1. Process Variability and Capability
Operations Management
Dr. Ron Lembke

2. Designed Size

3. Natural Variation

4. Theoretical Basis of Control Charts
95.5 % of values
within 2σ

5. Theoretical Basis of Control Charts
99.7 % of values
within 3σ

6. Skewness Lack of symmetry Pearson’s coefficient of skewness:
Positive Skew > 0
Skewness = 0
Negative Skew < 0

7. Kurtosis Amount of peakedness or flatness Kurtosis = 0 Kurtosis < 0

8. Heteroskedasticity
Sub-groups with different variances

9. Design Tolerances Design tolerance:
Determined by users’ needs
UTL -- Upper Tolerance Limit
LTL -- Lower Tolerance Limit
Eg: specified size +/ inches
No connection between tolerance and 
completely unrelated to natural variation.

10. Process Capability and 6
LTL
UTL
LTL
UTL 3
A “capable” process has UTL and LTL 3 or more standard deviations away from the mean, or 3σ.
99.7% (or more) of product is acceptable to customers

11. Process Capability
Capable Not Capable
LTL
UTL
LTL
UTL
LTL
UTL
LTL
UTL

12. Process Capability Specs: 1.5 +/- 0.01 Mean: 1.505 Std. Dev. = 0.002
Are we in trouble?

13. Process Capability Specs: 1.5 +/- 0.01 Mean: 1.505 Std. Dev. = 0.002
LTL = 1.5 – 0.01 = 1.49
UTL = = 1.51
Mean: Std. Dev. = 0.002
LCL = *0.002 = 1.499
UCL = = 1.511
Process
Specs
1.499
1.511
1.49
1.50
1.51

14. Capability Index
Capability Index (Cpk) will tell the position of the control limits relative to the design specifications.
Cpk ≥ 1.0, process is capable
Cpk < 1.0, process is not capable

15. Process Capability, Cpk
Tells how well parts produced fit into specs
Process
Specs 3 3
LTL
UTL

16. Process Capability Tells how well parts produced fit into specs
For our example:
Cpk= min[ 0.015/0.006, 0.005/0.006]
Cpk= min[2.5,0.833] = < 1 Process not capable

17. Process Capability: Re-centered
If process were properly centered
Specs: 1.5 +/- 0.01
LTL = 1.5 – 0.01 = 1.49
UTL = = 1.51
Mean: Std. Dev. = 0.002
LCL = *0.002 = 1.494
UCL = = 1.506
Process
Specs
1.49
1.494
1.506
1.51

18. If re-centered, it would be Capable
Since 1.67 > 1, process would now be Capable
Process
Specs
1.49
1.494
1.506
1.51

19. What if Not Capable? We can’t sell everything we make to these people
Find less choosy customers?
Measure & sort all for them?
Shifting the Mean
Not trivial, may be easy
Adjust a setting?
Reducing variability – Hard!
Process varies over time
Find and get rid of variability
Training workers, new equip?
LTL
UTL
LTL
UTL

20. Summary All processes exhibit variability
Reviewed some basic statistical concepts
Defined process capability measurement, Cpk
Difficulties of making a process capable