1. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
VISUALIZING VARIATION
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

2. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Capability Versus Control
Statistical Control
In Control
Out of Control
Capability
IDEAL
Capable
Not Capable
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

3. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Process Capability Ratio – A Standard Measure of How Good a Process Is.
A simple ratio:
Specification Width
Actual “Process Width”
Generally, the higher the better.
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

4. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Process Capability Ratio
Nominal
value
Process distribution
Lower
specification
Upper
specification 20 30
Minutes 25
Process is capable
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

5. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Process Capability
Nominal
value
Process distribution
Lower
specification
Upper
specification 20 30
Minutes 25
Process is capable
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

6. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Mean
Spec range=6σ
USL
LSL
Spec Range 6σ
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

7. Variability on Process Capability
Effects of Reducing
Variability on Process Capability
Nominal value
Six sigma
Four sigma
Two sigma
Lower
specification
Upper
specification
Mean
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

8. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
LSL
LCL
UCL
USL
LCL
UCL
LSL
USL 3σ 3σ 3σ 3σ
LCL
LSL
USL
UCL 3σ 3σ
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

9. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
PCR>1
Process Capability
(a)
LSL
USL 3σ 3σ
PCR=1
Nonconforming
Units
Nonconforming
Units
(b) 3σ 3σ
Figure Process Fallout
and the process
capability ratio (PCR).
USL
LSL
PCR<1
Nonconforming
Units
Nonconforming
Units
(c)
LSL
USL 3σ 3σ
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

10. LSL of Pressure Strength =200 psi
Glass container strength data (psi)
Sample
Data x R 1
265
205
263
307
220
252
102 2
268
260
234
299
215
255.2 84 3
197
286
274
243
231
242.2 89 4
267
281
214
318
269
104 5
346
317 22
258
276
287.8 6
300
208
187
264
271
246
113 7
280
242
321
228
266.2 93 8
250
293
273.4 49 9
254
294
223
263.4 71 10
308
235
283
277
272.6 73 11
200
328
296
261
128 12
290
266.8 55 13
221
176
248
227.8 87 14
334
272
286.8 69 15
262
245
301
268.8 56 16
253
287
270.4 34 17
337 18
278
275 28 19
270
298
272.2 48 20
257
210
251
253.4 70
264.06
77.3
Process Capability
Analysis using
Control Charts
LSL of Pressure Strength =200 psi
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

11. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Vane-opening Measurements
sample number x1 x2 x3 x4 x5 x r s 1 33 29 31 32
31.6 4 2 35 37
33.4 6 3 34 36 30
32.2 5
33.8 38 39 40
38.4 7 8
36.8 10 9 28 43 15 11
29.8 12 13 27 14
34.8
35.6 16
30.8 17
2.0000 18 19 25
28.2 20
33.32
5.8
2.345
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

12. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Not capable
Cp<1 T O W E R D S X C L N e
Cp=1
Merely capable
Acceptable
Cp=1.33
4σ level
Cp=1.66
Performing
6σ level
Tolerance
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

13. Process Fallout (In defective ppm) PCR One-Sided Specification
Two-Sided Specification
0.25
226,628
453,255
0.5
66,807
133,614
0.6
35,931
71,861
0.7
17,865
35,729
0.8
8,198
16,395
0.9
3,467
6,934 1
1,350
2,700
1.1
484
967
1.2
159
318
1.3 48 96
1.4 14 27
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

14. Recommended Minimum Values of the Process Capability ratio Two-sided
Specification
One-sided
Existing processes
1.33
1.25
New processes
1.50
1.45
Safely, strength, or critical parameter, existing process
Safely, strength, or critical parameter, new process
1.67
1.60
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

15. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Reasons for Poor Process Capability σ
LSL
USL
(a) σ
LSL
USL
(b)
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

16. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
LSL
USL
σ=2
• Cp does
not take
process
centering
into
Account
• It is a
measure of
potential
capability,
not
actual
capability
(a)
σ=2
(b)
σ=2
(c)
σ=2
(d)
σ=2
(e)
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

17. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Process Capability
What it is 1.
CAPABILITY
Short term
in control
Pooled std dev 2.
PRFORMANCE
Long term
Not in control
Total/overall std dev
compares
spec range
(tolerance)
to process
width
How
much
variability a
how close
process
centre is to
nearest
spec limit
How
centered b
or ENTITLEMENT is the theoretical best
hat a process can achieve.
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

18. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Bursting Strength for 100 Glass Containers
265
197
346
280
200
221
261
278
205
286
317
242
254
235
176
262
248
250
263
274
260
281
246
271
307
258
321
294
328
245
270
220
231
276
228
223
296
301
337
298
268
267
300
334
257
208
299
308
264
210
234
187
269
253
214
283
272
287
215
318
293
277
290
275
251
Specification=
Calculate
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

19. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Vane-opening Measurements
sample number x1 x2 x3 x4 x5 x r s 1 33 29 31 32
31.6 4 2 35 37
33.4 6 3 34 36 30
32.2 5
33.8 38 39 40
38.4 7 8
36.8 10 9 28 43 15 11
29.8 12 13 27 14
34.8
35.6 16
30.8 17
2.0000 18 19 25
28.2 20
33.32
5.8
2.345
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

20. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Batch
Viscosity
Percent solids
Lb.Gal 1 74
64.2
13.3 22
62.7 2 69
62.6
13.4 23 70
62.9 3 79
63.7 24 75
13.2 4
63.6 25 65
62.8 5 62
62.4 26 6
63.4 27 7 73
63.2 28 72 8 63 29 77 9 68
63.8
13.5 30 10
63.5 31 11 32 12
63.1 33
62.5 13 34 14 35 66
62.3
13.1 15 36 16 76
63.3 37 17
64.1 38 18 39 19 40 20
63.9 41 21
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

21. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Statistical Analysis - MSA
Terminology
– Measurement system:
the collection of operations, procedures, gauges and
other equipment, software and personnel used to
assign a number to a characteristic being measured;
the complete process used to obtain a measurement.
Material
Man
Method
MEASUREMENT SYSTEM
Machine
(Time)
Environment
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

22. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Sources of Variation
Product Variability
(Actual variability)
Measurement
Variability
Total Variability
(Observed variability)
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

23. Possible Source of Variation
Observed Process Variation
Actual Process Variation
Measurement Variation
Long-term Variation
Short-term Variation
Variation within
a Sample
Variation
due to Gage
Variation
due to Operators
Accuracy/Bias
Repeatability
Linearity
Stability
Reproducibility
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

24. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Types of Measurement Error that Affect Location
Measurement
System Bias
Accuracy
Linearity
Stability
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

25. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
What is GR&R?
Measurement Systems Analysis
How good is our
measurement system?
Total Variance
Measurement System
Variability —
Determined through
“R&R Study”
Process Variance
Measurement Variance
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

26. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Gauge R&R Allows Control of the
Measurement System
Components of Measurement System Variation
Variation due to gage
Variation due to operators
Operator
Operator by part
Reproducibility
Repeatability
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

27. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Types of Measurement Error that Affect Spread
Precision/Repeatability
Reproducibility
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

28. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
BIAS Is Definition
BIAS— the difference between the
observed average of the measurement and
the reference value. The reference-value is
the serves agreed-upon
Reference
value that as an Value
reference. The reference value can be
determined by averaging several
measurements with a higher level (e.g.,
metrology lab) of measuring equipment.
ACCURACY IS THE SAME AS BIAS
Reference
value
Observed Average Value
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

29. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Stability (drift) Definition
Stability — Is the total variation in the average value measurement
obtained with a measurement system
(test / gage ) on the same master parts when measuring a single
characteristic over an extended period.
Time-2
time
Time-1
Magnitude
time
Points to the frequency of mean center Calibration
Stability
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

30. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Repeatability Definition
Repeatability — The variation in
measurements
obtained with one measurement instrument
when used several times by one appraiser while
measuring the identical characteristic on
same part.
REPEATABILITY
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

31. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Part number
Measurements 1 2 X R 21 20
20.5 24 23
23.5 3 4 27 5 19 18
18.5 6 22 7
21.5 8 17 9 10 11 25 12 13 14 15 29 30
29.5 16 26
25.5
22.3
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

32. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Part Measurement Data (Single Operator or p=1)
Measurement
Part number 1 2
X-bar R 21 20
20.5 24 23
23.5 3 4 27 5 19 18
18.5
P=1
For n=2 is 1.128, USL=60; LSL=5
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

33. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Reproducibility Definition
Reproducibility — Is the
variation in the average of the Operator measurements made by different appraisers using the same measuring instrument when measuring the identical characteristic on the same part.
Operator-B
Operator-C
Operator-A
Reproducibility
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

34. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Repeatability-Equipment Variation (EV)
Trials 2
4.56 3
8.05
= x 4.56 =
n=number of parts
R=number of trials
Reproducibility-Appraiser Variation (AV)
Appraisers 2
3.65 3
2.70
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

35. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Repeatability &Reproducibility (R&R)
Parts 2
3.65 3
2.7 4
2.3 5
2.08 6
1.93 7
1.82 8
1.74 9
1.67 10
1.62
Part variation (PV)
Total variation(TV)
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

36. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Source of Variation
% Contribution
Total Gage Repeatability & Reproducibility
Repeatability
Reproducibility
Part-to-Part
Total Variation
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

37. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Gage R&R, Distinct Categories
Number of Distinct Categories = Eight
• This is the number of distinct categories this
measurement system can distinguish.
• The number of groups within your process
data that your measurement system
can discern.
Source of Variation
Varcomp
Total Gage R&R
1.90E-04
Repeatability
8.73E-05
Reproducibility
1.02E-04
Name
Part-to-Part
6.45E-03
Total Variation
6.64E-03
Distinct categories
Good Gage
>10
Marginal Gage
4 to 9
Reject Gage
<4
Dist Categories = Round down
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

38. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Assignment (Data are from Vardeman & Jobe (1999))
A study was conducted to investigate the precision of measuring the heights of 10
steel punches (in 10-3 inches) using a certain micrometer caliper. Three operators
were randomly selected for the the study. Ten parts with steel punches are randomly selected. Each operator measured each punch three time. Here are the data.
Row
Punch
Oper A
Oper B
Oper C 1
496
497 97 2
499
498 3 4 5 6
500 7 8 9 10 11 12 13 14
501 15
Row
Punch
Oper A
Oper B
Oper C 16 6
499
500
498 17 18
497 19 7
503 20 21
502 22 8
501 23 24 25 9 26 27 28 10
496 29
494 30
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

39. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Check This in Minitab
Thermal Impedance Data (c/w*100)for the Gauge R&R Experiment
Part Number
Inspector 1
Inspector 2
Inspector 3
Test 1
Test 2
Test 3 1 37 38 41 40 42 2 43 3 30 31 29 28 4 5 6 45 44 46 7 25 26 27 8 9 10 35 34
Prof. Indrajit Mukherjee, School of Management, IIT Bombay

40. Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Attribute Gage R&R Effectiveness
SCORING REPORT
Attribute Legend
1.Pass
2.Fail
DATE:
3/10/1996
NAME:
Allied Employee
PRODUCT:
3313 Spark Plug
SBU:
TEST CONDITIONS:
F&SP
Known Population
Operator #1
Operator #2
Operator #3
Y/N
Sample #
Attribute
Try #1
Try #2
Agree 1
Pass
Fail N 2 3 4 Y 5 6 7 8 9 10 11 12 13 14 15 _ 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Prof. Indrajit Mukherjee, School of Management, IIT Bombay