1. TM 720: Statistical Process Control
TM Lecture 10
Short Run SPC and Gage Reproducibility & Repeatability
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

2. TM 720: Statistical Process Control
Assignment:
Reading:
Finish Chapters 7 and 9
Sections 7.4 – 7.8
Sections 9 – 9.2
Assignment:
Access Excel Template for Individuals Control Charts:
Download Assignment 7 for practice
Use the data on the HW7 Excel sheet to do the charting, verify the control limits by hand calculations
Solutions for 6 and 7 will post on Thursday
Review for Exam II
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

3. TM 720: Statistical Process Control
Review
Shewhart Control charts
Are for sample data from an approximate Normal distribution
Three lines appear on all Shewhart Control Charts
UCL, CL, LCL
Two charts are used:
X-bar for testing for change in location
R or s-chart for testing for change in spread
We check the charts using 4 Western Electric rules
Attributes Control charts
Are for Discrete distribution data
Use p- and np-charts for tracking defective units
Use c- and u-charts for tracking defects on units
Use p- and u-charts for variable sample sizes
Use np- and c-charts with constant sample sizes
11/7/2018
TM 720: Statistical Process Control

4. TM 720: Statistical Process Control
Short Run SPC
Many products are made in smaller quantities than are practical to control with traditional SPC
In order to have enough observations for statistical control to work, batches of parts may be grouped together onto a control chart
This usually requires a transformation of the variable on the control chart, and a logical grouping of the part numbers (different parts) to be plotted.
A single chart or set of charts may cover several different part types
11/7/2018
TM 720: Statistical Process Control

5. TM 720: Statistical Process Control
DNOM Charts
Deviation from Nominal
Variable computed is the difference between the measured part and the target dimension
where: Mi is the measured value of the ith part
Tp is the target dimension for all of part number p
11/7/2018
TM 720: Statistical Process Control

6. TM 720: Statistical Process Control
DNOM Charts
The computed variable (xi) is part of a sub-sample of size n
xi is normally distributed
n is held constant for all part numbers in the chart group.
Charted variables are x and R, just as in a traditional Shewhart control chart, and control limits are computed as such, too:
11/7/2018
TM 720: Statistical Process Control

7. TM 720: Statistical Process Control
DNOM Charts
Usage:
A vertical dashed line is used to mark the charts at the point at which the part numbers change from one part type to the next in the group
The variation among each of the part types in the group should be similar (hypothesis test!)
Often times, the Tp is the nominal target value for the process for that part type
Allows the use of the chart when only a single-sided specification is given
If no target value is specified, the historical average (x) may be used in its’ place
11/7/2018
TM 720: Statistical Process Control

8. Standardized Control Charts
If the variation among the part types within a logical group are not similar, the variable may be standardized
This is similar to the way that we converted from any normally distributed variable to a standard normal distribution:
Express the measured variable in terms of how many units of spread it is away from the central location of the distribution
11/7/2018
TM 720: Statistical Process Control

9. Standardized Charts – x and R
Standardized Range:
Plotted variable is
where: Ri is the range of measure values for the ith sub-sample of this part type j
Rj is the average range for this jth part type
11/7/2018
TM 720: Statistical Process Control

10. Standardized Charts – x and R
Standardized x:
Plotted variable for the sample is
where: Mi is the mean of the original measured values for this sub-sample of the current part type (j)
Tj is the target or nominal value for this jth part type
11/7/2018
TM 720: Statistical Process Control

11. Standardized Charts – x and R
Usage:
Two options for finding Rj:
Prior History
Estimate from target σ:
Examples:
Parts from same machine with similar dimensions
Part families – similar part tolerances from similar setups and equipment
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TM 720: Statistical Process Control

12. Standardized Charts – Attributes
Standardized zi for Proportion Defective:
Plotted variable is
Control Limits:
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TM 720: Statistical Process Control

13. Standardized Charts – Attributes
Standardized zi for Number Defective:
Plotted variable is
Control Limits:
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TM 720: Statistical Process Control

14. Standardized Charts – Attributes
Standardized zi for Count of Defects:
Plotted variable is
Control Limits:
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TM 720: Statistical Process Control

15. Standardized Charts – Attributes
Standardized zi for Defects per Inspection Unit:
Plotted variable is
Control Limits:
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TM 720: Statistical Process Control

16. Gage Capability Studies
TM 720: Statistical Process Control
Gage Capability Studies
Ensuring an adequate gage and inspection system capability is an important consideration!
In any problem involving measurement the observed variability in product due to two sources:
Product variability - σ2product
Gage variability - σ2gage i.e., measurement error
Total observed variance in product:
σ2total = σ2product + σ2gage (system)
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

17. e.g. Assessing Gage Capability
TM 720: Statistical Process Control
e.g. Assessing Gage Capability
Following data were taken by one operator during gage capability study.
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

18. e.g. Assessing Gage Capability Cont'd
TM 720: Statistical Process Control
e.g. Assessing Gage Capability Cont'd
Estimate standard deviation of measurement error:
Dist. of measurement error is usually well approximated by the Normal, therefore
Estimate gage capability:
That is, individual measurements expected to vary as much as owing to gage error.
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

19. Precision-to-Tolerance (P/T) Ratio
TM 720: Statistical Process Control
Precision-to-Tolerance (P/T) Ratio
Common practice to compare gage capability with the width of the specifications
In gage capability, the specification width is called the tolerance band
(not to be confused with natural tolerance limits, NTLs)
Specs for above example: 32.5 ± 27.5
Rule of Thumb:
P/T  0.1  Adequate gage capability
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

20. Estimating Variance Components of Total Observed Variability
TM 720: Statistical Process Control
Estimating Variance Components of Total Observed Variability
Estimate total variance:
Compute an estimate of product variance
Since :
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

21. Gage Std Dev Can Be Expressed as % of Product Standard Deviation
TM 720: Statistical Process Control
Gage Std Dev Can Be Expressed as % of Product Standard Deviation
Gage standard deviation as percentage of product standard deviation :
This is often a more meaningful expression, because it does not depend on the width of the specification limits
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

22. Using x and R-Charts for a Gage Capability Study
TM 720: Statistical Process Control
Using x and R-Charts for a Gage Capability Study
On x chart for measurements:
Expect to see many out-of-control points
x chart has different meaning than for process control
shows the ability of the gage to discriminate between units (discriminating power of instrument)
Why? Because estimate of σx used for control limits is based only on measurement error, i.e.:
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

23. Using x and R-Charts for a Gage Capability Study
TM 720: Statistical Process Control
Using x and R-Charts for a Gage Capability Study
On R-chart for measurements:
R-chart directly shows magnitude of measurement error
Values represent differences between measurements made by same operator on same unit using the same instrument
Interpretation of chart:
In-control: operator has no difficulty making consistent measurements
Out-of-control: operator has difficulty making consistent measurements
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

24. Repeatability & Reproducibility: Gage R & R Study
TM 720: Statistical Process Control
Repeatability & Reproducibility: Gage R & R Study
If more than one operator used in study then measurement (gage) error has two components of variance:
σ2total = σ2product + σ2gage
σ2reproducibility + σ2repeatability
Repeatability:
σ2repeatability - Variance due to measuring instrument
Reproducibility:
σ2reproducibility - Variance due to different operators
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

25. TM 720: Statistical Process Control
Ex. Gage R & R Study
20 parts, 3 operators, each operator measures each part twice
Estimate repeatability (measurement error):
Use d2 for n = 2 since each range uses 2 repeat measures
Operator i xi Ri 1
22.30
1.00 2
22.28
1.25 3
22.10
1.20
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

26. TM 720: Statistical Process Control
Ex. Gage R & R Study Cont'd
Estimate reproducibility:
Differences in xi  operator bias since all operators measured same parts
Use d2 for n = 3 since Rx is from sample of size 3
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

27. TM 720: Statistical Process Control
Ex. Gage R & R Study Cont'd
Total Gage variability:
Gage standard deviation (measurement error):
P/T Ratio:
Specs: USL = 60, LSL = 5
Note:
Would like P/T < 0.1!
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

28. Comparison of Gage Capability Examples
TM 720: Statistical Process Control
Comparison of Gage Capability Examples
Gage capability is not as good when we account for both reproducibility and repeatability
Train operators to reduce σ2reproducability from
Since σ2repeatability = (largest component), direct effort toward finding another inspection device.
σ2 repeatability σ2 reproducibility
σ2 product
P/T
Single operator
0.8865
0.0967
Three operators
1.0195
0.1181
1.0263
0.1120
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

29. Gage Capability Based on Analysis of Variance
TM 720: Statistical Process Control
Gage Capability Based on Analysis of Variance
A gage R & R study is actually a designed experiment
Therefore ANOVA can be used to analyze the data from an experiment and to estimate the appropriate components of gage variability
Assume there are:
a parts
b operators
each operator measures every part n times
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

30. The measurements, yijk, are represented by a model
TM 720: Statistical Process Control
The measurements, yijk, are represented by a model
where
constant m – overall measurement mean
r.v. ti – effect from part differences
r.v. bj – effect from operator differences
r.v. tbij – joint effect of parts & operator differences
r.v. eijk – error from measuring instrument
with
i = part (i = 1, …, a)
j = operator (j = 1, …, b)
k = measurement (k = 1, …, n)
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

31. The Variance Components for the Gage R&R Study Using the Model
TM 720: Statistical Process Control
The Variance Components for the Gage R&R Study Using the Model
The variance of an observation yijk is
So:
is the variance from parts
is the variance from operators
is the joint variance from parts & operators
is the variance from measuring instrument
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

32. Repeatability & Reproducibility
TM 720: Statistical Process Control
Repeatability & Reproducibility
Reproducibility (Operators)
Repeatability (Measuring Device)
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

33. Gage R&R – ANOVA Method StatGraphics Output
TM 720: Statistical Process Control
Gage R&R – ANOVA Method StatGraphics Output
ANOVA Table
Source Sum Squares Df Mean Square F-Ratio P-Value
Oper
Part
Oper*Part
Residual
Total
Operator variable: Operator
Part variable: Part
Trial variable: Trial
Measurement variable: Measurement
3 operators 20 parts 2 trials
Estimated Estimated Percent
Sigma Variance of Total
Repeatability
Reproducibility
Interaction
R & R
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

34. Comparison of Gage Capability Examples
TM 720: Statistical Process Control
Comparison of Gage Capability Examples
σ2repeatability σ2reproducibility
σ2gage
P/T
Single operator
0.8865
0.0967
Three operators
(Tabular Method)
1.0195
0.1181
1.0263
0.1120
(ANOVA Method)
0.9958
1.0944
1.4797
0.1614
11/7/2018
TM 720: Statistical Process Control
(c) D.H. Jensen & R.C. Wurl

35. TM 720: Statistical Process Control
Questions & Issues
Topics for Exam II:
Shewhart Continuous Variable Control Charts
X-bar and R; X-bar and S-charts
Control Limits from samples or standards using table
Western Electric Rules
Shewhart-Like Discrete Variable Control Charts
P, NP, C, U-charts
Defectives vs. Defects; Variable or Constant Sample Sizes
Control Charts for Individual Measurements
X and Moving Range; Moving Average, EWMA, CUSUM
Short Run Statistical Process Control
DNOM and Standardized charts (continuous / discrete)
Gage Repeatability and Reproducibility
Control Chart Method – only!
11/7/2018
TM 720: Statistical Process Control