1. Process and Measurement System Capability Analysis
Chapter 8

2. Introduction Statistical Techniques
Process Capability Analysis Using a Histogram or a Probability Plot
Process Capability Analysis Ratios
Process Capability Analysis Using a Control Chart
Process Capability Analysis Using design of Experiments
Process Capability Analysis with Attribute Data
Gauge and Measurement System Capability Studies
Setting Specification Limits on discrete Components
Estimating the Natural Tolerance limits of a Process

3. Statistical Techniques
Process Capability = the uniformity of a process
Natural Tolerances= µ+3σ and µ-3σ = UNTL (Upper Natural Tolerance Limit) and LNTL (Lower Natural Tolerance Limit)
Normal Distribution includes 99.73% of the variable
Or 0.27% of the process output will fall outside natural tolerance limits
Process Capability Analysis a study to estimate PC (Process Capability)
Product Characterization- samples from the supplier with no idea of the process they used or made the product with.

4. Process Capability Analysis Using a Histogram or a Probability Plot
100 or more observations for stability
Steps to follow before data collection- Choose machine to be used, select process operating conditions, select operator and monitor the collection process
Use x bar and standard deviation for calculations od capability
Substitute x bar for µ and s for σ in the formula= x̄ + 3s

5. Histogram Example Cp and Cpk > 1 but < 1
Histogram Example Cp and Cpk > 1 but < 1.33 Taken from:

6. Probability Plotting
Alternative to Histogram which can determine the shape of the distribution
Advantage is divide the range into intervals which often produces reasonable results for small samples
Ranked data versus cumulative frequency on special paper with vertical scale with distribution as a straight line

7. Probability Plot Example Data not normally distributed so 3σ will not work Taken from: Authors Robert and Marilyn Hart

8. Capability Ratios
Cp is the USL (Upper Specified Limit) minus the LSL (Lower Specified Limit) divided by 6σ and measures potential capability
Cp = USL-LSL/ 6σ
Sample Standard deviation s or Rbar/D2 which is C-hatp=USL-LSL/6σ- hat
Off center process- Cp measure the spread of the specification relative to six sigma not of the account of where the mean is locate
Cpk = min{Cpu, Cpl}, Cpu= USL-µ/3σ, Cpl = µ-LSL/3σ
Cpk measures actual capability and is based on normal distribution of the process output

9. PCR Continued
Cpc = USL-LSL/6*sprt((Pi/2)*E|X-T|) T is the target =1/2(USL+LSL)
Cpm =USL-LSL/6t t-sqrd = E[(x-T)sqrd] then Cpm =USL-LSL/6*sqrt sigma sqrd+(mu-T) sqrd which also = Cp/sqrt 1+ξ sqrd
Confidence Intervals and Tests on PCRs
C-hat p or C-hat pk- Cp replace σ with s to find interval in Cpk = [1- Zalpha/2*sqrt1/9nc-hatsqrdpk=1/2(n-1))≤ Cpk if ≤C-hat pk change – to +
Cpc is USL-LSL/s*sqrt (pi0/2)*c

10. PCR continued Testing Hypothesis about PCRs
Ho: Cp=Cpo (or the process is not capable)
H1: Cp≥Cpo (or the process is capable)
Formulate the statistical test in the terms of C-hat p, so the we will reject Ho if C-hat p exceeds the critical value c.

11. PC Analysis Using a Control Chart
They do not necessarily display potential capability and do not address statistical control
Regard them as a primary technique of PC analysis.
2 charts can be used Attribute and Variable
Xbar and R should be used whenever possible because stronger and better information comes from them.
They allow both instantaneous and variability across time to be analyzed

12. PC Control Chart continued
Xbar Chart – Centerline= Xbarbar, UCL= Xbarbar+ Asub2Rbar, LCL= Xbarbar- Asub2Rbar
R Chart- Centerline Rbar, UCL= Dsub4*Rbar, LCL=Dsub3Rbar
Mu-hat = Xbarbar sigma-hat= Rbar/Dsub2

13. Xbar and R Chart Example Taken From: http://www. minitabknowledge
Xbar and R Chart Example Taken From: author Minitab.com

14. PC Analysis Using Designed Experiments
Is a systematic approach to varying the input control variables
Used in isolating and estimating the sources of variability in a process
Experiments can be designed to sample a various number of machines with several applicators to obtain an estimation of the variances

15. PC Analysis with Attribute Data
Process Performance is measured by this data
Nonconforming units or defectives, or nonconformities or defects
DPU (Defects Per Unit) = Total number of Defects/Total number of units
DPMO (Defects per million opportunities) = Total number of defects/ Number of units x Number of opportunities

16. Gauge and Measurement System Capability Studies
Basic Concepts 1. Determine how much of the variability is the gauge or instrument, 2. isolate those components of variability in the measuring system, 3. Assess the capability of the instrument or gauge
Second step in DMAIC problem solving process
Utilize the 2 R’s of the measurement system capability (MSC), Repeatability and reproducibility.
Another important aspect of MSC is Linearity- reflects the differences in observed accuracy and /or precision experienced over the range of measurements made by the system.

17. G&M continued
Stability and Bias 2 of the problems with linearity due to the calibration and maintenance issues.
Stability is the different levels of variability in different operating regimes
Bias reflects the difference between observed measurements and a true value from master or gold standard. Or from a different measurement technique known to produce accurate values
Uses Xbar and R charts

18. G&M continued
Precision to Tolerance ratio= P/T= k sigma-hat sub gauge/USL-LSL
Compares the estimate of gauge capability to the width of the specifications or the tolerance band (USL-LSL) for t he part that is being measured.
Discrimination Ratio for the gauge to capable the DR must exceed 4
Gauge precision and accuracy- precision is the variability in the measurement and the accuracy is the ability of the instrument to measure the true value.

19. The Analysis of Variance Mode
Measures the R&R (repeatability and Reproducibility) of and experiment.
Also known as ANOVA the standard model for a gauge R&R experiment
Can be used to estimate variance components by partitioning the variability in the measurements into component parts
SS-total= SS-parts+SS-operators+SS-p*o+SS-error

20. ANOVA continued
MS-p= SS-parts/p-1, MS-o=SS-operators/o-1, MS-po=SS-p*o/(p-1)(o- 1), MS-E+SS-error/po(n-1)
Expected value are; E(MS-p)= σsqrd + nσsqrd + bnσsqrd, E(MS-o)= σsqrd + nσsqrd + anσsqrd, E(MS-po) = σsqrd + nσsqrd-po, E(MS-E) = σsqrd
σ-hatsqrd= MS-E, σ-hatsqrd-po= MSpo-MSE/n, σ-hatsqrdo =Mso- Mspo/pn, σ-hatsqrdp= MSp-Mspo/on

21. Setting Specification Limits on discrete Components
Linear Combination is the variables of the dimension of a component where asubc is constants. The all follow a straight line.
Example: If the xi are normally and independently distributed with a mean or mu and variance sigma sqrd, then y is normally distributed with mean or mu = to summation of and mu and variance = to summation a sqrd and sigma sqrd
Nonlinear combination is a combination of means mu1, mu2,…mun, are the nominal dimensions associated with the components x1, x2, …xn and then by expanding the right-hand side of the equation about the mu’s.

22. Estimating the Natural Tolerance Limits of a Process
Uses concepts such as 100(1-alpha), mu+z sub alpha/2*sigma.
Tolerance limits based on the Normal Distribution= xbar+z sub alpha/2*s
Confidence limits are used to provide and interval estimate of the parameter of a distribution
Tolerance limits are used to indicate limits between which we can expect to find a specified proportion of a population.
Nonparametric Tolerance limits- interval that are based on the distribution of the extreme values in a sample from an arbitrary continuous distribution.

23. End
Questions?