line on the x chart)

7 Statistical Basis of the Charts IV the estimator of σ: 1. the standard deviation 2. the ranges of the m samples (the range method) Range of a sample of size n: The average range: x 1,..., x n R = x max x min R = mi=1 R i m In Chap. 4: relative range W : W = R σ

8 Statistical Basis of the Charts V Properties of relative range The parameters of the distribution of W are a function of sample size n E(W ) = d 2 An estimator of σ: ˆσ = R d 2 (d 2 : Appendix Table VI) R: the average range of the m preliminary samples ˆσ = R d 2 ( an unbiased estimator of σ)

9 Statistical Basis of the Charts VI 補 充 : the distribution of the sample range if x i i.i.d F(x), i = 1,..., n R = x max x min = x (n) x (1) If the samples are taken from N (0, 1) f R (r) = n(n+1) [Φ(x + r) Φ(x)] n 2 φ(x)φ(x+r)dx, r > 0 If the samples are taken from N (0, σ 2 ) W = R σ f R(r) The moments of the range R can be derived form the p.d.f.

10 Statistical Basis of the Charts VII the x control chart: [ ] σ µ Z α/2 n σ, µ + Z α/2 n Z α/2 = 3 ˆσ = R d 2 The parameters of the x chart: UCL = x + 3 d 2 n R Center Line = x LCL = x + 3 d 2 n R

11 Statistical Basis of the Charts VIII Define A 2 = 3 d 2 n

12 Statistical Basis of the Charts IX R chart: The center line: R An estimate of σ R : (Under normal distribution assumption) where d 3 = the s.d. of W R = W σ σ R = d 3 σ ˆσ R = d 3 R d 2

13 Statistical Basis of the Charts X The parameters of the R chart: UCL = R + 3d 3 R d 2 Center Line = R LCL = R 3d 3 R d 2 Assume D 3 = 1 3 d 3 d 2, D 4 = d 3 d 2

14 Statistical Basis of the Charts XI

15 Example I Example 6.1 Hard-bake process 25 samples, each of size 5 wafers It is best to begin with the R chart. Ri R = 25 = x = n = 5 Appendix Table VI D 3 = 0, D 4 = R chart: LCL = RD 3 = 0, UCL = RD 4 = Appendix Tale VI A 2 = x chart: LCL = x A 2 R = , UCL = x+a 2 R =

16 Example II

17 Example III R Chart for predata Group summary statistics UCL CL LCL Group Number of groups = 25 Center = StdDev = LCL = 0 UCL = Number beyond limits = 0 Number violating runs = 0

18 Example IV xbar Chart for predata Group summary statistics UCL CL LCL Group Number of groups = 25 Center = StdDev = LCL = UCL = Number beyond limits = 0 Number violating runs = 0

19 Example V The process is in control at the state levels and adopt the trial control limits for use in phase II, where monitoring of future production is of interest library(qcc) ex6_1=read.table("ex6_1.csv",header=t,sep=",") predata=ex6_1[1:25,] barx=mean(rowmeans(predata)) barr=mean(apply(predata,1,function(x) max(x)-min(x))) qcc(predata,type="r") qcc(predata,type="xbar")

20 Estimating Process Capability I Estimate the mean flow width of the resist: x = microns The process s.d.: ˆσ = R d 2 = microns The specification limits: 1.50 ± 0.50 microns The control chart data may be used to describe the capability of the process to produce wafers relative to these specifications. p =P{x < 1.00} + P{x > 2.00} =Φ( ) + 1 Φ( ) = about 0.035% (350 parts per million) of wafers produced will be outside of the specifications

21 Estimating Process Capability II ex61_xbar=qcc(predata,type="xbar") process.capability(ex61_xbar,spec.limits=c(1,2)) Process Capability Analysis for predata LSL Target USL Number of obs = 125 Center = StdDev = Target = 1.5 LSL = 1 USL = 2 Cp = 1.19 Cp_l = 1.21 Cp_u = 1.18 Cp_k = 1.18 Cpm = 1.19 Exp<LSL 0.015% Exp>USL 0.02% Obs<LSL 0% Obs>USL 0%

22 Estimating Process Capability III Process capability ratio (C p ; PCR) a quality characteristic with both upper and lower specification limits: C p = USL LSL 6σ Another method: the percentage uses up about p% of the specification band P = ( 1 C p ) 100%

23 Estimating Process Capability IV hard-bake process: Ĉ p = (0.1398) = 1.192> 1 the natural tolerance limits in the process are inside the lower and upper specification limits ( ) 1 ˆP = 100% = 83.89% Ĉ p

24 Revision of Control Limits and Center Lines I require periodic revision of the control limits and center lines every week every month every 20, 50, or 100 samples Replace the CL of the x chart with a target value ( x 0 ) If the R chart exhibits control, this can be helpful in shifting the process average to the desired value. (by a fairly simple adjustment of a manipulatable( 可 操 縱 的 ) variable) If the mean is not easily influenced by a simple process adjustment a complex and unknown function of several process variables and a target value x may not be helpful If R chart is out of control eliminate the out-of-control points, recompute a revised value of R

25 Phase II Operation I

26 Phase II Operation II

27 Phase II Operation III qcc(ex6_1[1:25,], type="xbar", newdata=ex6_1[26:45,]) qcc(ex6_1[1:25,], type="r", newdata=ex6_1[26:45,]) Examining control chart data: helpful to construct a run chart of the individual observation in each sample tier chart or tolerance diagram: box plots is usually a simple way to construct the tier diagram

28 Phase II Operation IV

29 CL, SL, NTL I There is no connection or relationship between { the control limits on the x and R charts the specification limits on the process Control limits: driven by the natural variability of the process (natural tolerance limits(ntl) of the process) UNTL, LNTL: 3σ above and below the process mean Specification limits: determined externally( 在 外 面 ); may be set by management, the manufacturing engineers, the customers etc.

30 CL, SL, NTL II CL and SL There is no mathematical or statistical relationship between the control limits and specification limits Control chart use control limits tolerance chart(individual observations) helpful to plot the specification limits

31 Rational Subgroups I x chart: monitors the average quality level in the process Samples should be selected: maximized the chances for shifts in the process average to occur between samples between-sample variability: variability in the process R chart: over time measures the variability within a sample within-sample variability: the instantaneous( 即 時 的 ) process variability at a given time

32 Rational Subgroups II Carefully examining how the control limits for the x and R charts are determined from past data The estimate of the process s.d. σ used in constructing the control limits is calculated from the variability within each sample reflects only within-sample variability mi=1 nj=1 s = (x ij x) 2 to estimate σ mn 1 σ will be overestimated combines both between-sample and within-sample variability

33 Guidelines for the Design of the Control Chart I x and R charts: 1. sample size( 樣 本 大 小 ) 2. control limit width( 管 制 界 線 寬 度 ) 3. frequency of sampling( 抽 樣 頻 率 ) Complete solution to know: ( 經 濟 考 量 ) the cost of sampling( 抽 樣 成 本 ) the costs of investigating and possibly correcting the process in response to out-of-control signal( 調 查 和 矯 正 失 控 製 程 成 本 ) the costs associated with producing a product that does not meet specifications( 製 品 不 合 格 成 本 )

34 Guidelines for the Design of the Control Chart II Some general guidelines that will aid in control chart design x chart: detect { large shifts (2σ or large) n = 4, 5, 6 small shifts n = 15 25(large sample size) smaller samples less risk of a process shift occurring while a sample is taken

35 Guidelines for the Design of the Control Chart III R chart: insensitive to shift in the process s.d. for small samples n = 5 about a 40% chance to detecting the shift σ 2σ large sample size (n > 10 or 12): more effective use a control chart for s or s 2 ( R chart)

36 Guidelines for the Design of the Control Chart IV Allocating sampling( 抽 樣 配 置 ) problem:choosing 1. the sample size 2. the frequency of sampling have only a limited number of resources to allocate to the inspection process available strategies: small, frequent samples: larger samples less frequently: n=5/every half hour favored by the current industry n = 20/every two hours

37 Guidelines for the Design of the Control Chart V The rate of production: influences the choice of sample size & sampling frequency Control Limits: Ex: 50,000 units per hour(high rates of production) 在 高 速 生 產 的 過 程, 在 同 一 時 間 收 集 n = 5 或 n = 20 不 會 造 成 太 大 的 差 異 若 檢 驗 成 本 不 高,high-speed production processes 通 常 會 監 測 較 大 的 樣 本 數 Usually, 3σ type I errors are very expensive to investigate as wide as 3.5σ out-of-control signals are quickly and easily investigated 2.5 or 2.75σ

38 Changing sample size on the x and R charts I Assume: n is constant from sample to sample How about n is not constant? the center line on the R chart is changed x and s charts making a permanent ( 固 定 性 的 ) change, i.e., n old nnew

39 Changing sample size on the x and R charts II Notations: R old = average range for the old sample size Rnew = average range for the new sample size n old = old sample size nnew = new sample size d 2 (old) = factors d 2 for the old sample size d 2 (new) = factors d 2 for the new sample size

40 Changing sample size on the x and R charts III x chart [ ] d2 (new) UCL = x + A 2 R (new) d 2 (old) old [ ] d2 (new) UCL = x A 2 R (new) d 2 (old) old R chart [ ] d2 (new) UCL = D 4 R (new) d 2 (old) old [ ] CL = Rnew d2 (new) = R d 2 (old) old [ ] d2 (new) UCL = D 3 R (new) d 2 (old) old

41 Changing sample size on the x and R charts IV Example 6.2 the hard-bake process in Example 6.1 good control n old = 5 reduce nnew = 3 The new control charts: [ Rnew = Type n R d2 A 2 Old New d 2 (new) d 2 (old) ] R old =

42 Changing sample size on the x and R charts V The new control limits on the x chart: [ ] d2 (new) UCL = x + A 2 R (new) d 2 (old) old = [ ] d2 (new) UCL = x A 2 R (new) d 2 (old) old = The new parameters for the R chart: [ ] d2 (new) UCL = D 4 R (new) d 2 (old) old = CL = Rnew = [ ] d2 (new) UCL = max{0, D 3 R (new) d 2 (old) old } = 0

43 Changing sample size on the x and R charts VI n 1. the width of the control limits on x chart ( σ n ) 2. the center line and the upper control limits ( d 2 when n )

44 Probability Limits on the x and R charts I Name of control limits: α = Z = 3.09 Western Europe: 0.001(= α/2) probability limits (one direction) United States: three-sigma limits; a multiple of the standard deviation of the statistic (k σ); x d normally distributed x chart: k = Z α/2 = 3.09 when α = 0.002

45 Probability Limits on the x and R charts II R chart: using the percentage points of the distribution of the relative range W = R/σ the subgroup size: n W = R σ Var(R) = σ Var(W ) P(σW (n) R σw (n)) = 1 α = (W α/2 (n), W 1 α/2 (n)) = (W 0.001, W (n)) Estimate σ by R/d 2

46 Probability Limits on the x and R charts III The and limits for R: (W (n)( R/d 2 ), W (n)( R/d 2 )) UCL = W (n)( R/d 2 ) = D R UCL = W (n)( R/d 2 ) = D R when 3 n 6, produce LCL 0

47 Charts Based on Standard Values I Possible to specify standard values for the process mean and standard deviation: Standards: µ and σ The x chart based on standard values UCL = µ + 3 σ n = µ + Aσ Center line = µ LCL = µ 3 σ n = µ Aσ

48 Charts Based on Standard Values II The R chart based on standard values σ = R/d 2 d 2 : the mean of the distribution of the relative range (E( R σ ) = d 2) σ R = d 3 σ (where d 3 = Var(W )) UCL = d 2 σ + 3d 3 σ = D 2 σ (D 2 = d 2 + 3d 3 ) Center line = d 2 σ LCL = d 2 σ 3d 3 σ = D 1 σ (D 1 = d 2 3d 3 )

49 Charts Based on Standard Values III Care: when standard values of µ and σ are given May be these standards are not really applicable( 適 當 的 ) to the process Standard value of σ seem to give more trouble than standard value of µ. If the process is really in control at some other mean and standard deviation, then the analyst may spend considerable effort looking for assignable causes that do not exist. In processes where the mean of the quality characteristic is controlled by adjustments to the machine, standard or target values of µ are sometimes helpful in achieving management goals with respect to process performance.

50 Interpretation of x and R I Interpreting patterns on the x chart: must determine whether or not the R chart is in control First eliminate the R chart assignable causes Never attempt to interpret the x chart when the R chart indicates an out-of-control condition

51 Interpretation of x and R II Case 1: Cyclic patterns x chart-systematic environmental changes: 溫 度 操 作 員 疲 勞 人 員 輪 班 或 機 器 輪 流 電 壓 變 動 R chart: 維 修 排 程 人 員 疲 勞 工 具 磨 損 Ex: the on-off cycle of a compressor( 壓 縮 機 ) in the filling machine-systematic variability

52 Interpretation of x and R III Case 2: Mixture pattern 特 徵 : with relative few points near the center line generated by two overlapping distributions generating the process overcontrol: adjustments too often Parallel machines: output product from several sources

53 Interpretation of x and R IV Case 3: Shift in process level New workers Changes in methods raw materials or machines a change in the skill

54 Interpretation of x and R V Case 4: Trend in process level gradual wearing( 逐 步 的 磨 損 ) 工 具 或 重 要 製 成 元 件 的 惡 化 人 員 疲 勞 季 節 影 響 : 溫 度 Monitoring and analyzing processes with trends: regression control chart

55 Interpretation of x and R VI Case 5: Stratification lack of natural variability incorrect calculation of control limits come from two different distribution: R will be incorrectly inflated( 膨 脹 ) causing the limits on the x chart to be too wide

56 Interpretation of x and R VII In interpreting patterns on the x and R charts: consider the two chart jointly If the underlying distribution is normal, x and R charts: statistically independent If there is correlation between x and R values: the underlying distribution is skewed Those analyses may be in error if specifications have been determined assume normality.

57 The Effect of Non-normality on x and R charts I Assumption: the underlying distribution of the quality characteristic is normal Interested in knowing the effect of departures from normality on x and R charts robustness?? Literature: Burr (1967) The usual normal theory control limit constants are very robust to the normality assumption and can be employed unless the population is extremely non-normal.

58 The Effect of Non-normality on x and R charts II Studies: the Uniform, right triangular, gamma, and two bimodal distributions In most cases, samples of size 4 or 5 are sufficient to ensure reasonable robustness to the normality assumption Worst cases: small values or r in Gamma distribution, ex: r = 1/2 or 1 actual α-risk if n 4 Normal distribution:

59 The Effect of Non-normality on x and R charts III The sampling distribution of R is not symmetric Symmetric 3σ control limits are only an approximation the actual α-risk on R chart: if n = 4 not the R chart is more sensitive to departure from normality than the x chart

60 The OC Function I The ability of the x and R charts to detect shift in process quality OC curve for an x control chart σ: assumed known and constant mean shifts: µ 0 = µ 1 = µ 0 + kσ (in-control) The probability of not detecting the shift on the first subsequent sample: (β-risk) β = P {LCL x UCL µ = µ 1 = µ 0 + kσ} ( x N (µ, σ 2 /n), Contol limits: µ 0 ± Lσ/ n) [ ] [ ] UCL (µ0 + kσ) LCL = Φ σ/ (µ0 + kσ) Φ n σ/ n = Φ [ L k n ] Φ [ L k n ]

61 The OC Function II L = 3, k = 2, n = 5 β = Φ[3 2 5] Φ[ 3 2 5] = the probability that such a shift will be detects on the first subsequence sample: 1 β = k = 1, n = 5 β = 0.75

62 The OC Function III β: the probability of not detecting the shift on the first subsequent sample 1 β: the probability that such a shift will be detects on the first subsequence sample The probability that the shift is detected on the second sample: β(1 β) = 0.75(0.25) = 0.19 The probability that the shift is detected on the rth subsequence sample: β r 1 (1 β)

63 The OC Function IV average run length: The expected number of samples taken before the shift is detected: (the expectation of the geometric distribution) ARL = r=1 rβ r 1 (1 β) = 1 1 β Ex: n = 5, k = 1 ARL = = 4 Small sample sizes often result in a relatively large β-risk

64 The OC Function V OC curve for the R chart The distribution of the relative range W = R/σ σ 0: in-control value of the standard deviation shift to a new value: σ 1 > σ 0 The probability of not detecting a shift on the first sample following the shift where λ = σ 1 σ 0 β = P { LCL R UCL σ 1 } (Contol limits: d 2σ 0 ± Lσ 0d 3) = P((d 2 3d 3)σ 0 R (d 2 + 3d 3)σ 0 σ 1) = P(λ 1 (d 2 3d 3) R σ 1 λ 1 (d 2 + 3d 3) σ 1)

65 The OC Function VI λ = σ 1 σ 2 = 2, n = 5 β 0.6 have only about a 40% chance of detecting the shift on each subsequent sample R chart is insensitive to small or moderate shifts for n = 4 6 Recommendation: use at least 20 to 25 preliminary subgroups in establishing x and R charts

66 The Average Run Length for the x chart I ARL = In-control: ARL 0 = 1 α Out-of-control: ARL 1 = 1 1 β 1 P(one point plots out of control) Average time to signal(ats): the average time to signal ATS = ARL h (h = intervals of sampling time) I : the expected number of individual units sampled: I = ARL n (n = the sample size)

67 The Average Run Length for the x chart II

68 Control Charts for x and s I x and s charts: 1. the sample size n is moderately large: n 10 or the sample size n is variable the unbiased estimator of σ 2 : ni=1 sample variance: s 2 (x i x) = n 1 s is not an unbiased estimator of σ: E(s) = c 4 σ, Var(s) = σ 1 c4 2(H.W.) c 4 = 2 Γ(n/2) n 1 Γ((n 1)/2) : a constant that depends on n

69 Control Charts for x and s II s chart: the standard value σ is given UCL = c 4 σ + 3σ 1 c4 2 = B 6σ Center line = c 4 σ LCL = c 4 σ 3σ 1 c4 2 = B 5σ s chart: σ is unknown Estimator of σ: s/c 4 where s = 1 m mi=1 s i UCL = s + 3 s c 4 1 c 2 4 = B 4 s Center line = s LCL = s 3 s c 4 1 c 2 4 = B 3 s

70 Control Charts for x and s III x chart Estimator of σ: s/c 4 where s = 1 mi=1 m s i s UCL = x + 3 = x + A 3 s c 4 n Center line = x s LCL = x 3 = x A 3 s c 4 n n i=1 if using s = (x i x) 2 n the definition of c 4, B 3, B 4, A 3 are altered( 改 變 ) Traditionally: preferred the R chart to the s chart the simplicity of calculating R from each sample

71 Control Charts for x and s IV Example 6.3: 活 塞 環 的 內 半 徑

72 Control Charts for x and s V Example 6.3: the piston ring inside diameter measurements( 活 塞 環 的 內 半 徑 ) m = 25, n = 5 x = , s = x chart: s chart: UCL = x + A 3 s = Center line = x = LCL = x A 3 s = UCL = B 4 s = Center line = s = LCL = B 3 s = 0 Estimation of σ: ˆσ = s c 4 = = 0.01

73 Control Charts for x and s VI

74 Control Charts for x and s VII

75 The x and s Control Charts with Variable Sample Size I easy to apply in cases where the sample sizes are variable n i : the number of observations in the ith sample the center line of x and s control charts: mi=1 [ n i x mi=1 i (n i 1)si 2 x = mi=1 s = n mi=1 i n i m ] 1/2 A 3, B 3, B 4 : depend on the sample size used in each individual subgroup x chart: s chart: UCL = x + A 3 s Center line = x LCL = x A 3 s UCL = B 4 s Center line = s LCL = B 3 s

76 The x and s Control Charts with Variable Sample Size II

77 The x and s Control Charts with Variable Sample Size III

78 The x and s Control Charts with Variable Sample Size IV Alternative: 1. using an average sample size n n i are not very different or in a presentation to management the average sample size may not be an integer 2. a modal(most common) sample size: 最 常 出 現 的 sample size n i 來 估 計 σ 的 值 有 17 個 n i = 5 average all the s i for which n i = 5 s = = ˆσ = s c 4 = = 0.01

79 The s 2 control Chart I s 2 chart UCL = s2 n 1 χ2 α/2,n 1 Center line = s 2 LCL = s2 n 1 χ2 1 α/2,n 1 (n 1)s2 σ 2 P χ 2 n 1 ( σ 2 χ 2 1 α/2,n 1 n 1 s 2 σ2 χ 2 ) α/2,n 1 = 1 α n 1

80 The Shewhart Control Chart for Individual Measurement I n = 1: Automated( 自 動 化 ) inspection and measurement technology is used Data: available relatively slowly; inconvenient to allow sample sizes of n > 1 Repeat measurements on the process differ only because of laboratory or analysis error Multiple measurements are taken on the same unit of product differ very little s.d. too small; Ex; 一 捲 紙 塗 料 的 厚 度 Individual measurements: transactional, business and service process, no basis for rational subgrouping

81 The Shewhart Control Chart for Individual Measurement II Control chart for individual units: Moving range control chart moving range of two successive observations: MR i = x i x i 1 n = 2 D 3 = 0, D 4 = UCL = D 4 MR = UCL = D 4 MR = 0

82 The Shewhart Control Chart for Individual Measurement III

83 The Shewhart Control Chart for Individual Measurement IV library(qcc) ex6_5 = c(310,288,297,298,307,303,294,297,308,306,294, 299,297,299,314,295,293,306,301,304) qcc(ex6_5, type = "xbar.one", plot = TRUE) Ex6_5_r = matrix(cbind(ex6_5[1:length(ex6_5)-1],ex6_5[2:length(ex6_5)]), ncol=2) qcc(ex6_5_r, type="r", plot = TRUE,title="R chart for Ex6_5")

84 The Shewhart Control Chart for Individual Measurement V The interpretation of the individuals control chart is similar to that of the ordinary x control chart. Sometimes a point will plot outside the control limits on both the individual chart and the moving range chart. a large value of x will also lead to a large value of the moving range most likely indicates that the mean is out of control not both the mean and the variance of the process are out of control

85 The Shewhart Control Chart for Individual Measurement VI Phase II Operation and Interpretation of the individual charts The individual measurements on the x chart are assumed to be uncorrelated, and any apparent pattern on this chart should be carefully investigated.

86 The Shewhart Control Chart for Individual Measurement VII ex6_5_new =c(305,282,305,296,314,295,287,301,298,311, 310,292,305,299,304,310,304,305,333,328) qcc(ex6_5, type = "xbar.one", plot = TRUE, newdata=ex6_5_new) Ex6_5_r_new = matrix(cbind(ex6_5_new[1:length(ex6_5_new)-1],ex6_5_new[2:length(ex6_5_new)]), ncol=2) Ex6_5_r_all =rbind(ex6_5_r,ex6_5_r_new) qcc(ex6_5_r_all[1:19,],type="r",newdata=ex6_5_r_all[20:38,],title="r chart for Ex6_5")

87 The Shewhart Control Chart for Individual Measurement VIII Some authorities: recommended not constructing and plotting the MR chart. The MR chart cannot really provide useful information about a shift in process variability. the careful in interpretation and relies primarily on the individual chart

88 The Shewhart Control Chart for Individual Measurement IX Crowder (1987b) The ARL 0 of the combined procedure will generally be much less than the ARL 0 of a standard Shewhart control chart when the process is in control. (type I error α ) results closer to the Shewhart ARL 0 if we use UCL = DMR, where 4 D 5 The ability of the individuals control chart to detect small shifts is very poor. Size of Shift β ARL 1 1σ σ σ

89 The Shewhart Control Chart for Individual Measurement X Dangerous: narrower limits ARL 0 but the occurrence of false alarms detecting small shifts in phase II with individual values: Chapter 9: the cumulative sum control chart or the EWMA control chart

90 The Shewhart Control Chart for Individual Measurement XI Individual control chart: ARL 0 is dramatically affected by non-normal data moderate departure from normality the control limits may be inappropriate for phase II process monitoring Methods: 1. determine the control limits based on the percentiles of the correct underlying distribution 2. transform the original variable to a new variable that is approximately normally distributed Important to check the normality assumption: the normal probability plot