ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Process and Measurement System Capability Analysis Introduction “Process capability” means broadly the ability of a process to achieve satisfactory performance, in light of its inherent variability. For example, the Process Capability Ratio (PCR) C p compares the difference between a process’s specification limits with the difference between its natural tolerance limits: USL − LSL C p = UNTL − LNTL . But specifications are not necessary to analyze process capability. 1 / 15 Process and Measurement System Capability Introduction
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Process Capability Analysis Using Graphics Histogram Histograms can be helpful in estimating process capability. Example: Bursting strength of glass containers Table 8.1 gives the bursting strengths (psi) of 100 glass containers. In R: burst <- read.csv("Data/Table-08-01.csv"); hist(burst$Strength) # To match Figure 8.2: hist(burst$Strength, right = FALSE, breaks = seq(from = 170, to = 350, by = 20)) 2 / 15 Process and Measurement System Capability Process Capability Analysis Using Graphics
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control The histogram is broadly similar to the normal density, so the data may be summarized by ¯ x = 264 . 1 , s = 32 . 0. The q-q plot is a better tool for comparing data with the normal distribution: qqnorm(burst$Strength) The q-q plot is very much what we would expect for a random sample from a normal distribution, so we estimate the natural tolerance limits µ ± 3 σ by 264 . 1 ± 3 × 32 . 0 = (168 . 1 , 360 . 1) . We expect 99.73% of containers to have strengths in this interval. 3 / 15 Process and Measurement System Capability Process Capability Analysis Using Graphics
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control The natural tolerance limits describe what to expect from this production process. Notes No specification limits are needed up to this point. However, if limits are given, they may be added to the histogram to allow a visual assessment of process capability. Probabilities like 99.73% are heavily dependent on the data being normally distributed. Small samples, even as large as 100, do not permit identification of small but important deviations from normality. For example, a sample of 100 from the t -distribution with 10 degrees of freedom may also look normal, but that distribution has almost 3 times the probability outside the natural tolerance limits. 4 / 15 Process and Measurement System Capability Process Capability Analysis Using Graphics
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Process Capability Ratios Recall USL − LSL C p = UNTL − LNTL = USL − LSL . 6 σ If the process is centered ( µ = 1 2 (USL + LSL)) then 1 2 (USL − LSL) USL − µ = σ σ = 3 C p . 5 / 15 Process and Measurement System Capability Process Capability Ratios
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control If the process is also normally distributed, the fraction nonconforming is � X − µ > USL − µ � P ( X > USL) + P ( X < LSL) = P σ σ � X − µ � < LSL − µ + P σ σ = P ( Z > 3 C p ) + P ( Z < − 3 C p ) = 2[1 − Φ(3 C p )] . So the process capability ratio determines the fraction of nonconforming items. 6 / 15 Process and Measurement System Capability Process Capability Ratios
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control For example, if C p ≥ 1, then at least 99.73% of items will conform to specification (2,700 ppm nonconformant). If C p ≥ 1 . 5, then at most 6.8 ppm of items will be nonconformant (“six-sigma” quality would be 3.4 ppm). 7 / 15 Process and Measurement System Capability Process Capability Ratios
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control One-sided process capability In some contexts, there may be only one specification limit. For the example of the glass containers, LSL = 200 psi, and there is no USL. The lower process capability ratio C pl is C pl = µ − LSL , 3 σ which is estimated by (¯ x − LSL) / (3 s ) = 0 . 67. 8 / 15 Process and Measurement System Capability Process Capability Ratios
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Again, if the process is normally distributed, the fraction nonconforming is � X − µ � < LSL − µ P ( X < LSL) = P σ σ = P ( Z < − 3 C pl ) = 1 − Φ(3 C pl ) . For the glass containers, the fraction of nonconforming containers, assuming normality, is 1 − Φ(3 C pl ) = 1 − Φ(2) = 2 . 28%, or around 22,800 ppm. 9 / 15 Process and Measurement System Capability Process Capability Ratios
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Off-center process capability If a process is not centered, the fraction nonconforming cannot be calculated only from C p , and is higher than it would be for a centered process with the same σ . A modified PCR is C pk = min( C pu , C pl ) , where C pl is the lower process capability ratio and C pu = USL − µ , 3 σ is the corresponding upper process capability ratio . 10 / 15 Process and Measurement System Capability Process Capability Ratios
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Clearly C p = 1 2( C pu + C pl ) , and hence C pk ≤ C p . For a normally distributed process, the fraction nonconforming is 1 − Φ(3 C pk ) + 1 − Φ[3(2 C p − C pk )] . The second term is always less than or equal to the first, and much smaller for a substantially off-center process. For example, if C p = 1 . 5 and C pk = 1, the two terms are 1 − Φ(3) = 1 . 35 × 10 − 3 and 1 − Φ(6) = 9 . 87 × 10 − 10 , respectively. 11 / 15 Process and Measurement System Capability Process Capability Ratios
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Other PCRs Various other efforts have been made to define PCRs that capture process capability when the process is not centered. Statistical inference All C p s are defined in terms of population parameters, so when they are estimated from sample data, the usual issues arise: Point estimates; Interval estimates; Testing hypotheses. 12 / 15 Process and Measurement System Capability Process Capability Ratios
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Process Capability Analysis Using a Control Chart The process capability ratios C p etc. do not distinguish between short-term variability, which is inherent to the process, and long-term variability, which may be attributed to assignable causes. A control chart can help to separate them, when the sample structure is known. 13 / 15 Process and Measurement System Capability Process Capability Analysis Using a Control Chart
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Example: bursting strengths The bursting strengths of glass containers were actually collected in m = 20 samples of size n = 5: burst$Sample <- rep(1:20, rep(5, 20)) library(qcc) burstG <- qcc.groups(burst$Strength, burst$Sample) summary(qcc(burstG, "R")) summary(qcc(burstG, "xbar")) 14 / 15 Process and Measurement System Capability Process Capability Analysis Using a Control Chart
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control The process appears to be in control, so the new statistics can be used to assess process capability: process.capability(qcc(burstG, "xbar"), c(LSL = 200, USL = NA)) # C_pl = 0.6425, 95% CI is (0.5495, 0.7355) Note that the output includes confidence intervals for the PCRs (in this case, only C pl can be calculated, because there is no USL). We can use the more conventional pooled variance estimate of σ 2 , instead of the range-based calculation: process.capability(qcc(burstG, "xbar", std.dev = "RMSDF"), spec.limits = c(LSL = 200, USL = NA)) 15 / 15 Process and Measurement System Capability Process Capability Analysis Using a Control Chart
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